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Align, Reason and Learn: Enhancing Medical Vision-and-Language Pre-training with Knowledge
Zhihong Chen
Shenzhen Research Institute of Big Data,
The Chinese University of Hong Kong,ShenzhenChina
[email protected]
,
Guanbin Li
Sun Yat-sen University,GuangzhouChina
[email protected]
and
Xiang Wan
Shenzhen Research Institute of Big Data,
The Chinese University of Hong Kong,ShenzhenChina
Pazhou Lab,GuangzhouChina
[email protected]
(2022)
Abstract.
Medical vision-and-language pre-training (Med-VLP) has received considerable attention owing to its applicability to extracting generic vision-and-language representations from medical images and texts.
Most existing methods mainly contain three elements: uni-modal encoders (i.e., a vision encoder and a language encoder), a multi-modal fusion module, and pretext tasks, with few studies considering the importance of medical domain expert knowledge and explicitly exploiting such knowledge to facilitate Med-VLP.
Although there exist knowledge-enhanced vision-and-language pre-training (VLP) methods in the general domain, most require off-the-shelf toolkits (e.g., object detectors and scene graph parsers), which are unavailable in the medical domain.
In this paper, we propose a systematic and effective approach to enhance Med-VLP by structured medical knowledge from three perspectives.
First, considering knowledge can be regarded as the intermediate medium between vision and language, we align the representations of the vision encoder and the language encoder through knowledge.
Second, we inject knowledge into the multi-modal fusion model to enable the model to perform reasoning using knowledge as the supplementation of the input image and text.
Third, we guide the model to put emphasis on the most critical information in images and texts by designing knowledge-induced pretext tasks.
To perform a comprehensive evaluation and facilitate further research, we construct a medical vision-and-language benchmark including three tasks.
Experimental results illustrate the effectiveness of our approach, where state-of-the-art performance is achieved on all downstream tasks.
Further analyses explore the effects of different components of our approach and various settings of pre-training.111The source code is available at https://github.com/zhjohnchan/ARL.
vision-and-language; knowledge-enhanced learning; multi-modal pre-training; medical analysis
††journalyear: 2022††copyright: acmcopyright††conference: Proceedings of the 30th ACM International Conference on Multimedia ; October 10–14, 2022; Lisboa, Portugal.††booktitle: Proceedings of the 30th ACM International Conference on Multimedia (MM ’22), October 10–14, 2022, Lisboa, Portugal††price: 15.00††isbn: 978-1-4503-9203-7/22/10††doi: 10.1145/3503161.3547948††submissionid: 814††ccs: Computing methodologies Artificial intelligence††ccs: Computing methodologies Machine learning††ccs: Computing methodologies Multi-task learning
1. Introduction
Medical data streams from various sources, among which vision and language are two critical ones.
It includes image data (e.g., radiography, magnetic resonance imaging, and computed tomography) and text data (e.g., radiology reports and medical texts).
Medical vision-and-language pre-training (Med-VLP) aims to jointly process data from these two modalities to learn generalizable multi-modal representations from large-scale medical image-text data.
It enables a vision-and-language model to address a wide range of medical vision-and-language tasks (e.g., medical visual question answering (Med-VQA), medical image-text classification (Med-ITC), and medical image-text retrieval (Med-ITR)), which can be crucial for alleviating the data scarcity problem in the medical field.
In the past few years, vision-and-language pre-training (VLP) has drawn sustaining attention (Chen et al., 2020; Kim et al., 2021; Su et al., 2019; Tan and Bansal, 2019; Lu et al., 2019; Li et al., 2020b; Cui et al., 2021) and achieved state-of-the-art performance on many vision-and-language tasks in the general domain.
In general, a VLP system consists of three elements: (i) uni-modal encoders (i.e., a vision encoder and a language encoder) that encode images and texts into image and text features, respectively; (ii) a multi-modal fusion module that performs the fusion of the encoded image and text features; (iii) pretext tasks (e.g., masked image modeling (MIM), masked language modeling (MLM), and image-text matching (ITM)) that assist the learning of VLP models.
More recently, some studies (Li et al., 2020a; Khare et al., 2021; Gong et al., 2021; Moon et al., 2021) applied VLP to the medical domain and significantly improved the performance for medical vision-and-language tasks (especially for Med-VQA).
These methods are superior in capturing the mappings between images and texts and thus enable the pre-trained models to understand the complicated cross-modal information.
For example, (Khare et al., 2021; Gong et al., 2021) proposed to perform the pre-training on medical image-text pairs to capture medical knowledge, and the evaluation on Med-VQA has demonstrated the validity of their proposed methods.
Although these methods have motivated the learning of image-text correspondences through well-designed model architectures and pretext tasks, most of them disregard the complementary information (i.e., knowledge) shared by different modalities and still lack the explicit knowledge modeling for Med-VLP.
Even in the general domain, there are only a few VLP studies (Li et al., 2020b; Yu et al., 2021; Chen et al., 2021; Cui et al., 2021) on incorporating external knowledge into the pre-training process.
For instance, ERNIE-ViL (Yu et al., 2021) constructed a scene graph from the input text to build the semantic connections between vision and language and emphasized the importance of keywords (e.g., objects, attributes, and relationships between objects) through the designs of pretext tasks.
ROSITA (Cui et al., 2021) used a unified scene graph shared by the input image and text to enhance the
semantic alignments between vision and language.
Similarly, KB-VLP (Chen et al., 2021) used object tags detected from images and knowledge graph embeddings extracted from texts to enhance the learning of knowledge-aware representations.
However, the aforementioned studies require off-the-shelf toolkits (e.g., object detectors and scene graph parsers), which are generally unavailable in the medical domain.
Furthermore, they might be limited in scalability as their performance depends heavily on the reliability of the object detectors or scene graph parsers.
Therefore, it is expected to have a better solution to exploit external knowledge more appropriately and systematically and further improve the generalization ability of Med-VLP methods.
In this paper, we propose a systematic approach to Med-VLP enhanced by structured expert domain knowledge from the Unified Medical Language System (Bodenreider, 2004) (UMLS), a large medical knowledge base containing many biomedical terminologies with the associated information, such as synonyms and categorical groupings.
To ensure the effectiveness and efficiency of our approach, structured knowledge is injected into the Med-VLP system from three perspectives:
(i) Aligning Through Knowledge: It uses knowledge as the intermediate medium between vision and language to align the image and text features encoded by the uni-modal encoders;
(ii) Reasoning Using Knowledge: It develops a knowledge-enhanced multi-modal fusion module to integrate knowledge into the interaction process of the image and text features;
(iii) Learning From Knowledge: It constructs knowledge-induced pretext tasks to assist the model in capturing underlying critical medical information of the images and texts to promote the medical vision-and-language understanding.
As a result, the proposed method is able to learn cross-modal domain-specific knowledge from large-scale medical image-text datasets and medical knowledge bases to promote the learning of semantically aligned and knowledge-aware image and text representations.
We perform the pre-training on three large-scale medical image-text datasets, i.e., ROCO (Pelka et al., 2018), MedICaT (Subramanian et al., 2020), and MIMIC-CXR (Johnson et al., 2019).
To verify the effectiveness of our approach and facilitate further research, we construct a medical vision-and-language understanding benchmark including three tasks (i.e., Med-VQA, Med-ITC, and Med-ITR).
Experimental results demonstrate the effectiveness of our approach, where state-of-the-art performance is achieved on all datasets.
2. Related Work
Vision-and-Language Pre-training (VLP)
Motivated by the success of the self-supervised pre-training recipe of BERT in NLP, there has been an increasing interest in developing VLP methods to address a wide range of vision-and-language tasks.
In general, VLP methods can be categorized with respect to three perspectives.
For the designs of the uni-modal encoders, different methods adopt different image features (e.g., region features (Li et al., 2019; Lu et al., 2019), patch embeddings (Kim et al., 2021; Li et al., 2021; Wang et al., 2021b), and grid features (Huang et al., 2020)) and distinct text features (e.g., statistic embeddings (Kim et al., 2021) and dynamic embeddings (Dou et al., 2021)).
For multi-modal fusion modules, existing methods can be classified into two categories (i.e., single-stream and dual-stream).
In specific, for the single-stream fusion, the models (Li et al., 2019; Chen et al., 2020; Su et al., 2019; Li et al., 2020b) use a single Transformer for early and unconstrained fusion between modalities; for the dual-stream fusion, the models (Tan and Bansal, 2019; Lu et al., 2019; Yu et al., 2021) adopt the co-attention mechanism to interact different modalities.
For pretext tasks, inspired by uni-modal pre-training schemes such as MLM (Devlin et al., 2019; Liu et al., 2019) and causal language modeling (Brown et al., 2020), existing studies explore a variety of pre-training tasks, including MLM (Li et al., 2019; Lu et al., 2019; Tan and Bansal, 2019), MIM (Lu et al., 2019; Chen et al., 2020), ITM (Li et al., 2019; Zhang et al., 2021), image-text contrastive (Li et al., 2021) and prefix language modeling (Wang et al., 2021b).
This paper adopts a purely Transformer-based backbone architecture using the dual-stream fusion with ViT-based grid features and BERT-based dynamic text features and three common pretext tasks (i.e., MLM, MIM, and ITM).
Medical Vision-and-Language Pre-training (Med-VLP)
Being one of the applications and extensions of VLP to the medical domain, Med-VLP aims to understand the content and relations between medical images and their corresponding texts.
It can be traced back to (Li et al., 2020a), which explored the performance of four vision-and-language models pre-trained in the general domain on a disease classification task.
Then MMBERT (Khare et al., 2021), PubMedCLIP (Eslami et al., 2021), and MedViLL (Moon et al., 2021) performed pre-training on medical image-text data before fine-tuning on the downstream tasks.
Compared with these studies, we design a more appropriate and systematic scheme for Med-VLP from four aspects (i.e., pre-training datasets, model designs, pre-training tasks, and evaluation benchmarks).
Knowledge-Enhanced Pre-training
For uni-modal pre-training in CV and NLP, many works have investigated how to incorporate knowledge into the pre-trained models.
According to the knowledge injection schemes, existing studies can be classified into four categories: embeddings combination (Zhang et al., 2019; Peters et al., 2019), data structure compatibility (Liu et al., 2020; Sun et al., 2020; He et al., 2020), knowledge supervision (Sun et al., 2019; Wang et al., 2021a), and neural-symbolic methods (Amizadeh et al., 2020).
For VLP, knowledge can be acquired from both the image and text modalities, and there are several works (Li et al., 2020b; Yu et al., 2021; Cui et al., 2021) studying to integrate knowledge into their methods.
ERNIE-ViL (Yu et al., 2021) built detailed semantic alignments between vision and language based on the scene graph parsed from the text.
ROSITA (Cui et al., 2021) proposed integrating extra cross-modal knowledge mappings to enhance the learning of semantic alignments between vision and language.
Different from them, we revisit existing knowledge-enhanced methods and propose to inject knowledge from three VLP-specific perspectives without requiring object detectors or scene graph parsers, which are unavailable in the medical domain.
3. The Proposed Approach
We follow the standard pre-train-and-fine-tune paradigm for medical vision-and-language understanding.
In the pre-training stage, the framework develops a variety of pretext tasks to train the Med-VLP model using medical image-text pairs.
In the fine-tuning stage, the pre-trained Med-VLP model is transferred to various medical vision-and-language downstream tasks.
An overview of the proposed approach is demonstrated in Figure 2, and the details of the general Med-VLP framework, the knowledge extraction process (shown in Figure 1), and the injection of knowledge into the general Med-VLP framework are introduced in the following subsections.
3.1. The General Med-VLP Framework
The general Med-VLP framework can be partitioned into three major components, i.e., the uni-modal encoders, the multi-modal fusion module, and the pretext tasks.
The overall description of the three components is detailed below.
Uni-modal Encoders
In the Med-VLP framework, there is a vision encoder and a language encoder, which encode the input image and text into image and text features, respectively.
For the vision encoder, we study the use of vision Transformer (Dosovitskiy et al., 2020) (ViT).
In ViT, an input image $\boldsymbol{I}\in\mathbb{R}^{H\times W\times C}$ is first segmented into patches $\{\boldsymbol{x_{1}^{v}},\boldsymbol{x_{2}^{v}},\ldots,\boldsymbol{x_{N_{v}}^{v}}\}$, where $H\times W$ is the image resolution, $C$ is the number of channels, $N_{v}$ is the number of patches, $\boldsymbol{x_{i}^{v}}\in\mathbb{R}^{P^{2}\times C}$ and $P\times P$ is the patch resolution.
Then the patches $\{\boldsymbol{x_{1}^{v}},\boldsymbol{x_{2}^{v}},\ldots,\boldsymbol{x_{N_{v}}^{v}}\}$ are flattened and linearly projected into patch embeddings through a linear transformation $\boldsymbol{E^{v}}\in\mathbb{R}^{P^{2}C\times D}$ and a special learnable token embedding $\boldsymbol{x_{I}^{v}}\in\mathbb{R}^{D}$ is prepended for the aggregation of visual information.
Therefore, the input image representations are obtained via summing up the patch embeddings and learnable 1D position embeddings $\boldsymbol{E^{v}_{pos}}\in\mathbb{R}^{(N_{v}+1)\times D}$:
(1)
$$\boldsymbol{X^{v}}=[\boldsymbol{x_{I}^{v}};\boldsymbol{x_{1}^{v}}\boldsymbol{E^{v}};\boldsymbol{x_{2}^{v}}\boldsymbol{E^{v}};\ldots;\boldsymbol{x_{N_{v}}^{v}}\boldsymbol{E^{v}}]+\boldsymbol{E^{v}_{pos}}.$$
Then $\boldsymbol{X^{v}}$ is fed into a Transformer model with $L_{v}$ Transformer layers.
Finally, we obtain the contextualized image representations $\boldsymbol{H^{v}}=[\boldsymbol{h^{v}_{I}};\boldsymbol{h^{v}_{1}};\boldsymbol{h^{v}_{2}};...;\boldsymbol{h^{v}_{N_{v}}}]$.
For the language encoder, we follow BERT (Devlin et al., 2019) to tokenize the input text to subword tokens $\{x_{1}^{l},x_{2}^{l},\ldots,x_{N_{l}}^{l}\}$ by WordPiece (Wu et al., 2016) and then represent subword tokens as $\{\boldsymbol{x_{1}^{l}},\boldsymbol{x_{2}^{l}},\ldots,\boldsymbol{x_{N_{l}}^{l}}\}$, where $\boldsymbol{x_{i}^{l}}\in\mathbb{R}^{V}$ are the one-hot form of $x_{i}^{l}$, $V$ is the vocabulary size, and $N_{l}$ is the number of tokens.
Subsequently, the tokens are linearly projected into embeddings through a linear transformation $\boldsymbol{E^{l}}\in\mathbb{R}^{V\times D}$.
Afterwards, a start-of-sequence token embedding $\boldsymbol{x_{T}^{l}}\in\mathbb{R}^{D}$ and a special boundary token embedding $\boldsymbol{x_{SEP}^{l}}\in\mathbb{R}^{D}$ are added to the text sequence.
Therefore, the input text representations are computed via summing up the token embeddings and text position embeddings $\boldsymbol{E^{l}_{pos}}\in\mathbb{R}^{(N_{l}+2)\times D}$:
(2)
$$\boldsymbol{X^{l}}=[\boldsymbol{x_{T}^{l}};\boldsymbol{x_{1}^{l}E^{l}};\ldots;\boldsymbol{x_{N_{l}}^{l}E^{l}};\boldsymbol{{x}_{SEP}^{l}}]+\boldsymbol{E^{l}_{pos}}.$$
Then $\boldsymbol{X^{l}}$ is fed into a Transformer model with $L_{l}$ Transformer layers.
Finally, we obtain the contextualized text representations $\boldsymbol{H^{l}}=[\boldsymbol{h^{l}_{T}};\boldsymbol{h^{l}_{1}};\boldsymbol{h^{l}_{2}};\ldots;\boldsymbol{h^{l}_{N_{l}}};\boldsymbol{h^{l}_{SEP}}]$.
Multi-model Fusion Module
We adopt the co-attention mechanism in the multi-modal fusion module to fuse the contextualized representations from images and texts.
In detail, the multi-modal fusion module consists of two Transformer models for vision and language, respectively, each of which is a stack of $L_{m}$ Transformer layers.
In each Transformer layer, there are three sub-layers, i.e., a self-attention sub-layer, a cross-attention sub-layer, and a feed-forward sub-layer.
The attention mechanism is applied in the self-attention and cross-attention sub-layers and is defined as
(3)
$$\text{ATTN}(\boldsymbol{Q},\boldsymbol{K},\boldsymbol{V})=\operatorname{softmax}\left(\boldsymbol{Q}\boldsymbol{K}^{\top}/\sqrt{D_{k}}\right)\cdot\boldsymbol{V},$$
where $\boldsymbol{Q}$, $\boldsymbol{K}$, and $\boldsymbol{V}$ are the query, key, value matrices linearly transformed from the corresponding input sequences, respectively, and $D_{k}$ is the dimension of $\boldsymbol{K}$.
In the self-attention sub-layer, the representations interact within modalities:
(4)
$$\displaystyle\boldsymbol{H^{vs}}$$
$$\displaystyle=\text{ATTN}(\boldsymbol{H^{v}},\boldsymbol{H^{v}},\boldsymbol{H^{v}}),$$
(5)
$$\displaystyle\boldsymbol{H^{ls}}$$
$$\displaystyle=\text{ATTN}(\boldsymbol{H^{l}},\boldsymbol{H^{l}},\boldsymbol{H^{l}}),$$
where $\boldsymbol{H^{vs}}$ and $\boldsymbol{H^{ls}}$ are the self-attention outputs for vision and language, respectively.
Then residual connections followed by layer normalization are employed to $\boldsymbol{H^{vs}}$ and $\boldsymbol{H^{ls}}$ and we denote the results as $\boldsymbol{H^{vs}}$ and $\boldsymbol{H^{ls}}$, respectively, for simplicity.
In the cross-attention sub-layer, the representations interact across modalities to integrate cross-modal information into their representations:
(6)
$$\displaystyle\boldsymbol{H^{vc}}=\text{ATTN}(\boldsymbol{H^{vs}},\boldsymbol{H^{ls}},\boldsymbol{H^{ls}}),$$
(7)
$$\displaystyle\boldsymbol{H^{lc}}=\text{ATTN}(\boldsymbol{H^{ls}},\boldsymbol{H^{vs}},\boldsymbol{H^{vs}}),$$
where $\boldsymbol{H^{vc}}$ and $\boldsymbol{H^{lc}}$ are the cross-attention outputs for vision and language, respectively.
Similarly, residual connections followed by layer normalization are employed to $\boldsymbol{H^{vc}}$ and $\boldsymbol{H^{lc}}$, and we denote the results as $\boldsymbol{H^{vc}}$ and $\boldsymbol{H^{lc}}$, respectively, for simplicity.
Finally, $\boldsymbol{H^{vc}}$ and $\boldsymbol{H^{lc}}$ are input to the feed-forward sub-layer (i.e., a multi-layer perceptron (MLP)) to obtain the multi-modal representations $\boldsymbol{Z^{v}}=[\boldsymbol{z^{v}_{I}};\boldsymbol{z^{v}_{1}};\boldsymbol{z^{v}_{2}};\ldots;\boldsymbol{z^{v}_{N_{v}}}]$ for vision and $\boldsymbol{Z^{l}}=[\boldsymbol{z^{l}_{T}};\boldsymbol{z^{l}_{1}};\boldsymbol{z^{l}_{2}};\ldots;\boldsymbol{z^{l}_{N_{l}}};\boldsymbol{z^{l}_{SEP}}]$ for language.
Pretext Tasks
Given the aforementioned structure (denoted as $\mathcal{M}_{\theta}$) with its parameters $\theta$, the Med-VLP framework develops various pretext tasks (e.g., masked language modeling (MLM), masked image modeling (MIM), and image-text matching (ITM)) to guide the learning of $\theta$.
Assuming there are $S$ pretext tasks, the learning process can be formalized as
(8)
$$\theta^{*},\theta_{1}^{*},...,\theta_{S}^{*}=\mathop{\arg\min}_{\theta,\theta_{1},...,\theta_{S}}\sum_{i=1}^{S}L_{i}(Y_{i},\mathcal{D}_{\theta_{i}}(\mathcal{M}_{\theta}(I,T)),$$
where $L_{i}$ are the loss functions of pretext tasks, $Y_{i}$ are the corresponding ground-truth labels, and $\mathcal{D}_{\theta_{i}}$ are the prediction heads with their parameters $\theta_{i}$.
3.2. Knowledge Extraction
Although knowledge graphs (KGs) have shown their effectiveness in many natural language processing (NLP) tasks (Zhang et al., 2019; Peters et al., 2019; Liu et al., 2020) and computer vision (CV) tasks (Marino et al., 2017; Wang et al., 2018), the existing Med-VLP methods rarely consider incorporating KGs to provide rich structured knowledge for better vision-and-language understanding.
Therefore, we propose to enhance Med-VLP by leveraging external domain expert knowledge from UMLS.
In doing so, we extract knowledge through two stages, as illustrated in Figure 1.
The first stage is to apply a named entity recognition, and linking tool ScispaCy (Neumann et al., 2019) to pre-process the texts in the pre-training corpus to link entities in the texts to the UMLS knowledge base for entity disambiguation.
Therefore, for each image-text pair, there is an entity sequence $ES=\{x_{1}^{e},x_{2}^{e},\ldots,x_{N_{es}}^{e}\}$ aligning to the token sequence $T=\{x_{1}^{l},x_{2}^{l},\ldots,x_{N_{l}}^{l}\}$, where $x_{i}^{e}$ are the extracted entities and $N_{es}$ is the length of the entity sequence.
Following (Diao et al., 2020), to record the position of the extracted entities, we adopt an entity matching matrix $\boldsymbol{P}\in\mathbb{R}^{N_{l}\times N_{es}}$, where each element is represented by
(9)
$$\boldsymbol{P}_{ij}=\begin{cases}1&x_{i}^{l}\in x_{j}^{e}\\
0&x_{i}^{l}\notin x_{j}^{e}\end{cases},$$
where $P$ is employed to assist the interaction between the text and the entity sequence in the knowledge injection process (as described in the next subsection).
After pre-processing all the texts, we can obtain an entity set $\mathcal{E}=\{e_{i}\}_{i=1}^{N_{e}}$ containing all the $N_{e}$ entities related to the pre-training corpus.
The second stage is to extract relevant knowledge graph triples from the UMLS knowledge base once both the head and tail entities of the triple are in the entity set $\mathcal{E}$.
We denote the extracted knowledge graph (i.e., a sub-graph of the UMLS knowledge base) as the knowledge set $\mathcal{G}=\{k_{i}=(h_{i},r_{i},t_{i})\}_{i=1}^{N_{g}}$, where $N_{g}$ is the number of knowledge graph triples, $k_{i}$ are the knowledge graph triples, and $h_{i}$, $r_{i}$ and $t_{i}$ represent the head entity, relation, and tail entity, respectively.
3.3. Knowledge Injection
To integrate knowledge into the general Med-VLP framework, first, we perform knowledge representation following two steps:
(i) We apply knowledge representation learning algorithm (e.g., TransE (Bordes et al., 2013)) to the knowledge graph $\mathcal{G}=\{k_{i}=(h_{i},r_{i},t_{i})\}_{i=1}^{N_{g}}$ to obtain the entity embeddings $\{\boldsymbol{e_{i}}\}_{i=1}^{N_{e}}$, where $\boldsymbol{e_{i}}\in\mathbb{R}^{D_{e}}$ and $D_{e}$ is the dimension of the entity embeddings;
(ii) We adopt Graph Neural Networks (e.g., Graph Attention Networks (Veličković et al., 2018)) to take account of the whole structure of the graph to aggregate local information in the graph neighborhood for each node, and obtain the entity representations (denoted as $\{\boldsymbol{e_{i}}\}_{i=1}^{N_{e}}$ for simplicity, where $\boldsymbol{e_{i}}\in\mathbb{R}^{D_{e}}$).
Afterwards, given the input image $I$ and text $T=\{x_{1}^{l},x_{2}^{l},\ldots,x_{N_{l}}^{l}\}$ with its corresponding entity sequence $ES=\{x_{1}^{e},x_{2}^{e},\ldots,x_{N_{es}}^{e}\}$, we develop three essential and systematic designs to inject knowledge from the following perspectives:
(i) Aligning Through Knowledge
Knowledge can be regarded as the intermediate medium between vision and language, where knowledge can be used as an explanation of the meaning behind both images and texts.
In most cases, entities in knowledge graph triples can be treated as anchor points that appear in the image and are mentioned in the accompanying text.
Motivated by this fact, we propose to align the image representations and the text representations from uni-modal encoders through knowledge.
Similar to (Li et al., 2021), it serves two purposes:
It improves the unimodal encoders to better understand the semantic meaning of images and texts;
It eases the learning of semantic alignments between images and texts.
Formally, given the aggregated image representation $\boldsymbol{h^{v}_{I}}$ and text representation $\boldsymbol{h^{l}_{T}}$, we calculate the image-knowledge and text-knowledge similarity followed by a sigmoid function:
(10)
$$\displaystyle p_{i}^{v}$$
$$\displaystyle=\text{sigmoid}(\boldsymbol{e_{i}}^{\top}\boldsymbol{W_{vk}}\boldsymbol{h^{v}_{I}}),i=1,...,N_{e},$$
(11)
$$\displaystyle p_{i}^{l}$$
$$\displaystyle=\text{sigmoid}(\boldsymbol{e_{i}}^{\top}\boldsymbol{W_{lk}}\boldsymbol{h^{l}_{T}}),i=1,...,N_{e},$$
where $\boldsymbol{W_{vk}}\in\mathbb{R}^{D_{e}\times D}$ and $\boldsymbol{W_{lk}}\in\mathbb{R}^{D_{e}\times D}$ are trainable weights for the linear transformation.
Therefore, the alignments for image-knowledge and text-knowledge are learned explicitly through minimizing the following functions:
(12)
$$\displaystyle L_{vk}$$
$$\displaystyle=-\sum_{i=1}^{N_{e}}(y_{i}\log p_{i}^{v}+\left(1-y_{i}\right)\log\left(1-p_{i}^{v}\right)),$$
(13)
$$\displaystyle L_{lk}$$
$$\displaystyle=-\sum_{i=1}^{N_{e}}(y_{i}\log p_{i}^{l}+\left(1-y_{i}\right)\log\left(1-p_{i}^{l}\right)),$$
where $y_{i}$ can be defined as
(14)
$$y_{i}=\begin{cases}1&e_{i}\in ES\\
0&e_{i}\notin ES\end{cases}.$$
Therefore, knowledge is employed as an intermediate medium to enhance and smooth image-text mappings by doing so.
(ii) Reasoning Using Knowledge
As the supplementation of the input image and text, knowledge can also be utilized to assist the reasoning of the Med-VLP model.
In doing so, we enhance the multi-modal fusion module by knowledge.
Formally, given the entity sequence $ES=\{x_{1}^{e},x_{2}^{e},...,x_{N_{es}}^{e}\}$, first, we extract its entity representations $\boldsymbol{H^{e}}=[\boldsymbol{h^{e}_{1}};\boldsymbol{h^{e}_{2}};\ldots;\boldsymbol{h^{e}_{N_{es}}}]$.
Second, we apply self-attention to the entity representations to encode the contextualized information:
(15)
$$\boldsymbol{H^{es}}=\text{ATTN}(\boldsymbol{H^{e}},\boldsymbol{H^{e}},\boldsymbol{H^{e}}),$$
where $\boldsymbol{H^{es}}$ is the self-attention outputs of entity representations.
Residual connections followed by layer normalization are employed to $\boldsymbol{H^{es}}$, and we denote the outputs as $\boldsymbol{H^{es}}$ for simplicity.
Third, to interact the entities with the image, since there is no available toolkits to structuralize the input image to construct mappings between image patches and entities, we directly perform cross-attention on $\boldsymbol{H^{es}}$ and $\boldsymbol{H^{vs}}$:
(16)
$$\boldsymbol{H^{ec}}=\text{ATTN}(\boldsymbol{H^{es}},\boldsymbol{H^{vs}},\boldsymbol{H^{vs}}),$$
where $\boldsymbol{H^{ec}}$ is the cross-attention outputs of entity representations.
Residual connections followed by layer normalization are employed to $\boldsymbol{H^{ec}}$, and we denote the outputs as $\boldsymbol{H^{ec}}$ for simplicity.
Fourth, since the mappings between the entities and the text are recorded by $\boldsymbol{P}$, we can use it to fuse $\boldsymbol{H^{ec}}$ and $\boldsymbol{H^{ls}}$ through:
(17)
$$\boldsymbol{\tilde{H}^{lc}}=\boldsymbol{P}\boldsymbol{H^{ec}}+\boldsymbol{H^{lc}},$$
where $\boldsymbol{\tilde{H}^{lc}}$ is the text representations encoded with the image and entity information.
Finally, we apply residual connections followed by layer normalization to $\boldsymbol{\tilde{H^{lc}}}$ and input it to the feed-forward sub-layer to complete the knowledge-enhanced multi-modal fusion.
In the meantime, $\boldsymbol{H^{ec}}$ is input to another feed-forward sub-layer to produce the representations $\boldsymbol{Z^{e}}$ for the next layer.
(iii) Learning From Knowledge
Knowledge can help us to induce more sophisticated pretext tasks to guide the model to learn more informative representations.
In our paper, we follow (Sun et al., 2019) to design a knowledge-induced mask generation strategy.
Specifically, when performing the MLM task given the input text $T=\{x_{1}^{l},x_{2}^{l},\ldots,x_{N_{l}}^{l}\}$ with its entity sequence $ES=\{x_{1}^{e},x_{2}^{e},\ldots,x_{N_{es}}^{e}\}$, we do not mask subword tokens in $T$ randomly.
Instead, we randomly sample entities from $ES$ and then mask consecutive spans of subword tokens belonging to the sampled entities.
Since entities can be abstract or have a physical existence, it can force the model to focus on critical medical information in both images and texts.
Therefore, knowledge can be injected into the Med-VLP framework in a systematic way through the above three designs.
4. Experimental Settings
4.1. Pre-training Setup
Datasets
In our experiments, we perform the pre-training on three datasets, which are described as follows:
•
ROCO (Pelka et al., 2018): a dataset of radiology figure-caption pairs from PubMed Central, an open-access biomedical literature database.
It has over 81,000 radiology images (from various imaging modalities) and their corresponding captions.
•
MedICaT (Subramanian et al., 2020): a dataset of medical figure-caption pairs also extracted from PubMed Central.
Different from ROCO, 75% of its figures are compound figures, including several sub-figures.
It contains over 217,000 images with their captions and inline textual references.
•
MIMIC-CXR (Johnson et al., 2019): the largest radiology dataset to date that consists of 473,057 chest X-ray images (in frontal or lateral views) and 206,563 reports from 63,478 patients from the Beth Israel Deaconess Medical Center.
For all the datasets, we exclude those samples with the length of their texts less than 3.
For ROCO and MedICaT, we filter non-radiology samples, and for MIMIC-CXR, we only keep images in the frontal view.
As for the dataset split, we adopt the official splits of ROCO and MIMIC-CXR.
For MedICaT, we randomly sample 1,000 image-text pairs for validation and 1,000 for test, and the remaining image-text pairs are used for training.
Implementation Details
For the uni-modal encoders, we use the vision encoder with CLIP-ViT-B (Radford et al., 2021) ($L_{v}=12$) and the language encoder with RoBERTa-base (Liu et al., 2019) ($L_{l}=12$).
For the multi-modal fusion module, we set the number of Transformer layers $L_{m}=6$, and the dimension of the hidden states $D=768$ with the number of heads set to 12.
For knowledge representation and injection, we set the dimension of the hidden states $D_{e}=256$.
For the pretext tasks, we adopt (knowledge-enhanced) MLM, MIM (He et al., 2021), and ITM, where the masking ratios of MLM and MIM are set to 15% and 75%, respectively.
For the optimization, the models are trained with AdamW optimizer (Loshchilov and Hutter, 2018) for 100,000 steps with the learning rates for the uni-modal encoders and the remaining parameters set to 1e-5 and 5e-5, respectively.
The warm-up ratio is set to 10%, and the learning rate is linearly decayed to 0 after warm-up.
Besides, we use center-crop to resize each image to the size of 288$\times$288.
4.2. Vision-and-Language Transfer Tasks
To evaluate the performance, we construct a medical vision-and-language understanding benchmark including three tasks.
The details of the tasks and fine-tuning strategies are described below.
Medical Visual Question Answering (Med-VQA)
This task requires the model to answer natural language questions about a medical image.
We adopt three publicly available Med-VQA datasets (i.e., VQA-RAD (Lau et al., 2018), SLACK (Liu et al., 2021b) and VQA-2019 (Abacha et al., 2019)), where VQA-RAD consists of 315 images and 3515 questions, SLACK contains 642 images and 14,028 questions, and VQA-2019 contains 4,200 images and 15,292 questions.
To fine-tune on this task, we regard it as a multi-label classification task and feed the concatenation of the image and text representations to a two-layer MLP to predict the corresponding answer.
During training, the models are trained with a binary cross-entropy loss with a batch size of 64.
Medical Image-Text Classification (Med-ITC)
This task aims to produce the classification label given an image-text pair.
We evaluate the performance on MELINDA (Wu et al., 2021), a Biomedical Experiment Method Classification dataset that contains 5,371 image-text pairs.
To fine-tune on this task, we learn a two-layer MLP on top of the concatenation of the image and text representations.
We train the models with a cross-entropy loss with a batch size of 16 over a maximum of 20 epochs.
Medical Image-Text Retrieval (Med-ITR)
The target of this task is to calculate a similarity score between an image and a text and then perform cross-modal retrieval.
There are two subtasks for this task, where image-to-text (I2T) retrieval requires retrieving the most relevant texts from a large pool of texts given an image, and vice versa for text-to-image (T2I) retrieval.
We conduct experiments on the ROCO dataset and measure both zero-shot and fine-tuned performance.
To fine-tune on this task, we initialize the similarity score head from the pre-trained ITM head.
The model is tuned with cross-entropy loss to maximize the scores on positive pairs with 15 random texts sampled as negative samples with a batch size of 256 over a maximum of 10 epochs.
During the evaluation, we sample 2,000 image-text pairs from the ROCO test set and report the results on the sampled 2,000 image-text pairs due to the large time complexity of the ranking process.222The time complexity of the ranking process is $O(N^{2})$, where $N$ is the sample number.
For all tasks, we use the AdamW optimizer with the learning rate set to 5e-6 and 2.5e-5 for the model backbone and prediction heads, respectively, and the warm-up ratio set to 10%.
For the evaluation metrics, we adopt accuracy for Med-VQA and Med-ITC, and Recall@K333Recall@K corresponds to whether the ground truth is included among top K results. (K=1, 5, 10) for Med-ITR.
5. Experimental Results
5.1. Main Results
We compare the proposed approach with existing methods on the same datasets, with their results reported in Table 1, 2, and 3.
There are several observations drawn from different aspects.
First, our approach achieves the best performance on all tasks, which confirms the validity of the proposed pre-training approach.
Second, for Med-VQA, our approach achieves significant improvements even compared with those pre-training methods (e.g., MTPT and MMBERT), which indicates the usefulness of incorporating knowledge into the pre-training process.
Third, for Med-ITC, the results of those strong baselines (i.e., RoBERTa, SciBERT, and ViL-BERT) are achieved by continued pre-training on the MELINDA dataset.
Our approach achieves better performance without such requirements, which indicates that an appropriate design can alleviate the need for continued pre-training on downstream datasets.
Fourth, for Med-ITR, the proposed approach achieves a substantial improvement compared with the state-of-the-art methods, where ViLT and METER are two strong baselines in the general domain.
This shows that it is necessary to design an appropriate approach (including pre-training data and methods) for the medical domain.
5.2. Ablation Studies
To illustrate the effectiveness of our proposed approach, we perform an ablation study on the three proposed knowledge injection designs.
The experiments are conducted on the VQA-RAD dataset, and the results are reported in Table 4.
We have the following observations.
First, for the model parameters, only the RK design brings extra 12M parameters ($\sim$3.4% of the whole model) while other designs do not add additional parameters, which justifies introducing knowledge to Med-VLP through our approach can be done with a small price.
Second, the results of pre-training with two designs (ID 5, 6, and 7) and one design (ID 2, 3, and 4) are consistently better than those of pre-training with one design (ID 2, 3, and 4) and without any knowledge injection design (ID 1), respectively.
This demonstrates the excellent compatibility and complementarity of our design perspectives, which is critical in a multi-component approach and allows us to develop more designs under such a framework.
Third, injecting the knowledge into the multi-modal fusion module (ID 3) achieves a significant improvement.
The reason behind this might be that knowledge (i.e., entities here) serves three functions:
(i) It smooths the interaction process of the image and text representations;
(ii) It provides information at a greater granularity than words;
(iii) It removes ambiguity between diverse words by linking to the knowledge base.
Fourth, performing the aligning process can further improve the performance of the RK design (ID 5), which can be explained by the fact that the aligning processing can produce better knowledge representations for the RK process.
Fifth, our full approach (ID 8) achieves the best performance, which confirms the effectiveness of the proposed framework for medical knowledge injection.
5.3. Qualitative Analysis
To further investigate the effectiveness of our approach, we perform qualitative analysis on some cases with their results shown in Figure 3.
For “aligning through knowledge”, in the input text of this case, there is an entity “Brain magnetic resonance
imaging” which links to the entity “C4028269: Nuclear magnetic resonance imaging brain” in the UMLS knowledge base.
The sub-figures (3(C) and 3(D)) show that the image and text representations produced by the uni-modal encoders have high similarity with the entity representation of “C4028269”, which implicitly pulls the image and text representations close (as shown in the sub-figure 3(E)).
For “reasoning using knowledge”, the sub-figures 3(H), 3(I), and 3(J) illustrate that using entities is beneficial for aligning the text with the image, where the learned entity-image attention mappings are better than the subword-image mappings.
The reason behind this is that entities are more complete semantic units.
In contrast, words (or subwords) have a smaller granularity than entities, making the correspondences between images and texts more challenging to learn.
For “learning from knowledge”, the masked medical entities are correctly recovered by the pre-trained model (as shown in the sub-figure 3(O)) since the knowledge-induced pretext task guides the model to put more emphasis on the medical knowledge-related information.
In addition, the masked and recovered images are also shown in the sub-figures 3(K) and 3(M), respectively, which shows the high quality of the image reconstruction.
In summary, these cases reveal that injecting knowledge through the three proposed designs is essential in modeling the hidden structures among the images and texts better to promote Med-VLP.
6. Conclusion
In this paper, we propose to pre-train the medical vision-and-language model with medical domain knowledge, where the knowledge is injected into the Med-VLP framework from three aspects: (i) aligning the image and text representations through knowledge before their interaction; (ii) treating knowledge as the supplementation of the input image and text to assist the reasoning during the multi-modal fusion process; (iii) utilizing knowledge to induce more sophisticated pretext tasks to guide the model put more emphasis on the critical medical information.
To perform a comprehensive evaluation and facilitate further research, we construct a medical vision-and-language understanding benchmark, including three tasks (i.e., Med-VQA, Med-ITC, and Med-ITR).
Experimental results on the downstream datasets demonstrate the effectiveness of our approach, which achieves state-of-the-art performance.
Further analyses investigate the effects of different components in our approach and show that our approach is able to better learn the correspondences between vision and language so as to produce more generic and effective vision-and-language representations.
Acknowledgement
This work is supported in part by the Chinese Key-Area Research and Development Program of Guangdong Province (2020B0101350001), in part by the Guangdong Basic and Applied Basic Research Foundation (2020B1515020048), in part by the National Natural Science Foundation of China (61976250), in part by the Guangzhou Science and technology project (No.202102020633), and is also supported by the Guangdong Provincial Key Laboratory of Big Data Computing, The Chinese University of Hong Kong, Shenzhen.
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Abstract
Low $x$ deep-inelastic $ep$ scattering data, taken in 1994 at the H1
detector at HERA, are analysed in the Breit frame of reference.
The evolution of the peak and width of the current hemisphere fragmentation
function is presented as a function of $Q$ and compared
with $e^{+}e^{-}$ results at equivalent centre of mass energies.
Differences between the average charged multiplicity and the multiplicity
of $e^{+}e^{-}$ annihilations at low energies are analysed.
Invariant energy spectra are compared with MLLA predictions.
Distributions of multiplicity are presented as functions of Bjorken-$x$
and $Q^{2}$, and KNO scaling is discussed.
DESY 97-108 ISSN 0418-9833
June 1997
Evolution of $ep$ Fragmentation and Multiplicity
Distributions in the Breit Frame
H1 Collaboration
C. Adloff${}^{35}$, S. Aid${}^{13}$, M. Anderson${}^{23}$, V. Andreev${}^{26}$, B. Andrieu${}^{29}$, V. Arkadov${}^{36}$, C. Arndt${}^{11}$, I. Ayyaz${}^{30}$, A. Babaev${}^{25}$, J. Bähr${}^{36}$, J. Bán${}^{18}$, Y. Ban${}^{28}$, P. Baranov${}^{26}$, E. Barrelet${}^{30}$, R. Barschke${}^{11}$, W. Bartel${}^{11}$, U. Bassler${}^{30}$, H.P. Beck${}^{38}$, M. Beck${}^{14}$, H.-J. Behrend${}^{11}$, A. Belousov${}^{26}$, Ch. Berger${}^{1}$, G. Bernardi${}^{30}$, G. Bertrand-Coremans${}^{4}$, R. Beyer${}^{11}$, P. Biddulph${}^{23}$, P. Bispham${}^{23}$, J.C. Bizot${}^{28}$, K. Borras${}^{8}$, F. Botterweck${}^{27}$, V. Boudry${}^{29}$, S. Bourov${}^{25}$, A. Braemer${}^{15}$, W. Braunschweig${}^{1}$, V. Brisson${}^{28}$, W. Brückner${}^{14}$, P. Bruel${}^{29}$, D. Bruncko${}^{18}$, C. Brune${}^{16}$, R. Buchholz${}^{11}$, L. Büngener${}^{13}$, J. Bürger${}^{11}$, F.W. Büsser${}^{13}$, A. Buniatian${}^{4}$, S. Burke${}^{19}$, M.J. Burton${}^{23}$, G. Buschhorn${}^{27}$, D. Calvet${}^{24}$, A.J. Campbell${}^{11}$, T. Carli${}^{27}$, M. Charlet${}^{11}$, D. Clarke${}^{5}$, B. Clerbaux${}^{4}$, S. Cocks${}^{20}$, J.G. Contreras${}^{8}$, C. Cormack${}^{20}$, J.A. Coughlan${}^{5}$, A. Courau${}^{28}$, M.-C. Cousinou${}^{24}$, B.E. Cox${}^{23}$, G. Cozzika${}^{9}$, D.G. Cussans${}^{5}$, J. Cvach${}^{31}$, S. Dagoret${}^{30}$, J.B. Dainton${}^{20}$, W.D. Dau${}^{17}$, K. Daum${}^{40}$, M. David${}^{9}$, C.L. Davis${}^{19,41}$, A. De Roeck${}^{11}$, E.A. De Wolf${}^{4}$, B. Delcourt${}^{28}$, M. Dirkmann${}^{8}$, P. Dixon${}^{19}$, W. Dlugosz${}^{7}$, C. Dollfus${}^{38}$, K.T. Donovan${}^{21}$, J.D. Dowell${}^{3}$, H.B. Dreis${}^{2}$, A. Droutskoi${}^{25}$, J. Ebert${}^{35}$, T.R. Ebert${}^{20}$, G. Eckerlin${}^{11}$, V. Efremenko${}^{25}$, S. Egli${}^{38}$, R. Eichler${}^{37}$, F. Eisele${}^{15}$, E. Eisenhandler${}^{21}$, E. Elsen${}^{11}$, M. Erdmann${}^{15}$, A.B. Fahr${}^{13}$, L. Favart${}^{28}$, A. Fedotov${}^{25}$, R. Felst${}^{11}$, J. Feltesse${}^{9}$, J. Ferencei${}^{18}$, F. Ferrarotto${}^{33}$, K. Flamm${}^{11}$, M. Fleischer${}^{8}$, M. Flieser${}^{27}$, G. Flügge${}^{2}$, A. Fomenko${}^{26}$, J. Formánek${}^{32}$, J.M. Foster${}^{23}$, G. Franke${}^{11}$, E. Gabathuler${}^{20}$, K. Gabathuler${}^{34}$, F. Gaede${}^{27}$, J. Garvey${}^{3}$, J. Gayler${}^{11}$, M. Gebauer${}^{36}$, R. Gerhards${}^{11}$, A. Glazov${}^{36}$, L. Goerlich${}^{6}$, N. Gogitidze${}^{26}$, M. Goldberg${}^{30}$, D. Goldner${}^{8}$, K. Golec-Biernat${}^{6}$, B. Gonzalez-Pineiro${}^{30}$, I. Gorelov${}^{25}$, C. Grab${}^{37}$, H. Grässler${}^{2}$, T. Greenshaw${}^{20}$, R.K. Griffiths${}^{21}$, G. Grindhammer${}^{27}$, A. Gruber${}^{27}$, C. Gruber${}^{17}$, T. Hadig${}^{1}$, D. Haidt${}^{11}$, L. Hajduk${}^{6}$, T. Haller${}^{14}$, M. Hampel${}^{1}$, W.J. Haynes${}^{5}$, B. Heinemann${}^{11}$, G. Heinzelmann${}^{13}$, R.C.W. Henderson${}^{19}$, H. Henschel${}^{36}$, I. Herynek${}^{31}$, M.F. Hess${}^{27}$, K. Hewitt${}^{3}$, K.H. Hiller${}^{36}$, C.D. Hilton${}^{23}$, J. Hladký${}^{31}$, M. Höppner${}^{8}$, D. Hoffmann${}^{11}$, T. Holtom${}^{20}$, R. Horisberger${}^{34}$, V.L. Hudgson${}^{3}$, M. Hütte${}^{8}$, M. Ibbotson${}^{23}$, Ç. İşsever${}^{8}$, H. Itterbeck${}^{1}$, A. Jacholkowska${}^{28}$, C. Jacobsson${}^{22}$, M. Jacquet${}^{28}$, M. Jaffre${}^{28}$, J. Janoth${}^{16}$, D.M. Jansen${}^{14}$, L. Jönsson${}^{22}$, D.P. Johnson${}^{4}$, H. Jung${}^{22}$, P.I.P. Kalmus${}^{21}$, M. Kander${}^{11}$, D. Kant${}^{21}$, U. Kathage${}^{17}$, J. Katzy${}^{15}$, H.H. Kaufmann${}^{36}$, O. Kaufmann${}^{15}$, M. Kausch${}^{11}$, S. Kazarian${}^{11}$, I.R. Kenyon${}^{3}$, S. Kermiche${}^{24}$, C. Keuker${}^{1}$, C. Kiesling${}^{27}$, M. Klein${}^{36}$, C. Kleinwort${}^{11}$, G. Knies${}^{11}$, T. Köhler${}^{1}$, J.H. Köhne${}^{27}$, H. Kolanoski${}^{39}$, S.D. Kolya${}^{23}$, V. Korbel${}^{11}$, P. Kostka${}^{36}$, S.K. Kotelnikov${}^{26}$, T. Krämerkämper${}^{8}$, M.W. Krasny${}^{6,30}$, H. Krehbiel${}^{11}$, D. Krücker${}^{27}$, A. Küpper${}^{35}$, H. Küster${}^{22}$, M. Kuhlen${}^{27}$, T. Kurča${}^{36}$, J. Kurzhöfer${}^{8}$, B. Laforge${}^{9}$, M.P.J. Landon${}^{21}$, W. Lange${}^{36}$, U. Langenegger${}^{37}$, A. Lebedev${}^{26}$, F. Lehner${}^{11}$, V. Lemaitre${}^{11}$, S. Levonian${}^{29}$, M. Lindstroem${}^{22}$, F. Linsel${}^{11}$, J. Lipinski${}^{11}$, B. List${}^{11}$, G. Lobo${}^{28}$, J.W. Lomas${}^{23}$, G.C. Lopez${}^{12}$, V. Lubimov${}^{25}$, D. Lüke${}^{8,11}$, L. Lytkin${}^{14}$, N. Magnussen${}^{35}$, H. Mahlke-Krüger${}^{11}$, E. Malinovski${}^{26}$, R. Maraček${}^{18}$, P. Marage${}^{4}$, J. Marks${}^{15}$, R. Marshall${}^{23}$, J. Martens${}^{35}$, G. Martin${}^{13}$, R. Martin${}^{20}$, H.-U. Martyn${}^{1}$, J. Martyniak${}^{6}$, T. Mavroidis${}^{21}$, S.J. Maxfield${}^{20}$, S.J. McMahon${}^{20}$, A. Mehta${}^{5}$, K. Meier${}^{16}$, P. Merkel${}^{11}$, F. Metlica${}^{14}$, A. Meyer${}^{11}$, A. Meyer${}^{13}$, H. Meyer${}^{35}$, J. Meyer${}^{11}$, P.-O. Meyer${}^{2}$, A. Migliori${}^{29}$, S. Mikocki${}^{6}$, D. Milstead${}^{20}$, J. Moeck${}^{27}$, F. Moreau${}^{29}$, J.V. Morris${}^{5}$, E. Mroczko${}^{6}$, D. Müller${}^{38}$, T. Walter${}^{38}$, K. Müller${}^{11}$, P. Murín${}^{18}$, V. Nagovizin${}^{25}$, R. Nahnhauer${}^{36}$, B. Naroska${}^{13}$, Th. Naumann${}^{36}$, I. Négri${}^{24}$, P.R. Newman${}^{3}$, D. Newton${}^{19}$, H.K. Nguyen${}^{30}$, T.C. Nicholls${}^{3}$, F. Niebergall${}^{13}$, C. Niebuhr${}^{11}$, Ch. Niedzballa${}^{1}$, H. Niggli${}^{37}$, G. Nowak${}^{6}$, T. Nunnemann${}^{14}$, M. Nyberg-Werther${}^{22}$, H. Oberlack${}^{27}$, J.E. Olsson${}^{11}$, D. Ozerov${}^{25}$, P. Palmen${}^{2}$, E. Panaro${}^{11}$, A. Panitch${}^{4}$, C. Pascaud${}^{28}$, S. Passaggio${}^{37}$, G.D. Patel${}^{20}$, H. Pawletta${}^{2}$, E. Peppel${}^{36}$, E. Perez${}^{9}$, J.P. Phillips${}^{20}$, A. Pieuchot${}^{24}$, D. Pitzl${}^{37}$, R. Pöschl${}^{8}$, G. Pope${}^{7}$, B. Povh${}^{14}$, S. Prell${}^{11}$, K. Rabbertz${}^{1}$, P. Reimer${}^{31}$, H. Rick${}^{8}$, S. Riess${}^{13}$, E. Rizvi${}^{21}$, P. Robmann${}^{38}$, R. Roosen${}^{4}$, K. Rosenbauer${}^{1}$, A. Rostovtsev${}^{30}$, F. Rouse${}^{7}$, C. Royon${}^{9}$, K. Rüter${}^{27}$, S. Rusakov${}^{26}$, K. Rybicki${}^{6}$, D.P.C. Sankey${}^{5}$, P. Schacht${}^{27}$, S. Schiek${}^{13}$, S. Schleif${}^{16}$, P. Schleper${}^{15}$, W. von Schlippe${}^{21}$, D. Schmidt${}^{35}$, G. Schmidt${}^{13}$, L. Schoeffel${}^{9}$, A. Schöning${}^{11}$, V. Schröder${}^{11}$, E. Schuhmann${}^{27}$, B. Schwab${}^{15}$, F. Sefkow${}^{38}$, A. Semenov${}^{25}$, V. Shekelyan${}^{11}$, I. Sheviakov${}^{26}$, L.N. Shtarkov${}^{26}$, G. Siegmon${}^{17}$, U. Siewert${}^{17}$, Y. Sirois${}^{29}$, I.O. Skillicorn${}^{10}$, T. Sloan${}^{19}$, P. Smirnov${}^{26}$, M. Smith${}^{20}$, V. Solochenko${}^{25}$, Y. Soloviev${}^{26}$, A. Specka${}^{29}$, J. Spiekermann${}^{8}$, S. Spielman${}^{29}$, H. Spitzer${}^{13}$, F. Squinabol${}^{28}$, P. Steffen${}^{11}$, R. Steinberg${}^{2}$, J. Steinhart${}^{13}$, B. Stella${}^{33}$, A. Stellberger${}^{16}$, J. Stier${}^{11}$, J. Stiewe${}^{16}$, U. Stößlein${}^{36}$, K. Stolze${}^{36}$, U. Straumann${}^{15}$, W. Struczinski${}^{2}$, J.P. Sutton${}^{3}$, S. Tapprogge${}^{16}$, M. Taševský${}^{32}$, V. Tchernyshov${}^{25}$, S. Tchetchelnitski${}^{25}$, J. Theissen${}^{2}$, G. Thompson${}^{21}$, P.D. Thompson${}^{3}$, N. Tobien${}^{11}$, R. Todenhagen${}^{14}$, P. Truöl${}^{38}$, G. Tsipolitis${}^{37}$, J. Turnau${}^{6}$, E. Tzamariudaki${}^{11}$, P. Uelkes${}^{2}$, A. Usik${}^{26}$, S. Valkár${}^{32}$, A. Valkárová${}^{32}$, C. Vallée${}^{24}$, P. Van Esch${}^{4}$, P. Van Mechelen${}^{4}$, D. Vandenplas${}^{29}$, Y. Vazdik${}^{26}$, P. Verrecchia${}^{9}$, G. Villet${}^{9}$, K. Wacker${}^{8}$, A. Wagener${}^{2}$, M. Wagener${}^{34}$, R. Wallny${}^{15}$, B. Waugh${}^{23}$, G. Weber${}^{13}$, M. Weber${}^{16}$, D. Wegener${}^{8}$, A. Wegner${}^{27}$, T. Wengler${}^{15}$, M. Werner${}^{15}$, L.R. West${}^{3}$, S. Wiesand${}^{35}$, T. Wilksen${}^{11}$, S. Willard${}^{7}$, M. Winde${}^{36}$, G.-G. Winter${}^{11}$, C. Wittek${}^{13}$, M. Wobisch${}^{2}$, H. Wollatz${}^{11}$, E. Wünsch${}^{11}$, J. Žáček${}^{32}$, D. Zarbock${}^{12}$, Z. Zhang${}^{28}$, A. Zhokin${}^{25}$, P. Zini${}^{30}$, F. Zomer${}^{28}$, J. Zsembery${}^{9}$, and
M. zurNedden${}^{38}$, ${}^{1}$ I. Physikalisches Institut der RWTH, Aachen, Germany${}^{a}$
${}^{2}$ III. Physikalisches Institut der RWTH, Aachen, Germany${}^{a}$
${}^{3}$ School of Physics and Space Research, University of Birmingham,
Birmingham, UK${}^{b}$
${}^{4}$ Inter-University Institute for High Energies ULB-VUB, Brussels;
Universitaire Instelling Antwerpen, Wilrijk; Belgium${}^{c}$
${}^{5}$ Rutherford Appleton Laboratory, Chilton, Didcot, UK${}^{b}$
${}^{6}$ Institute for Nuclear Physics, Cracow, Poland${}^{d}$
${}^{7}$ Physics Department and IIRPA,
University of California, Davis, California, USA${}^{e}$
${}^{8}$ Institut für Physik, Universität Dortmund, Dortmund,
Germany${}^{a}$
${}^{9}$ CEA, DSM/DAPNIA, CE-Saclay, Gif-sur-Yvette, France
${}^{10}$ Department of Physics and Astronomy, University of Glasgow,
Glasgow, UK${}^{b}$
${}^{11}$ DESY, Hamburg, Germany${}^{a}$
${}^{12}$ I. Institut für Experimentalphysik, Universität Hamburg,
Hamburg, Germany${}^{a}$
${}^{13}$ II. Institut für Experimentalphysik, Universität Hamburg,
Hamburg, Germany${}^{a}$
${}^{14}$ Max-Planck-Institut für Kernphysik,
Heidelberg, Germany${}^{a}$
${}^{15}$ Physikalisches Institut, Universität Heidelberg,
Heidelberg, Germany${}^{a}$
${}^{16}$ Institut für Hochenergiephysik, Universität Heidelberg,
Heidelberg, Germany${}^{a}$
${}^{17}$ Institut für Reine und Angewandte Kernphysik, Universität
Kiel, Kiel, Germany${}^{a}$
${}^{18}$ Institute of Experimental Physics, Slovak Academy of
Sciences, Košice, Slovak Republic${}^{f,j}$
${}^{19}$ School of Physics and Chemistry, University of Lancaster,
Lancaster, UK${}^{b}$
${}^{20}$ Department of Physics, University of Liverpool,
Liverpool, UK${}^{b}$
${}^{21}$ Queen Mary and Westfield College, London, UK${}^{b}$
${}^{22}$ Physics Department, University of Lund,
Lund, Sweden${}^{g}$
${}^{23}$ Physics Department, University of Manchester,
Manchester, UK${}^{b}$
${}^{24}$ CPPM, Université d’Aix-Marseille II,
IN2P3-CNRS, Marseille, France
${}^{25}$ Institute for Theoretical and Experimental Physics,
Moscow, Russia
${}^{26}$ Lebedev Physical Institute, Moscow, Russia${}^{f}$
${}^{27}$ Max-Planck-Institut für Physik,
München, Germany${}^{a}$
${}^{28}$ LAL, Université de Paris-Sud, IN2P3-CNRS,
Orsay, France
${}^{29}$ LPNHE, Ecole Polytechnique, IN2P3-CNRS,
Palaiseau, France
${}^{30}$ LPNHE, Universités Paris VI and VII, IN2P3-CNRS,
Paris, France
${}^{31}$ Institute of Physics, Czech Academy of
Sciences, Praha, Czech Republic${}^{f,h}$
${}^{32}$ Nuclear Center, Charles University,
Praha, Czech Republic${}^{f,h}$
${}^{33}$ INFN Roma 1 and Dipartimento di Fisica,
Università Roma 3, Roma, Italy
${}^{34}$ Paul Scherrer Institut, Villigen, Switzerland
${}^{35}$ Fachbereich Physik, Bergische Universität Gesamthochschule
Wuppertal, Wuppertal, Germany${}^{a}$
${}^{36}$ DESY, Institut für Hochenergiephysik,
Zeuthen, Germany${}^{a}$
${}^{37}$ Institut für Teilchenphysik,
ETH, Zürich, Switzerland${}^{i}$
${}^{38}$ Physik-Institut der Universität Zürich,
Zürich, Switzerland${}^{i}$
${}^{39}$ Institut für Physik, Humboldt-Universität,
Berlin, Germany${}^{a}$
${}^{40}$ Rechenzentrum, Bergische Universität Gesamthochschule
Wuppertal, Wuppertal, Germany${}^{a}$
${}^{41}$ Visitor from Physics Dept. University Louisville, USA
${}^{a}$ Supported by the Bundesministerium für Bildung, Wissenschaft,
Forschung und Technologie, FRG,
under contract numbers 6AC17P, 6AC47P, 6DO57I, 6HH17P, 6HH27I,
6HD17I, 6HD27I, 6KI17P, 6MP17I, and 6WT87P
${}^{b}$ Supported by the UK Particle Physics and Astronomy Research
Council, and formerly by the UK Science and Engineering Research
Council
${}^{c}$ Supported by FNRS-NFWO, IISN-IIKW
${}^{d}$ Partially supported by the Polish State Committee for Scientific
Research, grant no. 115/E-343/SPUB/P03/120/96
${}^{e}$ Supported in part by USDOE grant DE F603 91ER40674
${}^{f}$ Supported by the Deutsche Forschungsgemeinschaft
${}^{g}$ Supported by the Swedish Natural Science Research Council
${}^{h}$ Supported by GA ČR grant no. 202/96/0214,
GA AV ČR grant no. A1010619 and GA UK grant no. 177
${}^{i}$ Supported by the Swiss National Science Foundation
${}^{j}$ Supported by VEGA SR grant no. 2/1325/96
1 Introduction
In a previous publication [1] the H1 experiment analysed hadron
production in $ep$ deep–inelastic scattering (DIS) in the
Breit frame [2, 3] of reference. We showed similarities between
hadronic distributions in the ‘current’ hemisphere of this frame of reference
and those in a given single hemisphere of an $e^{+}e^{-}$ annihilation interaction.
A similar analysis [4] was performed by the ZEUS experiment. The
first H1 study analysed 1993 data resulting from $e^{-}p$ collisions. The 1994
run used incident positrons and this analysis has more than six times as many
events despite tighter cuts on event quality. Since the $Q^{2}$ values
investigated are still in general below the kinematic region where anything
other than virtual photon exchange is relevant, the results of the two
analyses are directly comparable.
It is the object of this analysis to study the evolution of the
fragmentation function and charged hadronic multiplicity distributions.
One aim is to continue the comparison with equivalent $e^{+}e^{-}$ results as a test
of the universality of properties of the partons found within a proton and
those of the quarks produced in pairs from the vacuum. Another aim is to show
sensitivity to the running of the QCD coupling constant and to leading order
(LO) QCD processes from consideration of such hadronic final states.
After a description of the relevant parts of the apparatus (section 2)
and a discussion of the treatment of the data (section 3), this paper describes
the Breit frame of reference (section 4) and extends (section 5) the earlier
analysis of the evolution of the fragmentation function with $Q^{2}$, where
$q^{2}=-Q^{2}$ is the square of the four-momentum transferred from the incident
lepton. The average charged multiplicity dependence on $Q$ (section 6) is
used to study the effects of LO QCD processes. For the first time we present
invariant energy spectra (section 7) from HERA and H1 multiplicity
distributions (section 8) in the Breit frame.
2 The H1 Detector
The H1 detector is described in detail elsewhere [5]. Here,
we give only a short description of the components more relevant to this
analysis. We employ a right-handed coordinate system with the
positive (forward) $z$ axis being in the direction of the incoming proton
beam. Polar angles are measured from this direction.
Momentum measurements of charged particles are provided, in the central
region of the apparatus, by two cylindrical and co-axial drift
chambers [6] for ($r,\phi$) measurement which have significantly fewer
dead cells compared with the situation in 1993. These detectors are
supplemented by two $z$-chambers. In the forward (proton) direction the
equivalent measurements are provided by three Radial and three Planar drift
chamber modules, the Forward Track Detector [7]. All these track
detectors are inside a uniform 1.15 T magnetic field. Track segments from all
devices are combined to give efficient detection and momentum measurement with
$\delta p/p^{2}$ $\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$1% /GeV for
most of the angular range used in this analysis, $10^{\circ}<\theta<160^{\circ}$.
In the polar angle range $4^{\circ}<\theta<153^{\circ}$ the trackers
are surrounded by a fine-grained liquid argon (LAr) sampling
calorimeter [8] with lead and steel absorber in the electromagnetic
and hadronic sections respectively. The calorimeter cells measure hadronic
energy flow and the energy of the scattered electron for high $Q^{2}$ events.
The LAr calorimeter is complemented by a backward electromagnetic,
lead-scintillator, calorimeter (BEMC [9]) covering the angular range
$151^{\circ}<\theta<176^{\circ}$. Behind this there is a double–layer scintillator
hodoscope which gives efficient background rejection by time of flight
measurement. The data are derived from two separate samples corresponding to
the scattered positron being triggered by, and detected in, either the BEMC or
the LAr calorimeter. The transition is at a $Q^{2}$ of about 100 GeV${}^{2}$ and these
samples are referred to in this paper as, respectively, low or high $Q^{2}$ data.
The trigger is already fully efficient at positron energies of 10 GeV.
3 Data Selection and Corrections
The full data sample, triggered as above, consists of some $150\,000$
events taken at an $ep$ centre of mass energy, $\sqrt{s}$, of 300 GeV. Event
kinematic variables used in this analysis are calculated using only the
scattered lepton, which gives both the best resolution in the chosen region
and clear freedom from bias on the hadronic system studied. Throughout, the
events are selected to have an identified scattered positron [10] with
an energy $E_{e}^{\prime}>14$ GeV.
To reject beam associated background, we demand that there be no veto
from the time of flight system and that a vertex is found for the event
within 30 cm of the nominal vertex position. The low (high) $Q^{2}$ data
referred to in the above section are selected to be within the limits
$12<Q^{2}<100$ GeV${}^{2}$ ($100<Q^{2}<8000$ GeV${}^{2}$). The dimensionless inelasticity
variable, $y=Q^{2}/xs$, is required to be in the range $0.05<y<0.6$. These
conditions ensure that contamination from mis-identified photoproduction events
is below the 1% level. Together these cuts remove $\sim$ 45% of the original
sample. In addition, a cut is placed to safely exclude a further $\sim$ 5%
of diffractive events which are not well-modelled with the DIS Monte Carlo
programs used in this analysis and which have no analogue in $e^{+}e^{-}$
interactions. We require that a total cluster energy of at least 0.5 GeV
should be observed in the region $4.4^{\circ}<\theta<15^{\circ}$.
Events are then selected in which a massless quark would be scattered
through $10^{\circ}<\theta<150^{\circ}$ using four momentum conservation with the
scattered positron. This is inside the acceptance of the H1 track detectors
and thus minimises corrections, at the cost of removing a further $\sim$ 24%
of events.
There is a source of error arising from QED radiation which comes
about because of an incorrect, typically overestimated, value of $Q^{2}$. The
error in the boost leads to a miscalculation of the direction of the Breit
frame axis. In most cases, this leads to an apparently empty, or at least
depleted, region of phase space where the scattered quark fragments are
expected. With improved statistics this has become a significant
effect. To reduce the size of the necessary QED corrections, the new analysis
utilises more severe cuts on the mass of the total hadronic system,
$W$, of $W^{2}>4400\,$ GeV${}^{2}$ both using the scattered lepton variables,
and separately from the hadronic system variables with the Jaquet Blondel
method [11].
Following all of the selection procedures there are $20\,810$ events in
the low $Q^{2}$ sample and $1\,250$ events at high $Q^{2}$.
In addition to these event selections there are also cuts
made to reject badly measured tracks. Any tracks in the central chambers
with transverse momentum below 150 MeV and those in the forward track detectors
with momentum below 500 MeV are removed. We also remove tracks failing minimum
requirements on the number of hits in a given chamber and the quality of the
track fit. In order to have only the
primary multiplicity, cuts are also made to exclude tracks not originating from
the interaction vertex. There remains a small excess contribution due to the
decays of short-lived strange particles.
To correct for this excess and for acceptance losses, we utilise the
DJANGO6 [12] Monte Carlo event generator. This combines a LO perturbative
QCD matrix element calculation and the colour dipole model of hadronisation
with a calculation of QED radiative effects. Radiative effects remaining after
our event selections have been corrected by comparing the results of Monte
Carlo calculations with and without the inclusion of QED radiative effects.
These corrections are at the $\sim$10%
level. The detector response is simulated using a program based
on the GEANT [13] package and the simulated events are reconstructed
and selected using exactly the same analysis chain as with real data.
The total bin-by-bin corrections made throughout this analysis are generally
well within $\pm$20% and vary smoothly in any given distribution.
The largest individual sources of systematic error for this analysis
originate in possible calibration errors of $\pm$1% for the BEMC and $\pm$3%
for the LAr electromagnetic calorimeter energy scales. These directly affect
the accuracy of Lorentz boosts and give rise to an uncertainty in the number
of tracks of $\sim$5% at low $Q^{2}$ and $\sim$8% at high $Q^{2}$, irrespective
of other kinematic selection. The corresponding systematic error in the Monte
Carlo derived
acceptance correction functions has been estimated to be $\sim$ 2% using
several different generators [14, 15]. Visual scans of real and simulated
data have ascertained that the efficiency of the track detectors is modelled
to an accuracy of better than 2% by our Monte Carlo simulations.
The full experimental details of how the corrected distributions of
this analysis are obtained from the data may be found in reference [16].
4 The Breit Frame of Reference
The $ep$ Breit frame is aligned with the hadronic centre of mass (HCM) but
boosted along a common $z$ direction such that the incident virtual photon has
zero energy, zero transverse momentum and a $z$ component of momentum $-Q$.
As with the laboratory frame of reference, we choose the positive $z$ axis to
be in the direction of the incoming proton. The negative
$z$ direction is referred to as the ‘current’ hemisphere of the
interaction. Our earlier Breit frame analysis [1] showed that
multiplicities in the current region of the Breit frame depend on $Q$ and
not on Bjorken $x$ ($\sim Q^{2}/W^{2}$), as opposed to the HCM where multiplicities
depend on the natural scale, $W$ [17]. In the naïve quark parton
model (QPM) the massless incoming quark has energy $Q/2$ and $z$ component
of momentum $+Q/2$, carrying an approximate fraction $x$ of the proton’s
momentum. After scattering it still has energy $Q/2$, with momentum $-Q/2$.
By comparison, we thus take the equivalent to the $e^{+}e^{-}$ centre of mass
energy, $E^{*}$, to be $Q$.
As compared with the HCM the Breit frame current hemisphere is
dominated by the fragments of the struck quark alone; the ‘spectator’ proton
remnants go entirely into the ‘target’ hemisphere, with much higher momentum.
There is excellent acceptance for the current hemisphere in the central H1
detector. For example, even in the highest $Q^{2}$ interval of this analysis
there is only an $\sim$8% contribution of tracks from the Forward Track
Detector. The HCM current hemisphere, in contrast, has an energy scale of
$W/2$ rather than $Q/2$ and generally
can not be seen in its entirety. This makes comparisons with a
complete $e^{+}e^{-}$ interaction hemisphere somewhat easier in the Breit frame.
When QCD corrections to the parton model are considered, the incident
parton carries, in general, a proportion of the incident proton’s momentum
that is larger than $x$ and has an energy larger than $Q/2$.
Furthermore, since the four-momentum of the photon is fixed the energy seen in
the current hemisphere may be either greater than or less than $Q/2$. In an
$e^{+}e^{-}$ interaction it is also the case that the energy in any given hemisphere
may be greater than or less than $E^{*}/2$. The effects of final state radiation
in $ep$ DIS and $e^{+}e^{-}$ interactions are similar, but other LO QCD processes which
affect DIS, initial state QCD radiation and boson gluon fusion (BGF), have no
equivalent in $e^{+}e^{-}$ annihilation. Note that, to order $\alpha_{s}$, i.e.
two exiting partons as well as the spectator system, DIS events may have
no energy in the current hemisphere when the incident parton has energy
above $Q$, other than that due to hadronisation [3]. Momentum
conservation ensures that this situation never exists in any given hemisphere
of an $e^{+}e^{-}$ interaction.
5 Evolution of the Fragmentation Function
The $ep$ Breit frame equivalent of the $e^{+}e^{-}$ scaled hadron momentum
$x_{p}=2p_{hadron}/E^{*}$ is $x_{p}=2p_{hadron}/Q$, where only hadrons in the
current hemisphere are considered. The event normalised distribution
$D(x_{p},Q^{2})=(1/N_{evts})\times dn_{tracks}^{\pm}/dx_{p}$, the fragmentation
function, characterises the processes by which partons shower and then
hadronise. In this paper, the intention is to present the spectra of
charged particles originating from the primary vertex following the
fragmentation of light quarks.
At high enough energies and for light enough final state particles,
fragmentation functions approximately scale and are “soft”, rising rapidly
at small $x_{p}$ and peaking near $x_{p}=0$. As the energy of the
initial parton increases, $D(x_{p},Q^{2})$ evolves into an even softer function
with increased probability for low $x_{p}$ hadrons at the expense of high $x_{p}$.
This scaling violation can be seen with the low and high $Q^{2}$ data
as a function of $x_{p}$ in Fig. 1(a) or in Fig. 2 where the
fragmentation function is plotted as a function of $Q$ for
different intervals of $x_{p}$. Typically, in this analysis, the intervals
of $x_{p}$ are much greater than the resolution ($\sim$6% rms). In these figures
the H1 data are compared with equivalent data from $e^{+}e^{-}$
experiments [18] plotted as a function of $E^{*}$. Most published
$e^{+}e^{-}$ results refer to full event multiplicities. Here, as elsewhere in this
analysis, track multiplicity data from $e^{+}e^{-}$ experiments have been halved to
correspond to the fragmentation of one timelike quark. Note that H1 as a
single experiment is able to measure the violation for a wide range of $Q$.
Fig. 2 also displays a prediction from a leading order QCD plus
hadronisation Monte Carlo (DJANGO6 [12]) calculation. The scaling
violation effect has a similar origin to the scaling violations in structure
functions, and with next to leading order calculations may be used as a test
of perturbative QCD [19].
When the fragmentation function is plotted as a function of the variable
$\xi=\ln(1/x_{p})$ the turn-over region is expanded. According to the
expectations of the Modified Leading Logarithmic Approximation (MLLA) and
Local Parton Hadron Duality (LPHD), $D(\xi)$ is Gaussian in the
neighbourhood of the peak [20]. Even the high statistics of this,
compared with our earlier, analysis see no significant deviation from this
Gaussian behaviour, as is clear in Fig. 1(b). Most track measurement
problems occur with particles of very high or very low momenta and these are
correlated with low and high $\xi$ respectively. To avoid any dependence on
these areas the fits of this analysis are restricted to a region within
one unit of the dimensionless $\xi$ on either side of the mean, but tests in
which this range is varied by $\pm$20% show no systematic difference.
The evolution of the fragmentation function may be summarised
by the $Q$, or $E^{*}$, dependence of the peak and width (dispersion)
values of the fitted Gaussian as is shown in Fig. 3 and in
Table 1. It is notable that the systematic effects discussed in
section 3 contribute only a total 2% uncertainty to the peak value and have
no significant effect on the width measurement. The results are
compatible with those published earlier, which had considerably lower precision,
as well as with those of an analysis [4] by the ZEUS collaboration.
The results are also compatible with various $e^{+}e^{-}$
experiments [18, 21, 22], where the relevant evolution variable is
the centre of mass energy, $E^{*}$. For this comparison and for the consideration
of multiplicities in the next section, these published $e^{+}e^{-}$ data have
been re-fitted by us to be in a directly comparable form. As well as the
previously mentioned factor of two to convert $e^{+}e^{-}$ event multiplicities to
quark multiplicities, we also reduce published total multiplicity results by
8% to account for $K^{0}$ and $\Lambda$ decay tracks [23, 24], and by a
variable factor below 3% to correct for the
increased multiplicity from $b$ quark fragmentation [25] in $e^{+}e^{-}$ .
Assuming gluon coherence MLLA/LPHD predicts a dependence of the
evolution of the peak and width of the distribution on the dimensionless
variable $Y=\ln(Q/\Lambda_{\hbox{\it eff}})$, where $\Lambda_{\hbox{\it eff}}$ is a scale parameter determining
a presumed cut–off of a parton shower. The prediction [20] gives the
peak,
$\xi_{peak}$ and the width, $\xi_{width}$, of the Gaussian approximation to be
$$\xi_{peak}=0.5Y+c_{2}\sqrt{Y}+{\cal K}$$
$$\xi_{width}=\sqrt{Y^{3\over 2}/2c_{1}}{\rm,}$$
where $c_{1}=\sqrt{36N_{c}/b}$ and $c_{2}=B\sqrt{{{1}\over{16}}b/N_{c}}$ with
$b={{11}\over{3}}N_{c}-{{2}\over{3}}N_{f}$ and
$B=({{11}\over{3}}N_{c}+{{2}\over{3}}N_{f}/N_{c}^{2})/b$ are constants dependent only
on the number of contributing colours, $N_{c}$, and flavours, $N_{f}$ within the
parton shower. For ease of comparison, we follow [21, 22] in the
assumption that this will be dominated by the light quarks and set $N_{f}=3$.
The term ${\cal K}$ contains higher order corrections and is expected to be
roughly constant and of order 1. The result of a simultaneous fit
($\chi^{2}$/NDF of 10/20) to the peak and width values obtained from the present
H1 data alone is shown as a solid line in Fig. 3 and yields the
values $\Lambda_{\hbox{\it eff}}=0.21\pm 0.02$ GeV and ${\cal K}=-0.43\pm 0.06$, in agreement with
the values $\Lambda_{\hbox{\it eff}}=0.21\pm 0.02$ GeV and ${\cal K}=-0.32\pm 0.06$ obtained in an
analysis [21] of the $\xi_{peak}$ evolution of combined $e^{+}e^{-}$ data.
6 The Average Charged Particle Multiplicity
The area underneath the fragmentation function is the average
charged multiplicity and this is given, after all corrections, in
Table 1. It is shown as a function of $Q$ in Fig. 4, with ZEUS
results [4] and with a curve which accurately
parameterises [24] the
average charged multiplicity from many $e^{+}e^{-}$ experiments as a function of $E^{*}$.
The error associated with this fit at any given energy is at the percent level.
There are predictions [26] that both the absolute
average charged DIS multiplicity and the shape of the evolution with $Q^{2}$
should agree with that of $e^{+}e^{-}$ experiments. As previous results indicated,
although there
was very good agreement between H1 and ZEUS [4] analyses and those of
$e^{+}e^{-}$ at high $Q(E^{*})$, the DIS analyses give smaller average charged
multiplicities at low values of $Q(E^{*})$. The new data confirm that this is a
large effect.
The discussion of section 4 indicated that the LO processes
which are present in $ep$ but not in $e^{+}e^{-}$ interactions can produce a
depletion of the current region of the Breit frame. Pure
QPM interactions or events with only final state radiation (i.e. $e^{+}e^{-}$
like events) might both be expected to have a current region hadronic state
characteristic of a quark recoiling with energy $\sim Q/2$ and with only
a $z$ component of momentum. BGF events with a depleted or empty current
hemisphere
or events with significant initial state QCD radiation would, on the contrary,
have a current region hadronic final state with significantly less energy
at an angle to the $z$ axis. However, hadronisation effects might well
mask these naïve expectations.
In order to investigate this experimentally, we add the four–momenta of
all calorimeter energy clusters in the current region of the Breit frame. The
energy component of the resultant four–momentum vector is defined to be
$E_{z<0}$ and the angle it makes with the $z$ axis to be $\Theta_{BF}$. We
then plot
$E_{z<0}/Q$ against $\cos\Theta_{BF}$, as shown in Fig. 5(a,b)
for the low and high $Q^{2}$ samples. From the above discussion we would expect
that $e^{+}e^{-}$ like events might cluster near ($\cos\Theta_{BF}=-1,E_{z<0}/Q=1/2$) in this plot, whereas events from the other LO processes would
either not appear in this plot or be spread away from this point, typically
towards lower energies and larger angles in the current hemisphere, that is to
the lower right-hand corner of this plot.
The high $Q^{2}$ sample, where $\alpha_{s}$ is small, $x$ is large, and the proton
has a
much smaller gluon content, shows obvious clustering near the (-1,$0.5$) point,
while this is much less evident at low $Q^{2}$.
Monte Carlo simulations qualitatively confirm the expectations but it
is hard to be quantitative, both because of the difficulty in strict
definitions of LO QCD event classes and because experimental resolution has a
large contribution to event migration on this plot. Nevertheless,
it is possible [27] to investigate the effect of choosing ever
more $e^{+}e^{-}$ like events by selecting according to the variable $R$, where
$R^{2}=(E_{z<0}/Q-0.5)^{2}+(\cos\Theta_{BF}+1)^{2}$. In order to compare with
$e^{+}e^{-}$ results, we divide $\left<n(R)\right>$, the average multiplicity of
events in an annulus of width $\Delta R=0.1$ at radius $R$, by the equivalent
single–hemisphere $e^{+}e^{-}$ multiplicity. There is a very small variation of $\left<Q\right>$ with $R$, so this $e^{+}e^{-}$ multiplicity is evaluated at an energy
corresponding to the average value of $Q$ of the H1 data in that annulus. The
behaviour of this ratio as a function of $R$ can be seen, with statistical
error bars only, in Fig. 6(a,b). The systematic errors corresponding
to the shaded region are as discussed in section 3. Events with a totally
empty current hemisphere have zero multiplicity but no defined value of $R$,
and are thus not included in these plots. As $R\to 0$ the average charged
hadronic multiplicity increases, possibly even beyond the corresponding
$e^{+}e^{-}$ values, although it is not inconsistent with $e^{+}e^{-}$ values given the
systematic errors involved. Our Monte Carlo simulations also show this general
behaviour.
The above result and discussions motivate a cut to remove extreme low
energy and off–axis events. This is shown as a solid line in
Fig. 5(a,b) joining the points (-1,0) and (0,0.5) in this plane and
is termed the ‘Breit frame energy flow selection’ in the rest of this paper. It
produces samples of $11\,735$ events and $1\,005$ events at low and high $Q^{2}$
respectively, which are less affected by the low–order QCD processes peculiar
to DIS.
This selection is by no means as exact as the technique of letting
$R\to 0$ but in circumstances where the data are already binned in other
variables, e.g. energy spectra or multiplicity distributions, gives an
indication of whether or not distributions move in the direction of closer
similarity between this data and $e^{+}e^{-}$ data.
As is clear from Table 1, the peak and width values of the
fragmentation function are insensitive to this energy flow selection procedure.
The average multiplicity, however, is sensitive and as expected increases
with this selection, especially at low $Q^{2}$.
7 Invariant Energy Spectra
The evolution of the fragmentation function at low energies, or high values of
$\xi$, may be better studied using the event–normalised “invariant”
spectrum, $(1/N_{evts})\times Edn_{tracks}^{\pm}/d^{3}p$. According to the
predictions of perturbative QCD based on the MLLA and LPHD, the
hadronic spectrum at low momentum should be nearly independent of the energy of
the parent parton [28]. Here we follow [28] in taking
$d^{3}p=4\pi p^{2}_{hadron}dp_{hadron}$
and by computing the energy, $E$, of each track as if it had a mass
$Q_{0}(=$270 MeV$\approx\Lambda_{QCD})$ given by a presumed cut-off mass for the
parton shower, i.e. $E=\sqrt{Q_{0}^{2}+p_{hadron}^{2}}$. The data are corrected
for acceptance with the same bin-by-bin method described in section 3. The
correction function is both smooth and close to unity, having an average value
of around 1.15. Although there is always a loss of track acceptance at low
laboratory momentum the boost reduces this loss in the Breit frame. This means
that $ep$ experiments can get closer to the critical low energy limit without
serious loss of acceptance, than can $e^{+}e^{-}$ experiments.
The results are shown in Fig. 7(a,b) where the indicated errors contain
contributions from statistical and (negligible) momentum measurement effects
as well as systematic effects that have already been discussed and which grow
slightly with track energy. The solid lines show MLLA/LPHD predictions at
the relevant $Q$ values for a simple quark behaviour using the calculations
of [28] which the authors show are in agreement with $e^{+}e^{-}$ data. It is
clear that at low $Q^{2}$ these predictions for the behaviour of quark fragments
are in disagreement with the data but are significantly closer to data
subject to the Breit frame energy flow selection described in the previous
section. This selection makes little difference at high $Q^{2}$, where agreement
is better anyway. The QCD predictions are very sensitive to the ‘running’ of the
strong coupling constant, $\alpha_{s}$, as the energy scale alters in the parton
shower. A prediction is shown for the same average energy, normalised to
interpolated $\left<n^{\pm}\right>$ multiplicities in $e^{+}e^{-}$ experiments, but
utilising a fixed value of $\alpha_{s}$ (set to 0.215, a value which roughly
describes the energy dependence of $\left<n^{\pm}\right>$ and the slope of the
first moment of the energy spectrum in $e^{+}e^{-}$ experiments [29]). The
data show that such an assumption is clearly untenable. This calculation is
only in leading order so this cannot yet be turned
around to compute a meaningful value of $\Lambda_{QCD}$.
Data subject to the energy flow selection are re-plotted in six intervals of
$Q^{2}$ in Fig. 7(c) in order to display the clear evidence for a common
limit at low particle energy independent of event momentum transfer.
8 Charged Particle Multiplicity Distributions
Charged hadron multiplicity distributions are considerably more
difficult to measure than average values. In this analysis we have tested a
number of ways of extracting true distributions from observed distributions. The best method is
found to be a simple bin-by-bin correction procedure on the multiplicity
distribution. This has, however, the problem of not modelling correlations
which then gives the method a dependence on the Monte Carlo generator.
We tried two further methods to solve this difficulty.
It is straightforward to use a Monte Carlo simulation to form a generated,
or ‘true’, to reconstructed distribution transfer matrix, which is
physics-generator independent. Inversion of this matrix in order to extract a
true
distribution from a raw input distribution has many problems due to the small
number of high multiplicity events, resulting in the matrix becoming singular
and unstable. We have also tried a method [17, 30] of
approximating the inverted matrix with its inverse and then re-weighting the
input Monte Carlo distribution in successive approximation iterations to
minimise dependence on the initial generated distribution.
The systematic errors of all three techniques have been estimated by
degrading the output distributions back to the ‘raw’ (reconstructed) data level
using the reliable ‘generated to reconstructed’ matrix. This can then be
compared with the original, real input distribution. We find that the larger
dependence on the generated distribution of the bin-by-bin correction method
is still offset by overall smaller systematic errors and the ability to get
meaningful corrections at high multiplicity when the statistics are low. We
thus choose this method to present our results. Quantitatively, the average
correction made per bin is 16%(32%) at low (high) $Q^{2}$ with an error of
method assessed as 10%(17%).
The resulting multiplicity distributions are
given in Table 2 and shown in Fig. 8 in the form of
$P(n)$, the probability of observing a charged multiplicity $n$.
These results are for the multiplicity distribution in the current hemisphere
of all events and do not include any energy flow selection. The bins of
$x$ and $Q^{2}$ are exclusive but, because of the event selections, are not
totally occupied at the edges.
This leads to only minor discrepancies in the average values of $x$ and $Q^{2}$
between neighbouring bins. In general the observation is that the results are
compatible with a gradual shift to higher multiplicities with increasing $Q^{2}$
but with only a small change as a function of $x$ which might be due to minor
bin edge effects. It is interesting to note the many zero-charged multiplicity
events, especially at low $Q^{2}$ where we might expect higher-order QCD
processes to dominate.
Integrating over $x$ gives smaller statistical errors and a
greater reach in multiplicity. The evolution with $Q$ of the shape of the
multiplicity distribution may be studied by normalising with the
average charged hadron multiplicity to give the KNO [31] form
$\Psi(z)=\left<n\right>\times P(n)$, where $z=n/\left<n\right>$. Equivalent
single-hemisphere $e^{+}e^{-}$ distributions [24, 32, 33] exhibit clear
scaling in this variable but only for high ($E^{*}>20$ GeV) energies.
In an analysis [17] of multiplicity distributions in a limited region
of the hadronic centre of mass frame, the H1 collaboration has also observed
good agreement with a common distribution as $W$ increases between 80 and
220 GeV. The results of this
analysis in the Breit frame, shown in Fig. 9(a), are for $Q$ values
much lower than the $W$ values and show a clear violation of KNO scaling,
as also reported by the ZEUS collaboration [4]. For reference, we
display two sets of lines joining $e^{+}e^{-}$ data at an energy close to our largest
$Q$ range [32] and also at LEP energies [24]. As with $e^{+}e^{-}$
data, the H1 low $Q^{2}$ distribution starts with a flatter curve and a large
contribution from low multiplicities. Then, as $Q^{2}$ increases, the data get
closer to the high energy scaling $e^{+}e^{-}$ distributions.
In Fig. 9(b) the distribution for low $Q^{2}$ events is compared
with results from the $e^{+}e^{-}$ JETSET [34] parton shower model, using
parameter settings determined by the DELPHI collaboration [35] at an
energy set to be close to the mean value of the low $Q^{2}$ data. The
data
distribution is again much flatter but a better agreement is achieved when
using the Breit frame energy flow selection discussed in section 6.
This KNO scaling violation is also seen in $ep$ DIS Monte Carlo generated
events, where it is to a large extent caused by leading
order perturbative QCD events, especially BGF events, which, as has already
been noted, have significantly lower multiplicities in the current hemisphere
of the Breit frame. It thus appears
that $ep$ and $e^{+}e^{-}$ data share a flattening of the KNO distribution at low
energies but that there is an additional
sensitivity to the effects of higher order QCD processes in the Breit frame
which do not contribute to $e^{+}e^{-}$ events.
9 Summary
The Breit frame fragmentation function analysis
has been repeated with the higher statistics of 1994 data and is
now extended to give multiplicity distributions as functions of both
$x$ and $Q^{2}$. Invariant energy spectra are shown to be incompatible with a
calculation utilising a constant value for $\alpha_{s}$. We discuss a
method of discriminating against events in kinematic regions dominated by
QCD processes which do not contribute to $e^{+}e^{-}$ interactions.
We show that some discrepancies between $ep$ Breit frame current hemisphere
and $e^{+}e^{-}$ average charged multiplicities, including an observed violation of
KNO scaling at low $Q^{2}$, decrease as conditions are progressively made more
comparable. With a selection which gives similar kinematic conditions to that
of a single hemisphere $e^{+}e^{-}$ annihilation, the $ep$ energy spectra can be shown
to be compatible with a common, $Q^{2}$-independent, low momentum limit.
In summary, the data
give strong support to the concept of quark fragmentation universality.
Acknowledgments
We are grateful to the HERA machine group whose outstanding
efforts have made and continue to make this experiment possible. We thank
the engineers and technicians for their work in constructing and now
maintaining the H1 detector, our funding agencies for financial support, the
DESY technical staff for continual assistance, and the DESY directorate for the
hospitality which they extend to the non–DESY members of the collaboration.
We are grateful to S. Lupia for kind assistance with obtaining MLLA/LPHD
predictions.
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Asymptotically exact theory of fiber-reinforced composite beams
K. C. Le${}^{a,b}$111Email: [email protected], T. M. Tran${}^{a,b}$
${}^{a}$Materials Mechanics Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam
${}^{b}$Faculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam
Abstract
An asymptotic analysis of the energy functional of a fiber-reinforced composite beam with a periodic microstructure in its cross section is performed. From this analysis the asymptotically exact energy as well as the 1-D beam theory of first order is derived. The effective stiffnesses of the beam are calculated in the most general case from the numerical solution of the cell and homogenized cross-sectional problems.
keywords:
fiber-reinforced, composite, beam, periodic microstructure, variational-asymptotic method.
††journal: Composite Structures
1 Introduction
Fibre reinforced composite (FRC) beams are widely used in civil, mechanical, and aerospace engineering due to their low weight, high strength, and good damping properties (Hodges, 1990; Chandra et al., 1999; Librescu and Song, 2005). The exact treatment of FRC beams within the 3D elasticity is only possible in a few exceptional cases due to their complicated microstructure (see e.g. (Rodriguez et al., 2001; Sendeckyj, 2016) and the references therein). For this reason, different approaches have been developed depending on the type of beams. If FRC beams are thick, no exact one-dimensional theory can be constructed, so only the numerical methods or approximate semi-analytical methods applied to three-dimensional elasticity theory make sense. However, if the FRC beams are thin, the reduction from the three-dimensional to the one-dimensional theory is possible. This dimension reduction can be made asymptotically exact in the limit when the thickness-to-length ratio of the beam goes to zero. The rigorous derivation of the asymptotically exact one-dimensional beam theory based on the variational-asymptotic method (VAM) developed by Berdichevsky (1979) was first performed in (Berdichevsky, 1981). His asymptotic analysis shows that the static (or dynamic) three-dimensional problem of the beam can be split into two problems: (i) the two-dimensional cross-sectional problem and (ii) the one-dimensional variational problem whose energy functional should be found from the solution of the cross-sectional problem. The latter has been solved both for anisotropic homogeneous beams and for inhomogeneous beams with the constant Poisson’s ratio (Berdichevsky, 1981). In addition to these findings, Berdichevsky (1981) has shown that the average energy as well as the extension and bending stiffnesses of FRC beams with piecewise constant Poisson’s ratio must be larger than those of FRC beams with constant Poisson’s ratio (see also (Muskhelishvili, 2013)). However, to our knowledge, the question of how the corrections in energy and stiffnesses depend on the difference in Poisson’s ratio of fibers and matrix of FRC beams remains still an issue. It should be noted that VAM has been further developed in connection with the numerical analysis of cross-sectional problems for geometrically nonlinear composite beams by Hodges, Yu and colleagues in a series of papers (Hodges et al., 1992; Yu et al., 2002, 2005; Yu and Blair, 2012). Note also that VAM has been used, among others, to derive the 2D theory of homogeneous piezoelectric shells (Le, 1986b), the 2D theory of purely elastic anisotropic and inhomogeneous shells (Berdichevsky, 2009), the 2D theory of purely elastic sandwich plates and shells (Berdichevsky, 2010a, b), the theory of smart beams (Roy et al., 2007), the theory of low and high frequency vibrations of laminate composite shells (Lee and Hodges, 2009a, b), and more recently, the theory of smart sandwich and functionally graded shells (Le and Yi, 2016; Le, 2017).
For FRC beams that have the periodic microstructure in the cross section, an additional small parameter appears in the cross-sectional problems: The ratio between the length of the periodic cell and the characteristic size of the cross section. This allows an additional asymptotic analysis of the cross-sectional problems to simplify them. By solving the cell problems imposed with the periodic boundary conditions according to the homogenization technique (Braides, 2002; Milton, 2003), one finds the effective coefficients in the homogenized cross-sectional problems, which can then be solved analytically or numerically. The aim of this paper is to derive and solve the cell and homogenized cross-sectional problems for unidirectional FRC beams whose cross section has the periodic microstructure with piecewise constant Poisson’s ratio. The solution of the cell problems found with the finite element method is used to calculate the asymptotically exact energy and the strain and bending stiffnesses in the 1-D theory of FRC beams. Thus, we determine the dependence of the latter quantities on the shape and volume fraction of the fibers and on the difference in Poisson’s ratio of fibers and matrix that solves the above mentioned issue.
The paper is organized as follows. After this short introduction the variational formulation of the problem is given in Section 2. Sections 3 and 4 are devoted to the multi-scaled asymptotic analysis of the energy functional of FRC beams leading to the cross-sectional and cell problems. In Section 5 the cell problems are solved by the finite element method. Section 6 provides the solutions of the homogenized cross-sectional problems. Section 7 present one-dimensional theory of FRC beams. Finally, Section 8 concludes the paper.
2 Variational formulation for FRC beams
Consider a FRC beam that occupies the domain $\mathcal{B}=\mathcal{A}\times(0,L)$ of the 3-D euclidean space in its undeformed state. Let $x_{3}\equiv x$ be the coordinate along the beam axis. The cross section of the beam in the $(x_{1},x_{2})$-plane, $\mathcal{A}$, consist of two separated 2-D sub-domains $\mathcal{A}^{m}$ and $\mathcal{A}^{f}$ such that the matrix occupies the domain $\mathcal{A}^{m}\times(0,L)$, while the uni-directional fibers occupy the domain $\mathcal{A}^{f}\times(0,L)$. We choose the origin of the $(x_{1},x_{2})$-coordinates so that it matches the centroid of $\mathcal{A}$. We assume that the fibers are periodically situated in the matrix and the bond between the fibers and the matrix is perfect (see Fig. 1 representing the cross section of the beam where $\mathcal{A}^{f}$ is the set of gray circles). We first consider the beam in equilibrium, whose deformation is completely determined by the displacement field ${\bf u}(\mathbf{x})$. Let, for simplicity, the edge $x=0$ of the beam be clamped, while at the other edge $x=L$ of the beam the traction $\mathbf{t}(x_{1},x_{2})$ be specified. Gibbs’ variational principle (Berdichevsky, 2009) states that the true displacement $\check{\mathbf{u}}(\mathbf{x})$ minimizes the energy functional
$$I[\mathbf{u}(\mathbf{x})]=\int_{\mathcal{B}}W(\mathbf{x},\boldsymbol{%
\varepsilon})\differential[3]{x}-\int_{\mathcal{A}}\mathbf{t}(x_{1},x_{2})%
\dotproduct\mathbf{u}(x_{1},x_{2},L)\differential[2]{x}$$
(1)
among all admissible displacements $\mathbf{u}(\mathbf{x})$ satisfying the boundary conditions
$$\mathbf{u}(x_{1},x_{2},0)=0.$$
Here, $\differential[3]{x}=\differential x_{1}\differential x_{2}\differential x$ is the volume element, $\differential[2]{x}=\differential x_{1}\differential x_{2}$ the area element, and the dot denotes the scalar product. Function $W(\mathbf{x},\boldsymbol{\varepsilon})$, called stored energy density, reads
$$W(\mathbf{x},\boldsymbol{\varepsilon})=\frac{1}{2}\lambda(\mathbf{x})(\tr%
\boldsymbol{\varepsilon})^{2}+\mu(\mathbf{x})\boldsymbol{\varepsilon}\mathbf{:%
}\boldsymbol{\varepsilon},$$
where $\boldsymbol{\varepsilon}$ is the strain tensor
$$\boldsymbol{\varepsilon}=\frac{1}{2}(\gradient\mathbf{u}+(\gradient\mathbf{u})%
^{T}).$$
The problem is to replace the three-dimensional energy functional by an approximate one-dimensional energy functional for a thin FRC beam, whose functions depend only on the longitudinal co-ordinate $x$. The possibility of reduction of the three- to the one-dimensional functional is related to the smallness of two parameters: (i) the ratio between the thickness $h$ of the cross section and the length $L$ of the beam, and (ii) the ratio between the size $c$ of the periodic cell and $h$. By using the variational-asymptotic method, the one-dimensional energy functional will be constructed below in which terms of the order $h/L$ and $\epsilon=c/h$ are neglected as compared with unity (the first-order or “classical” approximation). Formally, this corresponds to the limits $h\to 0$ and $\epsilon\to 0$.
In order to perform the variational-asymptotic analysis, it is convenient to use the index notation, with Greek indices running from 1 to 2 and denoting the component of vectors and tensors in the $(x_{1},x_{2})$-plane. Summation over repeated Greek indices is understood. Index 3 of the coordinate $x_{3}$, the displacement $u_{3}$, and the traction $t_{3}$ is dropped for short. To fix the domain of the transverse co-ordinates in the passage to the limit $h\to 0$, we introduce the dimensionless co-ordinates
$$y_{\alpha}=\frac{x_{\alpha}}{h},\quad y_{\alpha}\in\bar{\mathcal{A}},$$
and transform the energy functional of the beam to
$$I=\int_{0}^{L}\int_{\bar{\mathcal{A}}}h^{2}W(y_{\alpha},\boldsymbol{%
\varepsilon})\differential[2]{y}\differential{x}-\int_{\bar{\mathcal{A}}}h^{2}%
\mathbf{t}(y_{\alpha})\dotproduct\mathbf{u}(y_{\alpha},L)\differential[2]{y}.$$
(2)
The transverse coordinates $y_{\alpha}$ play the role of the “fast” variables as opposed to the slow variable $x$. Regarding $\mathbf{u}$ as function of $y_{\alpha}$ and $x$, we separate the fast variables from the slow one. Now $h$ enters the action functional explicitly through the components of the strain tensor $\boldsymbol{\varepsilon}$
$$\varepsilon_{\alpha\beta}=\frac{1}{h}u_{(\alpha;\beta)},\quad 2\varepsilon_{%
\alpha 3}=\frac{1}{h}u_{;\alpha}+u_{\alpha,x},\quad\varepsilon_{33}=u_{,x}.$$
(3)
Here and below, the semicolon preceding Greek indices denotes the derivatives with respect to $y_{\alpha}$, while the parentheses surrounding a pair of indices the symmetrization operation.
Besides $y_{\alpha}$ there are also much faster variables
$$z_{\alpha}=\frac{x_{\alpha}}{c}=\frac{y_{\alpha}}{\epsilon}$$
associated with the periodicity of the elastic moduli of this composite materials in the transverse directions leading to the fast oscillation of the stress and strain fields in these directions. In order to characterize these faster changes of the stress and strain fields in the transverse directions we will assume that the displacement field $\mathbf{u}$ is a composite function of $\mathbf{y}$ and $x$ such that
$$\mathbf{u}=\mathbf{u}(\mathbf{y},\mathbf{y}/\epsilon,x)=\mathbf{u}(\mathbf{y},%
\mathbf{z},x),$$
where $\mathbf{y},\mathbf{z}$ are two-dimensional vectors with the components $y_{\alpha}$, $z_{\alpha}$, respectively, and where $\mathbf{u}(\mathbf{y},\mathbf{z},x)$ is a doubly periodic function in $\mathbf{z}$ with the period 1. This is a typical multi-scale Ansatz to composite structures. The asymptotic analysis must therefore be performed twice, first in the limit $h\to 0$ to realize the dimension reduction, and then in the limit $\epsilon\to 0$ to homogenize the cross-sectional problems and solve them.
3 Dimension reduction
Before starting the asymptotic analysis of the energy functional in the limit $h\to 0$ let us transform the stored energy density to another form more convenient for this purpose (Le, 1999). We note that, among terms of $W(\mathbf{z},\boldsymbol{\varepsilon})$, the derivatives $w_{(\alpha;\beta)}/h$ in $\varepsilon_{\alpha\beta}$ and $w_{;\alpha}/h$ in $\varepsilon_{\alpha 3}$ are the main ones in the asymptotic sense. Therefore it is convenient to single out the components $\varepsilon_{\alpha\beta}$ and $\varepsilon_{\alpha 3}$ in the stored energy density. We represent the latter as the sum of three quadratic forms $W_{\parallel}$, $W_{\angle}$, and $W_{\perp}$ according to
$$W_{\parallel}=\min_{\varepsilon_{\alpha\beta},\varepsilon_{\alpha 3}}W,\quad W%
_{\angle}=\min_{\varepsilon_{\alpha\beta}}(W-W_{\parallel}),\quad W_{\perp}=W-%
W_{\parallel}-W_{\angle}.$$
The “longitudinal” energy density $W_{\parallel}$ depends only on $\varepsilon_{33}$ and coincides with $W$ when the stresses $\sigma_{\alpha\beta}$ and $\sigma_{\alpha 3}$ vanish; the “shear” energy density $W_{\angle}$ depends only on $\epsilon_{\alpha 3}$; the remaining part $W_{\perp}$ is called the “transverse” energy density. From the definitions of $W_{\parallel}$, $W_{\angle}$ and
$W_{\perp}$ one easily finds out that
$$\displaystyle W_{\parallel}=\frac{1}{2}E(\varepsilon_{33})^{2},\quad E=\frac{%
\mu(3\lambda+2\mu)}{\lambda+\mu},$$
$$\displaystyle W_{\angle}=2\mu\varepsilon_{\alpha 3}\varepsilon_{\alpha 3},$$
$$\displaystyle W_{\perp}=\frac{1}{2}\lambda(\varepsilon_{\alpha\alpha}+2\nu%
\varepsilon_{33})^{2}+\mu(\varepsilon_{\alpha\beta}+\nu\delta_{\alpha\beta}%
\varepsilon_{33})(\varepsilon_{\alpha\beta}+\nu\delta_{\alpha\beta}\varepsilon%
_{33}),$$
where $E$ is the Young modulus and $\nu=\lambda/2(\lambda+\mu)$ the Poisson ratio. Note that $E$, $\nu$ as well as Lame’s constants $\lambda$, $\mu$ are doubly periodic functions of the fast variables $\mathbf{z}$. Note also the following identities
$$\displaystyle W_{\angle}$$
$$\displaystyle=\sigma_{\alpha 3}\varepsilon_{\alpha 3},$$
$$\displaystyle W_{\perp}$$
$$\displaystyle=\frac{1}{2}(\sigma_{\alpha\beta}\varepsilon_{\alpha\beta}+\sigma%
_{33}\varepsilon_{33}-E(\varepsilon_{33})^{2}).$$
We could start the variational-asymptotic analysis in the limit $h\to 0$ with the determination of the set $\mathcal{N}$ according to its general scheme (Le, 1999). As a result, it would turn out that, at the first step, the function ${\bf u}$ does not depend on the transverse co-ordinates $\mathbf{y}$ (and $\mathbf{z}$): ${\bf u}={\bf v}(x)$; at the second step the function ${\bf u}^{\star}$ becomes a linear function of $\mathbf{y}$ and involves one more degree of freedom $\varphi(x)$ representing the twist angle; and at the next step ${\bf u}^{\star\star}$ is completely determined through ${\bf u}$ and $\varphi$. Thus, the set $\mathcal{N}$ according to the variational-asymptotic scheme consists of functions ${\bf v}(x)$ and $\varphi(x)$. We will pass over these long, but otherwise standard, deliberations and make a change of unknown functions according to
$$\begin{split}\displaystyle u_{\alpha}(\mathbf{y},\mathbf{z},x)=v_{\alpha}(x)-%
he_{\alpha\beta}\varphi(x)y_{\beta}+hw_{\alpha}(\mathbf{y},\mathbf{z},x),\\
\displaystyle u(\mathbf{y},\mathbf{z},x)=v(x)-hv_{\alpha,x}(x)y_{\alpha}+hw(%
\mathbf{y},\mathbf{z},x),\end{split}$$
(4)
where $e_{\alpha\beta}$ are the two-dimensional permutation symbols ($e_{11}=e_{22}=0,e_{12}=-e_{21}=1$). By redefining $\mathbf{v}(x)$ and $\varphi(x)$ if necessary we can impose on functions $w_{\alpha}$ and $w$ the following constraints
$$\begin{split}\displaystyle\langle w_{\alpha}\rangle=0,\quad e_{\alpha\beta}%
\langle w_{\alpha;\beta}\rangle=0,\\
\displaystyle\quad\langle w\rangle=0,\quad\langle.\rangle\equiv\int_{\bar{%
\mathcal{A}}}.\differential[2]{y}.\end{split}$$
(5)
According to these constraints $v_{\alpha}(x)$ and $v(x)$ describe the mean displacements of the beam, while $\varphi(x)$ corresponds to the mean rotation of its cross section. Equations (4) and (5) set up a one-to-one correspondence between $u_{\alpha},u$ and the set of functions $v_{\alpha},v,\varphi,w_{\alpha},w$ and determine the change in the unknown functions $\{u_{\alpha},u\}\to\{v_{\alpha},v,\varphi,w_{\alpha},w\}$.
Based on the Saint-Venant principle for elastic beams, we may assume that the domain occupied by the beam consists of the inner domain and two boundary layers near the edges of the beam with width of the order $h$ where the stress and strain states are essentially three-dimensional. Then functional (2) can be decomposed into the sum of two functionals, an inner one for which an iteration process will be applied, and a boundary layer functional. When searching for $w_{\alpha}$ and $w$ the boundary layer functional can be neglected in the first-order approximation. Therefore, the dimension reduction problem reduces to finding the minimizer $\check{w}_{\alpha}$ and $\check{w}$ of the inner functional that, in the limit $h\to 0$, can be identified with the functional (2) without the last term.
We now fix $v_{\alpha},v,\varphi$ and seek $w_{\alpha},w$. Substituting (4) into the action functional (2) with the last term being removed, we will keep in it the asymptotically principal terms containing $w_{\alpha},w$ and neglect all other terms. The estimations based on Eqs. (3) and (4) lead to the asymptotic formulas
$$\varepsilon_{\alpha\beta}=w_{(\alpha;\beta)},\quad 2\varepsilon_{\alpha 3}=w_{%
;\alpha}-he_{\alpha\beta}\Omega y_{\beta},\quad\varepsilon_{33}=\gamma+h\Omega%
_{\alpha}y_{\alpha},$$
(6)
where $\gamma$, $\Omega_{\alpha}$, and $\Omega$ are the measures of elongation, bending, and twist defined by
$$\gamma=v_{,x},\quad\Omega_{\alpha}=-v_{\alpha,xx},\quad\Omega=\varphi_{,x}.$$
According to equations (6) the partial derivatives of $w_{\alpha},w$ with respect to
$x$ do not enter the energy functional. As $x$ becomes the formal parameter, we may drop the integral over $x$ and reduce the determination of $w_{\alpha},w$ to the uncoupled minimization problems for every fixed $x$ of the functionals
$$\displaystyle I_{\perp}[w_{\alpha}]=h^{2}\langle\frac{1}{2}\lambda(\mathbf{z})%
[w_{\alpha;\alpha}+2\nu(\mathbf{z})(\gamma+h\Omega_{\sigma}y_{\sigma})]^{2}\\
\displaystyle+\mu(\mathbf{z})[w_{(\alpha;\beta)}+\nu(\mathbf{z})\delta_{\alpha%
\beta}(\gamma+h\Omega_{\gamma}y_{\gamma})][w_{(\alpha;\beta)}+\nu(\mathbf{z})%
\delta_{\alpha\beta}(\gamma+h\Omega_{\delta}y_{\delta})]\rangle,$$
(7)
$$I_{\angle}[w]=h^{2}\langle\frac{1}{2}\mu(\mathbf{z})(w_{;\alpha}-h\Omega e_{%
\alpha\beta}y_{\beta})(w_{;\alpha}-h\Omega e_{\alpha\gamma}y_{\gamma})\rangle.$$
(8)
The minima are searched among all admissible functions $w_{\alpha},w$ satisfying the constraints (5). Functionals (7) and (8) represent the transverse and shear strain energies, integrated over the cross section of the beam. They are positive definite and convex, so the existence of their minimizers $\check{w}_{\alpha},\check{w}$ is guaranteed. We shall see in the next Sections that the minimum of (7) is equal to zero if $\nu$ is equal for both matrix and fiber, while that of (8) is equal to $1/2C\Omega^{2}$, with $C$ the torsional rigidity.
4 Homogenization
Consider now the other limit $\epsilon\to 0$. In this limit $\mathbf{y}$ plays the role of the “slow” variable, while $\mathbf{z}=\mathbf{y}/\epsilon$ becomes the fast variable. We start with the cross-sectional problem of minimizing functional (8) among $w(\mathbf{y},\mathbf{z},x)$ satisfying the constraint (5)${}_{3}$, where $\mu(\mathbf{z})$, expressed in terms of the fast variable $\mathbf{z}=\mathbf{y}/\epsilon$, is a doubly periodic function with period 1. Since $x$ is fixed in this cross-sectional problem, we shall drop this formal variable of $w$ in this Section. Following the homogenization technique based also on the variational-asymptotic method (Berdichevsky, 2009), we look for the minimizer in the form
$$w(\mathbf{y},\mathbf{z})=\psi(\mathbf{y})+\epsilon\chi(\mathbf{z}),$$
(9)
where $\chi(\mathbf{z})$ is a doubly periodic function with period 1. Note that $\chi$ may depend on the slow variable $\mathbf{y}$, but this dependency is suppressed for short. In addition to the constraint (5)${}_{3}$ we may impose the following constraint on $\chi(\mathbf{z})$
$$\langle\!\langle\chi\rangle\!\rangle\equiv\int_{\mathcal{C}}\chi(\mathbf{z})%
\differential[2]{z}=0,$$
(10)
where $\mathcal{C}=(0,1)\times(0,1)$ is the unit periodic cell. In this case $\psi(\mathbf{y})$ can be interpreted as the average value of $w(\mathbf{y},\mathbf{z})$ over the cell. Although the second term in (9) is small compared to $\psi(\mathbf{y})$ and goes to zero in the limit $\epsilon\to 0$, it contribution to the shear strains as the gradient of $w(\mathbf{y},\mathbf{z})$ has the same order as the gradient of $\psi(\mathbf{y})$ as seen from the asymptotic formula
$$w_{;\alpha}=\psi_{;\alpha}+\chi_{|\alpha},\quad\alpha=1,2.$$
(11)
Here, the vertical bar followed by an index $\alpha$ denotes the partial derivative with respect to the corresponding fast variable $z_{\alpha}$.
Inserting (11) into the energy functional (8), we get
$$I_{\angle}[\psi(\mathbf{y}),\chi(\mathbf{y}/\epsilon)]=h^{2}\langle\frac{1}{2}%
\mu(\mathbf{y}/\epsilon)(\chi_{|\alpha}+\xi_{\alpha})(\chi_{|\alpha}+\xi_{%
\alpha})\rangle,$$
with $\xi_{\alpha}=\psi_{;\alpha}-h\Omega e_{\alpha\beta}y_{\beta}$ being the function of the “slow” variable $\mathbf{y}$. We replace the double integral $\langle.\rangle$ by the sum of double integrals over the cells. Then
$$I_{\angle}[\psi(\mathbf{y}),\chi(\mathbf{y}/\epsilon)]=h^{2}\sum_{n=1}^{N}\int%
_{\mathcal{C}_{n}}\frac{1}{2}\mu(\mathbf{y}/\epsilon)(\chi_{|\alpha}+\xi_{%
\alpha})(\chi_{|\alpha}+\xi_{\alpha})\differential[2]{y},$$
(12)
where $N$ is the total number of cells and $\mathcal{C}_{n}$ is the cell $((i-1)\epsilon,i\epsilon)\times((j-1)\epsilon,j\epsilon)$. We minimize this functional in two steps: (i) Fix $\psi(\mathbf{y})$ and minimize the functional among doubly periodic $\chi(\mathbf{y}/\epsilon)$, (ii) Minimize the obtained functional among $\psi(\mathbf{y})$. Because $\chi(\mathbf{y}/\epsilon)$ is doubly periodic with respect to $\mathbf{y}$, we can minimize each integral in the sum independently. Besides, as $\xi_{\alpha}(\mathbf{y})$ change slowly in one cell, we may regard them in each cell integral as constants equal to their value in the middle of the cell. It is convenient to change the variable $\mathbf{y}$ in the cell integrals to $\mathbf{z}=\mathbf{y}/\epsilon-(i-1,j-1)$. Then the minimization of cell integrals reduces to the following cell problem: Minimize the functional
$$\langle\!\langle\frac{1}{2}\mu(\mathbf{z})(\chi_{|\alpha}+\xi_{\alpha})(\chi_{%
|\alpha}+\xi_{\alpha})\rangle\!\rangle$$
(13)
among doubly periodic functions $\chi(\mathbf{z})$ satisfying the constraint (10), where, as before, $\langle\!\langle.\rangle\!\rangle$ is the double integral in $\mathbf{z}$ over the unit cell. We denote the obtained minimum by $\bar{W}_{\angle}(\xi_{\alpha})$ and call it the average shear energy density. Then, replacing the sum in (12) by the integral in the limit $\epsilon\to 0$, we arrive at the following homogenized cross-sectional problem: Minimize the average functional
$$\bar{I}_{\angle}[\psi(\mathbf{y})]=h^{2}\langle\bar{W}_{\angle}(\psi_{;\alpha}%
-h\Omega e_{\alpha\beta}y_{\beta})\rangle$$
(14)
among $\psi(\mathbf{y})$ satisfying the condition
$$\langle\psi\rangle=0.$$
The standard calculus of variations shows that the minimizer of (13) satisfies the equation
$$[\mu(\mathbf{z})(\chi_{|\alpha}+\xi_{\alpha})]_{|\alpha}=0$$
(15)
in the unit cell, where $\boldsymbol{\xi}$ can be regarded as the constant vector in the cell (since it does not depend on the fast variable $\mathbf{z}$). This equation is subjected to the periodic boundary condition and the constraint (10). Besides, as the shear modulus suffers a jump at the boundary $\partial_{i}$ between the fiber and the matrix, the continuity of the traction
$$[\![\mu(\mathbf{z})(\chi_{|\alpha}+\xi_{\alpha})]\!]\nu_{\alpha}=0$$
(16)
should be fulfilled at this internal boundary $\partial_{i}$, where $\boldsymbol{\nu}$ is the unit normal vector outward to the fiber and $[\![.]\!]$ denotes the jump.
After finding the minimizer $\check{\chi}(\mathbf{z})$ as a solution to the boundary-value problem (15)-(16), we substitute it back into functional (13) to calculate the average shear energy density
$$\displaystyle\bar{W}_{\angle}(\xi_{\alpha})$$
$$\displaystyle=\langle\!\langle\frac{1}{2}\mu(\mathbf{z})(\check{\chi}_{|\alpha%
}+\xi_{\alpha})(\check{\chi}_{|\alpha}+\xi_{\alpha})\rangle\!\rangle$$
$$\displaystyle=\langle\!\langle\frac{1}{2}\mu(\mathbf{z})(\check{\chi}_{|\alpha%
}+\xi_{\alpha})\check{\chi}_{|\alpha}\rangle\!\rangle+\langle\!\langle\frac{1}%
{2}\mu(\mathbf{z})(\check{\chi}_{|\alpha}+\xi_{\alpha})\xi_{\alpha}\rangle\!\rangle.$$
(17)
As the minimizer $\check{\chi}$ satisfies (15)-(16), the first integral must vanish. Using the constancy of $\xi_{\alpha}$, we reduce the second integral to
$$\bar{W}_{\angle}(\xi_{\alpha})=\frac{1}{2}\xi_{\alpha}\langle\!\langle\mu(%
\mathbf{z})(\check{\chi}_{|\alpha}+\xi_{\alpha})\rangle\!\rangle.$$
The integrand is the shear stress $\sigma_{3\alpha}$, so the integral gives the average shear stress $\bar{\sigma}_{3\alpha}$. On the other side, due to the constraint (10),
$$\langle\!\langle\check{\chi}_{|\alpha}+\xi_{\alpha}\rangle\!\rangle=\xi_{%
\alpha},$$
so $\xi_{\alpha}$ is the average engineering shear strain $\bar{\varepsilon}_{3\alpha}$. We define the effective elastic shear moduli $\mu^{*}_{\alpha\beta}$ by the linear equation
$$\bar{\sigma}_{3\alpha}=\langle\!\langle\mu(\mathbf{z})(\check{\chi}_{|\alpha}+%
\xi_{\alpha})\rangle\!\rangle=\mu^{*}_{\alpha\beta}\xi_{\beta}.$$
(18)
Note that the homogenized material must not necessarily be isotropic even if the components of the composite are. So, in general $\mu^{*}_{\alpha\beta}$ is a tensor of second order. With this we get the average shear energy density in terms of $\xi_{\alpha}=\psi_{;\alpha}-h\Omega e_{\alpha\beta}y_{\beta}$
$$\bar{W}_{\angle}(\xi_{\alpha})=\frac{1}{2}\mu^{*}_{\alpha\beta}\xi_{\alpha}\xi%
_{\beta}.$$
(19)
We want to show that the tensor $\mu^{*}_{\alpha\beta}$ in (18) is symmetric. Indeed, since $\bar{W}_{\angle}(\xi_{\alpha})$ in (19) is a quadratic form, we can replace $\mu^{*}_{\alpha\beta}$ there by the symmetric tensor $\mu^{*}_{(\alpha\beta)}=\frac{1}{2}(\mu^{*}_{\alpha\beta}+\mu^{*}_{\beta\alpha})$. If we substitute (19) into (17) and differentiate it with respect to $\xi_{\alpha}$, we get
$$\mu^{*}_{(\alpha\beta)}\xi_{\beta}=\langle\!\langle\mu(\mathbf{z})(\check{\chi%
}_{|\alpha}+\xi_{\alpha})\rangle\!\rangle=\mu^{*}_{\alpha\beta}\xi_{\beta},$$
which proves the symmetry of $\mu^{*}_{\alpha\beta}$.
We turn now to the cross-sectional problem (7). Before doing the asymptotic analysis for it in the limit $\epsilon\to 0$ let us prove the following remarkable property: The minimum of functional (7) is zero if the Poisson ratio $\nu$ is an equal constant for both fiber and matrix (Berdichevsky, 1981). Indeed, let us choose the minimizer in the form
$$\check{w}_{\alpha}=-\nu\gamma\delta_{\alpha\beta}y_{\beta}-\frac{1}{2}\nu a_{%
\alpha\beta\gamma}(y_{\beta}y_{\gamma}-\langle y_{\beta}y_{\gamma}\rangle/|%
\bar{\mathcal{A}}|),$$
(20)
where $|\bar{\mathcal{A}}|$ is the area of $\bar{\mathcal{A}}$ and
$$a_{\alpha\beta\gamma}=h(\delta_{\alpha\beta}\Omega_{\gamma}+\delta_{\alpha%
\gamma}\Omega_{\beta}-\delta_{\beta\gamma}\Omega_{\alpha}).$$
Note that the tensor $a_{\alpha\beta\gamma}$ is symmetric with the last two indices, and
$$\check{w}_{\alpha;\beta}=-\nu\gamma\delta_{\alpha\beta}-a_{\alpha\beta\gamma}y%
_{\gamma}.$$
Due to our choice of the origin, $\langle y_{\alpha}\rangle=0$. Therefore the field (20) satisfies the constraints (5)${}_{1,2}$. It is easy to check that the transverse energy evaluated at $\check{w}_{\alpha}$ vanishes identically. Since functional (7) is non-negative definite, its minimum is obviously zero in this case.
The asymptotic analysis of problem (7) for changeable $\nu$ is quite similar to that of problem (8). In this case the minimizer is sought in the form
$$w_{\alpha}(\mathbf{y},\mathbf{z})=\psi_{\alpha}(\mathbf{y})+\epsilon\chi_{%
\alpha}(\mathbf{z}),$$
where $\chi_{\alpha}$ satisfy the constraints
$$\langle\!\langle\chi_{\alpha}\rangle\!\rangle=0.$$
(21)
To determine $\chi_{\alpha}$ we need to solve the following cell problem: Minimize the functional
$$\displaystyle\langle\!\langle\frac{1}{2}\lambda(\mathbf{z})(\chi_{\alpha|%
\alpha}+\bar{\varepsilon}_{\alpha\alpha}+2\nu(\mathbf{z})\xi)^{2}\\
\displaystyle+\mu(\mathbf{z})(\chi_{(\alpha|\beta)}+\bar{\varepsilon}_{\alpha%
\beta}+\nu(\mathbf{z})\delta_{\alpha\beta}\xi)(\chi_{(\alpha|\beta)}+\bar{%
\varepsilon}_{\alpha\beta}+\nu(\mathbf{z})\delta_{\alpha\beta}\xi)\rangle\!\rangle$$
(22)
among doubly periodic functions $\chi_{\alpha}(\mathbf{z})$ satisfying the constraint (21), where
$$\bar{\varepsilon}_{\alpha\beta}=\psi_{(\alpha|\beta)},\quad\xi=\gamma+h\Omega_%
{\alpha}y_{\alpha}.$$
Note that this problem is quite similar to the problem of determining the effective thermal expansion of composite material with periodic microstructure, where $\xi$ plays the role of the temperature increase (Sigmund and Torquato, 1997). Let the minimum of functional (22) be $\bar{W}_{\perp}(\bar{\varepsilon}_{\alpha\beta},\xi)$. Then the determination of $\psi_{\alpha}$ reduces to minimizing the following average functional
$$\bar{I}_{\perp}[\psi_{\alpha}(\mathbf{y})]=h^{2}\langle\bar{W}_{\perp}(\psi_{(%
\alpha;\beta)},\gamma+h\Omega_{\alpha}y_{\alpha})\rangle$$
(23)
among $\psi_{\alpha}(\mathbf{y})$ satisfying the constraints
$$\langle\psi_{\alpha}\rangle=0,\quad e_{\alpha\beta}\langle\psi_{\alpha;\beta}%
\rangle=0.$$
(24)
The minimizer of functional (22) satisfy the equilibrium equations
$$[\lambda(\mathbf{z})(\chi_{\beta|\beta}+\bar{\varepsilon}_{\beta\beta}+\xi)]_{%
|\alpha}+[2\mu(\mathbf{z})(\chi_{(\alpha|\beta)}+\bar{\varepsilon}_{\alpha%
\beta})]_{|\beta}=0$$
in the unit cell, where $\bar{\varepsilon}_{\alpha\beta}$ and $\xi$ can be regarded as the constant tensor and scalar in the cell (since they do not depend on the fast variable $\mathbf{z}$). These equations are subjected to the periodic boundary conditions and the constraints (21). Besides, as the elastic moduli suffers jumps at the boundary $\partial_{i}$ between the fiber and the matrix, the continuity of the traction
$$[\![\lambda(\mathbf{z})(\chi_{\beta|\beta}+\bar{\varepsilon}_{\beta\beta}+\xi)%
\nu_{\alpha}+2\mu(\mathbf{z})(\chi_{(\alpha|\beta)}+\bar{\varepsilon}_{\alpha%
\beta})\nu_{\beta}]\!]=0$$
should be fulfilled there. Since the functional (22) is quadratic, its minimum must be a quadratic form of $\bar{\varepsilon}_{\alpha\beta}$ and $\xi$
$$\bar{W}_{\perp}(\bar{\varepsilon}_{\alpha\beta},\xi)=\frac{1}{2}C^{*}_{\alpha%
\beta\gamma\delta}\bar{\varepsilon}_{\alpha\beta}\bar{\varepsilon}_{\gamma%
\delta}+D^{*}_{\alpha\beta}\bar{\varepsilon}_{\alpha\beta}\xi+\frac{1}{2}F^{*}%
\xi^{2}.$$
(25)
Using the same arguments as in the previous case, we can prove the following symmetry properties of the effective moduli
$$\displaystyle C^{*}_{\alpha\beta\gamma\delta}=C^{*}_{\beta\alpha\gamma\delta}=%
C^{*}_{\alpha\beta\delta\gamma}=C^{*}_{\gamma\delta\alpha\beta},$$
$$\displaystyle D^{*}_{\alpha\beta}=D^{*}_{\beta\alpha}.$$
We also introduce the effective tensor of Poisson’s ratios by
$$\nu^{*}_{\alpha\beta}=C^{*(-1)}_{\alpha\beta\gamma\delta}D_{\gamma\nu},$$
where $C^{*(-1)}_{\alpha\beta\gamma\delta}$ is the tensor of elastic compliances defined by
$$C^{*(-1)}_{\alpha\beta\gamma\delta}C_{\gamma\delta\zeta\eta}=\delta_{\alpha(%
\zeta}\delta_{\beta\eta)}.$$
It is easy to see that $\nu_{\alpha\beta}$ is symmetric.
5 Numerical solution of the cell problems
It is convenient to rewrite functional (22) by changing the sign of $\chi_{\alpha}(\mathbf{y})$
$$\frac{1}{2}\langle\!\langle C_{\alpha\beta\gamma\delta}(\mathbf{z})(-\chi_{(%
\alpha|\beta)}+\bar{\varepsilon}_{\alpha\beta}+\alpha_{\alpha\beta}(\mathbf{z}%
)\xi)(-\chi_{(\gamma|\delta)}+\bar{\varepsilon}_{\gamma\delta}+\alpha_{\gamma%
\delta}(\mathbf{z})\xi)\rangle\!\rangle,$$
(26)
where
$$C_{\alpha\beta\gamma\delta}(\mathbf{z})=\lambda(\mathbf{z})\delta_{\alpha\beta%
}\delta_{\gamma\delta}+\mu(\mathbf{z})(\delta_{\alpha\gamma}\delta_{\beta%
\gamma}+\delta_{\alpha\delta}\delta_{\beta\gamma}),\quad\alpha_{\alpha\beta}(%
\mathbf{z})=\nu(\mathbf{z})\delta_{\alpha\beta}.$$
The minimizer of functional (26) satisfies the variational equation
$$\langle\!\langle C_{\alpha\beta\gamma\delta}(\mathbf{z})\chi_{(\gamma|\delta)}%
\delta\chi_{\alpha|\beta}\rangle\!\rangle=\langle\!\langle C_{\alpha\beta%
\gamma\delta}(\mathbf{z})(\bar{\varepsilon}_{\gamma\delta}+\alpha_{\gamma%
\delta}(\mathbf{z})\xi)\delta\chi_{\alpha|\beta}\rangle\!\rangle$$
(27)
for all doubly periodic functions $\delta\chi_{\alpha}$. Eq. (27) will be solved by the finite element method (Hughes, 2012; Andreasen, 2014). For this purpose it is convenient to change from tensor notation to matrix notation (Hughes, 2012), in which Eq. (27) becomes
$$\langle\!\langle\boldsymbol{\varepsilon}^{T}(\boldsymbol{\psi})\mathbf{C}(%
\mathbf{z})\boldsymbol{\varepsilon}(\boldsymbol{\chi})\rangle\!\rangle=\langle%
\!\langle\boldsymbol{\varepsilon}^{T}(\boldsymbol{\psi})\mathbf{C}(\mathbf{z})%
(\bar{\boldsymbol{\varepsilon}}+\boldsymbol{\alpha}(\mathbf{z})\xi)\rangle\!\rangle,$$
(28)
where $\boldsymbol{\psi}=\delta\boldsymbol{\chi}$, $\boldsymbol{\varepsilon}(\boldsymbol{\chi})$ is the strain vector given by
$$\boldsymbol{\varepsilon}(\boldsymbol{\chi})=(\chi_{1|1},\chi_{2|2},\chi_{1|2}+%
\chi_{2|1})^{T},$$
while
$$\mathbf{C}(\mathbf{z})=\lambda(\mathbf{z})\begin{bmatrix}1&1&0\\
1&1&0\\
0&0&0\end{bmatrix}+\mu(\mathbf{z})\begin{bmatrix}2&0&0\\
0&2&0\\
0&0&1\end{bmatrix},\quad\boldsymbol{\alpha}(\mathbf{z})=\nu(\mathbf{z})\begin{%
bmatrix}1\\
1\\
0\end{bmatrix}.$$
(29)
The discretization of this equation based on the bilinear isoparametric elements is standard. The periodic boundary conditions are imposed by identifying the nodes on opposite sides of the unit cell. This is implemented by using the matrix edofMat for a full, regular grid to index into a periodic version of the grid (Andreasen, 2014). The global stiffness matrix is
$$\mathbf{K}=\mathop{\vphantom{\sum}\mathchoice{\vbox{\hbox{\huge$\mathrm{A}$}}}%
{\vbox{\hbox{\Large$\mathrm{A}$}}}{\mathrm{A}}{\mathrm{A}}}\displaylimits_{e=1%
}^{N}(\mathbf{k}_{e}),\quad\mathbf{k}_{e}=\int_{A_{e}}\mathbf{B}^{T}_{e}%
\mathbf{C}_{e}\mathbf{B}_{e}\differential[2]{z},$$
where $\mathop{\vphantom{\sum}\mathchoice{\vbox{\hbox{\huge$\mathrm{A}$}}}{\vbox{%
\hbox{\Large$\mathrm{A}$}}}{\mathrm{A}}{\mathrm{A}}}\displaylimits$ is the assembly operator taken over the total number $N$ of finite elements. The matrix $\mathbf{B}_{e}$ is the element strain-displacement matrix, $A_{e}$ is the domain of element $e$, and $\mathbf{C}_{e}$ is the constitutive matrix for the element. The indicator matrix is introduced that specifies whether the element is in $\mathcal{A}^{m}$ ($n_{e}=1$) or in $\mathcal{A}^{f}$ ($n_{e}=2$). The piecewise constant $\mathbf{C}(\mathbf{z})$ from (29) takes value $\mathbf{C}_{e}$ in the element in accordance with this indicator matrix.
The discretization of the right hand side of (28) yields the loads $\mathbf{f}^{(i)}$
$$\mathbf{f}^{(i)}=\mathop{\vphantom{\sum}\mathchoice{\vbox{\hbox{\huge$\mathrm{%
A}$}}}{\vbox{\hbox{\Large$\mathrm{A}$}}}{\mathrm{A}}{\mathrm{A}}}%
\displaylimits_{e=1}^{N}(\mathbf{f}^{(i)}_{e}),\quad\mathbf{f}^{(i)}_{e}=\int_%
{A_{e}}\mathbf{B}^{T}_{e}\mathbf{C}_{e}\bar{\boldsymbol{\varepsilon}}^{(i)}%
\differential[2]{z},\quad i=1,2,3$$
which correspond to the average macroscopic strains
$$\bar{\boldsymbol{\varepsilon}}^{(1)}=(1,0,0)^{T},\quad\bar{\boldsymbol{%
\varepsilon}}^{(2)}=(0,1,0)^{T},\quad\bar{\boldsymbol{\varepsilon}}^{(3)}=(0,0%
,1)^{T},$$
(30)
and
$$\mathbf{f}^{(4)}=\mathop{\vphantom{\sum}\mathchoice{\vbox{\hbox{\huge$\mathrm{%
A}$}}}{\vbox{\hbox{\Large$\mathrm{A}$}}}{\mathrm{A}}{\mathrm{A}}}%
\displaylimits_{e=1}^{N}(\mathbf{f}^{(4)}_{e}),\quad\mathbf{f}^{(4)}_{e}=\int_%
{A_{e}}\mathbf{B}^{T}_{e}\mathbf{C}_{e}\boldsymbol{\alpha}_{e}\differential[2]%
{z},$$
which correspond to $\xi=1$, with $\boldsymbol{\alpha}_{e}$ being the constitutive vector from (29) that takes values in the element in accordance with the indicator matrix. The displacement fields are computed by solving the corresponding linear equations with four load-cases
$$\mathbf{K}\boldsymbol{\chi}^{(i)}=\mathbf{f}^{(i)},\quad i=1,2,3,4.$$
When the displacements are obtained, the elements of the effective matrix $\bf{C}^{*}$ are found as:
$$C^{*}_{ij}=\sum_{e=1}^{N}\int_{A_{e}}(\boldsymbol{\chi}_{e}^{0(i)}-\boldsymbol%
{\chi}_{e}^{(i)})^{T}\mathbf{k}_{e}(\boldsymbol{\chi}_{e}^{0(j)}-\boldsymbol{%
\chi}_{e}^{(j)})\differential[2]{z},\quad i,j=1,2,3,$$
where $\boldsymbol{\chi}_{e}^{0(i)}$ are the three displacement fields corresponding to the unit strains from (30), and $\boldsymbol{\chi}_{e}^{(i)}$ contains three columns corresponding to the three displacement fields resulting from globally enforcing the unit strains (30). The indices in parentheses refer to the column number. The components of the effective moduli $D^{*}_{i}$ ($D^{*}_{\alpha\beta}$) due to $\xi$ are computed according to
$$D^{*}_{i}=\sum_{e=1}^{N}\int_{A_{e}}(\boldsymbol{\alpha}_{e}-\mathbf{B}_{e}%
\boldsymbol{\chi}_{e}^{(4)})^{T}\mathbf{C}_{e}(\bar{\boldsymbol{\varepsilon}}^%
{i}-\mathbf{B}_{e}\boldsymbol{\chi}_{e}^{(i)})\differential[2]{z},\quad i=1,2,3.$$
Having $D^{*}_{i}$ we can also compute the effective “Poisson” ratios $\nu^{*}_{i}$ ($\nu^{*}_{\alpha\beta}$) as follows
$$\nu^{*}_{i}=(C^{*}_{ij})^{-1}D^{*}_{j}.$$
(31)
Finally, the effective coefficient $F^{*}$ is given by
$$F^{*}=\sum_{e=1}^{N}\int_{A_{e}}(\boldsymbol{\alpha}_{e}-\mathbf{B}_{e}%
\boldsymbol{\chi}_{e}^{(4)})^{T}\mathbf{C}_{e}(\boldsymbol{\alpha}_{e}-\mathbf%
{B}_{e}\boldsymbol{\chi}_{e}^{(4)})\differential[2]{z}.$$
The Matlab-code homogenizecs.m to solve the cell problem and compute the effective moduli which is a modification of the code homogenize.m written by Andreasen (2014) is presented in the Appendix.
For the FRC bar it will be shown in the next Section that the minimum of functional (23) is $\frac{1}{2}(F^{*}-\nu^{*}_{\alpha\beta}C^{*}_{\alpha\beta\gamma\delta}\nu^{*}_%
{\gamma\delta})h^{2}\langle\xi^{2}\rangle$. Therefore it makes sense to investigate the quantity $H^{*}=F^{*}-\nu^{*}_{\alpha\beta}C^{*}_{\alpha\beta\gamma\delta}\nu^{*}_{%
\gamma\delta}$ giving the correction to the stiffnesses on extension and bending. We take the microstructure of the composite in such a way that the cross sections of the fibers are ellipses of half-axes $r$ and $r/a$ placed in the middle of the unit quadratic periodic cells. Then the region occupied by one fiber in the unit cell is given by the equation
$$\frac{(z_{1}-1/2)^{2}}{r^{2}}+\frac{(z_{2}-1/2)^{2}}{(r/a)^{2}}\leq 1.$$
Note that for the circle with $a=1$ the effective Poisson ratio tensor is $\nu^{*}_{\alpha\beta}=\nu^{*}\delta_{\alpha\beta}$, however this property is no longer valid for ellipses with $a>1$. Assigning the index 1 to the matrix and index 2 to the fiber, we choose the Young moduli and Poisson ratios as follows: $E_{1}=1\,$GPa, $E_{2}=2\,$GPa, $\nu_{1}=0.1$, $\nu_{2}=0.4$. The plot of $H^{*}$ as function of the largest half-axis $r$ for four different aspect ratios of the ellipses $a=1,2,4,8$ is presented in Fig. 2. We see that the correction $H^{*}$ varies in the range $(0,0.03)\,$GPa which is not so large compared with the mean value of the Young moduli. Since the volume fraction of the fibers is $s_{2}=\pi r^{2}/a$, we can also plot $H^{*}$ as function of this volume fraction.
Let us now fix the largest half-axis of elliptical cross section of the fibers to be $r=0.4$, choose $E_{1}=1\,$GPa, $E_{2}=2\,$GPa, $\nu_{1}=0.3$, vary the Poisson ratio of the fiber $\nu_{2}$ in its admissible range between $-1$ and $0.5$, and plot $H^{*}$ as function of $\nu_{2}$. Looking at this plot shown in Fig. 3 we see that the larger the difference between the Poisson ratios, the larger is the correction $H^{*}$. For $\nu_{2}=0.3$ the correction $H^{*}$ vanishes as expected. For $\nu_{2}$ near $-1$ which is not unrealistic (Milton, 1992) the correction is more than one third of the mean Young modulus which can no longer be neglected.
The solution of the cell problem (13) by the finite element method is quite similar. Its solution can be obtained from the previous problem if we put $\lambda(\mathbf{z})=0$. Below we present the numerical solution for the fibers with circular cross sections periodically embedded in the matrix. The unit cell is a square of length 1 with the circle of radius $r$ in the middle representing the cross section of the fiber. In this case the effective tensor is $\mu^{*}_{\alpha\beta}=\mu^{*}\delta_{\alpha\beta}$. The plot of $\mu^{*}$ as function of $s_{2}=\pi r^{2}$ is presented in Fig. 4. For the numerical simulation we choose $\mu_{1}=1\,$GPa, $\mu_{2}=2\,$GPa.
6 Solution of the homogenized cross-sectional problems
Let us first consider the homogenized cross-sectional problem (23) with the average transverse energy being given by (25). It is convenient to present the latter formula in the form
$$\bar{W}_{\perp}(\bar{\varepsilon}_{\alpha\beta},\xi)=\frac{1}{2}C^{*}_{\alpha%
\beta\gamma\delta}(\bar{\varepsilon}_{\alpha\beta}+\nu^{*}_{\alpha\beta}\xi)(%
\bar{\varepsilon}_{\gamma\delta}+\nu^{*}_{\gamma\delta}\xi)+\frac{1}{2}H^{*}%
\xi^{2},$$
(32)
where $\xi=\gamma+h\Omega_{\alpha}y_{\alpha}$. Note that the second term in (32) does not depend on $\psi_{\alpha}(\mathbf{y})$, so we need to minimize functional (23) that contains just the first term. We want to show that the minimum of the functional (23) that contains just the first term among $\psi_{\alpha}(\mathbf{y})$ satisfying the constraints (24) is equal to zero. Indeed, following the same line of reasoning as in Section 4 we choose the minimizer in the form
$$\check{\psi}_{\alpha}=-\nu^{*}_{\alpha\beta}\gamma y_{\beta}-\frac{1}{2}a_{%
\alpha\beta\gamma}(y_{\beta}y_{\gamma}-\langle y_{\beta}y_{\gamma}\rangle/|%
\bar{\mathcal{A}}|),$$
where
$$a_{\alpha\beta\gamma}=h(\nu^{*}_{\alpha\beta}\Omega_{\gamma}+\nu^{*}_{\alpha%
\gamma}\Omega_{\beta}-\nu^{*}_{\beta\gamma}\Omega_{\alpha}).$$
It is easy to check that $\check{\psi}_{\alpha}$ satisfy the constraints (24) and that the first term evaluated at these functions vanishes identically, so
$$\min_{\psi_{\alpha}(\mathbf{y})\in\eqref{eq:comp17a}}\bar{I}_{\perp}[\psi_{%
\alpha}(\mathbf{y})]=\frac{1}{2}H^{*}h^{2}\langle\xi^{2}\rangle.$$
Adding the average longitudinal energy density $h^{2}\langle W_{\parallel}\rangle=\frac{1}{2}\bar{E}\langle\xi^{2}\rangle$ to this average transverse energy density and integrating $\xi^{2}=(\gamma+h\Omega_{\alpha}y_{\alpha})^{2}$ over the cross section, we find the final energy density of extension and bending of FRC beam in the form
$$\Phi(\gamma,\Omega_{\alpha})=\frac{1}{2}(\bar{E}+H^{*})(|\mathcal{A}|\gamma^{2%
}+I_{\alpha\beta}\Omega_{\alpha}\Omega_{\beta}),$$
where
$$\bar{E}=\langle\!\langle E\rangle\!\rangle=s_{1}E_{1}+s_{2}E_{2},\quad I_{%
\alpha\beta}=h^{4}\langle y_{\alpha}y_{\beta}\rangle.$$
Thus, $H^{*}$ is the correction to the stiffnesses on extension and bending of FRC beam. To see how this correction changes the stiffnesses on extension and bending of FRC beam we plot both $\bar{E}$ and $\bar{E}+H^{*}$ as function of the volume fraction of fibers with circular cross sections placed in the middle of the quadratic periodic cell (see Fig. 5). The chosen elastic moduli are: $E_{1}=1\,$GPa, $E_{2}=2\,$GPa, $\nu_{1}=0.1$, $\nu_{2}=0.4$. The correction is about 3 percent of $\bar{E}$ in this case. Note that this correction becomes one third of $\bar{E}$ if $\nu_{2}$ is near $-1$.
We now turn to the minimization problem (14). First of all it is easy to see that the constraint $\langle\psi\rangle=0$ does not affect the minimum value of (14), because the latter is invariant with respect to the change of unknown function $\psi\to\psi+c$, with $c$ being a constant. By such the change one can always achieve the fulfillment of the constraint $\langle\psi\rangle=0$. The minimizer $\check{\psi}$ of $\bar{I}_{\angle}$ is proportional to $h\Omega$. Therefore the torsional rigidity $C$ can be calculated by
$$C=h^{4}\inf_{\bar{\psi}}\,\langle\mu^{*}\delta_{\alpha\beta}(\bar{\psi}_{|%
\alpha}-e_{\alpha\sigma}y_{\sigma})(\bar{\psi}_{|\beta}-e_{\beta\kappa}y_{%
\kappa})\rangle,$$
(33)
where $\bar{\psi}=\psi/h\Omega$. The solution to this problem is well-known for the elliptical and rectangular cross sections (Le, 1999). For the FRC beam with the elliptical cross section described by the equation
$$\displaystyle c_{\alpha\beta}y_{\alpha}y_{\beta}\leq 1,$$
where $c_{\alpha\beta}$ are the components of a positive definite symmetric second-rank tensor, the torsional rigidity reads
$$C=\frac{4\mu^{*}}{c^{2}}c_{\mu\lambda}c_{\lambda\tau}I_{\mu\tau},$$
(34)
where $I_{\mu\tau}=h^{4}\langle y_{\mu}y_{\tau}\rangle$ are the moments of inertia of the cross section.
In the co-ordinate system associated with the principal axes of the ellipse
$$\displaystyle c_{11}=\frac{h^{2}}{b_{1}^{2}},\quad c_{22}=\frac{h^{2}}{b_{2}^{%
2}},\quad c_{12}=0,$$
$$\displaystyle I_{11}=\frac{1}{4}\pi b_{1}^{3}b_{2},\quad I_{22}=\frac{1}{4}\pi
b%
_{2}^{3}b_{1},\quad I_{12}=0,$$
where $b_{1},b_{2}$ are the half-lengths of the major and minor axes. Substituting these formulas into (34) we get finally
$$\displaystyle C=\frac{\pi\mu^{*}b_{1}^{3}b_{2}^{3}}{b_{1}^{2}+b_{2}^{2}}=\frac%
{4\mu^{*}}{(I^{-1})_{\alpha\alpha}},$$
where $(I^{-1})_{\alpha\beta}$ is a tensor inverse to $I_{\alpha\beta}$.
It is easy to see that (34) is also the solution to the problem (33) for a hollow elliptical cross section
$$\lambda^{2}\leq c_{\alpha\beta}y_{\alpha}y_{\beta}\leq 1,\quad\lambda<1.$$
Similar calculations give
$$\displaystyle C=\frac{\pi\mu^{*}(1-\lambda^{4})b_{1}^{3}b_{2}^{3}}{b_{1}^{2}+b%
_{2}^{2}}.$$
For the rectangular cross section of width $a$ and height 1, where $a<1$, the solution is found in form of an infinite series (Le, 1999). The torsional rigidity is given by the formula
$$\displaystyle C=\mu^{*}ch^{4}a^{3},$$
where
$$\displaystyle c=\frac{1}{3}-\frac{64a}{\pi^{5}}\sum_{n=0}^{\infty}\frac{\tanh(%
\pi(2n+1)/2a)}{(2n+1)^{5}}.$$
For other cross sections the problem (33) can be solved by the finite element method.
7 1-D beam theory
Summarizing the results obtained in Sections 3-6, we can now reduce the 3-D problem of equilibrium of the FRC beam to the following 1-D variational problem: Minimize the energy functional
$$J[\mathbf{v},\varphi]=\int_{0}^{L}\Phi(\gamma,\Omega_{\alpha},\Omega)%
\differential{x}-\mathbf{f}\dotproduct\mathbf{v}(L)+m_{\alpha}v_{\alpha,x}(L)-%
m\varphi(L)$$
among functions $\mathbf{v}$ and $\varphi$ satisfying the kinematic boundary conditions
$$\mathbf{v}(0)=0,\quad v_{\alpha,x}(0)=0,\quad\varphi(0)=0.$$
In this functional the stored energy density $\Phi(\gamma,\Omega_{\alpha},\Omega)$ is given by
$$\Phi(\gamma,\Omega_{\alpha},\Omega)=\frac{1}{2}(\bar{E}+H^{*})(|\mathcal{A}|%
\gamma^{2}+I_{\alpha\beta}\Omega_{\alpha}\Omega_{\beta})+\frac{1}{2}C\Omega^{2},$$
while the resultant forces and moments acting at $x=L$ are equal to
$$\displaystyle\mathbf{f}$$
$$\displaystyle=\int_{\mathcal{A}}\mathbf{t}(x_{1},x_{2})\differential[2]{x},$$
$$\displaystyle m_{\alpha}$$
$$\displaystyle=\int_{\mathcal{A}}x_{\alpha}t(x_{1},x_{2})\differential[2]{x},%
\quad m=\int_{\mathcal{A}}e_{\alpha\beta}x_{\alpha}t_{\beta}(x_{1},x_{2})%
\differential[2]{x}.$$
The standard calculus of variations shows that $\mathbf{v}$ and $\varphi$ satisfy the equilibrium equations
$$T_{,x}=0,\quad M_{\alpha,xx}=0,\quad M_{,x}=0,$$
(35)
where
$$T=\frac{\partial\Phi}{\partial\gamma}=(\bar{E}+H^{*})|\mathcal{A}|\gamma,\quad
M%
_{\alpha}=\frac{\partial\Phi}{\partial\Omega_{\alpha}}=(\bar{E}+H^{*})I_{%
\alpha\beta}\Omega_{\beta},\quad M=\frac{\partial\Phi}{\partial\Omega}=C\Omega.$$
Besides, the following boundary conditions must be fulfilled at $x=L$:
$$T(L)=f,\quad M_{\alpha}(L)=m_{\alpha},\quad M_{\alpha,x}=f_{\alpha},\quad M=m.$$
(36)
Using the technique of Gamma-convergence (Braides, 2002; Milton, 2003), one can prove that the solution of (35)-(36) converges to the minimizer of (1) in the energetic norm as $h/L\to 0$ and $\epsilon=c/h\to 0$.
It is easy to extend this one-dimensional theory to the dynamics of FRC beam, where functions $\mathbf{v}$ and $\varphi$ depend on $x$ and $t$. One need just to include into the one-dimensional functional the kinetic energy density which, after the dimension reduction and homogenization in accordance with (4), takes the form
$$\Theta=\frac{1}{2}\bar{\rho}(|\mathcal{A}|\dot{\mathbf{v}}\dotproduct\dot{%
\mathbf{v}}+I_{\alpha\alpha}\dot{\varphi}^{2}),$$
(37)
with $\bar{\rho}=\langle\!\langle\rho\rangle\!\rangle=s_{1}\rho_{1}+s_{2}\rho_{2}$ being the mass density averaged over the unit cell. Hamilton’s variational principle for 1-D beam theory states that, among all admissible functions $\mathbf{v}(x,t)$ and $\varphi(x,t)$ satisfying the initial and end conditions as well as the kinematic boundary conditions, the true displacement $\check{\mathbf{v}}$ and rotation $\check{\varphi}$ are the extremal of the action functional
$$J[\mathbf{v},\varphi]=\int_{t_{0}}^{t_{1}}\int_{0}^{L}(\Theta-\Phi)%
\differential{x}\differential{t}+\int_{t_{0}}^{t_{1}}(\mathbf{f}\dotproduct%
\mathbf{v}(L)-m_{\alpha}v_{\alpha,x}(L)+m\varphi(L))\differential{t}.$$
The Euler equations become
$$\bar{\rho}|\mathcal{A}|\ddot{v}=T_{,x},\quad\bar{\rho}|\mathcal{A}|\ddot{v}_{%
\alpha}=M_{\alpha,xx},\quad\bar{\rho}I_{\alpha\alpha}\ddot{\varphi}=M_{,x}.$$
These equations are subjected to the boundary conditions (37) and the initial conditions. Similar to the static case, it can be proved that the solution of this 1-D dynamic theory converges to the solution of the 3-D theory in the limits $h/L\to 0$ and $\epsilon=c/h\to 0$ and for low frequency vibrations (Le, 1999).
8 Conclusion
It is shown in this paper that the rigorous first order approximate 1-D theory of thin FRC beams can be derived from the exact 3-D elasticity theory by the variational-asymptotic method. The developed finite element code can be used to solve cross-sectional problems with arbitrary elastic moduli of the fibers and matrix as well as arbitrary distributions and shapes of fibers. The extension of this multi-scaled asymptotic analysis to curved and naturally twisted FRC beams is straightforward (Le, 1999). As seen from (Le, 1999), the extension, bending, and torsion modes of beams are coupled in that case. The VAM combined with the finite element cross-sectional analysis can also be applied to the nonlinear FRC beams with periodic microstructure in the spirit of (Yu et al., 2002, 2005; Yu and Tang, 2007). Finally, let us mention the FRC beams with randomly distributed fibers under torsion analyzed in (Barretta et al., 2015). The analysis of extension and bending of such beams requires the solution of the plane strain problem which is quite challenging.
Appendix: Matlab-code
⬇
1function [CH,DH,FH] = homogenizecs(lx, ly, lambda, mu, phi, x)
2%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
3% lx = Unit cell length in x-direction.
4% ly = Unit cell length in y-direction.
5% lambda = Lame’s first parameter for both materials. Two entries.
6% mu = Lame’s second parameter for both materials. Two entries.
7% phi = Angle between horizontal and vertical cell wall. Degrees
8% x = Material indicator matrix. Size used to determine nelx/nely
9%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
10%% INITIALIZE
11% Compute contraction ratios
12nu = [lambda(1)/(2*(lambda(1) + mu(1)))...
13 lambda(2)/(2*(lambda(2) + mu(2)))];
14% Deduce discretization
15[nely, nelx] = size(x);
16% Stiffness matrix consists of two parts, one belonging to lambda and
17% one belonging to mu. Same goes for load vector
18dx = lx/nelx; dy = ly/nely;
19nel = nelx*nely;
20[keLambda, keMu, feLambda, feMu, feXiLambda, feXiMu] = elementMatVec(dx/2, dy/2, phi);
21% Node numbers and element degrees of freedom for full (not periodic) mesh
22nodenrs = reshape(1:(1+nelx)*(1+nely),1+nely,1+nelx);
23edofVec = reshape(2*nodenrs(1:end-1,1:end-1)+1,nel,1);
24edofMat = repmat(edofVec,1,8)+repmat([0 1 2*nely+[2 3 0 1] -2 -1],nel,1);
25%% IMPOSE PERIODIC BOUNDARY CONDITIONS
26% Use original edofMat to index into list with the periodic dofs
27nn = (nelx+1)*(nely+1); % Total number of nodes
28nnP = (nelx)*(nely); % Total number of unique nodes
29nnPArray = reshape(1:nnP, nely, nelx);
30% Extend with a mirror of the top border
31nnPArray(end+1,:) = nnPArray(1,:);
32% Extend with a mirror of the left border
33nnPArray(:,end+1) = nnPArray(:,1);
34% Make a vector into which we can index using edofMat:
35dofVector = zeros(2*nn, 1);
36dofVector(1:2:end) = 2*nnPArray(:)-1;
37dofVector(2:2:end) = 2*nnPArray(:);
38edofMat = dofVector(edofMat);
39ndof = 2*nnP; % Number of dofs
40%% ASSEMBLE STIFFNESS MATRIX
41% Indexing vectors
42iK = kron(edofMat,ones(8,1))’;
43jK = kron(edofMat,ones(1,8))’;
44% Material properties in the different elements
45lambda = lambda(1)*(x==1) + lambda(2)*(x==2);
46mu = mu(1)*(x==1) + mu(2)*(x==2);
47nu = nu(1)*(x==1) + nu(2)*(x==2);
48% The corresponding stiffness matrix entries
49sK = keLambda(:)*lambda(:).’ + keMu(:)*mu(:).’;
50K = sparse(iK(:), jK(:), sK(:), ndof, ndof);
51%% LOAD VECTORS AND SOLUTION
52% Assembly three load cases corresponding to the three strain cases
53sF = feLambda(:)*lambda(:).’+feMu(:)*mu(:).’;
54lambda_nu = lambda.*nu;
55mu_nu = mu.*nu;
56sF=[sF; feXiLambda(:)*lambda_nu(:).’ + feXiMu(:)*mu_nu(:).’];
57iF = repmat(edofMat’,4,1);
58jF = [ones(8,nel); 2*ones(8,nel); 3*ones(8,nel); 4*ones(8,nel)];
59F = sparse(iF(:), jF(:), sF(:), ndof, 4);
60% Solve (remember to constrain one node)
61chi(3:ndof,:) = K(3:ndof,3:ndof)\F(3:ndof,:);
62chi_4=chi(:,4);
63%% HOMOGENIZATION
64% The displacement vectors corresponding to the unit strain cases
65chi0 = zeros(nel, 8, 3);
66% The element displacements for the three unit strains
67chi0_e = zeros(8, 4);
68ke = keMu + keLambda; % Here the exact ratio does not matter, because
69fe = feMu + feLambda; % it is reflected in the load vector
70fe = [fe feXiLambda+feXiMu];
71chi0_e([3 5:end],:) = ke([3 5:end],[3 5:end])\fe([3 5:end],:);
72chi0_4_e=nu(:)*chi0_e(:,4)’;
73% epsilon0_11 = (1, 0, 0)
74chi0(:,:,1) = kron(chi0_e(:,1)’, ones(nel,1));
75% epsilon0_22 = (0, 1, 0)
76chi0(:,:,2) = kron(chi0_e(:,2)’, ones(nel,1));
77% epsilon0_12 = (0, 0, 1)
78chi0(:,:,3) = kron(chi0_e(:,3)’, ones(nel,1));
79CH = zeros(3);
80DH = [0; 0; 0];
81cellVolume = lx*ly;
82sumLambda = ((chi0_4_e(:,:)-chi_4(edofMat))*...
83 keLambda).*(chi0_4_e(:,:)-chi_4(edofMat));
84sumMu = ((chi0_4_e(:,:)-chi_4(edofMat))*keMu).*...
85 (chi0_4_e(:,:)-chi_4(edofMat));
86sumLambda = reshape(sum(sumLambda,2), nely, nelx);
87sumMu = reshape(sum(sumMu,2), nely, nelx);
88FH = 1/cellVolume*sum(sum(lambda.*sumLambda+mu.*sumMu));
89for i = 1:3
90 sumLambda = ((chi0_4_e(:,:)-chi_4(edofMat))*...
91 keLambda).*(chi0(:,:,i)-chi(edofMat+(i-1)*ndof));
92 sumMu = ((chi0_4_e(:,:)-chi_4(edofMat))*keMu).*...
93 (chi0(:,:,i)-chi(edofMat+(i-1)*ndof));
94 sumLambda = reshape(sum(sumLambda,2), nely, nelx);
95 sumMu = reshape(sum(sumMu,2), nely, nelx);
96 DH(i) = 1/cellVolume*sum(sum(lambda.*sumLambda+mu.*sumMu));
97 for j = 1:3
98 sumLambda = ((chi0(:,:,i) - chi(edofMat+(i-1)*ndof))*keLambda).*...
99 (chi0(:,:,j) - chi(edofMat+(j-1)*ndof));
100 sumMu = ((chi0(:,:,i) - chi(edofMat+(i-1)*ndof))*keMu).*...
101 (chi0(:,:,j) - chi(edofMat+(j-1)*ndof));
102 sumLambda = reshape(sum(sumLambda,2), nely, nelx);
103 sumMu = reshape(sum(sumMu,2), nely, nelx);
104 % Homogenized elasticity tensor
105 CH(i,j) = 1/cellVolume*sum(sum(lambda.*sumLambda + mu.*sumMu));
106 end
107end
108disp(’--- Homogenized elasticity CH ---’); disp(CH)
109disp(’--- Homogenized elasticity DH ---’); disp(DH)
110disp(’--- Homogenized elasticity FH ---’); disp(FH)
111%% COMPUTE ELEMENT STIFFNESS MATRIX AND FORCE VECTOR (NUMERICALLY)
112function [keLambda, keMu, feLambda, feMu, feXiLambda, feXiMu] = elementMatVec(a, b, phi)
113% Constitutive matrix contributions
114CMu = diag([2 2 1]); CLambda = zeros(3); CLambda(1:2,1:2) = 1;
115% Two Gauss points in both directions
116xx=[-1/sqrt(3), 1/sqrt(3)]; yy = xx;
117ww=[1,1];
118% Initialize
119keLambda = zeros(8,8); keMu = zeros(8,8);
120feLambda = zeros(8,3); feMu = zeros(8,3); feXiLambda = zeros(8,1);feXiMu = zeros(8,1);
121L = zeros(3,4); L(1,1) = 1; L(2,4) = 1; L(3,2:3) = 1;
122for ii=1:length(xx)
123for jj=1:length(yy)
124% Integration point
125x = xx(ii); y = yy(jj);
126% Differentiated shape functions
127dNx = 1/4*[-(1-y) (1-y) (1+y) -(1+y)];
128dNy = 1/4*[-(1-x) -(1+x) (1+x) (1-x)];
129% Jacobian
130J = [dNx; dNy]*[-a a a+2*b/tan(phi*pi/180) 2*b/tan(phi*pi/180)-a; ...
131-b -b b b]’;
132detJ = J(1,1)*J(2,2) - J(1,2)*J(2,1);
133invJ = 1/detJ*[J(2,2) -J(1,2); -J(2,1) J(1,1)];
134% Weight factor at this point
135weight = ww(ii)*ww(jj)*detJ;
136% Strain-displacement matrix
137G = [invJ zeros(2); zeros(2) invJ];
138dN = zeros(4,8);
139dN(1,1:2:8) = dNx;
140dN(2,1:2:8) = dNy;
141dN(3,2:2:8) = dNx;
142dN(4,2:2:8) = dNy;
143B = L*G*dN;
144% Element matrices
145keLambda = keLambda + weight*(B’ * CLambda * B);
146keMu = keMu + weight*(B’ * CMu * B);
147% Element loads
148feLambda = feLambda + weight*(B’ * CLambda * diag([1 1 1]));
149feMu = feMu + weight*(B’ * CMu * diag([1 1 1]));
150feXiLambda = feXiLambda + weight*(B’ * CLambda* [1; 1; 0]);
151feXiMu=feXiMu+ weight*(B’ * CMu* [1; 1; 0]);
152end
153end
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Thermalization in many-particle quantum walks
Dibwe Pierrot Musumbu
[email protected]
Center for Theoretical Physics of the Polish Academy of Science, Al. Lotnikow 32/46, 02-668 Warsaw, Poland
Maria Przybylska
[email protected]
Institute of Physics, University of Zielona Góra, Licealna 9, 65–417 Zielona Góra, Poland
Andrzej J. Maciejewski
[email protected]
Institute of Astronomy, University of Zielona Góra, Licealna 9, PL-65–417 Zielona Góra, Poland.
Abstract
Many-particles quantum walks of particles obeying Bose statistics moving on graphs of various topologies are introduced. A single coin tossing commands the conditional shift operation over the whole graph. Vertices particle densities, the mean values of the phase space variables, second order spatial correlation and counting statistics are evaluated and simulated. Evidence of an universal dynamics is presented.
Many-particle, bosons, thermalization, quantum walks
pacs: 03.65.Ge,02.30.Ik,42.50.Pq
I Introduction
Classical random walks have proven to be a powerful tool in both physics and mathematics through their numerous applications in
algorithmics: Markov processes, Monte Carlo simulations, etc…Recent developments of quantum computation and quantum information raises
renewed interests directed toward advantages in quantum algorithms Aharonov:01 ; Nayak:01 ; Milburn:01 ; Childs:01 and aiming to adapt random walks to the quantum world.
Problems such as bosons sampling Seshadreesan:01 , development of quantum metrology, simulations of many-body quantum systems Knill:01 , waveguides arrays Peruzzo:01 ; Hamilton:01 ; DeNicola:01 and many others, have driven the newly developed quantum walks toward the use of many quantum walkers.
In recent developments, the studies on quantum walks are also directed toward understanding the dynamics of many-particle systems on lattices
laflamme:01 ; laflamme:02 ; Mayer01 ; Xue02 ; Sheridan:01 . In that regards, studying many-particles on graph implies exploring individual vertices dynamics and how much they affect the entire system. Furthermore, looking at the evolution of both individual vertices and the entire system enables to explore how long the system can be confined in a certain regime. This is similar to measuring how fast the quantum walk spreads or how confined the quantum walk stays in a small neighborhood Aharonov:01 .
In this work we study many-particle quantum walks on closed graphs: the cyclic graph, the double hexagon graph and the Petersen graph. We simulate discrete time quantum walks on these graphs for twelve bosons on ten vertices. We focus on the universality of dynamics appearing in the counting statistics and we analyze how vertices occupation numbers evolve in time. We observe the universal behavior of the counting statistics that is independent of the types of graph and the initial conditions of the systems as long as the number of vertices of the graphs is the same and the number of quantum walkers remains unchanged.
We recall that the quantum walk is a unitary process and therefore, its probability distribution and many other observables do not converge julia:02 . Nevertheless
we must precise that a step in discrete time quantum walks is a two stage operation involving the state space and an additional degree of freedom known as the coin
degrees Childs04 . This puts a restriction on the unitarity in discrete quantum walks but it can be overcomes as suggested in Watrous:01 . Consequently, we do not expect the dynamics into our systems to converge. We control how long the quantum walks is confined on a vertex and how this affects the evolution of the entire quantum walks. We use the vertices particle distribution, the multimode phase space dynamics, the vertices counting statistics as well as the second order position correlation functions to study time evolution of our systems. In the counting statistics we
observed a change of regime.
The title of this article introduce the concept of thermalization in many-particle quantum walks.
In classical systems thermalization is associated with the time evolution of physical observables such as momentum, energy etc…
toward a Boltzmann distribution independently of theirs initial conditions. This behavior is explained by classical physics
in the following way. Almost all particle trajectories independently of theirs initial conditions, quickly begin to look alike
because theirs dynamics is governed by nonlinear equations that drive them to explore the constant-energy manifold ergodically.
However, if the classical system possesses additional first integrals that are functionally independent of the hamiltonian and
each other, then the phase curves of the system of particles are confined to a highly restricted region of
the energy manifold. Hence, the statistical predictions fail and the system does not thermalize Gallavotti:99 . But the time evolution of isolated quantum systems is linear and theirs spectra are discrete Krylov:01 . Moreover in isolated quantum systems, the conditions directing conserved quantities to provide independent constraints on the relaxation dynamics are not well understood Rigol01 ; Rigol02 ; Rigol03 . Roughly speaking,
thermalization in isolated quantum systems can be understood as a time evolution to a state that belongs to a proper subspace of the Hilbert space of the system. The thermalization of a quantum dynamics corresponds with an observable that mean values behave according to the prediction of an appropriate statistical mechanical ensemble (represented by a certain density matrix). In our simulations
of the quantum walks on the three systems, we have observed such a behavior for the particles counting statistics. We need to precise that we are analyzing the quantum thermalization with respect to the eigenstates thermalization hypothesis Deutsch ; Srednicki . This means that the dimension of the fraction of the Hilbert space containing the thermalized state remains constant in time. We want to avoid the generalities by reserving a more detailed analysis in the conclusion.
This article is planned as follows. We start by introducing the mathematical formulation of quantum walks using identical particles in the spatial field representation in order to construct the graph many-particle state (GMP state). Next we construct the conditional shift operator needed for the implementation of the shared coin’s many-particle quantum walks using the topology of the graph. After, we present the results of the simulations for many-particle quantum walks. In each case, we present the evolution of the vertices particles distributions, multimode phase space dynamics, the second order position correlation functions and the vertices counting statistics. The last section contains our conclusions and gives a short outlook.
II The graph many-particle state
Let us consider a family of operators $\widehat{\mathbf{\Uppsi}}^{\dagger}{\mbox{\small{(}}x\mbox{\small{)}}}$ acting on the vacuum state $|\Omega\rangle$ such that they create position eigenvector $|x\rangle$:
$$\widehat{\mathbf{\Uppsi}}^{\dagger}{\mbox{\small{(}}x\mbox{\small{)}}}|\Omega\rangle=|x\rangle.$$
(1)
This field operator creates a particle at position $x$. Similarly we have the family of operators $\widehat{\mathbf{\Uppsi}}{\mbox{\small{(}}x\mbox{\small{)}}}$ called the field annihilation operators defined as
$$\widehat{\mathbf{\Uppsi}}{\mbox{\small{(}}x\mbox{\small{)}}}|x\rangle=|\Omega\rangle.$$
(2)
In this work, we use discrete field operators therefore when considering the integration over space we will use summation over all the position $x$. We recall that the field operators satisfying a set of axioms were introduced by Jordan and Wigner Wigner01 in order to describe identical particles. Those axioms lead to two types of fields describing the two types of particles: bosons and fermions. The creation $\widehat{\mathbf{\Uppsi}}^{\dagger}{\mbox{\small{(}}x_{\mbox{\tiny{$2$}}}\mbox{\small{)}}}$ and the annihilation $\widehat{\mathbf{\Uppsi}}{\mbox{\small{(}}x_{\mbox{\tiny{$1$}}}\mbox{\small{)}}}$
and the field operator satisfy a commutator algebra,
$$\displaystyle[\widehat{\mathbf{\Uppsi}}{\mbox{\small{(}}x_{\mbox{\tiny{$1$}}}\mbox{\small{)}}},\,\widehat{\mathbf{\Uppsi}}^{\dagger}{\mbox{\small{(}}x_{\mbox{\tiny{$2$}}}\mbox{\small{)}}}]$$
$$\displaystyle=$$
$$\displaystyle\delta(x_{\mbox{\tiny{$1$}}}-x_{\mbox{\tiny{$2$}}}),$$
(3)
$$\displaystyle[\widehat{\mathbf{\Uppsi}}^{\dagger}{\mbox{\small{(}}x_{\mbox{\tiny{$1$}}}\mbox{\small{)}}},\,\widehat{\mathbf{\Uppsi}}^{\dagger}{\mbox{\small{(}}x_{\mbox{\tiny{$2$}}}\mbox{\small{)}}}]$$
$$\displaystyle=$$
$$\displaystyle 0,$$
(4)
$$\displaystyle[\widehat{\mathbf{\Uppsi}}{\mbox{\small{(}}x_{\mbox{\tiny{$1$}}}\mbox{\small{)}}},\,\widehat{\mathbf{\Uppsi}}{\mbox{\small{(}}x_{\mbox{\tiny{$2$}}}\mbox{\small{)}}}]$$
$$\displaystyle=$$
$$\displaystyle 0,$$
(5)
for bosons, and the anti commutator algebra
$$\displaystyle\{\widehat{\mathbf{\Uppsi}}{\mbox{\small{(}}x_{\mbox{\tiny{$1$}}}\mbox{\small{)}}},\,\widehat{\mathbf{\Uppsi}}^{\dagger}{\mbox{\small{(}}x_{\mbox{\tiny{$2$}}}\mbox{\small{)}}}\}$$
$$\displaystyle=$$
$$\displaystyle\delta(x_{\mbox{\tiny{$1$}}}-x_{\mbox{\tiny{$2$}}}),$$
(6)
$$\displaystyle\{\widehat{\mathbf{\Uppsi}}^{\dagger}{\mbox{\small{(}}x_{\mbox{\tiny{$1$}}}\mbox{\small{)}}},\,\widehat{\mathbf{\Uppsi}}^{\dagger}{\mbox{\small{(}}x_{\mbox{\tiny{$2$}}}\mbox{\small{)}}}\}$$
$$\displaystyle=$$
$$\displaystyle 0,$$
(7)
$$\displaystyle\{\widehat{\mathbf{\Uppsi}}{\mbox{\small{(}}x_{\mbox{\tiny{$1$}}}\mbox{\small{)}}},\,\widehat{\mathbf{\Uppsi}}{\mbox{\small{(}}x_{\mbox{\tiny{$2$}}}\mbox{\small{)}}}\}$$
$$\displaystyle=$$
$$\displaystyle 0,$$
(8)
for fermions.
Suppose that we want to populate the vertex at the position $x$ with $n$ particles. Mathematically we can write
$${\mbox{\large{(}}\widehat{\mathbf{\Uppsi}}^{\dagger}{\mbox{\small{(}}x\mbox{\small{)}}}\mbox{\large{)}}}^{n}|\Omega\rangle=\sqrt{(n_{x})!}|n_{x}\rangle,$$
(9)
where $|n_{x}\rangle$ is the eigenstate of the operator $\hat{\bf n}_{x}=\widehat{\mathbf{\Uppsi}}^{\dagger}{\mbox{\small{(}}x\mbox{\small{)}}}\widehat{\mathbf{\Uppsi}}{\mbox{\small{(}}x\mbox{\small{)}}}$. For the moment we are only
interested by the number of particles at vertex $x$, any other property of particles such as modes are irrelevant.
Since we will shift particles between different vertices therefore in a discrete setting, the position of a vertex will be indexed ${x}_{\alpha}$ in order to distinguish different vertices. In that regards, when considering two vertices with different number of particles, we can write $|{n}_{x_{\alpha\mbox{\tiny{$1$}}}}\rangle$ and $|{n}_{x_{\alpha\mbox{\tiny{$2$}}}}\rangle$. For the sake of notations simplification we will use:
$|{n}_{\alpha_{\mbox{\tiny{$1$}}}}\rangle$ instead $|{n}_{x_{\alpha\mbox{\tiny{$1$}}}}\rangle$ and $|{n}_{\alpha_{\mbox{\tiny{$2$}}}}\rangle$ instead $|{n}_{x_{\alpha\mbox{\tiny{$2$}}}}\rangle$.
We consider $N$ indistinguishable bosons distributed on a $M$-vertices graph. The Hilbert space of such a system is spanned by
$$D\mbox{\small{(}}\mbox{\small{$N,M$}}\mbox{\small{)}}=\bigg{(}{\mbox{\small{$M+N-1$}}\atop{\mbox{\small{$N$}}}}\bigg{)}$$
(10)
vectors $|{\bf n}\rangle=|n_{1},\ldots,n_{M}\rangle=|{n}_{\alpha_{\mbox{\tiny{$1$}}}}\rangle\otimes|{n}_{\alpha_{\mbox{\tiny{$2$}}}}\rangle\otimes\ldots\otimes|{n}_{\alpha_{\mbox{\tiny{$M$}}}}\rangle$ of magnitude $N=\sum_{\alpha}^{M}n_{\alpha}$. In quantum optics and quantum matter, laser light have been used to confine cold atoms on lattices. In such a case the word configuration is used to indicate a specific way of arranging cold atoms on lattice Immanuel . We use the same word configuration to indicate the vectors $|{\bf n}\rangle$. The components of the $|{\bf n}\rangle$’s represent the position occupation numbers $n_{\alpha}$ of individual vertices. The sum of the $n_{\alpha}$’s over the entire graph is equal to the total number of quantum walkers $N$. Individual possible configuration will be denoted as $|{\bf n}_{\ell}\rangle$, where $\ell=1,\ldots,D\mbox{\small{(}}\mbox{\small{$N,M$}}\mbox{\small{)}}$.
The set of all configuration is called Hilbert space of configurations. In general, a state of a $M$-vertices graph
containing $N$ particles can be represented as
$$\big{|}\Psi\big{\rangle}=\sum_{\ell}C_{\ell}|{\bf n}_{\ell}\rangle,$$
(11)
where $|{\bf n}_{\ell}\rangle$ generate a basis of the Hilbert space of many-particles on a graph. We use Dirac bra ket notation for simplicity.
In addition, we need an auxiliary Hilbert space (the so called coin’s Hilbert space) spanned by the vectors specifying the directions of the edges connected to any given vertex.
The number of edges ending at a vertex is called the vertex degree of this vertex. The degree of a graph is the largest of its vertices degrees Reinhard:01 . For a graph of degree $d$, the
coin’s Hilbert space is spanned by the basis $\{|v_{{}_{1}}\rangle,\ldots,|v_{d}\rangle\}$. These $d$ vectors are also known as
chiralities Ambainis02 of the coin and we will elaborate more on them in the next section. The tensor product of the coin’s Hilbert space and the configurations Hilbert space is equivalent to attaching a single coin to a configuration $|{\bf n}_{\ell}\rangle$. Therefore, a single coin is shared by all particles under the fixed configuration. The GMP states in Eq. (11) augmented with the coins chiralities
becomes
$$|\Psi_{r}\rangle=\sum_{\ell}\sum_{i}\frac{C_{i\ell}^{r}}{\mathcal{K}_{r}}|v_{i},\,{\bf n}_{\ell}\rangle,$$
(12)
where $|v_{i},\,{\bf n}_{\ell}\rangle=|v_{i}\rangle\otimes|{\bf n}_{\ell}\rangle$.
The state $|\Psi_{\mbox{\tiny{r}}}\rangle$ represents the system of $N$ particles distributed on $M$ vertices of the graph, augmented with the directional amplitudes. We use the abbreviation ’GMP state’ to indicate $|\Psi_{\mbox{\tiny{r}}}\rangle$. The index $r$ indicates the time step of the quantum walks and
$\mathcal{K}_{r}$ is the normalization constant
$$[\mathcal{K}_{r}]^{2}=\sum_{\ell}\sum_{i}|C_{i\ell}^{r}|^{2}.$$
(13)
Since $|n_{\alpha}\rangle$ can be created from the vacuum $|\Omega\rangle$, see Eq. (9), we have constructed the GMP state using field operator scheme Wigner:01 .
Now we want to equip the vertices populations with some additional properties. To this end we recall that every particle
created at a given vertex is characterized by its mode ${\upeta}$. Let us consider a position field operator $\widehat{\mathbf{\Uppsi}}^{\dagger}{\mbox{\small{(}}{x}_{\alpha}\mbox{\small{)}}}$ and the related Fock creation operator ${\hat{\bf b}}_{\upeta}^{\dagger}$ as well as the conjugated operators $\widehat{\mathbf{\Uppsi}}{\mbox{\small{(}}{x}_{\alpha}\mbox{\small{)}}}$ and ${\hat{\bf b}}_{\upeta}$:
$$\widehat{\mathbf{\Uppsi}}^{\dagger}{\mbox{\small{(}}{x}_{\alpha}\mbox{\small{)}}}=\sum_{\upeta}\frac{e^{-i{\varphi}_{\upeta}{x}_{\alpha}}}{\sqrt{N}}{\hat{\bf b}}_{\upeta}^{\dagger}\,\,\,{\mbox{and}}\,\,\,\widehat{\mathbf{\Uppsi}}{\mbox{\small{(}}{x}_{\alpha}\mbox{\small{)}}}=\sum_{\upeta}\frac{e^{i{\varphi}_{\upeta}{x}_{\alpha}}}{\sqrt{N}}{\hat{\bf b}}_{\upeta},$$
(14)
where $\varphi_{\upeta}$ denotes the phase of a mode $\upeta$ and ${x}_{\alpha}$ is the position. The boson Fock operators obey the following commutation rules
$$\big{[}{\hat{\bf b}}_{\upeta_{\mbox{\tiny{1}}}},\,{\hat{\bf b}}^{\dagger}_{\upeta_{\mbox{\tiny{2}}}}\big{]}=\delta_{{\upeta_{\mbox{\tiny{1}}}}{\upeta_{\mbox{\tiny{2}}}}},\,\,\,\big{[}{\hat{\bf b}}_{\upeta_{\mbox{\tiny{1}}}},\,{\hat{\bf b}}_{\upeta_{\mbox{\tiny{2}}}}\big{]}=0,\,\,\,\big{[}{\hat{\bf b}}^{\dagger}_{\upeta_{\mbox{\tiny{1}}}},\,{\hat{\bf b}}^{\dagger}_{\upeta_{\mbox{\tiny{2}}}}\big{]}=0.$$
(15)
The boson Fock representation enables us to rewrite the GMP state in Eq.
(12) in multimode boson Fock representation as:
$$|\Psi_{r}\rangle=\sum_{\ell}\sum_{j}\frac{C_{j\ell}^{r}}{\mathcal{K}_{r}}|v_{j}\rangle\bigotimes_{\alpha=1}^{M}{1\over{\sqrt{(n_{\alpha})!}}}\bigg{[}\sum_{\upeta}\frac{e^{-i{\varphi}_{\upeta}{x}_{\alpha}}}{\sqrt{N}}{\hat{\bf b}}_{\upeta}^{\dagger}\bigg{]}^{n_{\alpha}}|\Omega\rangle,$$
(16)
where $|\Omega\rangle$ is the vacuum state.
III The conditional shift operator
Before entering in the design of our conditional shift operator we need to give some remarks about distinction between continuous time quantum walks and discrete time quantum walks. Continuous time quantum walks are analogous to classical diffusion process where the stochastic matrix is substituted by the adjacency matrix of the graph and the probabilities are replaced by the amplitudes. This makes the continuous quantum walks as the process described by the Schrödinger equation where the Hamiltonian is the adjacency matrix Childs04 . Since the adjacency matrix is real and symmetric, the time evolution is unitary. Moreover it was proved that there is no discrete time process that is translationally invariant on a $d$-dimensional graph Meyer:01 ; Meyer:02 . Concerning our many-particle quantum walks, the conditional shift operator is the product of the field annihilation and creation operators, the adjacency matrix and the coin operator (Hadamard operator) as we will show in this section. During the implementation of the quantum walks, the action of the annihilation and creation operators in the conditional shifting of many-particles doesn’t preserve the length of configurations and therefore destroys the unitarity. For this reason we need a step by step normalization of the GMP state. We also notice that the dimension of the Hilbert space of the discrete time quantum walks is greater than that of a continuous time quantum walks.
As it was already noted by several authors, the structure of the GMP state suggests that the operator implementation of the quantum walks
is a two stages operation. The first stage is performed using the coin’s tossing operation.
It is defined by a coin tossing operator that is unitary operator acting on the coin’s Hilbert space. After the coin tossing the particle shifting is performed with respect to the selected direction. In a
single particle quantum walk on a line, according to Nayak:01 ; julia:01 ; julia:02 , the coin tossing is performed using the Hadamard gate
$${\bf H}={1\over{\sqrt{2}}}\left(\begin{array}[]{cc}1&1\\
1&-1\end{array}\right).$$
(17)
On a line the coin Hilbert space is two dimensional (i.e. $d=2$) and spanned by vectors $|v_{1}\rangle$ and $|v_{2}\rangle$ that correspond to the left-hand side move and right-hand move respectively. Considering the vectors $|v_{1}\rangle=\big{(}{\mbox{\tiny{1}}\atop{\mbox{\tiny{0}}}}\big{)}$ and $|v_{2}\rangle=\big{(}{\mbox{\tiny{0}}\atop{\mbox{\tiny{1}}}}\big{)}$ of the coins Hilbert space, the coin tossing respectively transforms $|v_{1}\rangle$ and $|v_{2}\rangle$ in following ways
$$\displaystyle{\bf H}|v_{1}\rangle$$
$$\displaystyle=$$
$$\displaystyle{1\over{\sqrt{2}}}(|v_{1}\rangle+|v_{2}\rangle),$$
(18)
$$\displaystyle{\bf H}|v_{2}\rangle$$
$$\displaystyle=$$
$$\displaystyle{1\over{\sqrt{2}}}(|v_{1}\rangle-|v_{2}\rangle).$$
(19)
This coin tossing operation is combined with the particle shifting operation where the shift is performed using the following operator
$${\bf S}=|v_{2}\rangle\langle v_{2}|\otimes\sum_{\nu\in\mathbb{Z}}|\nu-1\rangle\langle\nu|+|v_{1}\rangle\langle v_{1}|\otimes\sum_{\nu\in\mathbb{Z}}|\nu+1\rangle\langle\nu|.$$
(20)
This kind of shifting is well constructed for a single-particle quantum walk on the line where the structure of the
graph is such that adjacent vertices are also successive vertices. In such a setup the projector
$|\nu+1\rangle\langle\nu|\equiv|\nu+1\rangle\langle 0|\cdot|0\rangle\langle\nu|$
is equivalent to the product $\widehat{\mathbf{\Uppsi}}^{\dagger}{\mbox{\small{(}}{x}_{\nu+1}\mbox{\small{)}}}\widehat{\mathbf{\Uppsi}}{\mbox{\small{(}}{x}_{\nu}\mbox{\small{)}}}$ of field creation and annihilation operators. This line constructed scheme of coin tossing and particle shifting can be extended to other types of graphs and many-particle shifting. Bearing that in mind, we define the coin’s tossing operator ${\bf H}_{d}$ as a $d$-order Hadamard operator given by the following formula
$${\bf H}_{d}=\sum_{j,k}h_{jk}|v_{j}\rangle\langle v_{k}|\,\,\,\,\,j,\,k\,\in\,1,2,3,\ldots,d.$$
(21)
These $h_{\jmath k}$ are normalized roots of the unity of order $d$.
The operation on the coin’s Hilbert space provides the direction to the quantum walk. In addition, we need the shifting operation which implements the
movement of quantum walkers from theirs previous positions (one configuration) to their next positions (other configuration) at each step. Such an operation over the graph uses the property of
graph connectivity and the field creation and annihilation operations. In fact, particles can only be shifted between adjacent vertices, therefore
the adjacency matrix is involved in the shifting operation. For the $M$-vertices graph, the adjacency matrix
is a $M$-dimension square matrix with entries
$$\displaystyle A_{\mu\nu}$$
$$\displaystyle=$$
$$\displaystyle\begin{cases}1&~{}~{}\text{when $\mu$ and $\nu$ are connected,}\\
0&~{}~{}\text{when $\mu$ and $\nu$ are not connected.}\end{cases}$$
In addition to the adjacency matrix, the field creation and annihilation
operators interfere every time when a particle is shifted from vertex $\mu$ to vertex $\nu$. In other words, the annihilation operator acts on vertex
$\mu$ removing one occupant while the creation operator acts on vertex $\nu$ adding one more occupant.
For example for the cyclic graph in Fig.1a, the adjacency matrix is a $10\times 10$ matrix ${\bf A}=\big{(}A_{\nu\mu}\big{)}$; $\mu,\,\nu\,=\,1,\ldots,10$ of the form
$${\bf A}=\left(\begin{array}[]{ccccccc}0&1&\cdots&0&1\\
1&0&\cdots&0&0\\
\vdots&\vdots&\ddots&\vdots&\vdots\\
0&0&\cdots&0&1\\
1&0&\cdots&1&0\end{array}\right).\vspace*{-0.2cm}$$
(22)
Graphs considered in quantum walks are undirected and quantum particles on them can spread in all directions. For the cyclic graph in Fig.1a. the quantum walks on each edge $(\mu,\,\nu)$ are realized both ways clockwise and counterclockwise and this property enables to write its
adjacency matrix as the sum of its two mutually transposed components
$${\bf A}={\bf A}_{\mbox{\tiny{L}}}+{\bf A}_{\mbox{\tiny{L}}}^{\mbox{\tiny{T}}},$$
(23)
where
$${\bf A}_{\mbox{\tiny{L}}}=\left(\begin{array}[]{ccccccc}0&0&\cdots&0&1\\
1&0&\cdots&0&0\\
\vdots&\vdots&\ddots&\vdots&\vdots\\
0&0&\cdots&0&0\\
0&0&\cdots&1&0\end{array}\right).\vspace*{-0.2cm}$$
(24)
${\bf A}_{\mbox{\tiny{L}}}$ is the adjacency matrix for directed cyclic graph corresponding to counterclockwise walks on this graph.
Applying this property of the adjacency matrix to the vertex particle annihilation and creation operations enables to write the shift operator for the
cyclic graph in Fig.1a in the following form
$${\bf S}=\mathcal{C}^{\dagger}{\bf A}_{\mbox{\tiny{L}}}\mathcal{C}\otimes|v_{1}\rangle\langle v_{1}|+\mathcal{C}^{\dagger}{\bf A}_{\mbox{\tiny{L}}}^{\mbox{\tiny{T}}}\mathcal{C}\otimes|v_{2}\rangle\langle v_{2}|,$$
(25)
where:
$$\begin{cases}\mathcal{C}&=\big{(}\widehat{\mathbf{\Uppsi}}{\mbox{\small{(}}{x}_{\mbox{\tiny{$1$}}}\mbox{\small{)}}},\ldots,\widehat{\mathbf{\Uppsi}}{\mbox{\small{(}}{x}_{\mbox{\tiny{$M$}}}\mbox{\small{)}}}\big{)}^{\mbox{\tiny{T}}},\\
\mathcal{C}^{\dagger}&=\big{(}\widehat{\mathbf{\Uppsi}}^{\dagger}{\mbox{\small{(}}{x}_{\mbox{\tiny{$1$}}}\mbox{\small{)}}},\ldots,\widehat{\mathbf{\Uppsi}}^{\dagger}{\mbox{\small{(}}{x}_{\mbox{\tiny{$M$}}}\mbox{\small{)}}}\big{)}.\end{cases}$$
(26)
The similar decomposition of the adjacency matrix as in Eq. (23) is valid for the line graph. For two degrees undirected graphs, two directions of quantum walks correspond to two components of the adjacency matrix ${\bf A}$ in Eq. (22). In general let us consider a $d$ degrees graph, each direction induces an automorphism of the graph. The automorphism of graph are elements of the permutation group defined on the graph. On any given vertex $\mu$ we define an permutation in direction $k$, $P^{k}\mbox{\small{(}}\mu\mbox{\small{)}}\equiv\nu$. The set of all permutations $P^{k}\mbox{\small{(}}\mu\mbox{\small{)}}\equiv\nu$ over the entire graph defines a class of equivalence in the permutation group. Such a class of equivalence is a graph. The adjacency matrix $(A_{\nu\mu}^{k})$ of such a graph corresponding to direction $k$ is a component of the adjacency matrix of the graph $(A_{\nu\mu})$. For example for the Petersen graph given in Fig.1c one can define four directions i.e. $d=4$. The first direction $\mu\rightarrow\nu$ if $\mu-\nu=-1$ and the second direction $\mu\rightarrow\nu$ if $\mu-\nu=+1$. The third direction $\mu\rightarrow\nu$ if $\mu-\nu=-5$, the fourth direction $\mu\rightarrow\nu$ if $\mu-\nu=+5$. The undirected move on each direction generates two mutually transposed components of the adjacency matrix. Thus for the Petersen graph we can have four components ${\bf A}^{k}=(A_{\nu\mu}^{k})$ such that: ${\bf A}^{1}=({\bf A}^{2})^{\mbox{\tiny{T}}}$ and ${\bf A}^{3}=({\bf A}^{4})^{\mbox{\tiny{T}}}$. In general the action of the permutation group over the graph splits the adjacency matrix into $d$ components.
Therefore we generalize the shift operator given in Eq. (25) to the following form
$${\bf S}=\sum_{\mu,\nu}\sum_{k}\widehat{\mathbf{\Uppsi}}^{\dagger}{\mbox{\small{(}}{x}_{\nu}\mbox{\small{)}}}A_{\nu\mu}^{k}\widehat{\mathbf{\Uppsi}}{\mbox{\small{(}}{x}_{\mu}\mbox{\small{)}}}\otimes|v_{k}\rangle\langle v_{k}|.$$
(27)
Here $A_{\nu\mu}^{k}$ indicates the action of the permutation group of the graph on its adjacency matrix. Such an operation associates an element $A_{\nu\mu}$ with the chirality $k$ of the coins. Consequently action of permutation group operation coordinates the choice of vertices on which the boson
annihilation and boson creation act. On the line or the cyclic graph in Fig.1a, the permutation becomes just the transposition. The product of the Hadamard operator ${\bf H}_{d}$ defined in Eq. (21) and the shift operator ${\bf S}$ in Eq.(27), gives the conditional shift operator $\widehat{\mathcal{S}}$
$$\widehat{\mathcal{S}}=\sum_{\mu,\nu}\sum_{k}\widehat{\mathbf{\Uppsi}}^{\dagger}{\mbox{\small{(}}{x}_{\nu}\mbox{\small{)}}}A_{\nu\mu}^{k}\widehat{\mathbf{\Uppsi}}{\mbox{\small{(}}{x}_{\mu}\mbox{\small{)}}}\otimes\sum_{j}h_{jk}|v_{j}\rangle\langle v_{k}|.$$
(28)
In the case of the cyclic graph shown in Fig.1a, we have $d=2,\,\,M=10,\,\,h_{11}=h_{12}=h_{12}={1\over{\sqrt{2}}}$ and $h_{22}=-{1\over{\sqrt{2}}}$. In such a case the matrix transposition
operation is the only symmetry splitting the adjacency matrix therefore ${\bf A}_{\mbox{\tiny{L}}}=\big{(}A_{\nu\mu}^{\mbox{\tiny{1}}}\big{)}$ and ${\bf A}_{\mbox{\tiny{L}}}^{\mbox{\tiny{T}}}=\big{(}A_{\nu\mu}^{\mbox{\tiny{2}}}\big{)}$ with $\mu,\nu=1,\ldots,10$.
It follows
$$\widehat{\mathcal{S}}=\sum_{\mu,\nu=1}^{10}\sum_{k=1}^{2}\widehat{\mathbf{\Uppsi}}^{\dagger}{\mbox{\small{(}}{x}_{\nu}\mbox{\small{)}}}A_{\nu\mu}^{k}\widehat{\mathbf{\Uppsi}}{\mbox{\small{(}}{x}_{\mu}\mbox{\small{)}}}\otimes\sum_{j=1}^{2}h_{jk}|v_{j}\rangle\langle v_{k}|.$$
(29)
Suppose that at a $r^{\mbox{\tiny{th}}}$ step such a system is represented by a GMP state in Eq.(16). Then one step later it becomes
$$|\Psi_{r+{\mbox{\tiny{1}}}}\rangle=\widehat{\mathcal{S}}|\Psi_{r}\rangle=\sum_{\ell_{\mbox{\tiny{1}}}}\sum_{j}\frac{C_{j\ell_{\mbox{\tiny{1}}}}^{r+1}}{\mathcal{K}_{r+1}}\big{|}v_{j}{\bf n}_{\ell_{\mbox{\tiny{1}}}}\big{\rangle},$$
(30)
where
$$\displaystyle C_{j\ell_{\mbox{\tiny{1}}}}^{r+1}$$
$$\displaystyle=$$
$$\displaystyle\sum_{\mu,\nu}\sum_{k}\frac{h_{jk}C_{k\ell}^{r}}{\mathcal{K}_{r}}\mbox{\large{$f$}}_{\mu\nu}^{k},$$
(31)
$$\displaystyle|{\bf n}_{\ell_{\mbox{\tiny{1}}}}\rangle$$
$$\displaystyle\sim$$
$$\displaystyle{\bf S}\,|{\bf n}_{\ell}\rangle,$$
(32)
$$\displaystyle\mbox{\large{$f$}}_{\mu\nu}^{k}$$
$$\displaystyle=$$
$$\displaystyle A_{\nu\mu}^{k}h_{jk}\sqrt{{n}_{\mu}({n}_{\nu}+1)},$$
(33)
$$\displaystyle[\mathcal{K}_{r+1}]^{2}$$
$$\displaystyle=$$
$$\displaystyle\sum_{\ell_{\mbox{\tiny{1}}}}\sum_{j}|C_{j\ell_{\mbox{\tiny{1}}}}^{r+1}|^{2}.$$
(34)
The term $\mbox{\large{$f$}}_{\mu\nu}^{k}$ contains residuals of field operators actions and depends on graph’s adjacency matrix. It shows how the graph structure is encoded in the GMP state
during the evolution. In addition, at every step the GMP state must be normalized that is why the normalization constant $\mathcal{K}_{r}$ is step $r$ dependent. Considering the GMP state obtained in Eq.(30) we observe that the conditional shifting plays the role in redefining the amplitude of the configuration over the GMP state. The triple sum in the recursion relations in Eq.(31) defines the relation between the amplitudes $C_{j\ell_{\mbox{\tiny{1}}}}^{r+1}$ and $C_{k\ell}^{r}$ during the step implementation and couples all the parent configurations ${\bf n}_{\ell}$ involved in the induction of the new configurations ${\bf n}_{\ell_{\mbox{\tiny{1}}}}$ that constitute the evolved GMP state. In other words, the whole dynamics of many-particle quantum walks becomes walks over the configurations Hilbert space.
In addition, mathematical expression of the evolution of amplitudes can be used as indicators of interfering
parent configurations dragging the system in the unexplored regions of the configurations Hilbert space or trapping the system in a specific subspace of this Hilbert space.
IV Results and conclusions
The probability of reaching a specific configuration ${\bf n}_{\ell}$ at the step $r$ is given by
$$P_{\ell}^{r}=\sum_{j}\bigg{|}\frac{C_{j\ell}^{r}}{\mathcal{K}_{r}}\bigg{|}^{2}.$$
(35)
It is also the joint probability of the $M$-tuples occupation numbers $n_{\alpha}$’s over the graph.
The $q^{\mbox{\tiny{th}}}$ moment of the occupation number on a vertex $\alpha$ is given by:
$$\langle(n_{\alpha}\mbox{\small{(}}r\mbox{\small{)}})^{q}\rangle=\sum_{\ell}(n_{\alpha})^{q}P_{\ell}^{r},$$
(36)
where $\langle\cdot\rangle$ denote the mean value.
These moments are computationally accessible for any value of $q$. Frequently only $q=1,2$ and $3$ are used. The first moment represents the vertex expected particles occupation numbers $\langle n_{\alpha}\mbox{\small{(}}r\mbox{\small{)}}\rangle$. The $\langle n_{\alpha}\mbox{\small{(}}r\mbox{\small{)}}\rangle$ are also directly calculable using the GMP state after a specific number of steps $r$ by evaluating
$$\langle n_{\alpha}\mbox{\small{(}}r\mbox{\small{)}}\rangle=\langle\Psi_{r}|\widehat{\mathbf{\Uppsi}}^{\dagger}{\mbox{\small{(}}{x}_{\alpha}\mbox{\small{)}}}\widehat{\mathbf{\Uppsi}}{\mbox{\small{(}}{x}_{\alpha}\mbox{\small{)}}}|\Psi_{r}\rangle=\sum_{\ell}\sum_{j}\bigg{|}\frac{C_{j\ell}^{r}}{\mathcal{K}_{r}}\bigg{|}^{2}n_{\alpha}.$$
(37)
The Figs. 1(a), 1(b) and 1(c), present the expected vertices particle distribution $\langle n_{\alpha}\mbox{\small{(}}r\mbox{\small{)}}\rangle$ for a selected number of steps. We considered all three systems starting with an initial state of the form
$$|\Psi_{\mbox{\tiny{0}}}\rangle=\frac{C_{11}^{\mbox{\tiny{0}}}}{\mathcal{K}_{\mbox{\tiny{0}}}}|v_{\mbox{\tiny{1}}}{\bf n}_{\mbox{\tiny{1}}}\rangle+\frac{C_{22}^{\mbox{\tiny{0}}}}{\mathcal{K}_{\mbox{\tiny{0}}}}|v_{\mbox{\tiny{2}}}{\bf n}_{\mbox{\tiny{2}}}\rangle,$$
(38)
that is a linear combination of two configurations with the same amplitudes $\frac{C_{11}^{\mbox{\tiny{0}}}}{\mathcal{K}_{\mbox{\tiny{0}}}}={-i\over\sqrt{2}}$ and $\frac{C_{22}^{\mbox{\tiny{0}}}}{\mathcal{K}_{\mbox{\tiny{0}}}}={1\over\sqrt{2}}$. We also considered $M=10$ vertices and $N=12$ particles initially distributed on a single vertex in two initial configurations: $|{\bf n}_{\mbox{\tiny{1}}}\rangle=|0,0,N,0,0,0,0,0,0,0\rangle$ and $|{\bf n}_{\mbox{\tiny{2}}}\rangle=|0,0,0,0,N,0,0,0,0,0\rangle$.
In addition to the vertices particle density, we also evaluate the phase space evolution, the second order spatial correlations and the vertices counting statistics.
This scheme of many-particle quantum walks offers the possibility of accessing all types of correlations. The phase space is described by evaluating the position operator $\langle{x}_{\upeta}\mbox{\small{(}}r\mbox{\small{)}}\rangle=\langle\Psi_{r}|\hat{\bf x}_{\upeta}|\Psi_{r}\rangle$
$$\langle{x}_{\upeta}\mbox{\small{(}}r\mbox{\small{)}}\rangle=\sum_{\ell}\sum_{j}\bigg{|}\frac{C_{j\ell}^{r}(n_{\alpha})!}{\mathcal{K}_{r}n_{\alpha}^{n_{\alpha}}}\bigg{|}^{2}\sum_{\{{n}_{\upeta}\}}\prod_{\upeta}\cos[{\varphi}_{\upeta}{x}_{\alpha}],$$
(39)
the momentum operator $\langle{p}_{\upeta}\mbox{\small{(}}r\mbox{\small{)}}\rangle=\langle\Psi_{r}|\hat{\bf p}_{\upeta}|\Psi_{r}\rangle$
$$\langle{p}_{\upeta}\mbox{\small{(}}r\mbox{\small{)}}\rangle=\sum_{\ell}\sum_{j}\bigg{|}\frac{C_{j\ell}^{r}(n_{\alpha})!}{\mathcal{K}_{r}n_{\alpha}^{n_{\alpha}}}\bigg{|}^{2}\sum_{\{{n}_{\upeta}\}}\prod_{\upeta}\sin[{\varphi}_{\upeta}{x}_{\alpha}],$$
(40)
and the energy operator $\langle E_{\upeta}\mbox{\small{(}}r\mbox{\small{)}}\rangle=\langle\Psi_{r}|\hat{\bf E}_{\upeta}|\Psi_{r}\rangle$
$$\displaystyle\langle{E}_{\upeta}\mbox{\small{(}}r\mbox{\small{)}}\rangle$$
$$\displaystyle=$$
$$\displaystyle\sum_{\ell}\sum_{j}\bigg{|}\frac{C_{j\ell}^{r}(n_{\alpha})!}{\mathcal{K}_{r}n_{\alpha}^{n_{\alpha}}}\bigg{|}^{2}$$
(41)
$$\displaystyle\quad\times\sum_{\{{n}_{\upeta}\}}\prod_{\upeta}\bigg{(}{n}_{\upeta}+{1\over 2}-\cos[2{\varphi}_{\upeta}{x}_{\alpha}]\bigg{)},$$
calculated in each mode and at every step. These three quantities in Eqs. (39, 40 and 41) represent the phase space coordinates in each mode. We recall that these operators are respectively defined by: $\hat{\bf x}_{\upeta}={1\over\sqrt{2}}(\hat{\bf b}_{\upeta}+\hat{\bf b}_{\upeta}^{\dagger})$, $\hat{\bf p}_{\upeta}={i\over\sqrt{2}}(\hat{\bf b}_{\upeta}-\hat{\bf b}_{\upeta}^{\dagger})$, and $\hat{\bf E}_{\upeta}={1\over 2}\hat{\bf p}_{\upeta}^{\dagger}\hat{\bf p}_{\upeta}$. The constants such as the mass, the frequency and $\hbar$ are set to 1. The Figs. 1(d), 1(e) and 1(f) present the results of 400 steps simulations where the color bar is the energy $\langle{E}_{\upeta}\mbox{\small{(}}r\mbox{\small{)}}\rangle$ in Eq (41).
Looking at the position second order correlations at step $r$, we use the spacial second order correlation function given by
$$g_{(\alpha_{\mbox{\tiny{1}}},\alpha_{\mbox{\tiny{2}}},r)}^{\mbox{\tiny{(2)}}}=\frac{\langle\widehat{\mathbf{\Uppsi}}^{\dagger}{\mbox{\small{(}}{x}_{\alpha_{\mbox{\tiny{1}}}},r\mbox{\small{)}}}\widehat{\mathbf{\Uppsi}}^{\dagger}{\mbox{\small{(}}{x}_{\alpha_{\mbox{\tiny{2}}}},r\mbox{\small{)}}}\widehat{\mathbf{\Uppsi}}{\mbox{\small{(}}{x}_{\alpha_{\mbox{\tiny{2}}}},r\mbox{\small{)}}}\widehat{\mathbf{\Uppsi}}{\mbox{\small{(}}{x}_{\alpha_{\mbox{\tiny{1}}}},r\mbox{\small{)}}}\rangle}{\langle\widehat{\mathbf{\Uppsi}}^{\dagger}{\mbox{\small{(}}{x}_{\alpha_{\mbox{\tiny{2}}}},r\mbox{\small{)}}}\widehat{\mathbf{\Uppsi}}{\mbox{\small{(}}{x}_{\alpha_{\mbox{\tiny{2}}}},r\mbox{\small{)}}}\rangle\langle\widehat{\mathbf{\Uppsi}}^{\dagger}{\mbox{\small{(}}{x}_{\alpha_{\mbox{\tiny{1}}}},r\mbox{\small{)}}}\widehat{\mathbf{\Uppsi}}{\mbox{\small{(}}{x}_{\alpha_{\mbox{\tiny{1}}}},r\mbox{\small{)}}}\rangle}.$$
(42)
These quantities are used to explore the influence of other vertices on a fixed vertex occupation number dynamics.
The walks on the configurations Hilbert space are controlled by vertex occupation numbers. In Figs. 2(a) – 2(l), we present the results obtained for the $g_{(\alpha_{\mbox{\tiny{1}}},\alpha_{\mbox{\tiny{2}}},r)}^{\mbox{\tiny{(2)}}}$ for 400 steps of 12 quantum walkers over the three graphs in Fig. 1 where we present
the $\mbox{30}^{\mbox{\tiny{th}}}$ step (Figs. 2(a) – 2(c)), the $\mbox{50}^{\mbox{\tiny{th}}}$ step (Figs. 2(d) – 2(f)), the $\mbox{100}^{\mbox{\tiny{th}}}$ step (Figs. 2(g) – 2(i)) and the $\mbox{400}^{\mbox{\tiny{th}}}$ step (Figs. 2(j) – 2(l)).
There are various methods of exploring the evolution of the configurations Hilbert space. In this case we are focusing on microscopic evolutions by looking at a specific vertex $\alpha$. To such a vertex we join a counting $n_{\alpha}$. This is equivalent to fine tuning the detector to record a count only when $n_{\alpha}$ walkers report on vertex $\alpha$. This counting combines the combinatoric problem of distributing of $N$ indistinguishable bosons into $M$ vertices to the probability of a given configuration as in Eq. (35). The probability of finding $n_{\alpha}$ bosons on vertex $\alpha$ considered as the combinatorial problem of distributing of $N$ indistinguishable bosons into $M$ vertices is given by: ${D\mbox{\tiny{(}}\mbox{\tiny{$N$}}-n_{\alpha},\mbox{\tiny{$M$}}\mbox{\tiny{)}}\over{MD\mbox{\tiny{(}}\mbox{\tiny{$N,M$}}\mbox{\tiny{)}}}}$ where $D\mbox{\tiny{(}}\cdot,\cdot\mbox{\tiny{)}}$ is defined in Eq.(10)
We define the probability $P_{n_{\alpha}}^{r}$ of finding $n_{\alpha_{\mbox{\tiny{1}}}}=n_{\alpha}$ particles on vertex $\alpha$ at step $r$ as:
$$P_{n_{\alpha}}^{r}=\sum_{j}\frac{(C_{j\ell}^{r})^{*}C_{j\ell}^{r}}{[\mathcal{K}_{r}]^{2}M}\frac{D\mbox{\small{(}}N-n_{\alpha},M-1\mbox{\small{)}}}{D\mbox{\small{(}}N,M\mbox{\small{)}}}.$$
(43)
Considering the evolution of vertex by vertex counting statistics we observe a change of regime in the time step evolution counting statistics. This change appears in all the three graphs but at different time steps. For example for all these three systems, when the quantum walks start from the initial conditions in Eq.(38) the change of regime occurs after $r=94$ steps for the cyclic graph in Fig. 1a, after $r=70$ steps for the double hexagon graph in Fig. 1b and after $r=48$ steps for the Petersen graph in Fig. 1c. Before the change of regime the number of the counts grows quickly specially for the small number of particles on a vertex see Figs. 3(a)–3(d), Figs. 3(b)–3(e) and Figs. 3(c)–3(f). After the change of regime the counting statistics remains consistently the same for all the three graphs and for different initial conditions. During our simulations we also monitored what fraction of the configuration Hilbert space was effectively contributing in the GMP state with nonzero amplitudes. Let us simply name it the effective Hilbert space. Moreover, the observation of regime change in the counting statistics time evolution corresponds with the change of behavior of the dimension of the effective Hilbert space. Before the change of the regime this dimension grows quickly e.g for the cyclic graph from $7900$ at step $r=30$ to $68632$ at step $r=50$ (Corresponding numbers for the double hexagon graph are: $14507$, $115052$). At the change of regime step, the dimension of the effective Hilbert space reaches one of the two critical values $146860$ or $147070$ and at the subsequent steps oscillates between these two values. We say that this behavior is universal because it is present in different types of graphs and for different initial conditions as long as the number of vertices over the considered graphs is the same and the number quantum walkers remains the same. In this work we have present only the results
for one initial condition in Eq. (38) shown in Figs. 3(a)-3(l). We observed the same phenomenon for other initial conditions. It means that the vertices counting statistics shows universal behavior after the change of regime step for each system. It must also be noted that the change of regime step depends on the initial conditions for the designated graph in Fig.1.
Phenomenologically it can be observed that during the change of regime, the systems enter into a two degrees of freedom phase. In other words, during successive steps, the GMP states in Eq. (30) related to the three graphs evolve between two subspaces of the configuration Hilbert space of dimensions 146860 and 147070, respectively. Looking at generic many-particle quantum systems, it is conjectured that thermalization is possible at the level of individual eigenstates Deutsch ; Srednicki . Rigol et al in works Rigol01 ; Rigol03 ; Rigol02 on thermalization in generic isolated quantum systems suggest that despite the fact that an isolated quantum system doesn’t thermalize like a classical system, it is possible to identify an observable for which the mean values behave such as it is predicted using an appropriate statistical mechanical ensemble (represented by a certain density matrix). Authors of Rigol02 use the evolution of particles-mode distribution $n_{\upeta}$ to monitor numerically the thermal relaxation. In this work, we have chosen the vertices counting statistics to monitor the time evolution of the system. Under the hypothesis of eigenstate thermalization, the system evolves from an initial state with non-thermalized eigenstates to a final state with thermalized eigenstates. Therefore it is expected that individual many-particle eigenstates thermalize without affecting the dimension of the effective configuration Hilbert space. Consequently, under the hypothesis of eigenstates thermalization the GMP state during the shared coins many-particle quantum walks on closed graphs evolves toward a subspace of the configuration Hilbert space called effective Hilbert space of desired dimension. However, we are observing a cyclic changes of the dimension of the effective configuration Hilbert space for systems on graphs in Figs. 1 where some eigenstates coalesce during the evolution from the effective Hilbert space of dimension $147070$ to the effective Hilbert space of dimension $146860$ and split during the reverse evolution.
Moreover, we notice that the dimension of the effective Hilbert space oscillates between values that are roughly a third of dimension of the configurations Hilbert space. Then under the hypothesis of ergodicity, there must exist additional conserved physical observables including the Hamiltonian that are functionally independent each other. In such a case the evolution of the system will be restricted to a subspace of the configurations Hilbert space and the uniform distribution for the many-particle quantum walks becomes inaccessible. Such cases have been experimentally observed like in Fermi-Pasta-Ulam numerical experiment FerUlam01 . Nevertheless, looking at the evolution of the second order spacial correlations in Figs.2(j)–2(l), one can observe that the quantum walks have a finite confinement time and they are slowly evolving toward a uniform distribution in effective Hilbert space, see Fig 1(c). In conclusion, we conjecture that these systems become integrable Toshiya01 ; JHarnard01 ; Izergin:01 after the regime change steps. To understand the limits of this conjecture, it is useful to explore other observables involved in the dynamics. Here we have in mind higher order correlations that might be of relevance as well as other observables directly connected to the microscopic evolution of the configurations Hilbert space.
This research was supported by grant No. DEC-2011/02/A/ST1/00208 of National Science Centre of Poland.
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Uncertainty and Analyticity
Vladimir V. Kisil
School of Mathematics,
University of Leeds,
Leeds LS2 9JT,
England,
Web: http://www.maths.leeds.ac.uk/~kisilv/
[email protected]
(Date:: December 1, 2013)
Abstract.
We describe a connection between minimal uncertainty states and
holomorphy-type conditions on the images of the respective wavelet
transform. The most familiar example is the Fock–Segal–Bargmann
transform generated by the Gaussian, however, this also occurs under
more general assumptions.
Key words and phrases:Quantum mechanics, classical mechanics, Heisenberg
commutation relations, observables, Heisenberg group,
Fock–Segal–Bargmann space, $\mathrm{SU}(1,1)$, Hardy space
1991 Mathematics Subject Classification: Primary 81P05; Secondary 22E27
On leave from Odessa University.
\BibSpecAlias
incollectioninproceedings
1. Introduction
There are two and a half main examples of reproducing kernel spaces of
analytic function. One is the Fock–Segal–Bargmann (FSB) space and
others (one and a half)—the Bergman and Hardy spaces. The first
space is generated by the Heisenberg
group \citelist[Kisil11c]*§ 7.3 [Folland89]*§ 1.6,
two others—by the group $\mathrm{SU}(1,1)$ [Kisil11c]*§ 4.2 (this
explains our way of counting).
Those spaces have the following properties, which make their study
particularly pleasant and fruitful:
(i)
There is a group, which acts transitively on functions’
domain.
(ii)
There is a reproducing kernel.
(iii)
The space consists of holomorphic functions.
Furthermore, for FSB space there is the following property:
iv.
The reproducing kernel is generated by a function, which minimises
the uncertainty for coordinate and momentum observables.
It is known, that a transformation group is responsible for the
appearance of the reproducing
kernel [AliAntGaz00]*Thm. 8.1.3. This paper shows that the last
two properties are equivalent and connected to the group as well.
2. The Uncertainty Relation
In quantum mechanics [Folland89]*§ 1.1, an observable
(self-adjoint operator on a Hilbert space $\mathcal{H}{}$) $A$ produces the
expectation value $\bar{A}$ on a state (unit vector)
$\phi\in\mathcal{H}{}$ by $\bar{A}=\left\langle A\phi,\phi\right\rangle$. Then, the dispersion can
be evaluated as follow:
(1)
$$\Delta_{\phi}^{2}(A)=\left\langle(A-\bar{A})^{2}\phi,\phi\right\rangle=\left%
\langle(A-\bar{A})\phi,(A-\bar{A})\phi\right\rangle=\left\|(A-\bar{A})\phi%
\right\|^{2}.$$
The following theorem links obstructions of exact measurements with
non-commutativity of observables.
Theorem 1 (The Uncertainty relation).
If $A$ and $B$ are self-adjoint operators on a Hilbert space $\mathcal{H}{}$, then
(2)
$$\textstyle\left\|(A-a)u\right\|\left\|(B-b)u\right\|\geq\frac{1}{2}\left|\left%
\langle(AB-BA)u,u\right\rangle\right|,$$
for any $u\in\mathcal{H}{}$ from the domains of $AB$ and $BA$ and $a$,
$b\in\mathbb{R}{}$. Equality holds precisely when $u$ is a
solution of $((A-a)+\mathrm{i}r(B-b))u=0$ for some real $r$.
Proof.
The proof is well-known [Folland89]*§ 1.3, but
it is short, instructive and relevant for the following discussion,
thus we include it in full. We start from simple algebraic
transformations:
(3)
$$\displaystyle\left\langle(AB-BA)u,u\right\rangle$$
$$\displaystyle=$$
$$\displaystyle\left\langle(A-a)(B-b)-(B-b)(A-a))u,u\right\rangle$$
$$\displaystyle=$$
$$\displaystyle\left\langle(B-b)u,(A-a)u\right\rangle-\left\langle(A-a))u,(B-b)u\right\rangle$$
$$\displaystyle=$$
$$\displaystyle 2\mathrm{i}\Im\left\langle(B-b)u,(A-a)u\right\rangle$$
Then by the Cauchy–Schwartz inequality:
$$\textstyle\frac{1}{2}\left\langle(AB-BA)u,u\right\rangle\leq\left|\left\langle%
(B-b)u,(A-a)u\right\rangle\right|\leq\left\|(B-b)u\right\|\left\|(A-a)u\right\|.$$
The equality holds if and only if $(B-b)u$ and $(A-a)u$ are
proportional by a purely imaginary scalar.
∎
The famous application of the above theorem is the following
fundamental relation in quantum
mechanics. Recall [Folland89]*§ 1.2, that the one-dimensional
Heisenberg group $\mathbb{H}^{1}{}$ consists of points
$(s,x,y)\in\mathbb{R}^{3}{}$, with the group law:
(4)
$$\textstyle(s,x,y)*(s^{\prime},x^{\prime},y^{\prime})=(s+s^{\prime}+\frac{1}{2}%
(xy^{\prime}-x^{\prime}y),x+x^{\prime},y+y^{\prime}).$$
This is a nilpotent step two Lie group. By the Stone–von Neumann
theorem [Folland89]*§ 1.5, any infinite-dimensional
unitary irreducible representation of $\mathbb{H}^{1}{}$ is unitary
equivalent to the Schrödinger representation ${\rho_{\hslash}}$ in
$\mathcal{L}_{2}{}(\mathbb{R}{})$ parametrised by the Planck constant
$\hslash\in\mathbb{R}{}\setminus\{0\}$. A physically consistent form
of ${\rho_{\hslash}}$ is [Kisil10a]*(3.5):
(5)
$$[{\rho_{\hslash}}(s,x,y){f}\,](q)=e^{-2\pi\mathrm{i}\hslash(s+xy/2)-2\pi%
\mathrm{i}xq}\,{f}(q+{\hslash}y).$$
Elements of the Lie algebra $\mathfrak{h}_{1}$, corresponding to the
infinitesimal generators $X$ and $Y$ of one-parameters subgroups
$(0,t/(2\pi),0)$ and $(0,0,t)$ in $\mathbb{H}^{1}{}$, are
represented in (5) by the
(unbounded) operators $M$ and $D$ on
$\mathcal{L}_{2}{}(\mathbb{R}{})$:
(6)
$$\textstyle M=-\mathrm{i}q,\quad D=\hslash\frac{d}{dq},\quad\text{with the
commutator}\quad[M,D]=\mathrm{i}\hslash I.$$
In quantum mechanics, $f(q)\in\mathcal{L}_{2}{}(\mathbb{R}{})$ is
interpreted as a wave function of a particle, $M$ and $D$ are the
observables of coordinate and momentum.
Corollary 2 (Heisenberg–Kennard uncertainty relation).
For the coordinate $M$ and momentum $D$ observables we have the
Heisenberg–Kennard uncertainty relation:
(7)
$$\Delta_{\phi}(M)\cdot\Delta_{\phi}(D)\geq\frac{h}{2}.$$
The equality holds if and only if
$\phi(x)=e^{-cx^{2}}$, $c\in\mathbb{R}_{+}{}$ is the vacuum state
in the Schrödinger model.
Proof.
The relations follows from the commutator $[M,D]=\mathrm{i}\hslash I$,
which, in turn, is the representation of the Lie algebra
$\mathfrak{h}_{n}$ of the Heisenberg group. The minimal uncertainty
state in the Schrodinger representation is a solution of the
differential equation: $(M-\mathrm{i}rD)\phi=0$, or, explicitly:
(8)
$$(M-\mathrm{i}rD)\phi=-\mathrm{i}\left(q+r{\hslash}\frac{d}{dq}\right)\phi(q)=0.$$
The solution is the Gaussian $\phi(x)=e^{-cq^{2}}$,
$c=\frac{1}{2r\hslash}$. For $c>0$, this function is in
$\mathcal{L}_{2}{}(\mathbb{R}{})$.
∎
It is common to say that the Gaussian $\phi(q)=e^{-cq^{2}}$ represents
the ground state, which minimises the uncertainty of coordinate and
momentum.
3. Wavelet transform and analyticity
3.1. Induced wavelet transform
The following object is common in quantum
mechanics [Kisil02e], signal processing, harmonic
analysis [Kisil12d], operator theory [Kisil12b, Kisil13a]
and many other areas [Kisil11c]. Therefore, it has various
names [AliAntGaz00]: coherent states, wavelets, matrix
coefficients, etc. In the most fundamental
situation [AliAntGaz00]*Ch. 8, we start from an irreducible
unitary representation ${\rho}$ of a Lie group $G$ in a Hilbert
space $\mathcal{H}{}$. For a vector $f\in\mathcal{H}{}$ (called mother wavelet, vacuum
state, etc.), we define the map $\mathcal{W}_{f}$ from $\mathcal{H}{}$ to a space
of functions on $G$ by:
(9)
$$[\mathcal{W}_{f}v](g)=\tilde{v}(g)=\left\langle v,{\rho}(g)f\right\rangle.$$
Under the above assumptions, $\tilde{v}(g)$ is a bounded continuous
function on $G$. The map $\mathcal{W}_{f}$ intertwines ${\rho}(g)$
with the left shifts:
(10)
$$\mathcal{W}_{f}\circ{\rho}(g)=\Lambda(g)\circ\mathcal{W}_{f},\qquad\text{ %
where }\Lambda(g):\tilde{v}(g^{\prime})\mapsto\tilde{v}(g^{-1}g^{\prime}).$$
Thus, the image $\mathcal{W}_{f}\mathcal{H}{}$ is invariant under the left shifts on
$G$. If ${\rho}$ is square integrable and $f$ is
admissible [AliAntGaz00]*§ 8.1, then $\tilde{v}(g)$ is
square-integrable with respect to the Haar measure on $G$. At this
point, none of admissible vectors has an advantage over others.
It is common [Kisil11c]*§ 5.1, that there exists a subgroup
$H\subset G$ and a respective $f\in\mathcal{H}{}$ such that ${\rho}(h)f=\chi(h)f$ for some character $\chi$ of $H$. In this case, it
is enough to know values of $\tilde{v}(\mathsf{s}(x))$, for any
continuous section $\mathsf{s}$ from the homogeneous space $X=G/H$ to
$G$. The map $v\mapsto\tilde{v}(x)=\tilde{v}(\mathsf{s}(x))$
intertwines ${\rho}$ with the representation ${\rho_{\chi}}$ in a
certain function space on $X$ induced by the character $\chi$ of
$H$ [Kirillov76]*§ 13.2. We call the map $\mathcal{W}_{f}:v\mapsto\tilde{v}$ the induced wavelet
transform [Kisil11c]*§ 5.1.
For example, if $G=\mathbb{H}^{1}{}$, $H=\{(s,0,0)\in\mathbb{H}^{1}{}:\ s\in\mathbb{R}{}\}$ and its character
$\chi_{\hslash}(s,0,0)=e^{2\pi\mathrm{i}\hslash s}$, then any vector
$f\in\mathcal{L}_{2}{}(\mathbb{R}{})$ satisfies ${\rho_{\hslash}}(s,0,0)f=\chi_{\hslash}(s)f$ for the
representation (5). Thus, we still
do not have a reason to prefer any admissible vector to
others.
3.2. Right shifts and analyticity
To discover some preferable mother wavelets, we use the following a
general result from [Kisil11c]*§ 5. Let $G$ be a locally
compact group and ${\rho}$ be its representation in a Hilbert
space $\mathcal{H}{}$. Let $[\mathcal{W}_{f}v](g)=\left\langle v,{\rho}(g)f\right\rangle$ be the
wavelet transform defined by a vacuum state $f\in\mathcal{H}{}$. Then, the
right shift $R(g):[\mathcal{W}f](g^{\prime})\mapsto[\mathcal{W}f](g^{\prime}g)$ for
$g\in G$ coincides with the wavelet transform
$[\mathcal{W}_{f_{g}}](g^{\prime})=\left\langle v,{\rho}(g^{\prime})f_{g}\right\rangle$ defined by the vacuum
state $f_{g}={\rho}(g)f$. In other words, the covariant transform
intertwines right shifts on the group $G$ with the
associated action ${\rho}$ on vacuum states,
cf. (10):
(11)
$$R(g)\circ\mathcal{W}_{f}=\mathcal{W}_{{\rho}(g)f}.$$
Although, the above observation is almost trivial, applications of the following
corollary are not.
Corollary 3 (Analyticity of the wavelet transform, [Kisil11c]*§ 5).
Let $G$ be a group and $dg$ be a measure on $G$. Let
${\rho}$ be a unitary representation of $G$, which can be
extended by integration to a vector space $V$ of functions or
distributions on $G$. Let a mother wavelet $f\in\mathcal{H}{}$ satisfy the
equation
$$\int_{G}a(g)\,{\rho}(g)f\,dg=0,$$
for a fixed distribution $a(g)\in V$. Then any wavelet transform
$\tilde{v}(g)=\left\langle v,{\rho}(g)f\right\rangle$ obeys the condition:
(12)
$$D\tilde{v}=0,\qquad\text{where}\quad D=\int_{G}\bar{a}(g)\,R(g)\,dg,$$
with $R$ being the right regular representation of $G$.
Some applications (including discrete one) produced by the $ax+b$ group
can be found in [Kisil12d]*§ 6. We turn to the Heisenberg
group now.
Example 4 (Gaussian and FSB transform).
The Gaussian $\phi(x)=e^{-cq^{2}/2}$ is a null-solution of the
operator $\hslash cM-\mathrm{i}D$. For the centre $Z=\{(s,0,0):\ s\in\mathbb{R}{}\}\subset\mathbb{H}^{1}{}$, we define the section
$\mathsf{s}:\mathbb{H}^{1}{}/Z\rightarrow\mathbb{H}^{1}{}$ by
$\mathsf{s}(x,y)=(0,x,y)$. Then, the corresponding induced wavelet
transform is:
(13)
$$\tilde{v}(x,y)=\left\langle v,{\rho}(\mathsf{s}(x,y))f\right\rangle=\int_{%
\mathbb{R}{}}v(q)\,e^{\pi\mathrm{i}\hslash xy-2\pi\mathrm{i}xq}\,e^{-c(q+{%
\hslash}y)^{2}/2}\,dq.$$
The infinitesimal generators $X$ and $Y$ of one-parameters subgroups
$(0,t/(2\pi),0)$ and $(0,0,t)$ are represented through the right
shift in (4) by
$$\textstyle dR(X)=-\frac{1}{4\pi}y\partial_{s}+\frac{1}{2\pi}\partial_{x},\quad
dR%
(Y)=\frac{1}{2}x\partial_{s}+\partial_{y}.$$
For the representation induced
by the character $\chi_{\hslash}(s,0,0)=e^{2\pi\mathrm{i}\hslash s}$ we
have $\partial_{s}=2\pi\mathrm{i}\hslash I$. Cor. 3 ensures that the operator
(14)
$$\hslash c\cdot dR(X)+\mathrm{i}\cdot dR(Y)=-\frac{\hslash}{2}(2\pi x+{\mathrm{%
i}\hslash c}y)+\frac{\hslash c}{2\pi}\partial_{x}+\mathrm{i}\partial_{y}$$
annihilate any $\tilde{v}(x,y)$ from (13).
The integral (13) is known as
Fock–Segal–Bargmann (FSB) transform and in the most common case
the values $\hslash=1$ and $c=2\pi$ are used. For these,
operator (14) becomes $-\pi(x+\mathrm{i}y)+(\partial_{x}+\mathrm{i}\partial_{y})=-\pi z+2\partial_{%
\bar{z}}$ with
$z=x+\mathrm{i}y$. Then
the function $V(z)=e^{\pi z\bar{z}/2}\,\tilde{v}(z)=e^{\pi(x^{2}+y^{2})/2}\,\tilde{v}(x,y)$
satisfies the Cauchy–Riemann equation $\partial_{\bar{z}}V(z)=0$.
This example
shows, that the Gaussian is a preferred vacuum state (as producing
analytic functions through FSB transform) exactly for the same reason
as being the minimal uncertainty state: the both are derived from the
identity $(\hslash cM+\mathrm{i}D)e^{-cq^{2}/2}=0$.
3.3. Uncertainty and analyticity
The main result of this paper is a generalisation of the previous
observation, which bridge together Cor. 3 and
Thm. 1. Let $G$, $H$, ${\rho}$ and
$\mathcal{H}{}$ be
as before. Assume, that the homogeneous space
$X=G/H$ has a (quasi-)invariant measure
$d\mu(x)$ [Kirillov76]*§ 13.2. Then, for a function (or a
suitable distribution) $k$ on $X$ we can define the integrated
representation:
(15)
$${\rho}(k)=\int_{X}k(x){\rho}(\mathsf{s}(x))\,d\mu(x),$$
which is (possibly, unbounded) operators on (possibly, dense subspace
of) $\mathcal{H}{}$. In particular, $R(k)$ denotes the
integrated right shifts, for $H=\{e\}$.
Theorem 5.
Let $k_{1}$ and $k_{2}$ be two distributions on
$X$ with the respective integrated representations
${\rho}(k_{1})$ and ${\rho}(k_{1})$. The following are
equivalent:
(i)
A vector $f\in\mathcal{H}{}$ satisfies the identity
$$\Delta_{f}({\rho}(k_{1}))\cdot\Delta_{f}({\rho}(k_{2}))=\left|\left\langle[{%
\rho}(k_{1}),{\rho}(k_{1})]f,f\right\rangle\right|.$$
(ii)
The image of the wavelet transform $\mathcal{W}_{f}:v\mapsto\tilde{v}(g)=\left\langle v,{\rho}(g)f\right\rangle$ consists of functions
satisfying the equation $R(k_{1}+\mathrm{i}rk_{2})\tilde{v}=0$ for
some $r\in\mathbb{R}{}$, where $R$ is the integrated
form (15) of the right regular representation
on $G$.
Proof.
This is an immediate consequence of a combination of
Thm. 1 and Cor. 3.
∎
Example 4 is a particular case of this theorem
with $k_{1}(x,y)=\delta^{\prime}_{x}(x,y)$ and $k_{2}(x,y)=\delta^{\prime}_{y}(x,y)$
(partial derivatives of the delta function), which represent vectors
$X$ and $Y$ from the Lie algebra $\mathfrak{h}_{1}$. The next
example will be of this type too.
3.4. Hardy space
Let $\mathrm{SU}(1,1)$ be the group of $2\times 2$ complex matrices of the
form $\begin{pmatrix}\alpha&\beta\\
\bar{\beta}&\bar{\alpha}\end{pmatrix}$ with the unit determinant
$\left|\alpha\right|^{2}-\left|\beta\right|^{2}=1$. A standard basis in the
Lie algebra $\mathfrak{su}_{1,1}$ is
$$A=\frac{1}{2}\begin{pmatrix}0&-\mathrm{i}\\
\mathrm{i}&0\end{pmatrix},\quad B=\frac{1}{2}\begin{pmatrix}0&1\\
1&0\end{pmatrix},\quad Z=\begin{pmatrix}\mathrm{i}&0\\
0&-\mathrm{i}\end{pmatrix}.$$
The respective one-dimensional subgroups consist of matrices:
$$e^{tA}=\begin{pmatrix}\cosh\frac{t}{2}&-\mathrm{i}\sinh\frac{t}{2}\\
\mathrm{i}\sinh\frac{t}{2}&\cosh\frac{t}{2}\end{pmatrix},\ e^{tB}=\begin{%
pmatrix}\cosh\frac{t}{2}&\sinh\frac{t}{2}\\
\sinh\frac{t}{2}&\cosh\frac{t}{2}\end{pmatrix},\ e^{tZ}=\begin{pmatrix}e^{%
\mathrm{i}t}&0\\
0&e^{-\mathrm{i}t}\end{pmatrix}.$$
The last subgroup—the maximal compact subgroup of $\mathrm{SU}(1,1)$—is
usually denoted by $K$. The commutators of the basis elements are
(16)
$$[Z,A]=2B,\qquad[Z,B]=-2A,\qquad[A,B]=-\frac{1}{2}Z.$$
Let $\mathbb{T}{}$ denote the unit circle in $\mathbb{C}{}$ with
the rotation-invariant measure. The
mock discrete representation of $\mathrm{SU}(1,1)$ [Lang85]*§ VI.6 acts
on $\mathcal{L}_{2}{}(\mathbb{T}{})$ by unitary transformations
(17)
$$[{\rho_{1}}(g)f](z)=\frac{1}{(\bar{\beta}z+\bar{\alpha})}\,f\left(\frac{{%
\alpha}z+\beta}{\bar{\beta}z+\bar{\alpha}}\right),\qquad g^{-1}=\begin{pmatrix%
}\alpha&\beta\\
\bar{\beta}&\bar{\alpha}\end{pmatrix}.$$
The respective derived representation of the $\mathfrak{su}_{1,1}$
basis is:
$$\textstyle d{\rho^{A}_{1}}=\frac{\mathrm{i}}{2}(z+(z^{2}+1)\partial_{z}),\quad
d%
{\rho^{B}_{1}}=\frac{1}{2}(z+(z^{2}-1)\partial_{z}),\quad d{\rho^{Z}_{1}}=-%
\mathrm{i}I-2\mathrm{i}z\partial_{z}.$$
Thus, $d{\rho^{B+\mathrm{i}A}_{1}}=-\partial_{z}$ and the function
$f_{+}(z)\equiv 1$ satisfies $d{\rho^{B+\mathrm{i}A}_{1}}f_{+}=0$. Recalling
the commutator $[A,B]=-\frac{1}{2}Z$ we note that
${\rho_{1}}(e^{tZ})f_{+}=e^{\mathrm{i}t}f_{+}$. Therefore, there is the
following identity for dispersions on this state:
$$\textstyle\Delta_{f_{+}}(d{\rho^{A}_{1}})\cdot\Delta_{f_{+}}(d{\rho^{B}_{1}})=%
\frac{1}{2},$$
with the minimal value of uncertainty among all eigenvectors of the operator
${\rho_{1}}(e^{tZ})$.
Furthermore, the vacuum state $f_{+}$ generates the induced wavelet
transform for the subgroup $K=\{e^{tZ}\,\mid\,t\in\mathbb{R}{}\}$. We
identify $\mathrm{SU}(1,1)/K$ with the open unit disk $D=\{w\in\mathbb{C}{}\,\mid\,\left|w\right|<1\}$ \citelist[Kisil11c]*§ 5.5 [Kisil13a]. The map
$\mathsf{s}:\mathrm{SU}(1,1)/K\rightarrow\mathrm{SU}(1,1)$ is defined as
$\mathsf{s}(w)=\frac{1}{\sqrt{1-\left|w\right|^{2}}}\begin{pmatrix}1&w\\
\bar{w}&1\end{pmatrix}$. Then, the induced wavelet transform is:
$$\displaystyle\tilde{v}(w)=\left\langle v,{\rho_{1}}(\mathsf{s}(w))f_{+}\right\rangle$$
$$\displaystyle=\frac{1}{2\pi\sqrt{1-\left|w\right|^{2}}}\int_{\mathbb{T}{}}%
\frac{v(e^{\mathrm{i}\theta})\,d\theta}{1-we^{-\mathrm{i}\theta}}$$
$$\displaystyle=\frac{1}{2\pi\mathrm{i}\sqrt{1-\left|w\right|^{2}}}\int_{\mathbb%
{T}{}}\frac{v(e^{\mathrm{i}\theta})\,de^{\mathrm{i}\theta}}{e^{\mathrm{i}%
\theta}-w}.$$
Clearly, this is the Cauchy integral up to the factor
$\frac{1}{\sqrt{1-\left|w\right|^{2}}}$, which presents the conformal
metric on the unit disk. Similarly, we can consider the operator
$d{\rho^{B-\mathrm{i}A}_{1}}=z+z^{2}\partial_{z}$ and the function
$f_{-}(z)=\frac{1}{z}$ simultaneously solving the equations
$d{\rho^{B-\mathrm{i}A}_{1}}f_{-}=0$ and ${\rho_{1}}(e^{tZ})f_{-}=e^{-\mathrm{i}t}f_{-}$. It produces the integral with the conjugation Cauchy kernel.
Finally, we can calculate the operator (12)
annihilating the image of the wavelet transform. In the coordinates
$(w,t)\in(\mathrm{SU}(1,1)/K)\times K$, the restriction to the induced
subrepresentation is, cf. [Lang85]*§ IX.5:
$$\displaystyle\mathfrak{L}^{B-\mathrm{i}A}{}=\exp(2t\mathrm{i})(-\frac{1}{2}w+(%
1-\left|w\right|^{2})\partial_{\bar{w}}).$$
Furthermore, if $\mathfrak{L}^{B-\mathrm{i}A}{}\tilde{v}(w)=0$, then
$\partial_{\bar{w}}(\sqrt{1-w\bar{w}}\cdot\tilde{v}(w))=0$. That is,
$V(w)=\sqrt{1-w\bar{w}}\cdot\tilde{v}(w)$ is a holomorphic
function on the unit disk.
Similarly, we can treat representations of $\mathrm{SU}(1,1)$ in the space of
square integrable functions on the unit disk. The irreducible
components of this representation are isometrically
isomorphic [Kisil11c]*§ 4–5 to the weighted Bergman spaces of
(purely poly-)analytic functions on the unit, cf. [Vasilevski99].
References |
Photon-phonon-photon transfer in optomechanics
Andrey A. Rakhubovsky
Radim Filip
Department of Optics, Palacký University, 17. Listopadu 12, 771 46 Olomouc, Czech Republic
(January 14, 2021)
Abstract
We consider transfer of a highly nonclassical quantum state through an optomechanical system. That is we investigate a protocol consisting of sequential upload, storage and reading out of the quantum state from a mechanical mode of an optomechanical system. We show that provided the input state is in a test-bed single-photon Fock state, the Wigner function of the recovered state can have negative values at the origin, which is a manifest of nonclassicality of the quantum state of the macroscopic mechanical mode and the overall transfer protocol itself. Moreover, we
prove that the recovered state is quantum non-Gaussian for wide range of setup parameters. We verify that current electromechanical and optomechanical experiments can test this complete transfer of single photon.
1 Introduction
Quantum optomechanics [1, 2] can reversibly interconnect radiation to mechanical oscillators at the level of individual quanta [3, 4, 5, 6]
This connection has both fundamental and also practical consequences.
At fundamental side, it controllably converts a single photon to single phonon and then, back to individual photons.
It proves fundamental aspect that the information carried by the physical state of single quantum can be transferred between totally different physical systems.
Also, it says message to quantum thermodynamics [7], that minimal portion of energy can substantially change its form.
At practical side, such the high-fidelity photon-phonon-photon transducers can be used to sense mechanical motion at levels of single phonon level.
It can stimulate quantum metrology of mechanical motion [8] beyond the current state of the art.
Gaussian quantum entanglement [9] and squeezed states [10] between microwave radiation and mechanical oscillator have been experimentally generated and read out using resolved sideband regime.
However, Gaussian states are not sufficient resources for advanced quantum technology.
It requires non-Gaussian quantum states [11] to open full space of the applications.
Single photon state is basic non-Gaussian resource for optical technology, therefore the single phonon states are as well basic for non-Gaussian quantum optomechanics.
A mutual interconversion between a photon and a phonon is thus an important proof of this fundamental correspondence.
A single photon is simultaneously a very good test-bed to verify quantum nature of the physical interface between ligth and mechanics.
Recently, nonclassical correlations of phonon with photon at optical frequencies have been unconditionally generated [12]. This opens future possibilities to exploit optomechanics operated at single phonon level.
The transducers between optical, mechanical and microwave modes have been considered [13, 14] and realized in continuous-wave regime [15, 16, 17, 18].
However, the continuous wave regime with steady states is not suitable when quantum processing and metrology runs in time slots.
In this case, all operations are pulsed with optimized temporal shapes.
Theory proposals of photon-to-phonon transfer in the pulsed [19, 20, 21, 22] and stroboscopic regime [23] have been recently analyzed.
Also squeezed light improved read-out of mechanical state has been separately investigated [24].
In this paper, we analyze complete photon-phonon-photon coherent transfer for microwave and optical experiments in the pulsed and red side-band resolved regime.
We investigate in detail possibility to transfer the negativity of Wigner function and more general quantum non-Gaussianity through an optomechanical system in contrast to studies of engineering of a nonclassical state of a mechanical mode [25, 26].
Transfer of both negativity and non-Gaussianity is much more demanding than the transfer of Gaussian entanglement.
We verify that the electromechanical platforms [9, 27] with microwave radiation are already feasible for such transduction preserving negativity of Wigner function.
The optomechanical experiments require coupling higher than critical to reach this possibility.
To preserve quantum non-Gaussianity of single photon, optical experiment [12] is however already sufficient.
Our analysis can be extended to other systems [28].
We numerically verify all the predictions using analysis beyond adiabatic elimination of intracavity field.
We use single photon as a basic test bed for photon-phonon-photon channel, however, it verifies basic quality of coherent transfer for other non-Gaussian states.
Our results therefore open path to use photon-phonon-photon transducer for many future experiments in quantum metrology, quantum nonlinear optomechanics and quantum thermodynamics.
2 Summary of the main results
It has been shown previously [30] that in presence of a strong red-detuned coherent pulse, the quantum states of the mechanical mode of an optomechanical cavity and the resonant with the cavity pulse can in principle swapped. That is, the quantum state of an incident pulse is transferred to the mechanical mode and the state of the latter is mapped to the reflected light. The scheme analyzed here uses this state swap twice: first time to upload a highly nonclassical (single-photon) quantum state to the mechanical mode and second time to read the evolved state out. We were inspired by the quality of state read out in experiment [9].
First, the pulse in single-photon state in the optical mode described by annihilation operator $A^{\text{in}}$ is shone upon the cavity at its resonance frequency $\omega_{\mathrm{cav}}$. A strong classical coherent pulse with amplitude $\alpha_{1}$ at frequency $\omega_{\mathrm{cav}}-\omega_{m}$ enables the state swap, so that the mechanical state after the pulse reads [30, 24]
$$a_{m}(\tau)=\sqrt{T_{1}}A^{\text{in}}+\sqrt{1-T_{1}}a_{m}(0),$$
(1)
where $T_{1}=1-\exp(-2g_{0}^{2}N_{1}\tau_{1}/\kappa)$ is the transmittivity associated with the partial state swap. Here $g_{0}$ is the single-photon optomechanical coupling strength enhanced by the mean intracavity photon number $N_{1}=|\alpha_{1}|^{2}$. If the interaction time $\tau_{1}$ or optomechanical coupling $g_{0}\sqrt{N_{1}}$ is sufficient, the optical state is perfectly mapped to the mechanical mode. In practice, high optomechanical coupling may be inaccessible, or cause unwanted heating due to absorption in mirrors. The temporal duration $\tau_{1}$ of the pulse is as well limited by mechanical decoherence.
After the upload, the mechanical state is left to evolve for time $\tau_{\mathrm{s}}$. During this time the mechanical environment admixes thermal noise to the quantum state of the mechanical mode. This effect can be described by an effective transmittance $\delta=e^{-\gamma\tau_{\mathrm{s}}}$, where $\gamma$ is the mechanical damping rate. This admixture happens as well during the upload stage (and consequent readout), but we omit it for the state swap stages as this effect can be reduced by shortening the pulses durations.
Finally, another state swap is performed to read the mechanical state out. In a full analogy with Eq. 1 the state of the output optical mode reads
$$A^{\text{out}}=\sqrt{T_{2}}a_{m}(\tau_{1}+\tau_{\mathrm{s}})-\sqrt{1-T_{2}}A^{%
\text{vac}},$$
(2)
where $A^{\text{vac}}$ describes a vacuum mode incident to the cavity, $T_{2}=1-\exp(-2g_{0}^{2}N_{2}\tau_{2}/\kappa)$, $N_{2}=|\alpha_{2}|^{2}$.
The complete channel from the input optical state with annihilation operator $A^{\text{in}}$ to the output one with $A^{\text{out}}$ will be degraded by optical losses caused by e.g. imperfect coupling to the optomechanical cavity. The optical loss performs admixture of vacuum to the signal mode and is characterized by transmittivity $\eta=\kappa_{e}/\kappa$, where $\kappa$ and $\kappa_{e}$ are respectively total cavity decay rate and the cavity decay rate due to the coupling to the waveguide.
All the steps constituting the channel thus can be described by known transmittivities and therefore the quadratures of the output optical state can be expressed in terms of quadratures $Q^{\text{in}}$ of input state and quadratures of added noise $Q^{\text{N}}$
$$\displaystyle Q^{\text{out}}=\sqrt{\mathcal{T}}Q^{\text{in}}+\sqrt{1-\mathcal{%
T}}Q^{\text{N}},\quad Q=X,Y,$$
(3)
$$\displaystyle X^{k}=A^{k}{}^{\dagger}+A^{k},\quad Y^{k}=i(A^{k}{}^{\dagger}-A^%
{k}).$$
(4)
Here the transmittivity equals $\mathcal{T}=T_{1}T_{2}\eta^{2}\delta$.
The noisy mode $Q^{\text{N}}$ in the equation above comprises optical noises coming from imperfect coupling, thermal mechanical noise and the initial mechanical state. We assume all these modes to be zero-mean Gaussian noises with known variances. From the latter we construct the covariance matrix of $Q^{\text{N}}$ which is a diagonal $2\times 2$ matrix with the diagonal elements equal
$$V_{N}=1+\frac{T_{2}\eta}{1-\mathcal{T}}\Big{[}2n_{0}\delta(1-T_{1})+2n_{%
\mathrm{th}}(1-\delta)\Big{]}\ \stackrel{{\scriptstyle n_{0}=n_{\mathrm{th}}}}%
{{=}}\ 1+2n_{\mathrm{th}}\frac{\eta T_{2}(1-T_{1}\delta)}{1-\mathcal{T}}.$$
(5)
Here we assumed that the mechanical mode and its environment are initially in thermal states with occupation $n_{0}$ and $n_{\mathrm{th}}$ respectively and all the auxiliary optical modes are in vacuum.
In order to transfer negativity of the Wigner function of a single-photon state, the variance Eq. 5 should satisfy $V_{\mathrm{N}}<V_{\mathrm{N}}^{\text{neg}}\equiv\mathcal{T}/(1-\mathcal{T})$. Our goal is thus to increase the transmittivity $\mathcal{T}$ preserving moderate added noise variance $V_{\mathrm{N}}$. From the expression for $\mathcal{T}$ it is evident that it can be increased by increasing either of $T_{1}$ and $T_{2}$. However, increasing $T_{1}$ not only increases $\mathcal{T}$ but decreases $V_{\mathrm{N}}$ by diminishing the ratio of the initial thermal mechanical state in the output. Increase of $T_{2}$ is not so crucially important and it causes increase of both $\mathcal{T}$ and $V_{\mathrm{N}}$. These simple considerations can be proven by calculating partial derivatives
$$\displaystyle\left(\frac{\partial V_{\mathrm{N}}}{\partial\mathcal{T}}\right)_%
{T_{2}}$$
$$\displaystyle=-2n_{0}\frac{1-\delta\eta^{2}T_{2}}{\eta(1-\mathcal{T})^{2}}+2n_%
{\mathrm{th}}\frac{(1-\delta)\eta T_{2}}{(1-\mathcal{T})^{2}}\ \stackrel{{%
\scriptstyle n_{0}=n_{\mathrm{th}}}}{{=}}\ -2n_{\mathrm{th}}\frac{1-\eta^{2}T_%
{2}}{\eta(1-\mathcal{T})^{2}}\leqslant 0,$$
(6)
$$\displaystyle\left(\frac{\partial V_{\mathrm{N}}}{\partial\mathcal{T}}\right)_%
{T_{1}}$$
$$\displaystyle=2n_{0}\frac{1-T_{1}}{\eta(1-\mathcal{T})^{2}T_{1}}+2n_{\mathrm{%
th}}\frac{1-\delta}{\delta\eta(1-\mathcal{T})^{2}T_{1}}\ \stackrel{{%
\scriptstyle n_{0}=n_{\mathrm{th}}}}{{=}}\ 2n_{\mathrm{th}}\frac{1-T_{1}\delta%
}{T_{1}\eta\delta(1-\mathcal{T})^{2}}\geqslant 0.$$
(7)
The principal dependence of $V_{\mathrm{N}}$ on $\mathcal{T}$ is shown for different transmittances $T_{1,2}$ in Fig. 2. The mechanical mode is considered to be at equilibrium with the environment at $n_{0}=n_{\mathrm{th}}=1$. Even this low occupation creates a demand for a very efficient state upload, so that increasing $T_{1}$ with constant $T_{2}$ helps to suppress the added noise dramatically (dashed lines). It is worth noting that for certain values of $T_{1}$ the successful transfer of the negativity is impossible regardless of $T_{2}$. Note, it is evident that is much easier to reach entanglement preserving channel than to transfer negative Wigner function. From the Fig. 2 that shows the analysis following the full solution (see Section 3) we conclude that the current electromechanical setups [9, 27] are capable of the transfer of the negativity of the Wigner function provided there is a single-photon Fock state at input.
A major impediment to the successful photon-phonon-photon transfer preserving the negativity of Wigner function is the initial occupation of the mechanical mode. The influence of the mechanical bath during the state swap stages can in principle be suppressed by taking shorter pulses, however, the nonzero initial occupation has to be compensated for either by precooling of the mechanical mode or by a very efficient upload. In spite of our considered experimental scheme the precooling happens to be an additional resource that can relax the requirements to the upload of the quantum state stage. The considerations without the additional precooling therefore happen to be more conservative as such regime is more challenging experimentally and it requires the upload efficiencies close to unity. These efficiencies, however, are realistic at least in the domain of state-of-the-art electromechanics [9]. There is therefore an open way to experimentally test photon-phonon-photon channel preserving negativity of Wigner function.
We should note that in order to transmit negativity of the Wigner function, one should have $\mathcal{T}>1/2$ which places restrictions on each of individual transmittances, particularly on the efficiency of the coupling, $\eta$. That is, critically coupled systems with $\eta=0.5$ are incapable of transferring the negativity regardless of the effectiveness of state upload and readout. For these systems we consider another measure of nonclassicality, namely the quantum non-Gaussianity. A convenient criterion [31, 32] allows to determine whether a state can not be represented as a mixture of Gaussian states. The criterion establishes an upper boundary for the probability of the single photon state $p_{1}$ for a fixed probability of vacuum $p_{0}$. This boundary can be parametrized as follows:
$$p_{0}=\frac{e^{-e^{r}\sinh r}}{\cosh r},\quad p_{1,G}=\frac{e^{4r}-1}{4}\frac{%
e^{-e^{r}\sinh r}}{\cosh^{3}r},$$
(8)
with $r\in[0,+\infty)$. To evaluate the non-Gaussian properties of the restored state of the field we therefore need to evaluate the probabilities $p_{0}$ and $p_{1}$ corresponding to this state and certify if they belong to the region accessible to Gaussian states. The latter is shaded at Fig. 3. Experimentally both probabilities can be determined from homodyne detection scheme used to verify negativity of Wigner function [31]. Alternatively, single photon counters can be used [33].
In order to evaluate $p_{0}$ and $p_{1}$, we use our estimates for the transmittance $\mathcal{T}$ and added noise $V_{\mathrm{N}}$ to calculate the Wigner function of the recovered state, from which we find the needed probabilities. Results of this calculation are visualized for recent electro- and optomechanical experiments at Fig. 3. The demonstration of quantum non-Gaussianity transferred by a photon-phonon-photon channel is feasible with both platforms. The required efficiencies, particularly ones of the readout, are significantly lower for the electromechanics than that needed for the transfer of negativity of Wigner function. Note that for the weakly coupled optomechanical system, the transfer of negativity is impossible regardless of the efficiencies of the state swap. Due to the initial occupation of the mechanical modes a very efficient state upload is required.
3 Methods
For a rigorous analysis that accounts for the imperfections properly, we start with the linearised Heisenberg-Langevin equations for the optomechanical system with a strong resonantly red-detuned pump. In terms of annihilation operators $a_{c}$ and $a_{m}$ of optical and mechanical modes respectively the equations read [24]:
$$\displaystyle\dot{a}_{c}$$
$$\displaystyle=-\kappa a_{c}-iga_{m}+\sqrt{2\kappa}\left(\sqrt{\eta}a^{\text{in%
}}(t)+\sqrt{1-\eta}a^{\text{vac}}(t)\right),$$
(9)
$$\displaystyle\dot{a}_{m}$$
$$\displaystyle=-\frac{\gamma}{2}a_{m}-iga_{c}+\sqrt{\gamma}a^{\text{th}}(t).$$
(10)
Here $\gamma$ is mechanical viscous damping coefficient, $a^{\text{in}}$ and $a^{\text{vac}}$ are the input and auxiliary vacuum optical modes respectively, and $a^{\text{th}}$ describes the Markovian noises from the mechanical environment. The coupling equals $g=g_{k}\equiv g_{0}\sqrt{N_{k}}$ for $k$-th pulse and $g=0$ during the storage between the pulses. With help of the input-output relation
$$a_{2}^{\text{out}}(t)=-a_{2}^{\text{in}}(t)+\sqrt{2\kappa_{e}}a_{c}(t),$$
(11)
we are able to construct the relation connecting the bosonic operator of the input optical mode $A^{\text{in}}$ with the output $A^{\text{out}}$ in analogy with Eq. 3.
The bosonic operators are defined as
$$A^{\text{k}}=\dfrac{\int_{0}^{\tau}a^{\text{k}}(t)f^{\text{k}}(t)dt}{\sqrt{%
\int_{0}^{\tau}|f^{\text{k}}(t)|^{2}dt}},\quad\text{ k = in,out }.$$
(12)
The pulse shapes $f^{\text{in}}(t)$ and $f^{\text{out}}(t)$ are determined from the Eqs. 9 and 10. Generally expressions for these functions are quite complicated hence we do not write those explicitly here. In the limit of high $\kappa\gg g,\gamma$, where the cavity mode can be approximately eliminated [30],
$$A^{\text{in}}=\sqrt{\frac{2G_{1}}{e^{2G_{1}\tau_{1}}-1}}\int_{\tau_{1}}\mathrm%
{d}t\>a_{1}^{\text{in}}(t)e^{G_{1}t},\quad A^{\text{out}}=\sqrt{\frac{2G_{2}}{%
1-e^{-2G_{2}\tau_{2}}}}\int_{\tau_{2}}\mathrm{d}t\>a_{2}^{\text{out}}(t)e^{-G_%
{2}t},$$
(13)
with $G_{i}=g_{0}^{2}N_{i}/\kappa$.
Knowing the statistics of each individual constituent of the noisy mode $Q^{\text{N}}$, we can evaluate the variance of the added noise: $\text{var}\>{X^{\text{N}}}=\text{var}\>{Y^{\text{N}}}=V_{\mathrm{N}}$ e.g., following the steps outlined in [34], Appendix B. It is visualized as a function of the transmittivity $V_{\mathrm{N}}(\mathcal{T})$ in Fig. 2.
The effective transmittivities $T_{1}$ and $T_{2}$ associated with respectively upload and readout stages present a convenient way to parametrize this dependence. Each of these transmittivities can be increased via increase of either corresponding pump power (intracavity photon number) or pulse duration. The latter way is accompanied by increase of the mechanical environment impact. In the Fig. 2 we assume the coupling close to the highest reported and change $\tau_{1,2}$ to produce change of $T_{1,2}$. We use the following dimensionless parameters to produce the plots: for the blue lines (as reported in [9]): $\tau_{1}=\tau_{2}\leq 100/\kappa$, $g_{1}=g_{2}=0.25\kappa$, $\tau_{s}=3/\kappa$, $\gamma=2\times 10^{-4}\kappa$, $n_{\mathrm{th}}=n_{0}=20$, $\eta=0.83$. For the green lines (following [27]) $\tau_{1}=\tau_{2}\leq 10^{3}/\kappa$, $g_{1}=g_{2}=0.07\kappa$, $\tau_{s}=100/\kappa$, $\gamma=2\times 10^{-5}\kappa$, $n_{\mathrm{th}}=n_{0}=18$, $\eta=0.91$.
Evaluation of the transfer of quantum nongaussianity requires computation of the probability of vacuum and of single photon. If the restored optical quantum state is $\rho_{f}$, these probabilities equal $p_{0}=\langle 0|\rho_{f}|0\rangle$ and $p_{1}=\langle 1|\rho_{f}|1\rangle$ and can be determined by homodyne tomography [31]. The quantum state $\rho_{f}$ is found as a result of a single photon $\left|1\right\rangle$ passing a virtual beamsplitter with transmittance $\mathcal{T}$ that adds noise with variance $V_{\mathrm{N}}$. The curves in Fig. 3 are produced for same numerical parameters as Fig. 2. To produce Fig. 3 we use parameters reported in [12], $\tau_{1}\leq 3\times 10^{3}/\kappa$, $\tau_{2}\leq 2\times 10^{3}/\kappa$, $g_{1}=g_{2}=0.04\kappa$, $\tau_{s}=100/\kappa$, $\gamma=7\times 10^{-6}\kappa$, $n_{th}=n_{0}=1$, $\eta=0.5$ and $\eta=0.25$ respectively for yellow and purple lines.
Acknowledgements
The authors acknowledge finanacial support from Grant No. GB14-36681G of the Czech Science Foundation. A.R acknowledges institutional support of the Faculty of Science, Palacký University.
Author contributions
Both A.R. and R.F. contributed jointly to theory. R.F. psoposed the theoretical concept. A.R. carried out the detailed calculations. Both authors contributed to the interpretation of the work and the writing of the manuscript.
Additional Information
The authors declare no competing financial interests.
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Observation of a dynamical QCD string
L. Ya. Glozman
Institute for Physics,
University of Graz, Universitätsplatz 5, A-8010 Graz, Austria
Abstract
Mesons constructed from the quark propagators without the
lowest-lying eigenmodes of the Dirac operator reveal not only restored
$SU(2)_{L}\times SU(2)_{R}$ chiral and $U(1)_{A}$ symmetries, but actually a
higher symmetry. All possible chiral and $U(1)_{A}$ multiplets for the
states of the same spin are degenerate, i.e., the energy of the observed quantum
levels does not depend on the spin orientation of quarks in the system and
their parities. The quark-spin independence of the energy levels implies
absence of the magnetic interactions in the system. The ultrarelativistic
quark-antiquark system with only the color-electric interactions can
be interpreted (or defined) as a dynamical QCD string.
pacs: 11.30.Rd, 12.38.Gc, 11.25.-w
I Introduction
With the static color charges one observes on the lattice
a color-electric flux tube B , which, if the distance
between the static quarks is large, can be approximated as a string.
This string is nondynamical (in the sense that its ends are fixed).
In the light quark systems a crucial aspect is a relativistic
motion of quarks at the ends of a possible string. There is no
consistent theory of the dynamical QCD string with quarks at the ends.
It is even apriori unclear whether such a picture has something to do with
reality. In this case the chiral symmetry as well as its
dynamical braking should be relevant.
Both confinement and chiral symmetry breaking dynamics
are important for the hadronic mass generation. Their
interplay is responsible for a rather complicated structure
of the hadron spectra in the light quark sector. A large
degeneracy is seen in the highly excited mesons G1 ; G2 ,
which is, however, absent in the observed spectrum below 1.8 GeV.
The low-lying hadron spectra should be strongly affected by the
chiral symmetry breaking dynamics.
In order to disentangle the confinement physics from the
chiral symmetry breaking dynamics we remove on the lattice
from the valence quark
propagators the lowest-lying quasi-zero modes of the Dirac operator
keeping at the same time the gluon gauge configurations intact
LS ; GLS ; DGL . Indeed,
the quark condensate of the vacuum is related to
a density of the lowest quasi-zero eigenmodes of the
Dirac operator C :
$$<0|\bar{q}q|0>=-\pi\rho(0).$$
(1)
We subtract from the valence quark propagators their lowest-lying
chiral modes, which are a tiny part of the full amount of modes,
$$S_{RD(k)}=S_{Full}-\sum_{i=1}^{k}\,\frac{1}{\lambda_{i}}\,|\lambda_{i}\rangle%
\langle\lambda_{i}|\\
,$$
(2)
where $\lambda_{i}$ and $|\lambda_{i}\rangle$ are the eigenvalues and
the corresponding eigenvectors of the
Dirac operator. Given these reduced quark propagators we study the existence
of hadrons and
their masses upon reduction of the lowest-lying modes responsible for the
chiral symmetry breaking.
II Lattice technology
In contrast to refs. LS ; GLS , where a chirally not
invariant Wilson lattice Dirac operator was employed, we adopt now
DGL
a manifestly chiral-invariant overlap Dirac operator N .
The quark propagators were generously provided by the JLQCD
collaboration KEK ; Aoki:2012pma ; Noaki:2008iy . For some technical details of the work
we refer the reader to ref. DGL .
III Observations, symmetries and their string interpretation
At this first stage of the study we have investigated
the ground state and the excited states of all possible $\bar{q}q$
isovector
$J=1$ mesons, i.e., $\rho(1^{--})$, $a_{1}(1^{++})$, $b_{1}(1^{+-})$.
An exponential decay of the correlation function is interpreted as
a physical state and its mass is extracted. The quality of the
exponential decay improves with the reduction of the lowest modes.
The evolution of masses of the ground and excited states
upon reduction of the low-lying modes is shown in Fig. 1.
At the truncation energy about 50 MeV an onset of a degeneracy of the
states , $\rho,\rho^{\prime},a_{1},b_{1}$, as well as a degeneracy of their excited
states is seen. This degeneracy indicates a symmetry.
All possible multiplets of the $SU(2)_{L}\times SU(2)_{R}$ group for the $J=1$
mesons are shown in Table 1 G1 ; G2 . In the chirally symmetric world
there
must be two independent $\rho$-mesons that belong to different
chiral representations. The first one is a member of the $(0,1)+(1,0)$
representation. It can be created from the vacuum only by the operators
that have the same chiral structure, e.g. by the vector-isovector current
$\bar{q}\gamma^{i}\vec{\tau}q$.
Its chiral partner is the axial vector meson, $a_{1}$, that is created by
the axial-vector isovector current. When chiral symmetry is restored this $\rho$-meson
must be degenerate with the $a_{1}$ state. Another $\rho$-meson, along with
its chiral partner $h_{1}$, is a member of the
$(1/2,1/2)_{b}$ representation. It can be created only by the
operators that have the same
chiral structure, e.g. by $\bar{q}\sigma^{0i}\vec{\tau}q$.
In the real world (i.e. with broken chiral symmetry)
each $\rho$-state (i.e. $\rho$ and $\rho^{\prime}$) is a mixture of these two
representations and they are well split. Upon subtraction of 10 lowest
modes we observe two independent degenerate $\rho$-mesons. One of them
couples only to the vector current and does not couple to the
$\bar{q}\sigma^{0i}\vec{\tau}q$ operator. The other $\rho$ meson - the
other way around. This means that one of these degenerate $\rho$-states
belongs to the $(0,1)+(1,0)$ multiplet and the other one is a member
of the $(1/2,1/2)_{b}$ representation.
A degeneracy of the $(0,1)\oplus(1,0)$ $\rho$-meson with the $a_{1}$ meson
is a clear signal of the chiral $SU(2)_{L}\times SU(2)_{R}$ restoration.
Consequently, a similar degeneracy should be observed in all other chiral pairs
from Table 1.
The $U(1)_{A}$ symmetry transforms the $b_{1}$ state into the $(1/2,1/2)_{b}$
$\rho$-meson G1 ; G2 . Their degeneracy indicates a restoration of
the $U(1)_{A}$ symmetry. We conclude
that simultaneously both $SU(2)_{L}\times SU(2)_{R}$ and $U(1)_{A}$ symmetries
get restored.
The restored $SU(2)_{L}\times SU(2)_{R}\times U(1)_{A}$ symmetry requires a
degeneracy of four mesons that belong to $(1/2,1/2)_{a}$
and $(1/2,1/2)_{b}$ chiral multiplets G1 ; G2 , see Table 1.
This symmetry does not require, however, a degeneracy of these four states with other
mesons, in particular with $a_{1}$ and its chiral partner $\rho$. We clearly
see the latter degeneracy. This implies that there is some higher symmetry,
that includes
$SU(2)_{L}\times SU(2)_{R}\times U(1)_{A}$ as a subgroup. This higher symmetry requires a
degeneracy of all eight mesons from the Table 1.
All these eight mesons can be combined to a reducible $\bar{q}q$
representation, which is a product of two fundamental chiral
quark representations CJ :
$$\displaystyle[(0,1/2)+(1/2,0)]\times[(0,1/2)+(1/2,0)]=$$
$$\displaystyle(0,0)+(1/2,1/2)_{a}+(1/2,1/2)_{b}+[(0,1)+(1,0)]\;.$$
(3)
They exhaust all possible chiralities of quarks and antiquarks, i.e.,
their spin orientations,
as well as possible spatial and charge parities for non-exotic mesons.
The observed degeneracy of all these eight mesons suggests that the higher
symmetry, mentioned above, should combine
the $U(1)_{A}$ and the $SU(2)_{L}\times SU(2)_{R}$ rotations in the isospin
space with the
$SU(2)_{S}$ spin-symmetry. The latter one is due to independence of the energy
on the orientations
of the quark spins. The higher symmetry group that combines
all eight mesons from the Table 1 into one multiplet
of dimension 16 should be
$SU(2\cdot N_{f})$.
The quark-spin independence of the energy levels implies that there
are no magnetic interactions in the system, i.e., the spin-orbit
force, the color-magnetic (hyperfine) and tensor interactions
are absent G3 .
The energy of the system is entirely due to interactions of the color
charges via the color-electric field and due to a relativistic motion
of the system. We interpret (or define) such a system as a dynamical QCD
string. Note a significant qualitative difference with the case of a motion of
an electrically charged fermion in a static electric field or with a relative motion
of two fermions with the electric charge.
In the latter cases there exist
spin-orbit and spin-spin forces that are a manifestation of the magnetic interaction in the system.
In our case such a magnetic interaction is absent, which implies that both
quarks are at rest with respect to the electric field (that
moves together with
quarks). It is this circumstance that suggests to interpret (define) our system as a
dynamical QCD string.
The observed radial levels at the truncation energy 65 MeV,
at which we see the onset of the symmetry,
are approximately equidistant and can be described through
the simple relation
$$E_{n_{r}}=(n_{r}+1)\hbar\omega,~{}~{}~{}n_{r}=0,1,...$$
(4)
The extracted value of the radial
string excitation quantum
amounts to $\hbar\omega=(900\pm 70)$ MeV.
At the moment we cannot exclude, however,
the quadratic relation $E_{n_{r}}^{2}\sim(n_{r}+1)$,
because the excited level can be shifted up due to rather
small finite lattice volume.
There is an interesting aspect of the dynamical
QCD string that crucially distinguishes it from the
Nambu-Goto open string. The energy of the the Nambu-Goto open bosonic
string is determined by its orbital angular momentum $L$, $M^{2}\sim L$.
For the dynamical QCD string that contains chiral quarks at the ends
the orbital angular momentum $L$ of the relative motion
is not conserved GN .
For instance, two orthogonal $\rho$-mesons at the same energy
level are represented by the mutually orthogonal fixed superpositions of the
$S$- and $D$-waves.
$$\displaystyle\displaystyle|(0,1)+(1,0);1~{}1^{--}\rangle$$
$$\displaystyle=$$
$$\displaystyle\sqrt{\tfrac{2}{3}}\,|1;{}^{3}S_{1}\rangle+\sqrt{\tfrac{1}{3}}\,|%
1;{}^{3}D_{1}\rangle,$$
$$\displaystyle\displaystyle|(1/2,1/2)_{b};1~{}1^{--}\rangle$$
$$\displaystyle=$$
$$\displaystyle\sqrt{\tfrac{1}{3}}\,|1;{}^{3}S_{1}\rangle-\sqrt{\tfrac{2}{3}}\,|%
1;{}^{3}D_{1}\rangle.$$
Acknowledgments
I am very grateful to M. Denissenya and C. B. Lang
for our common efforts in lattice simulations. The JLQCD collaboration
is acknowledged for their suggestion to use their overlap gauge
configurations and
quark propagators. This work is partially supported by
the Austrian Science
Fund (FWF) through the grant P26627-N16.
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Testing of quantum nonlocal correlations under constrained free will and imperfect detectors
Abhishek Sadhu
[email protected]
Light and Matter Physics Group, Raman Research Institute, Bengaluru, Karnataka 560080, India
Siddhartha Das
[email protected]
Center for Security, Theory and Algorithmic Research, Centre of Quantum Science and Technology, International Institute of Information Technology, Hyderabad, Gachibowli, Telangana 500032, India
Abstract
In this work, we deal with relaxation of two central assumptions in the standard locally realistic hidden variable (LRHV) inequalities: free will in choosing of measurement settings and presence of perfect detectors at the measurement devices. Violation of LRHV inequalities by quantum correlations imply that given correlations are quantum nonlocal. However, in an adversarial situation, there could be a hidden variable introducing bias in the selection of measurement settings but observers with no access to hidden variable could be unaware of the bias. Whereas in practice, detectors don’t have perfect efficiency. A main focus of this paper is introduction of the framework where given a quantum state with nonlocal behavior under constrained free will, we can determine the threshold values of detector parameters (detector inefficiency and dark counts) such that the detectors are robust enough to certify nonlocality. We also introduce a new class of LRHV inequalities with constrained free will and discuss their implications in testing of quantum nonlocal correlations.
I Introduction
The randomness acts as a primitive resource for various desirable information processing and computational tasks, e.g., cryptography shannon1945mathematical ; bennett1984proceedings ; carl2006denial ; das2021universal ; primaatmaja2022security , statistical sampling yates1946review ; dix2002chance , probabilistic algorithms rabin1980probabilistic ; leon1988probabilistic ; sipser1983complexity , genetic algorithms holland1992adaptation ; morris1998automated ; deaven1995molecular , simulated annealing algorithms ram1996parallel ; fleischer1995simulated ; kirkpatrick1983optimization , neural networks bishop1994neural ; harrald1997evolving ; zhang2000neural , etc. Generation and certification of private randomness also form major goal of cryptographic protocols shannon1945mathematical ; acin2012randomness ; colbeck2012free ; RNGColbeck_2011 ; das2021universal ; primaatmaja2022security . The quantum theory allows for the generation and certification of true randomness einstein1935can ; bell1964einstein ; PhysRevLett.67.661 ; colbeck2012free ; das2021universal ; primaatmaja2022security , which otherwise seems to be impossible within the classical theory.
In a seminal paper bell1964einstein , J.S. Bell showed that the statistical predictions of the quantum mechanics cannot be explained by local realistic hidden variable (LRHV) theories. Several LRHV inequalities, also called Bell-type inequalities, that are based on two physical assumptions— existence of local realism and no-signalling criterion einstein1935can , have been derived since then (see e.g., brunner2014bell ; HSD15 and references therein). Violation of these LRHV inequalities brunner2014bell by quantum systems would imply presence of quantum nonlocal correlations that are not explainable by any LRHV theories. Hence, the randomness generated from the behavior of quantum systems with nonlocal correlations can be deemed to exhibit true randomness. Presence of these nonlocal correlations allow for the generation and certification of randomness and secret key in a device-independent way PhysRevLett.67.661 ; QKDAcin2006 ; acin2016certified ; kaur2021upper ; primaatmaja2022security ; zapatero2022advances ; wooltorton2022tight .
It is known that the experimental verification of the Bell’s inequality requires additional assumptions which could lead to incurring loopholes such as locality loophole bell2004speakable , freedom-of-choice (measurement independence or free will) loophole clauser1974experimental ; bell1985exchange , fair-sampling loophole (detection loophole) pearle1970hidden ; clauser1974experimental . In recent major breakthroughs hensen2015loophole ; giustina2015significant ; shalm2015strong ; big2018challenging , incompatibility of quantum mechanics with LRHV theories has been demonstrated by considerably loophole-free experiments showing violation of Bell’s inequality by quantum states with quantum nonlocal correlations. In an experimental setup, locality assumption requires space like separation between the measurement events einstein1935can ; bell1964einstein which leads to locality loophole aspect1982experimental ; weihs1998violation . In a Bell experiment with photonic setup, the detection events of the two parties are identified as belonging to the same pair, by observing whether the difference in the time of detection is small larsson2014bell . It makes possible for an adversary to fake the observed quantum correlations by exploiting the time difference between the detection events of the parties larsson2004bell giving rise to the detection loophole, which can be closed by using predefined coincidence detection windows larsson2004bell .
The free will assumption 111It should be noted that the free will assumption mentioned in this paper is also called measurement independence. This assumption relates to the possible correlations between the choice of measurement settings for the two parties, which can affect the observed experimental statistics. in the Bell-type inequality states that the parties (users) can choose the measurement settings freely or, use uncorrelated random number generators. In an experimental demonstration presented in big2018challenging human random decisions were used to choose the measurement settings in an attempt to close this loophole. In practice, the detectors are imperfect which introduce dark counts eisaman2011invited and detection inefficiencyeisaman2011invited . In this paper, we discuss implications of bias in the freedom-of-choice and non-ideal detectors on the LRHV tests of quantum nonlocal correlations.
To the best of our knowledge, the assumption of free will was relaxed in hall2010local using a distance measure based quantification of the measurement dependence. It was shown that the Bell-CHSH inequality can be violated by sacrificing equal amount of free will for both parties. This result was extended to the scenario of parties having different amounts of free will in friedman2019relaxed and for one of the parties in banik2012optimal . In an alternate approach, measurement dependence was quantified in putz2014arbitrarily ; putz2016measurement by bounding the probability of choosing of the measurement settings to be in a given range. Following this approach, tests for nonlocality has been constructed putz2014arbitrarily ; zhao2022tilted . These inequalities has been applied to randomness amplification protocols kessler2020device ; zhao2022tilted . However, the consideration of imperfect detectors in the implication of these measurement-dependent LRHV inequalities are still lacking as the above mentioned works assumed perfect detection while allowing for relaxation in measurement dependence.
A main focus of this work is to study the implications of imperfect detectors and constrained free will on the test of quantum nonlocal correlations. We adapt the approach discussed in sauer2020quantum to model imperfect detectors for the Bell experiment as a sequential application of a perfectly working inner box followed by a lossy outer box. The inner box contains a quantum source whose behavior is nonlocal under constrained free will, i.e., violates certain measurement dependent LRHV inequality. The outer box separately introduces detector inefficiency and dark counts for each party. Using this model, we determine the threshold values of the detector parameters that make detectors robust for testing of quantum nonlocality under constrained free will (e.g., see Fig. 1 with details in Section IV). Next for the scenario of perfect detectors, we compare the implications two different approaches presented in putz2014arbitrarily and hall2010local to quantify measurement dependence (a) by bounding the probability of choosing the measurement settings $x$ (for Alice’s side) and $y$ (for Bob’s side) conditioned on a hidden variable $\lambda$ to be in the range $[l,1-3l]$ putz2014arbitrarily and (b) by using a distance measure $M$ to quantify measurement settings distinguishability hall2010local . This comparison is made in the 2 (party) - 2 (measurement settings per party) - 2 (outcome per measurement) scenario and their effects on the certification of the nonlocality. We also introduce a new set of measurement dependent LRHV (MDL) inequalities by introducing distance-based measurement dependent quantity in adapted AMP tilted Bell inequalityacin2012randomness and discuss implications and trade-off between measurement dependence parameters and tilted parameters for the certification of quantum nonlocal correlations.
Throughout this paper we limit our discussions to LRHV for bipartite quantum systems. The structure of this paper is as follows. In Section II, we introduce the framework of locality and measurement dependence, which is a constraint that limits the free will of the user. We present a model where the user is tricked by an adversary into thinking that they have freedom-of-choice for the measurement. In Section III, we consider and compare two different approaches to quantify the measurement dependence in terms of parameters $l$ and $M$ mentioned earlier. We determine the critical values of these parameters necessary for the certification of nonlocality in the measurement dependence settings and compare them with amount of violation obtained for the Bell-CHSH inequality, tilted Hardy relations and the tilted Bell inequalities. In Section IV, We determine the threshold values of the detector parameters namely inefficiency and dark count probability such that the detectors are robust enough to certify nonlocality in presence of constrained free will. In Section V, we introduce a new set of LRHV inequalities adapted from the AMP tilted Bell inequality using distance measures based measurement dependence quantities. We use these inequalities to observe the effect of relaxing the free will assumption for either parties on the certification of quantum nonlocal correlations.
II The adverserial role in choice of measurement settings
Formally a quantum behavior of a device with bipartite state $\rho_{AB}$ is given by its quantum representation $\{p(ab|xy)\}$, where $p(ab|xy)=\operatorname{Tr}[\Gamma^{x}_{a}\otimes\Gamma^{y}_{b}\rho]$ with $\{\Gamma^{x}_{a}\}_{a}$ and $\{\Gamma^{y}_{b}\}_{b}$ being Positive Operator Valued Measures (POVMs) for each input $(x,y)$ to the device, for $x\in\mathcal{X}$ and $y\in\mathcal{Y}$. If a state $\psi_{AB}=\outerproduct{\psi}{\psi}_{AB}$ is pure, then we may simply represent it as a ket $\ket{\psi}_{AB}$ instead of density operator $\psi_{AB}$. Henceforth, the abbreviation MDL inequalities would stand for measurement dependent LRHV inequalities.
We consider the Bell scenario where two parties, Alice and Bob, share a bipartite quantum state $\rho_{AB}$. Each party can choose to perform one of two measurements available to them, i.e., $|\mathcal{X}|=2=|\mathcal{Y}|$. We attribute POVM $\{\Lambda^{x}_{a}\}_{x}$ to Alice and $\{\Lambda^{y}_{b}\}_{y}$ to Bob. Each of these measurements can have two outcomes. We denote measurement outcomes for Alice and Bob by $a$ and $b$, respectively. The statistics of the measurement outcomes in the experiment can then be described by the probability distribution $\textbf{P}=\{p(ab|xy)\}$ which is also termed as behavior. In this framework, let there exist a hidden variable $\lambda$ belonging to some hidden-variable space, $\Lambda.$ The probability distribution of the outputs conditioned on the inputs can then be expressed in the form of
$$p(ab|xy)=\sum_{\lambda\in\Lambda}p(ab|xy\lambda)p(\lambda|xy).$$
(1)
The hidden variable $\lambda$ (distribution according to $p(\lambda)$) can provide an explanation of the observed experimental (measurement) statistics. In each run of the experiment, there exists a fixed $\lambda$ that describes the outcome of the experimental trial following the distribution $p(ab|xy\lambda)$. After multiple runs of the experiment, the output statistics are described by sampling from the distribution $p(\lambda|xy)$.
In an adversarial scenario, Alice and Bob can believe that they are choosing all the settings with equal probability, i.e., $p(xy)=\frac{1}{4}~{}\forall~{}(x,y)$ while an adversary biases their choice in the scale $\lambda$. The adversary can distribute the settings chosen by Alice and Bob according to
$$p(xy)=\sum_{\lambda\in\Lambda}p(xy|\lambda)p(\lambda).$$
(2)
In Eq. (2), let $\lambda$ take two values, $\lambda_{1}$ and $\lambda_{2}$, whose probability distributions are given as $p\text{$(\lambda_{1}$)}=\sin^{2}(\theta_{\lambda})$ and $p\text{$(\lambda_{2}$)}=\cos^{2}(\theta_{\lambda})$, respectively. In the simplest scenario $x$ and $y$ can each take values $0$ or $1$ and there are four possible ways in which the measurement settings can be chosen by Alice and Bob, i.e., $(x,y)\in\{(0,0),(0,1),(1,0),(1,1)\}.$ Let the probability of choosing the measurement setting $(x,y)$ conditioned on the hidden variable be distributed according to Table 1.
To observe the effect of the conditional probability distribution for choice of $x$ and $y$ from Table 1 on $p(xy)$, consider the following examples:
(i)
For the case of $\theta_{s_{2}}=0.6847$, $\phi_{s_{2}}=1.2491$, $\phi_{s_{1}}=0.8861$, $\delta_{s_{2}}=1.2491$, $\theta_{\lambda}=0.785398$, $\theta_{s_{1}}=0.50413$, and $\delta_{s_{1}}=0.175353$. It can be seen that this choice of parameters set $p(xy)=0.25~{}\forall~{}(x,y)$.
(ii)
For the case of $\theta_{s_{2}}=0.5796$, $\phi_{s_{2}}=1.2491$, $\phi_{s_{1}}=0.6847$, $\delta_{s_{2}}=0.75$, $\theta_{\lambda}=0.57964$, $\theta_{s_{1}}=0.793732$, and $\delta_{s_{1}}=1.06395$. It can be seen that this choice of parameters set $p(xy)=0.25~{}\forall~{}(x,y)$.
The above examples show that by proper choice of parameters in Table 1, the adversary can trick Alice and Bob into thinking that they have free will in choosing the measurement setting. In the presence of a bias in the choice of measurement settings in the $\lambda$ scale which Alice and Bob are unaware of, the following constraints blasiak2021violations can be imposed on the conditional joint probability distribution:
a.
The signal locality, i.e., no-signalling, assumption imposes the factorisability constraint on the conditional joint probability distribution,
$$p(ab|xy\lambda)=p(a|x\lambda)p(b|y\lambda).$$
(3)
b.
The measurement independence, i.e., freedom-of-choice or free will, assumption requires that $\lambda$ does not contain any information about $x$ and $y$ which is equivalent to stating
$$\displaystyle p(\lambda|xy)=p(\lambda)$$
or equivalently,
$$\displaystyle p(xy|\lambda)=p(xy).$$
(4)
The relaxation of the measurement independence assumption, which we call measurement dependence indicates an adversary can infer the values of $x$ and $y$ by knowing $\lambda$.
III Quantifying measurement dependence
In this section, we first review two different approaches considered in putz2014arbitrarily ; zhao2022tilted and hall2010local ; banik2012optimal ; friedman2019relaxed to quantify the measurement dependence. We then compare these two approaches and observe their effects on the tests of quantum nonlocal behaviors.
III.1 Review of MDL inequalities from prior works
We review approach discussed in putz2014arbitrarily ; putz2016measurement to quantify measurement dependence by bounding the probability of choice of measurement settings conditioned on a hidden variable to be in a specific range (Section III.1.1). Then we review approach discussed in hall2010local ; banik2012optimal ; friedman2019relaxed to quantify measurement dependence using a distance measure (Section III.1.2).
III.1.1 Bound on the probability of choosing the measurement settings
In the works putz2014arbitrarily ; putz2016measurement , we observe that the probability of Alice and Bob to choose measurement settings $x$ and $y$ conditioned on $\lambda$ can be bounded in the form
$$l\leq p(x,y|\lambda)\leq h,$$
(5)
where $0\leq l\leq p(x,y|\lambda)\leq h\leq 1$. If Alice and Bob each chooses from two possible values of measurement settings, then $l=h=0.25$ corresponds to the complete measurement independence and $l,h\neq 0.25$ represents a bias in the choice of measurement settings.
In the 2 (user) - 2 (measurement settings per user) - 2 (outcome per measurement) scenario with $a,b\in\{+1,-1\}$ and $x,y\in\{0,1\}$, it was shown in putz2014arbitrarily ; putz2016measurement that all the measurement dependent local correlations satisfy the PRBLG MDL inequality
$$lp(++00)-h(p(+-01)+p(-+10)+p(++11))\leq 0.$$
(6)
A two-dimensional slice in the non-signalling space is shown in Fig. 2 (figure from putz2014arbitrarily ). In the figure, the quantum set is bounded by the green line and the set of non-signalling correlations lie within the black triangle.
In Fig. 2, the red dotted line corresponds to Eq. (6) with $h=1-3l$. If we set $h=1-3l,$ the PRBLG MDL inequality tilts from the Bell-CHSH inequality ($l=0.25$) to the non-signalling border ($l=0$). For $h=1-3l$, Eq. (6) modifies to the form
$$\displaystyle lp(++00)-(1-3l)(p(+-01)+p(-+10)$$
$$\displaystyle+p(++11))\leq 0.$$
(7)
We note that if Alice and Bob believes they have complete measurement dependence, i.e., $p(xy)=0.25\hskip 3.0pt\forall(x,y),$ then Eq. (7) reduces to the form
$$\displaystyle lp(++|00)-(1-3l)(p(+-|01)$$
$$\displaystyle+p(-+|10)+p(++|11))\leq 0.$$
(8)
It follows from zhao2022tilted that invoking the PRBLG MDL inequality (7) in the tilted Hardy test zhao2022tilted , we obtain the ZRLH MDL inequality. The ZRLH MDL inequality takes the form of
$$\displaystyle l[p(++|00)+wp(--|00)-\max\{0,w\}]$$
$$\displaystyle-(1-3l)[p(+-|01)+p(-+|10)+p(++|11)]\leq 0,$$
(9)
where $w$ is the tilting parameter taking real numbers in the range, $w\in(-0.25,1).$ We call the behaviors that violate Eqs. (8) and (9) as quantum nonlocal in the presence of measurement dependence.
III.1.2 Distance measure to quantify measurement distinguishability
We discussed in Section II that the experimental statistics described by the joint probability distribution $p(ab|xy)$ can be explained by $\lambda\in\Lambda$ in the following form,
$$p(ab|xy)=\int{\rm d}\lambda p(ab|xy\lambda)p(\lambda|xy).$$
(10)
The assumption of the measurement independence constrains the probability distribution of measurement settings via
$$p(\lambda|xy)=p(\lambda).$$
(11)
Eq. (11) implies that no extra information about $\lambda$ can be obtained from the knowledge of $x$ and $y$. This is equivalent to saying
$$p(xy|\lambda)=\frac{p(\lambda|xy)p(xy)}{p(\lambda)}=p(xy).$$
(12)
Eq. (12) implies that Alice and Bob has complete freedom in choosing the measurement settings $x$ and $y$ respectively. If $x\in U\equiv\{x_{1},x_{2}\}$ and $y\in V\equiv\{y_{1},y_{2}\}$, measurement dependence implies $p(\lambda|x_{1},y_{1})\neq p(\lambda|x_{2},y_{2})$. Distinguishability between $p(\lambda|x_{1},y_{1})$ and $p(\lambda|x_{2},y_{2})$ can be quantified using a distance measure defined in hall2010local ,
$$M=\int{\rm d}\lambda|p(\lambda|x_{1},y_{1})-p(\lambda|x_{2},y_{2})|.$$
(13)
We can express the probability to successfully distinguish $p(\lambda|x_{1},y_{1})$ and $p(\lambda|x_{2},y_{2})$ based on the knowledge of $\lambda$ as
$$F=\frac{1}{2}\left(1+\frac{M}{2}\right).$$
(14)
When $M=0$, we have $F=\frac{1}{2}$. This implies that the probability to distinguish the pairs of measurement settings is $\frac{1}{2}$, and the hidden variable does not have any information about the measurement settings. This is consistent with maximum free will that Alice and Bob has while choosing the measurement settings. For $M=2$, we have $F=1$. This implies that the probability to distinguish the pairs of measurement settings is upper bounded by $1$, and the hidden variable has complete information about the measurement settings. The local degrees of measurement dependence for Alice and Bob as introduced in hall2010local is given by
$$M_{1}\equiv\max\left\{\int{\rm d}\lambda\absolutevalue{p(\lambda|x_{1},y_{1})-p(\lambda|x_{2},y_{1})},\int{\rm d}\lambda\absolutevalue{p(\lambda|x_{1},y_{2})-p(\lambda|x_{2},y_{2})}\right\},$$
(15)
$$M_{2}\equiv\max\left\{\int{\rm d}\lambda\absolutevalue{p(\lambda|x_{1},y_{1})-p(\lambda|x_{1},y_{2})},\int{\rm d}\lambda\absolutevalue{p(\lambda|x_{2},y_{1})-p(\lambda|x_{2},y_{2})}\right\},$$
(16)
$M_{1}$ quantifies the measurement dependence for Alice’s settings keeping Bob’s settings fixed, and similarly the other way round for $M_{2}.$
The above parameters will be useful in deriving the bounds on the AMP tilted Bell inequalities acin2012randomness in the Section V.
III.2 Comparison and discussion on implications of different measurement dependent LRHV inequalities
In this section, we check for the allowed values of measurement dependence parameter $l$ to ensure nonlocality of quantum behaviors that are used to obtain randomness.
Consider the behavior in Table 2 that violates the AMP tilted Bell inequality (34). In the limit of $\alpha\rightarrow\infty,$ close to two bits of randomness can be obtained from this behavior by violating the AMP tilted Bell inequality acin2012randomness . For this behavior, the PRBLG MDL inequality (8) reduces to
$$l\leq\frac{3\alpha t+\alpha\cos(2\phi)t-2\sqrt{2}\alpha\sin^{2}(\phi)+\sqrt{2}(\cos(4\phi)-1)}{10\alpha t+4\alpha\cos(2\phi)\left(t+\sqrt{2}\right)-2\sqrt{2}\left(\alpha+3\sin^{2}(2\phi)\right)}$$
(17)
where $t=\sqrt{-\frac{\cos(4\phi)}{\alpha^{2}}+\frac{1}{\alpha^{2}}+2}.$ In the limit of $\alpha\rightarrow\infty,$ from Eq. (17) we have $l\leq 0.25.$
We see that in the limit of $\alpha\rightarrow\infty$ the behavior in Table 2 does not violate any PRBLG MDL inequality. For $\alpha=1$ (which is equivalent to Bell-CHSH inequality) and $\phi=\frac{\pi}{4}$ (the maximum violation) the behavior behavior in Table 2 violates the family of PRBLG MDL inequalities with $l>0.2023.$
We observe in Fig. 3 that for a fixed value of $\alpha$, the allowed values of $\phi$ from Table 2 that violate the PRBLG MDL inequality given by Eq. (8) increases with the increase in $l$. Also as $\alpha$ increases, for a particular value of $l$, there is a decrease in the number of the allowed values of $\phi$ from Table 2 that violate the AMP tilted Bell inequality.
It was shown in zhao2022tilted that the state $\psi_{g}=\outerproduct{\psi_{g}}{\psi_{g}}$ (18) and measurement settings $\{A_{0}^{g},A_{1}^{g},B_{0}^{g},B_{1}^{g}\}$ can be used to obtain $1$ bit of local randomness and close to $1.6806$ bits of global randomness, where
$$\ket{\psi_{g}}=\cos(\frac{\theta}{2})\ket{00}-\sin(\frac{\theta}{2})\ket{11},$$
(18)
$$\displaystyle A_{0}^{g}=B_{0}^{g}=$$
$$\displaystyle\frac{-\sqrt{2}\sin(\theta)\sqrt{\sin(\theta)}}{(2-\sin(\theta))\sqrt{\sin(\theta)+1}}\sigma_{x}$$
(19)
$$\displaystyle+\frac{-(\sin(\theta)+2)\sqrt{1-\sin(\theta)}}{(2-\sin(\theta))\sqrt{\sin(\theta)+1}}\sigma_{z},$$
$$\displaystyle A_{1}^{g}=B_{1}^{g}=\frac{\sqrt{2}\sqrt{\sin(\theta)}}{\sqrt{\sin(\theta)+1}}\sigma_{x}-\frac{\sqrt{1-\sin(\theta)}}{\sqrt{\sin(\theta)+1}}\sigma_{z},$$
(20)
such that $\theta=\sin^{-1}{(3-\sqrt{4w+5})}$ and $A_{x}^{g},B_{y}^{g}$ for $x,y\in\{0,1\}$ denote measurement settings for Alice and Bob, respectively.
Observation 1.
Let us consider the behavior given by the state $\psi_{g}$ (18) and measurement settings $\{A_{0}^{g},A_{1}^{g},B_{0}^{g},B_{1}^{g}\}$. For this behavior, the PRBLG MDL inequality given by Eq. (8) reduces to
$$\displaystyle\frac{2l}{(\xi-1)^{2}}\left(31\xi+w\left(4\xi-34\right)-69\right)$$
$$\displaystyle+\frac{2l(\xi-5)\sqrt{\xi-2}}{\sqrt{4-\xi}(\xi-1)}\sqrt{-4w+6\xi-13}$$
$$\displaystyle-\frac{2(1-3l)}{1-\xi}\bigg{(}8w-10\xi+\frac{\sqrt{\xi-2}\sin\left(2\sin^{-1}\left(3-\xi\right)\right)}{\sqrt{4-\xi}}$$
$$\displaystyle+22\bigg{)}\leq 0,$$
(21)
where $\xi=\sqrt{4w+5}.$ On inspection we see that for all $w\in(-0.25,1),$ Eq. (21) reduces to $l\leq 0.$ The behavior specified by the state $\psi_{g}$ and measurement settings $\{A_{0}^{g},A_{1}^{g},B_{0}^{g},B_{1}^{g}\}$ is nonlocal for $l>0$.
For the given behavior, the ZRLH MDL inequality (9) reduces to
$$\displaystyle\frac{1}{2\left(\xi-1\right)}(2l\left(-4w+7\xi-15\right)+$$
$$\displaystyle\frac{(1-3l)\sqrt{\xi-2}\sin\left(2\sin^{-1}\left(3-\xi\right)\right)}{\sqrt{4-\xi}}+8w-10\xi+22)$$
$$\displaystyle-l\max(0,w)\leq 0.$$
(22)
On inspection we see that for all $w\in(-0.25,1),$ Eq. (22) reduces to $l\leq 0$.
That is, the behavior given by state $\psi_{g}$ and measurement settings $\{A_{0}^{g},A_{1}^{g},B_{0}^{g},B_{1}^{g}\}$ violates both PRBLG and ZRLH MDL inequalities and its quantum nonlocality can be certified for all possible $l>0$.
IV Imperfect detector and constrained free will
We model the detection units of Alice and Bob using a two-box approach following sauer2020quantum . There is an inner box containing a quantum source generating bipartite quantum states whose behavior is nonlocal under constrained free will but assuming that detectors are perfect. Nonlocality of the quantum behavior in the inner box is tested by violation of a given MDL inequality. The output of the inner box is quantum nonlocal behavior that violates a given MDL inequality. An outer box introduces the detector imperfections, namely the detection inefficiency and dark counts. The quantum nonlocal behavior obtained from the inner box gets mapped at the outer box to the output behavior with detector imperfection parameters. The output behavior then undergoes an LRHV test based on which we are able to determine threshold values of detection inefficiency and dark counts such that the given quantum nonlocal behavior can still be certified to be nonlocal with imperfect detectors. A deviation from a two-box approach in sauer2020quantum is the introduction of the measurement dependence assumption to the working of the inner box. Alice and Bob has access only to the input settings and the outputs of the outer box. We assume that either party (Alice and Bob) has access to two identical detectors that can distinguish between the orthogonal outputs.
The measurement outcomes for the inner box are labeled as $a^{id},b^{id}$, respectively. We note that $a^{id}$ and $b^{id}$ can each take values from the set $\{+,-\}.$ Introducing non-unit detection efficiency, $0\leq\eta\leq 1$, and non-zero dark count probability, $0\leq\delta\leq 1$ in the outer box, the ideal two outcome scenario becomes a 4-outcome scenario with the addition of no-detection event $\Phi$, and the dark-count event $\chi.$ These events are defined in the following way:
$\Phi:$
One particle is sent to the party and none of the two detectors of that party click.
$\chi:$
One particle is sent to the party and both the detectors of the party click.
We label the measurement outcomes for Alice and Bob obtained from the outer box as $a^{ob},b^{ob}$, respectively. We note that $a^{ob}$ and $b^{ob}$ can each take values from the set $\{+,-,\Phi,\chi\}.$ Furthermore, we assume that Alice and Bob’s detection units have identical values for $\eta$ and $\delta$. The conditional probability of observing the outcome $t$ from the outer box conditioned on observing $t^{id}$ in the ideal scale is given by $p(t|t^{id})$ with $t\in\{a^{ob},b^{ob}\}$ and $t^{id}\in\{a^{id},b^{id}\}.$ The observed joint probabilities can then be expressed as sauer2020quantum
$$p(a^{ob}b^{ob}|xy)=\sum_{a^{id},b^{id}}p(a^{ob}|a^{id})p(b^{ob}|b^{id})p(a^{id}b^{id}|xy).$$
(23)
We relax the free will assumption in the inner box by introducing the hidden variable $\lambda\in\Lambda$. Considering this assumption, $p(a^{id}b^{id}|xy)$ takes the form of
$$p(a^{id}b^{id}|xy)=\sum_{\lambda\in\Lambda}p(a^{id}b^{id}|xy\lambda)p(\lambda|xy).$$
(24)
The hidden variable, $\lambda$ (distributed according to $p(\lambda)$) provides an explanation of the observed experimental statistics of the inner box. The distribution of settings that are chosen by Alice and Bob depends on $\lambda$ via the following relation,
$$p(xy)=\sum_{\lambda\in\Lambda}p(xy|\lambda)p(\lambda).$$
(25)
If we impose the locality condition from Eq. (3) on the experimental statistics of the inner box, we arrive at the following factorisability constraint,
$$p(a^{id}b^{id}|xy\lambda)=p(a^{id}|x\lambda)p(b^{id}|y\lambda).$$
(26)
Also, if we impose the measurement independence assumption from Eq. (4), we arrive at the following constraint,
$$p(xy|\lambda)=p(xy)\text{(or equivalently, }p(\lambda|xy)=p(\lambda)).$$
(27)
The output statistics of the outer box for imperfect detectors can depend on the output of the inner box in the following four ways sauer2020quantum :
i)
No particle is detected on either of the detectors and no dark count detection event takes place. We can then write the following:
$$p(a|a^{id})=(1-\eta)(1-\delta)^{2}.$$
(28)
ii)
No particle is detected by the detector that should have detected it and a dark count takes place in the other detector. We can then write the following:
$$p(a|a^{id})=(1-\eta)(1-\delta)\delta.$$
(29)
iii)
Either the particle is detected by one of the detectors and a dark count takes place in the other detector or the particle is not detected and dark counts take place in both the detectors. We can then write the following:
$$\displaystyle p(a|a^{id})$$
$$\displaystyle=\eta\delta+(1-\eta)\delta^{2}$$
(30)
$$\displaystyle=\delta[1-(1-\eta)(1-\delta)].$$
iv)
Either the particle is detected and no dark count takes place or the particle is not detected and a dark count takes place in the detector in which the particle should have been registered. We can then write the following:
$$\displaystyle p(a|a^{id})$$
$$\displaystyle=\eta(1-\delta)+(1-\eta)\delta^{2}$$
(31)
$$\displaystyle=(1-\delta)[1-(1-\eta)(1-\delta)].$$
The quantum nonlocal behavior obtained from the inner box after getting mapped to the output behavior with detector imperfection parameters remain nonlocal if it violates the inequality sauer2020quantum
$$\displaystyle p(++|00)+p(++|01)+p(++|10)-p(++|11)$$
$$\displaystyle-p_{A}(+|0)-p_{B}(+|0)\leq 0,$$
(32)
where $p_{A}(o|s)$ and $p_{B}(o|s)$ are the probabilities of Alice and Bob to obtain the outcome $o$ on measuring $s.$
At first, let us assume there is a quantum source in the inner box that generating a bipartite quantum state whose behavior $\{p(a^{id},b^{id}|xy)\}$ violates the PRBLG MDL inequality given by Eq. (8) assuming that detectors are perfect. The behavior $\{p(a^{id},b^{id}|xy)\}$ obtained from the inner box gets mapped at the outer box to the behavior $\{p(a^{ob},b^{ob}|xy)\}$ with the introduction of the detector inefficiency $\eta$ and dark count probability $\delta$. The behavior $\{p(a^{ob},b^{ob}|xy)\}$ is then inserted in Eq. (32) to obtain the critical detector parameters using Algorithm 1. For a fixed value of $\delta$ we obtain the minimum value of $\eta$ that violates Eq. (32) using Algorithm 1. We abbreviate left-hand side as LHS.
We plot the critical values of $\eta$ and $\delta$ obtained using Algorithm 1 in Fig. 1.
Observation 2.
From Fig. 1, we see that the minimum value of $\eta$ for a given $\delta$ is the most demanding for $l=0$ and decreases as $l$ increases. For a fixed value of $l$, the minimum value of $\eta$ increases monotonically with the dark count probability. We note that for $\delta=0,$ we have $\eta=0.667$ for all the values of $l$.
We next assume there is a quantum source in the inner box that generating a bipartite quantum state whose behavior $\{p(a^{id},b^{id}|xy)\}$ violates the ZRLH MDL inequality given by Eq. (9) assuming that detectors are perfect. The behavior $\{p(a^{id},b^{id}|xy)\}$ obtained from the inner box gets mapped at the outer box to the behavior $\{p(a^{ob},b^{ob}|xy)\}$ with the introduction of the detector inefficiency $\eta$ and dark count probability $\delta$. The behavior $\{p(a^{ob},b^{ob}|xy)\}$ is then inserted in Eq. (32) to obtain the critical detector parameters using Algorithm 1. For a fixed value of $\delta$ we obtain the minimum value of $\eta$ that violates Eq. (32) using Algorithm 1. We abbreviate left-hand side as LHS.
We plot the critical values of $\eta$ and $\delta$ from Algorithm 2 for $w=0$ in Fig. (4).
Observation 3.
The ZRLH MDL inequality given by Eq. (9) reduces to the PRBLG MDL inequality Eq. (8) for $w=0$. For $l=0$, Eq. (9) and Eq. (8) reduce to the same form and is independent of $w$. For these two cases, we observe similar dependence for the threshold detector parameters as can be seen from Fig. 4. Also, we observe from Fig. 4 that for a fixed value of $l$, the minimum detection efficiency increases monotonically with the dark count probability.
We comment here that using Algorithm 2, one can calculate the detector requirements for other values of $(w,l)$ not mentioned in this section. Next, we consider the state $\ket{\psi_{g}}$ and measurement settings $\{A_{0}^{g},A_{1}^{g},B_{0}^{g},B_{1}^{g}\}$ with $\theta=1.13557.$ This choice of state and measurement settings was shown to ensure $1.6806$ bits of global randomness in zhao2022tilted .
Observation 4.
Consider the quantum source in the inner box generates bipartite quantum state $\ket{\psi_{g}}.$ Let Alice and Bob choose the measurement settings $\{A_{0}^{g},A_{1}^{g},B_{0}^{g},B_{1}^{g}\},$ with $\theta=1.13557.$ With this choice of state and measurement settings we obtain the behavior $\{p(a^{id},b^{id}|xy)\}$ assuming that the detectors are perfect. The output behavior $\{p(a^{ob},b^{ob}|xy)\}$ obtained from the outer box is evaluated using Eq. (23). The behavior $\{p(a^{ob},b^{ob}|xy)\}$ is nonlocal if we have a violation of the inequality
$$\displaystyle\delta\eta[\delta(9.50424\delta-4\delta^{2}-9.00848)+4.25636]$$
$$\displaystyle+\eta^{2}[\delta(2\delta^{3}-5.50424\delta^{2}+5.82473\delta-3.13675)$$
$$\displaystyle+0.816258]+2\delta(\delta-1)(\delta^{2}-\delta+1)$$
$$\displaystyle-0.752119\eta\leq 0.$$
(33)
The allowed values of the pair $(\eta,\delta)$ for which $p(a^{ob},b^{ob}|xy)$ is quantum nonlocal shown in Fig. 5.
In Fig. 5 we observe that for $\delta=0$ the minimum detection efficiency $\eta_{\text{crit}}\approx 0.92$ is required to ensure $p(a^{ob},b^{ob}|xy)$ is quantum nonlocal. We also observe that minimum values of $\eta$ increases monotonically with increase in $\delta$.
V The Tilted Bell inequality
The AMP tilted Bell inequality is given as acin2012randomness
$$I_{\alpha}^{\beta}:=\beta\langle x_{1}\rangle+\alpha\langle x_{1}y_{1}\rangle+\alpha\langle x_{1}y_{2}\rangle+\langle x_{2}y_{1}\rangle-\langle x_{2}y_{2}\rangle\leq\beta+2\alpha.$$
(34)
It plays an important role in (a) demonstrating the inequivalence between the amount of certified randomness and the amount of nonlocality acin2012randomness , (b) self-testing of all bipartite pure entangled states coladangelo2017all , (c) protocol for device-independent quantum random number generation with sublinear amount of quantum communication bamps2018device , (d) unbounded randomness certification from a single pair of entangled qubits with sequential measurements curchod2017unbounded .
In the following proposition, we obtain the bound on $I_{\alpha}^{\beta}$ when the measurement independence assumption is relaxed (see Appendix A for proof).
Proposition 1.
The AMP tilted Bell expression $I_{\alpha}^{\beta}$ in the presence of locality and the relaxed measurement independence is bounded by
$$I_{\alpha}^{\beta}\leq\beta+2\alpha+\min\{\alpha(M_{1}+\min\{M_{1},M_{2}\})+M_{2},2\},$$
(35)
where $M_{1}$ and $M_{2}$ are the measurement dependence parameters for Alice and Bob.
V.1 Comparison of one-sided and two-sided measurement dependence to ensure quantum representation
Let Alice and Bob have free will in choosing the measurement settings, then the maximum violation of $I_{\alpha}^{\beta}$ obtained by quantum nonlocal behaviors is given by acin2012randomness
$$I_{\alpha}^{\beta}\leq 2\sqrt{(1+\alpha^{2})(1+\frac{\beta^{2}}{4})}.$$
(36)
As direct consequences of Proposition 1, we have following corollaries.
Corollary 1.
When $M_{1}$ = $M_{2}$ = $M$, the quantum nonlocal behaviors that maximally violate Eq. (34) with the amount of violation given by Eq. (36), remains nonlocal for
$$M<\frac{-2\alpha-\beta+\sqrt{(1+\alpha^{2})(4+\beta^{2})}}{1+2\alpha}.$$
(37)
For $\alpha=1\text{ and }\beta=0$ (the Bell-CHSH scenario), Eq. (37) reduces to $M<\frac{2}{3}(\sqrt{2}-1)\approx 0.276$ and is consistent with the observation in friedman2019relaxed .
Corollary 2.
When $M_{1}=0$ and $M_{2}=M$, the quantum nonlocal behaviors that maximally violate of Eq. (34) with the amount of violation given by Eq. (36), remains nonlocal for
$$M<-2\alpha-\beta+\sqrt{(1+\alpha^{2})(4+\beta^{2})}$$
(38)
For $\beta=0\text{ and }\alpha=1$ (the Bell-CHSH scenario.), Eq. (38) reduces to $M<2(\sqrt{2}-1)\approx 0.828$ and is consistent with the observation in friedman2019relaxed .
Corollary 3.
When $M_{1}=M$ and $M_{2}=0$, the quantum nonlocal behaviors that maximally violate of Eq. (34) with the amount of violation given by Eq. (36), remains nonlocal for
$$M<\frac{1}{\alpha}(-2\alpha-\beta+\sqrt{(1+\alpha^{2})(4+\beta^{2})})$$
(39)
For $\beta=0\text{ and }\alpha=1$ (the Bell-CHSH scenario.), Eq. (39) reduces to $M<2(\sqrt{2}-1)\approx 0.828$ and is consistent with the observation in friedman2019relaxed .
We consider $\beta=0$ and $\alpha\geq 1$ and plot in Fig. 6, the upper bound on $M$ as a function of $\alpha$ for MDL behaviors. The values of $M$ and $\alpha$ from Fig. 6 ensures that the quantum nonlocal behaviors that maximally violate Eq. (34) remains nonlocal in the presence of measurement dependence.
We observe in Fig. 6 that for $\beta=0$ and $\alpha\geq 1$, the introduction of one-sided measurement dependence allows for higher values of $M$ (implying lower degree of freedom-of-choice) as compared to introducing both-sided measurement dependence. Also, for the case of one-sided measurement dependence, Alice can have higher values of $M$ as compared to Bob.
V.2 Bounds on the measurement dependence for testing nonlocality
We observe in Proposition 1 that in the presence of relaxed measurement dependence, the behaviors $\{p(ab|xy)\}$, that violates Eq. (40) are nonlocal.
$$I_{\alpha}^{\beta}\leq\beta+2\alpha+\min\{\alpha(M_{1}+\min\{M_{1},M_{2}\})+M_{2},2\}$$
(40)
In the following, we discuss some of the cases where Eq. (40) can be violated.
a.
We consider a situation when both Alice and Bob have the same measurement dependence, we have $M_{1}=M_{2}=M$. For this case, violating Eq. (40) requires
$$I_{\alpha}^{\beta}>\beta+2\alpha+\min\{(2\alpha+1)M,2\}.$$
(41)
If $(2\alpha+1)M\geq 2$, $I_{\alpha}^{\beta}$ reaches the no-signalling boundary. Whereas, if $(2\alpha+1)M<2$, then Eq. (41) reduces to inequality
$$M<\frac{I_{\alpha}^{\beta}-\beta-2\alpha}{2\alpha+1}.$$
(42)
b.
We consider the situation when only Bob has measurement dependence, we have $M_{1}=0,M_{2}=M.$ For this case, violating Eq. (40) requires
$$I_{\alpha}^{\beta}>\beta+2\alpha+\min\{M,2\}.$$
(43)
If $M=2$, $I_{\alpha}^{\beta}$ reaches the no-signalling boundary. Whereas, if $M<2$, then Eq. (43) to the inequality
$$M<I_{\alpha}^{\beta}-\beta-2\alpha.$$
(44)
c.
We consider the situation when only Alice has measurement dependence, we have $M_{1}=M,M_{2}=0.$ For this case, violating Eq. (40) requires
$$I_{\alpha}^{\beta}>\beta+2\alpha+\min\{\alpha M,2\}.$$
(45)
If $\alpha M\geq 2$, $I_{\alpha}^{\beta}$ reaches the no-signalling boundary. Whereas if $\alpha M<2$, then Eq. (45) reduces to the inequality
$$M<\frac{I_{\alpha}^{\beta}-\beta-2\alpha}{\alpha}.$$
(46)
We note that for $\alpha=1$ and $\beta=0,$ the values of $M_{1}$ and $M_{2}$ that violates Eq. (40) are in agreement with that obtained in friedman2019relaxed .
Consider there exists some behavior $\mathbf{P}^{\prime}=\{p^{\prime}(ab|xy)\}$ that violates Eq. (34) with the amount of violation given by $I_{\alpha}^{\beta}=I^{\prime}(\alpha,\beta,\mathbf{P}^{\prime})$. For $\alpha=1,\beta=8$, we show in Fig. 7 the possible values of measurement dependence parameters $M_{1}$ and $M_{2}$ for which the amount of violation of Eq. (34) given by $I^{\prime}$ cannot be described by a deterministic measurement dependent local model. The plots for the other combinations of $(\alpha,\beta)$ is given in Appendix B.
VI Discussion
In this paper, two different approaches in quantifying measurement dependence via parameters $l$ and $(M_{1},M_{2})$ from standard literature is presented and a bound on these parameters for certifying nonlocality of different behaviors is obtained. It is observed that the behavior certifying close to 2 bits of randomness remains measurement dependent nonlocal only in the limit of complete measurement independence, i.e., in the limit of $l\rightarrow 0.25.$ The behavior that provides 1.6806 bits of global randomness from the violation of the tilted Hardy relations is measurement dependent nonlocal for arbitrarily small values of $l.$ This motivates further study on obtaining an inequality that provides close to two bits of global randomness in the limit of arbitrarily low measurement dependence, a problem we leave for future study. Deviating from the conventional approach of assuming perfect detectors, we present a framework to determine the threshold values of the detector parameters that are robust enough to certify nonlocality of given quantum behaviors. This is an important step towards experimentally obtaining nonlocality in the presence of relaxed measurement independence. For an illustration, we presented the critical requirements for generating 1.6806 bits of tilted-Hardy certified global randomness. The detector parameters obtained from this study is expected to have important applications in experimentally implementing various information processing tasks that rely on quantum nonlocality as a resource.
The modified analytical bound on the AMP tilted Bell inequality in terms of $M_{1}$ and $M_{2}$ has been obtained. Using the analytical bound, the bounds on $M_{1}$ and $M_{2}$ to ensure quantumness and nonlocality are observed. It is observe that one-sided measurement dependence is more advantageous from the point of view of the user as compared to two-sided measurement dependence. The analytical bound obtained is expected to have applications in self-testing of quantum states and other device independent information processing protocols like randomness generation and secure communication.
For future work, it would be interesting to see implications of measurement dependence in multipartite Bell-type inequalities brunner2014bell ; HSD15 for the certification of multipartite nonlocality and device-independent conference keys RMW18 ; holz2020genuine ; HWD22 .
Note— We noticed the related work “Quantum nonlocality in presence of strong measurement dependence” by Ivan Šupić, Jean-Daniel Bancal, and Nicolas Brunner recently posted as arXiv:2209.02337.
Acknowledgements.
The authors thank Karol Horodecki for discussions. A.S. acknowledges the PhD fellowship from the Raman Research Institute, Bangalore. A.S. thanks the International Institute of Information Technology, Hyderabad for the hospitality during his visit in the summer of 2022 where part of this work was done.
Appendix A Calculating the modified local bound for the AMP tilted Bell inequality
We consider the AMP tilted-Bell inequality introduced in acin2012randomness and given by,
$$I_{\alpha}^{\beta}=\beta\langle x_{1}\rangle+\alpha\langle x_{1}y_{1}\rangle+\alpha\langle x_{1}y_{2}\rangle+\langle x_{2}y_{1}\rangle-\langle x_{2}y_{2}\rangle$$
(47)
The determinism assumption states that the measurement outcomes are deterministic functions of the choice of settings and the hidden variable $\lambda,$ i.e., $a=A(x,\lambda)$ and $b=B(x,\lambda)$ with $p(a|x\lambda)=\delta_{a,A(x,\lambda)}$ and $p(b|y\lambda)=\delta_{b,B(y,\lambda)}.$ If we assume that the choice of the measurement settings on the side of Alice and Bob can depend on some hidden variable, $\lambda$, the correlations of Alice and Bob can be expressed as
$$\langle xy\rangle=\int{\rm d}\lambda p(\lambda|x,y)A(x,\lambda)B(y,\lambda).$$
(48)
The correlation function of Alice can also be written as
$$\langle x\rangle=\int{\rm d}\lambda p(\lambda|x)A(x,\lambda).$$
(49)
Applying Eq. (48) and Eq. (49) to Eq. (47) we have
$$\displaystyle I_{\alpha}^{\beta}=\beta\int$$
$$\displaystyle{\rm d}\lambda p(\lambda|x_{1})A(x_{1},\lambda)$$
(50)
$$\displaystyle+\alpha\int{\rm d}\lambda p(\lambda|x_{1},y_{1})A(x_{1},\lambda)B(y_{1},\lambda)$$
$$\displaystyle+\alpha\int{\rm d}\lambda p(\lambda|x_{1},y_{2})A(x_{1},\lambda)B(y_{2},\lambda)$$
$$\displaystyle+\int{\rm d}\lambda p(\lambda|x_{2},y_{1})A(x_{2},\lambda)B(y_{1},\lambda)$$
$$\displaystyle-\int{\rm d}\lambda p(\lambda|x_{2},y_{2})A(x_{2},\lambda)B(y_{2},\lambda).$$
Adding and subtracting the terms
$$\displaystyle\alpha\int{\rm d}\lambda p(\lambda|x_{1},y_{2})A(x_{1},\lambda)B(y_{1},\lambda)\text{ and}$$
$$\displaystyle\int{\rm d}\lambda p(\lambda|x_{2},y_{2})A(x_{2},\lambda)B(y_{1},\lambda)$$
to Eq. (50) we obtain
$$\displaystyle I_{\alpha}^{\beta}$$
$$\displaystyle=\beta\int{\rm d}\lambda p(\lambda|x_{1})A(x_{1},\lambda)$$
(51)
$$\displaystyle\qquad+\alpha\int{\rm d}\lambda A(x_{1},\lambda)B(y_{1},\lambda)\bigg{[}p(\lambda|x_{1},y_{1})$$
$$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad-p(\lambda|x_{1},y_{2})\bigg{]}$$
$$\displaystyle\qquad+\int{\rm d}\lambda A(x_{2},\lambda)B(y_{1},\lambda)\bigg{[}p(\lambda|x_{2},y_{1})$$
$$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad-p(\lambda|x_{2},y_{2})\bigg{]}$$
$$\displaystyle\qquad+\alpha\int{\rm d}\lambda p(\lambda|x_{1},y_{2})\bigg{[}A(x_{1},\lambda)B(y_{2},\lambda)$$
$$\displaystyle\qquad\qquad\qquad\qquad\qquad+A(x_{1},\lambda)B(y_{1},\lambda)\bigg{]}$$
$$\displaystyle\qquad-\int{\rm d}\lambda p(\lambda|x_{2},y_{2})\bigg{[}A(x_{2},\lambda)B(y_{2},\lambda)$$
$$\displaystyle\qquad\qquad\qquad\qquad-A(x_{2},\lambda)B(y_{1},\lambda)\bigg{]}.$$
We observe that Eq. (51) is bounded by $I_{\alpha}^{\beta}\leq\max[T_{1}]+\max[T_{2}]+\max[T_{3}]$. We also note that as $p(\lambda|x_{1})$ is a normalised probability distribution, $\int{\rm d}\lambda p(\lambda|x_{1})=1$. This implies that the maximum value of the quantity $\beta\int{\rm d}\lambda p(\lambda|x_{1})A(x_{1},\lambda)$ is given by $\beta$ when $A(x_{1},\lambda)$ is set to $1$. With these observations, $T_{1}$ can be simplified as
$$\displaystyle T_{1}=$$
$$\displaystyle\beta\int{\rm d}\lambda p(\lambda|x_{1})A(x_{1},\lambda)$$
(52)
$$\displaystyle+\alpha\int{\rm d}\lambda A(x_{1},\lambda)B(y_{1},\lambda)\bigg{[}p(\lambda|x_{1},y_{1})$$
$$\displaystyle\qquad\qquad-p(\lambda|x_{1},y_{2})\bigg{]}\leq\beta+\alpha M_{2}.$$
where we have set $B(y_{1},\lambda)=1.$ To evaluate the maximum value of $T_{2}$ we set $A(x_{2},\lambda)=1$ and obtain
$$\displaystyle T_{2}=$$
$$\displaystyle\int{\rm d}\lambda A(x_{2},\lambda)B(y_{1},\lambda)\bigg{[}p(\lambda|x_{2},y_{1})-p(\lambda|x_{2},y_{2})\bigg{]}$$
$$\displaystyle\leq M_{2}.$$
(53)
We evaluate $T_{3}$ as follows,
$$\displaystyle T_{3}$$
$$\displaystyle=$$
$$\displaystyle\alpha\int{\rm d}\lambda p(\lambda|x_{1},y_{2})\bigg{[}A(x_{1},\lambda)B(y_{2},\lambda)+A(x_{1},\lambda)B(y_{1},\lambda)\bigg{]}$$
$$\displaystyle-\int{\rm d}\lambda p(\lambda|x_{2},y_{2})\bigg{[}A(x_{2},\lambda)B(y_{2},\lambda)-A(x_{2},\lambda)B(y_{1},\lambda)\bigg{]}$$
$$\displaystyle=$$
$$\displaystyle\int{\rm d}\lambda A(x_{1},\lambda)B(y_{2},\lambda)\bigg{[}\alpha p(\lambda|x_{1},y_{2})$$
(54)
$$\displaystyle\qquad\qquad\qquad\qquad-p(\lambda|x_{2},y_{2})\frac{A(x_{2},\lambda)}{A(x_{1},\lambda)}\bigg{]}$$
$$\displaystyle+\int{\rm d}\lambda A(x_{1},\lambda)B(y_{1},\lambda)\bigg{[}\alpha p(\lambda|x_{1},y_{2})$$
$$\displaystyle\qquad\qquad\qquad\qquad+p(\lambda|x_{2},y_{2})\frac{A(x_{2},\lambda)}{A(x_{1},\lambda)}\bigg{]}.$$
To get the maximum value of $T_{3}$, we set the values of $A(x_{1},\lambda),B(y_{1},\lambda),A(x_{1},\lambda),B(y_{2},\lambda)$ to one. This then implies,
$$\displaystyle T_{3}=$$
$$\displaystyle\int{\rm d}\lambda\bigg{[}\alpha p(\lambda|x_{1},y_{2})-p(\lambda|x_{2},y_{2})\bigg{]}$$
(55)
$$\displaystyle+\int{\rm d}\lambda\bigg{[}\alpha p(\lambda|x_{1},y_{2})+p(\lambda|x_{2},y_{2})\bigg{]}.$$
We evaluate the first term of the Eq. (55) as
$$\displaystyle\int{\rm d}\lambda\bigg{[}\alpha p(\lambda|x_{1},y_{2})-p(\lambda|x_{2},y_{2})\bigg{]}$$
$$\displaystyle=$$
$$\displaystyle\int{\rm d}\lambda\bigg{[}\alpha p(\lambda|x_{1},y_{2})-\alpha p(\lambda|x_{2},y_{2})+\alpha p(\lambda|x_{2},y_{2})$$
(56)
$$\displaystyle\qquad-p(\lambda|x_{2},y_{2})\bigg{]}$$
$$\displaystyle=$$
$$\displaystyle\int{\rm d}\lambda\bigg{[}\alpha\bigg{(}p(\lambda|x_{1},y_{2})-p(\lambda|x_{2},y_{2})\bigg{)}$$
(57)
$$\displaystyle\qquad+(\alpha-1)p(\lambda|x_{2},y_{2})\bigg{]}$$
$$\displaystyle=$$
$$\displaystyle\alpha\int{\rm d}\lambda\bigg{[}p(\lambda|x_{1},y_{2})-p(\lambda|x_{2},y_{2})\bigg{]}$$
(58)
$$\displaystyle\qquad+(\alpha-1)\int{\rm d}\lambda p(\lambda|x_{2},y_{2}).$$
In Eq. (58) we note that
$$\alpha\int{\rm d}\lambda\bigg{(}p(\lambda|x_{1},y_{2})-p(\lambda|x_{2},y_{2})\bigg{)}\leq\alpha M_{1}.$$
We note that as $p(\lambda|x_{2},y_{2})$ is a normalised probability distribution, $\int{\rm d}\lambda p(\lambda|x_{2},y_{2})=1.$ This then implies
$$\int{\rm d}\lambda\bigg{[}\alpha p(\lambda|x_{1},y_{2})-p(\lambda|x_{2},y_{2})\bigg{]}\leq\alpha M_{1}+\alpha-1,$$
(59)
which on simplification reduces to
$$\int{\rm d}\lambda\bigg{[}\alpha p(\lambda|x_{1},y_{2})-p(\lambda|x_{2},y_{2})\bigg{]}\leq\alpha(M_{1}+1)-1.$$
(60)
We proceed in the same way for the second term in Eq. (55) as follows,
$$\displaystyle\int{\rm d}\lambda\bigg{[}\alpha p(\lambda|x_{1},y_{2})+p(\lambda|x_{2},y_{2})\bigg{]}$$
$$\displaystyle=$$
$$\displaystyle\int{\rm d}\lambda\bigg{[}\alpha p(\lambda|x_{1},y_{2})+\alpha p(\lambda|x_{2},y_{2})-\alpha p(\lambda|x_{2},y_{2})$$
(61)
$$\displaystyle\qquad+p(\lambda|x_{2},y_{2})\bigg{]}$$
$$\displaystyle=$$
$$\displaystyle\int{\rm d}\lambda\alpha\bigg{[}p(\lambda|x_{1},y_{2})+p(\lambda|x_{2},y_{2})\bigg{]}$$
(62)
$$\displaystyle\qquad-(\alpha-1)\int{\rm d}\lambda p(\lambda|x_{2},y_{2})$$
$$\displaystyle=$$
$$\displaystyle\alpha\int{\rm d}\lambda\bigg{[}p(\lambda|x_{1},y_{2})+p(\lambda|x_{2},y_{2})\bigg{]}$$
(63)
$$\displaystyle\qquad+(1-\alpha)\int{\rm d}\lambda p(\lambda|x_{2},y_{2}).$$
We note that as $p(\lambda|x_{1},y_{2})$, $p(\lambda|x_{2},y_{2})$ are normalised probability distributions, $\int{\rm d}\lambda p(\lambda|x_{1},y_{2})=1$ and $\int{\rm d}\lambda p(\lambda|x_{2},y_{2})=1.$ This then implies
$$\int{\rm d}\lambda\bigg{[}\alpha p(\lambda|x_{1},y_{2})+p(\lambda|x_{2},y_{2})\bigg{]}=\alpha+1.$$
(64)
If we insert Eq. (60) and Eq. (64) in Eq. (55), we obtain
$$\displaystyle T_{3}\leq\alpha(M_{1}+2).$$
(65)
Now combining Eq. (52), Eq. (53) and Eq. (65), we have the bound on $I_{\alpha}^{\beta}$ as
$$\displaystyle I_{\alpha}^{\beta}$$
$$\displaystyle\leq T_{1}+T_{2}+T_{3}$$
(69)
$$\displaystyle\leq\beta+\alpha M_{2}+M_{2}+\alpha(M_{1}+2)$$
$$\displaystyle\leq\beta+2\alpha+\alpha M_{1}+(\alpha+1)M_{2}$$
$$\displaystyle\leq\beta+2\alpha+\alpha(M_{1}+M_{2})+M_{2}.$$
We again start by considering Eq. (50) as
$$\displaystyle I_{\alpha}^{\beta}=$$
$$\displaystyle\beta\int{\rm d}\lambda p(\lambda|x_{1})A(x_{1},\lambda)$$
(70)
$$\displaystyle\qquad+\alpha\int{\rm d}\lambda p(\lambda|x_{1},y_{1})A(x_{1},\lambda)B(y_{1},\lambda)$$
$$\displaystyle\qquad+\alpha\int{\rm d}\lambda p(\lambda|x_{1},y_{2})A(x_{1},\lambda)B(y_{2},\lambda)$$
$$\displaystyle\qquad+\int{\rm d}\lambda p(\lambda|x_{2},y_{1})A(x_{2},\lambda)B(y_{1},\lambda)$$
$$\displaystyle\qquad-\int{\rm d}\lambda p(\lambda|x_{2},y_{2})A(x_{2},\lambda)B(y_{2},\lambda).$$
Adding and subtracting the terms $\alpha\int{\rm d}\lambda p(\lambda|x_{2},y_{1})A(x_{1},\lambda)B(y_{1},\lambda)$ and $\alpha\int{\rm d}\lambda p(\lambda|x_{2},y_{2})A(x_{1},\lambda)B(y_{2},\lambda)$ to Eq. (50) we obtain
$$\displaystyle I_{\alpha}^{\beta}$$
$$\displaystyle=\beta\int{\rm d}\lambda p(\lambda|x_{1})A(x_{1},\lambda)$$
(71)
$$\displaystyle\qquad+\alpha\int{\rm d}\lambda A(x_{1},\lambda)B(y_{1},\lambda)\bigg{[}p(\lambda|x_{1},y_{1})$$
$$\displaystyle\qquad\qquad\qquad\qquad\qquad-p(\lambda|x_{2},y_{1})\bigg{]}$$
$$\displaystyle\qquad+\alpha\int{\rm d}\lambda A(x_{1},\lambda)B(y_{2},\lambda)\bigg{[}p(\lambda|x_{1},y_{2})$$
$$\displaystyle\qquad\qquad\qquad\qquad\qquad-p(\lambda|x_{2},y_{2})\bigg{]}$$
$$\displaystyle\qquad+\int{\rm d}\lambda A(x_{2},\lambda)B(y_{1},\lambda)\bigg{[}p(\lambda|x_{2},y_{1})$$
$$\displaystyle\qquad\qquad\qquad-\frac{B(y_{2},\lambda)}{B(y_{1},\lambda)}p(\lambda|x_{2},y_{2})\bigg{]}$$
$$\displaystyle\qquad+\alpha\int{\rm d}\lambda A(x_{1},\lambda)B(y_{1},\lambda)\bigg{[}p(\lambda|x_{2},y_{1})$$
$$\displaystyle\qquad\qquad\qquad+\frac{B(y_{2},\lambda)}{B(y_{1},\lambda)}p(\lambda|x_{2},y_{2})\bigg{]}.$$
We observe that Eq. (71) is bounded by $I_{\alpha}^{\beta}\leq\max[t_{1}]+\max[t_{2}]+\max[t_{3}]$. We also note that as $p(\lambda|x_{1})$ is a normalised probability distribution, $\int{\rm d}\lambda p(\lambda|x_{1})=1$. This implies that the maximum value of the quantity $\beta\int{\rm d}\lambda p(\lambda|x_{1})A(x_{1},\lambda)$ then takes the value of $\beta$ when $A(x_{1},\lambda)$ is set to $1$. With these observations, $t_{1}$ can be simplified as,
$$\displaystyle t_{1}$$
$$\displaystyle=\beta\int{\rm d}\lambda p(\lambda|x_{1})A(x_{1},\lambda)$$
(72)
$$\displaystyle\qquad+\alpha\int{\rm d}\lambda A(x_{1},\lambda)B(y_{1},\lambda)\bigg{[}p(\lambda|x_{1},y_{1})$$
$$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad-p(\lambda|x_{2},y_{1})\bigg{]}$$
$$\displaystyle\leq\beta+\alpha M_{1}.$$
Similarly, $t_{2}$ can be simplified as
$$\displaystyle t_{2}$$
$$\displaystyle=\alpha\int{\rm d}\lambda A(x_{1},\lambda)B(y_{2},\lambda)\bigg{[}p(\lambda|x_{1},y_{2})$$
(73)
$$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad-p(\lambda|x_{2},y_{2})\bigg{]}$$
$$\displaystyle\leq\alpha M_{1},$$
and for $t_{3},$ we have the expression
$$\displaystyle t_{3}$$
$$\displaystyle=\int{\rm d}\lambda A(x_{2},\lambda)B(y_{1},\lambda)\bigg{[}p(\lambda|x_{2},y_{1})$$
(74)
$$\displaystyle\qquad\qquad\qquad-\frac{B(y_{2},\lambda)}{B(y_{1},\lambda)}p(\lambda|x_{2},y_{2})\bigg{]}$$
$$\displaystyle\qquad+\alpha\int{\rm d}\lambda A(x_{1},\lambda)B(y_{1},\lambda)\bigg{[}p(\lambda|x_{2},y_{1})$$
$$\displaystyle\qquad\qquad\qquad+\frac{B(y_{2},\lambda)}{B(y_{1},\lambda)}p(\lambda|x_{2},y_{2})\bigg{]}.$$
We set the values of $A(x_{1},\lambda),B(y_{1},\lambda),A(x_{2},\lambda),B(y_{2},\lambda)$ to one and obtain
$$\displaystyle t_{3}$$
$$\displaystyle=\int{\rm d}\lambda\bigg{[}p(\lambda|x_{2},y_{1})-p(\lambda|x_{2},y_{2})\bigg{]}$$
(75)
$$\displaystyle\qquad+\alpha\int{\rm d}\lambda\bigg{[}p(\lambda|x_{2},y_{1})+p(\lambda|x_{2},y_{2})\bigg{]}.$$
The first term in Eq, (75) is bounded by
$$\int{\rm d}\lambda\bigg{[}p(\lambda|x_{2},y_{1})-p(\lambda|x_{2},y_{2})\bigg{]}\leq M_{2}.$$
(76)
We note that as $p(\lambda|x_{2},y_{1})$ and $p(\lambda|x_{2},y_{2})$ are normalised probability distribution, $\int{\rm d}\lambda p(\lambda|x_{2},y_{1})=1$ and $\int{\rm d}\lambda p(\lambda|x_{2},y_{2})=1.$ Eq. (75) then takes the form of
$$t_{3}\leq M_{2}+2\alpha.$$
(77)
Now combining the values of $t_{1},$ $t_{2}$ and $t_{3}$, we have the bound on $I_{\alpha}^{\beta}$ as
$$\displaystyle I_{\alpha}^{\beta}$$
$$\displaystyle\leq t_{1}+t_{2}+t_{3}$$
(80)
$$\displaystyle\leq\beta+\alpha M_{1}+\alpha M_{1}+M_{2}+2\alpha$$
$$\displaystyle\leq\beta+2\alpha+2\alpha M_{1}+M_{2}.$$
We note that the bound on $I_{\alpha}^{\beta}$ must be the minimum of Eqs. (69) and (80) and takes the form of
$$I_{\alpha}^{\beta}\leq\beta+2\alpha+\alpha[M_{1}+\min\{M_{1},M_{2}\}]+M_{2}.$$
(81)
We observe that for the case of $\beta=0$ and $\alpha=1,$
$$I_{1}^{0}\leq 2+M_{1}+M_{2}+\min\{M_{1},M_{2}\},$$
(82)
which is in agreement with that obtained in friedman2019relaxed . We note that the maximum value that $I_{\alpha}^{\beta}$ [Eq. (47)] can take is $\beta+2\alpha+2$ (We get this bound by setting $\langle x_{1}\rangle=1$, $\langle x_{1}y_{1}\rangle=1$, $\langle x_{1}y_{2}\rangle=1$, $\langle x_{2}y_{1}\rangle=1$, $\langle x_{2}y_{2}\rangle=-1$) and arrive at
$$I_{\alpha}^{\beta}\leq\beta+2\alpha+\min\{\alpha(M_{1}+\min\{M_{1},M_{2}\})+M_{2},2\}.$$
(83)
Appendix B Bounds on the measurement dependence to certify nonlocality
We observe in Proposition 1 that in the presence of relaxed measurement dependence, the behaviors $\{p(ab|xy)\}$, that violates Eq. (84) are nonlocal.
$$I_{\alpha}^{\beta}\leq\beta+2\alpha+\min\{\alpha(M_{1}+\min\{M_{1},M_{2}\})+M_{2},2\}$$
(84)
Consider there exists some behavior $\mathbf{P}^{\prime}=\{p^{\prime}(ab|xy)\}$ that violates Eq. (34) with the amount of violation given by $I_{\alpha}^{\beta}=I^{\prime}(\alpha,\beta,\mathbf{P}^{\prime})$. We show in Fig. 8 and Fig. 9 the possible values of measurement dependence parameters $M_{1}$ and $M_{2}$ for which the amount of violation of Eq. (34) given by $I^{\prime}$ cannot be described by a deterministic measurement dependent local model. We observe that Fig. (8) is in agreement with that obtained in friedman2019relaxed .
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Topological semigroups and universal spaces related to
extension dimension
A. Chigogidze
Department of Mathematics and Statistics,
University of Saskatchewan, McLean Hall, 106 Wiggins Road,
Saskatoon, SK, S7N 5E6, Canada
[email protected]
,
A. Karasev
Department of Mathematics and Statistics,
University of Saskatchewan, McLean Hall, 106 Wiggins Road,
Saskatoon, SK, S7N 5E6, Canada
[email protected]
and
M. Zarichnyi
Department of Mechanics and Mathematics, Lviv State University,
Universitetska 1, 290602 Lviv, Ukraine
[email protected]
(Date:: )
Abstract.
It is proved that there is no structure of left
(right) cancelative semigroup on $[L]$-dimensional universal space
for the class of separable compact spaces of extensional
dimension $\leq[L]$. Besides, we note that the homeomorphism group of
$[L]$-dimensional space whose nonempty open sets are universal for
the class of separable compact spaces of extensional
dimension $\leq[L]$ is totally disconnected.
Key words and phrases:Extension dimension, universal space, semigroup
1991 Mathematics Subject Classification: 54H15, 54F45, 55M10
The first author was partially supported by NSERC research grant.
1. Preliminaries
Let $L$ be a CW-complex and $X$ a Tychonov space. The Kuratowski notation $X\tau L$ means that, for any continuous map
$f\colon A\to L$ defined on a closed subset $A$ of $X$, there
exists an extension $\bar{f}\colon X\to L$ onto X. This notation
allows us to define the preorder relation $\preceq$ onto the class
of CW-complexes: $L\preceq L^{\prime}$ iff, for every Tychonov space $X$,
$X\tau L$ implies $X\tau L^{\prime}$ [1].
The preorder relation $\preceq$ naturally generates the equivalence
relation $\sim$: $L\sim L^{\prime}$ iff $L\preceq L^{\prime}$ and $L^{\prime}\preceq L$. We
denote by $[L]$ the equivalence class of $L$.
The following notion is introduced by A. Dranishnikov (see, [5] and [4]). The
extension dimension of a Tychonov space $X$ is less than or
equal to $[L]$ (briefly, $\mathop{\mathrm{ext-dim}}(X)\leq[L]$) if $X\tau L$.
We say that a Tychonov space $Y$ is said to be a universal space for
the class of compact
metric spaces $X$ with $\mathop{\mathrm{ext-dim}}(X)\leq[L]$ if $Y$ contains a topological copy
of every compact
metric space $X$ with $\mathop{\mathrm{ext-dim}}(X)\leq[L]$. See [1] and [2] for
existence of universal spaces.
In what follows we will need the following statement which appears in [3] as Lemma 3.2.
Proposition 1.1.
Let $i_{0}=\min\{i:\pi_{i}(L)\neq 0\}$. Then $\mathop{\mathrm{ext-dim}}(S^{i_{0}})\leq[L]$.
2. Main theorem
Recall that a semigroup $S$ (whose operation is denoted as multiplication)
is called a left cancelation
semigroup if $xy=xz$ implies $y=z$ for every $x,y,z\in S$.
Theorem 2.1.
Let $L$ be a connected CW-complex and Let $Y$ be a universal space for the class of compact
metric spaces $X$ with $\mathop{\mathrm{ext-dim}}(X)\leq[L]$. If $\mathop{\mathrm{ext-dim}}(Y)=[L]$, then there
is no structure of left (right) cancelation semigroup on $Y$
compatible with its topology.
Proof.
Suppose the contrary and let $Y$ be a left cancelation semigroup.
Let $\alpha(\coprod_{j=1}^{\infty}S^{i_{0}}_{j})$ be the Alexandrov
compactification of the countable topological sum of copies $S^{i_{0}}_{j}$ of
the sphere
$S^{i_{0}}$, where $i_{0}=\min\{i:\pi_{i}(L)\neq 0\}$. By the countable sum
theorem for extension dimension and Proposition 1.1, $\mathop{\mathrm{ext-dim}}(\alpha(\coprod_{j=1}^{\infty}S^{i_{0}}_{j}))\leq[L]$ and, since $Y$ is universal,
$Y$ contains a copy of
$\alpha(\coprod_{j=1}^{\infty}S^{i_{0}}_{j})$. We will assume that
$\alpha(\coprod_{j=1}^{\infty}S^{i_{0}}_{j})\subset Y$. Besides, since
$\mathop{\mathrm{ext-dim}}(Y)\geq[S^{1}]$, we see that $Y$ contains an arc $J$. Let $a,b$ be
endpoints of $J$. There exists $j_{0}$ such that $aS^{i_{0}}_{j_{0}}\cap bS^{i_{0}}_{j_{0}}=\emptyset$. By Proposition 1.1, there exists
a map $f\colon aS^{i_{0}}_{j_{0}}\cup bS^{i_{0}}_{j_{0}}\to L$ such that
$f|aS^{i_{0}}_{j_{0}}$ is a constant map and $f|bS^{i_{0}}_{j_{0}}$ is not
null-homotopic. Extend map $f$ to a map $\bar{f}\colon Y\to L$. Let
$g\colon[0,1]\to J$ be a homeomorphism, then the map $F\colon S^{i_{0}}_{j_{0}}\times[0,1]\to L$, $\bar{f}(x,t)=g(t)x$, is a homotopy
that contradicts to the fact that $f|bS^{i_{0}}_{j_{0}}$ is not
null-homotopic.
∎
The homeomorphism group $\mathrm{Homeo}(X)$ of a space $X$ is
endowed with the compact-open topology.
Theorem 2.2.
Suppose $\mathop{\mathrm{ext-dim}}(X)=[L]$ and every nonempty open subset of $X$ is
universal for the class of separable metric spaces $X$ with
$\mathop{\mathrm{ext-dim}}(X)\leq[L]$. Then the homeomorphism group $\mathrm{Homeo}(X)$ is
totally disconnected.
Proof.
Suppose the contrary. Let $h\in\mathrm{Homeo}(X)$,
$h\neq\mathrm{id}_{X}$. There exists $x\in X$ such that $h(x)\neq x$
and, therefore, there exists a neighborhood $U$ of $x$ such that
$h(U)\cap U=\emptyset$. Since $U$ is universal for the class of
separable metric spaces $X$ with $\mathop{\mathrm{ext-dim}}(X)\leq[L]$, there exists an
embedding of $S^{i_{0}}$ into $U$, where $i_{0}$ is as in Proposition
1.1. We may suppose that $S^{i_{0}}\subset U$. There exists a
map $f\colon S^{i_{0}}\cup h(S^{i_{0}})\to L$ such that the restriction
$f|S^{i_{0}}$ is not null-homotopic while the restriction
$f|h(S^{i_{0}})$ is null-homotopic. Since $\mathop{\mathrm{ext-dim}}(X)\leq[L]$, there exists an
extension $\bar{f}\colon X\to L$ of the map $f$. The set
$$W=\{g\in\mathrm{Homeo}(X):\bar{f}|g(S^{i_{0}})\text{ is not null-homotopic }\}$$
is an open and
closed subset of $\mathrm{Homeo}(X)$. We see that $W$ is a neighborhood
of unity that does not contain $h$.
∎
3. Open problems
Note that the case $L=S^{n}$ corresponds to the case of covering dimension.
In this case, the topology of homeomorphism groups of some universal spaces has been
investigated by many authors (see the survey [6]).
In particular, it is known (see [8] and [7]) that the homeomorphism
group of the $n$-dimensional Menger compactum $M^{n}$ (note that $M^{n}$ satisfies the conditions of
Theorem 2.2 with $L=S^{n}$) is one-dimensional.
Let $[L]\geq[S^{1}]$ and $X$ be as in Theorem 2.2. Is
$\dim(\mathrm{Homeo}(X))\geq 1$?
Another version: Is there $X$ that satisfies the conditions of
Theorem 2.2 and such that $\dim(\mathrm{Homeo}(X))\geq 1$?
References
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A. Chigogidze, Cohomological dimension of Tychonov spaces. Topology Appl. 79 (1997), no. 3, 197–228.
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A. Chigogidze, V. Valov, Universal metric spaces and extension dimension.
Geometric topology: Dubrovnik
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A. Chigogidze, M. Zarichnyi, On absolute extensors modulo a complex. Topology Appl. 86 (1998), no. 2,
169–178.
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A. Dranishnikov, J. Dydak, Extension dimension and extension types. Tr. Mat. Inst. Steklova 212 (1996),
Otobrazh. i Razmer., 61–94; translation in Proc. Steklov Inst. Math. 1996, no. 1 (212), 55–88
[5]
A. N. Dranishnikov, The Eilenberg-Borsuk theorem for mappings in an arbitrary
complex. (Russian) Mat. Sb. 185
(1994), no. 4, 81–90.
[6]
Chigogidze, Alex; Kawamura, Kazuhiro; Tymchatyn, E. D. Menger manifolds. Continua (Cincinnati, OH, 1994),
37–88, Lecture Notes in Pure and Appl. Math., 170, Dekker, New York, 1995.
[7]
L. G. Oversteegen, E. D. Tymchatyn, On the dimension of certain totally disconnected spaces. Proc. Amer. Math.
Soc. 122 (1994), no. 3, 885–891.
[8]
B. L. Brechner, On the dimensions of certain spaces of homeomorphisms.
Trans. Amer. Math. Soc. 121 (1966)
516–548. |
A Panoramic 3D Vision System from Multiple RGBD Sensors
Hang Liu${}^{1}$, Hengyu Li${}^{1*}$, Xiahua Liu${}^{2}$, Jun Luo${}^{1}$, Shaorong Xie${}^{1}$, Yu Sun${}^{1,3}$
${}^{1}$Hang Liu, Hengyu Li, Jun Luo, Shaorong Xie with the School of Mechatronic Engineering and Automation, Shanghai University, China. Hengyu Li is the corresponding author. [email protected]@shu.edu.cn${}^{2}$Xiahua Liu with the Department of Engineering Mathematics, University of Bristol.${}^{3}$Yu Sun with the Department of Mechanical and Industrial Engineering, University of
Toronto, Canada, and the Shanghai University. [email protected] work was supported by the National Natural Science Foundation of China (grant numbers 61525305 and 61625304), the Shanghai Natural Science Foundation (grant numbers 17ZR1409700 and 18ZR1415300), and the basic research project of Shanghai Municipal Science and Technology Commission (grant number 16JC1400900).
Abstract
This letter presents a panoramic 3D vision system that was designed specifically for mapping the indoor environment. The system consists of multiple RGBD sensors that cover a 360${}^{\circ}$ field of view, simplifying the reconstruction of indoor environments. A new extrinsic calibration approach is proposed to estimate the relative poses between the RGBD sensors. This approach relies on well-matched keypoints from a descriptor-based pattern, in this case, of a minimally overlapping field of view between sensors. A distributed capturing system was also designed to provide panoramic RGBD frames in real time. Experiments validate the accuracy of the proposed calibration approach and the efficiency of the proposed panoramic 3D vision system for indoor mapping.
I INTRODUCTION
In recent years, indoor scene reconstruction and robot navigation have attracted much attention with the advent of low-cost and efficient depth and color (RGBD) devices such as the Microsoft Kinect, Intel RealSense, and Structure Sensor. The depth sensors of these devices can provide a depth map with a VGA resolution (640x480) at video-rate (e.g., 30 Hz) using efficient light-coding technology that avoids the challenging task of dense 3D reconstruction from color images. The 3D models reconstructed using these depth sensors have been used to generate more realistic 3D content for virtual reality (VR) and help align the rendered virtual objects with real scenes for augmented reality (AR). Furthermore, the direct depth sensing capability of these depth sensors are particularly suitable for helping robots manipulate and navigate in an unknown environment.
With an RGBD sensor, the simultaneous localization and mapping (SLAM)-based approach is mainly used for fusing the point cloud frames to reconstruct the indoor scenes [1, 2, 3]. However, hundreds or even thousands of frames must be captured in state-of-the-art SLAM systems to reconstruct a common indoor environment such as a room or an office [4] because of two problems:
1) The field of view (FOV) of depth sensors is limited; thus, only a small part of the scene is represented in a single frame. The Kinect, for example, has a horizontal FOV of 57${}^{\circ}$, which is much smaller than the horizontal 240${}^{\circ}$ FOV of the Hokuyo URG-04LX-UG01, a laser scanner with a similar range and measurement accuracy compared with the Kinect [5].
2) To track the poses of depth sensors to effectively fuse multiple point cloud frames, consecutive captured frames should be captured to have sufficient scene overlap. Typically, more than ninety percent of overlap is required, which further increases the number of frames for reconstruction.
A solution to the problem is to use a multicamera setup in which the cameras face different directions to sample different sections of the environment. A typical multicamera setup consists of multiple color cameras [6, 7] or RGBD cameras [8, 9].
In this letter, we focus on a panoramic 3D vision system that consists of multiple RGBD sensors for indoor mapping. A new extrinsic calibration method that relies on a descriptor-based pattern to provide well-matched keypoints is proposed to estimate the relative poses between the RGBD cameras with minimal overlapping FoV. A distributed capturing system based on local area network (LAN) is also proposed to provide multiple RGBD frames in real time. The experiments using real scenes validate these features of our study.
II RELATED WORK
Panoramic cameras cover a wider field of view and therefore are applied in mapping systems to provide better localization constraints that can improve the accuracy of structure and motion estimation. One type of panoramic camera is the central catadioptric camera [10], which uses a combination of lenses and mirrors placed in a carefully arranged configuration to capture a much wider FOV. This type of camera can capture 360${}^{\circ}$ with a single camera and therefore is widely used in SLAM systems [11, 12]. Unfortunately, this type of camera has a poor and varying spatial resolution, lowering the measurement accuracy of a single camera pixel [13].
Another approach is to use a multicamera panoramic camera system. Although there are some complications from using multiple cameras, such as additional image data and more difficult calibration, the main advantage of a multicamera system is that it enables the construction of a high resolution 360${}^{\circ}$ image. Studies using this type of multicamera systems appear in [14, 15].
SLAM systems that contain only color cameras can only compute sparse or semidense maps. However, applications such as augmented reality or obstacle avoidance for robots require a dense map of the environment. Thus, some range sensors are introduced in panoramic SLAM systems from laser range finders [11, 16] to stereo matching with triangulation [17] and structured light RGBD systems [18, 19]. Laser approaches are expensive and computing the depth from stereo image pairs is time-consuming. The structured light RGBD systems are low cost and have acceptable measuring accuracy in close ranges, which are suitable for direct 3D SLAM systems in indoor environments.
Complications can occur with the use of multiple cameras, such as more difficult calibration and additional image data. Fernandez-Moral et al. proposed to calibrate multiple RGBD sensors by finding and matching planes [8] or lines [20]. However, new algorithms for extracting and matching these planes or lines require redesign. The reason why classical extrinsic calibration strategies such as the chessboard-based method and the keypoints-based method cannot be applied is because the overlapping FoV between RGBD sensors cannot meet the overlap requirements of these methods. Chessboard-based methods require that chessboards be completely visible in the views of all cameras that must be calibrated. If the overlapping FoV is too small (e.g., thirty percent overlap), the overlapping image regions cannot provide correctly matched keypoints. Because the $N$ number of cameras in a panoramic vision system provide $N$ times the image data, it is also important to find effective ways to efficiently capture and transfer all the data.
In the following, we give the details of our calibration approach. Then, the multi-RGBD camera setup and a LAN-based distributed capturing system are presented in section IV. In addition, we analyze the precision of the proposed calibration approach and show the efficiency of indoor scene reconstruction using the proposed panoramic 3D vision system. Finally, the conclusions are summarized.
III EXTRINSIC CALIBRATION
In this section, we address the problem of estimating extrinsic calibration (i.e., relative poses) between RGBD sensors that have little overlapping FoV. Fig. 1 shows a setup of the proposed panoramic 3D vision system that consists of 12 Kinect v1 sensors. All Kinects are vertically positioned for a more compact design. The Kinect v1 sensor has an angular field of view (FoV) of 43${}^{\circ}$ from the vertical. The overlap FoV of two neighboring sensors is only approximately 30 percent of the vertical FoV of each sensor.
We propose to solve the calibration problem of multiple RGBD sensor with little overlapping FoV using a feature descriptor-based calibration pattern [21], which can provide robust and accurate matched feature points in this case of minimal overlapping FoV. Based on these matched 2D feature points and depth maps from the depth sensors, we construct two 3D point sets to estimate poses by bundle adjustment. Then, a pose graph optimization method is used to refine the estimated poses.
III-A Initial estimation of poses
The descriptor-based calibration pattern is composed of several noise images at different scales in accordance with the mechanism of SIFT/SURF. Compared with natural scenes, this pattern contains a high number of detectable features that can be easily detected by a camera at varying distances. Thus, the descriptor-based pattern can provide many and more accurately matched keypoints between two cameras in the case of minimal overlapping FoV.
In Fig. 2, the detected SURFT keypoints of such a pattern is represented by green circles. Fig. 2 shows the feature detection results of a nature scene. There are many more keypoints in Fig. 2. Fig. 2 and Fig. 2 show the matched keypoints of a pair of images captured by two cameras with an approximately thirty percent overlapping FoV. Well-matched keypoints are connected by green lines; poorly matched keypoints are connected by red lines. The matching results are generated by the Flann-based descriptor matcher implemented in OpenCV, where matched keypoints are accepted only if its descriptor distance is less than 2.0 times the minimum descriptor distances. Fig. 2 shows that the descriptor-based pattern provides more well-matched keypoints. In this work, the keypoints detector can only be applied to the overlapping regions of neighboring images to reduce the possibility of mismatch.
Then, based on these matched keypoints, we must find their corresponding depth values from the depth maps generated by the depth camera in the RGBD sensor to construct 3D point sets to estimate the poses. Due to the different spatial positions and intrinsic parameters of the IR camera of the depth sensor and of the color camera in the RGBD sensor, the depth map is not aligned with the color image. Because the depth and color cameras share enough overlapping FoV to detect the entire chessboard, we use the highly accurate chessboard calibration method to obtain the relative poses between the color and depth camera and their intrinsic parameters to align the depth map with the color image.
Let $[u,v]$ represent the coordinates of a keypoint, $Z$ represents its depth value in $[u,v]$ of corresponding aligned depth map, and $[X,Y,Z]^{T}$ represent the 3D point of $[u,v]$ in the color camera coordinate system. According to the pinhole camera model, the values of $X$ and $Y$ can be calculated as
$$\begin{array}[]{l}X{\rm{=}}(u-{u_{0}})Z/{f_{x}},\\
Y=(v-{v_{0}})Z/{f_{y}}.\end{array}$$
(1)
We obtain two 3D point sets $\{p_{i}\}$, $\{p^{\prime}_{i}\}$; $i=1,2,...,N$, based on $(\ref{eq1})$. $p_{i}$ and $p^{\prime}_{i}$ are $3\times 1$ column matrices. The relative poses between these two 3D point sets can be found by minimizing
$$\mathop{\min}\limits_{\xi}=\frac{1}{2}\sum\limits_{i=1}^{n}{\left\|{\left({{{p%
^{\prime}}_{i}}-\exp(\xi\hat{~{}}){p_{i}}}\right)}\right\|},$$
(2)
where $\xi\in se(3)$ is a vector with six dimensions that represents the camera pose, $\hat{~{}}$ maps $\xi$ to a matrix with four mentions ${R^{4\times 4}}$. $\xi\hat{~{}}$ is mapped to $T\in SE(3)$ by the exponential map
$exp()$. We use bundle adjustment to jointly solve all camera poses [22]. The derivative with respect to the camera pose of an error element is given by
$$-(\exp(\xi\hat{~{}})p)\hat{~{}}$$
(3)
where $\hat{~{}}$ denotes the mapping from ${\exp(\xi\hat{~{}})p}$ represented by $[x,y,z]^{T}$ to corresponding antisymmetric matrix
$\left[{\begin{array}[]{*{20}{c}}0&{-z}&y\\
z&0&{-x}\\
{-y}&x&0\end{array}}\right]$, the detailed derivation process can be found in the appendices.
III-B Pose Graph Optimization
Assume that there are $N$ RGBD sensors in the panoramic 3D vision system. Let $T_{i,j}$ denote the estimated relative poses between two adjacent RGBD sensors, $i=1,2,...,N$; $j=i+1$, when $i$ equals $N$, $j$ is 1. When the pose of the first RGBD sensor is set to $[0,0,0,1]^{T}$, we can calculate the pose $x_{i}$ for each RGBD sensor based on $T_{i,j}$. Due to the pose estimation error, the pose that is calculated by ${T_{N,1}}{x_{N}}$ is not equal to $[0,0,0,1]^{T}$. Fortunately, this panoramic setup of RGBD sensors provides a definite loop-closure constraint for optimizing the estimated poses. We can solve this problem by using the popular pose graph optimization method in SLAM [22].
According to pose graph optimization theory, the problem can be solved by finding the minimum of a function of this form:
$${x^{*}}=\arg{\kern 1.0pt}{\kern 1.0pt}\mathop{\min}\limits_{x}\sum\limits_{(i,%
j)\in S}{r_{i,j}^{T}}(x){\Lambda_{i,j}}r{}_{i,j}(x),$$
(4)
where $x={\left({x_{1}^{T},{\kern 1.0pt}...,x_{n}^{T}}\right)^{T}}$ is a vector of poses, $r{}_{i,j}$ is the residual of the predicted and observed relative poses between the $i-$th and $j-$th node, ${\Lambda_{i,j}}$ denotes the measurement information matrix, and $S$ represents the set of edges that connect the nodes.
IV EXPERIMENTS AND RESULTS
In this section, we evaluate the accuracy of the proposed extrinsic calibration method and show the efficiency of constructing indoor environments using the panoramic 3D vision system.
IV-A System Setup
We designed a camera rig that consists of twelve Kinect v1 RGBD sensors mounted in a radial configuration (see Fig. 1). We also designed a distributed system built on top of a local area network (LAN) to capture the RGBD frames from all twelve sensors in real time. The distributed capturing system (see Fig. 3) comprises twelve Raspberry Pi single board computers, a gigabit switch and a high-performance computer. Raspberry Pi is used to obtain the RGBD frames from each Kinect and send the data to the high-performance computer through the LAN using User Datagram Protocol. The high-performance computer is used to receive and process the RGBD frames. Both the depth and color frames are set to a size of $640\times 480$; the values of a single pixel in the depth frame and color frame have a size of 2 bytes and 3 bytes, respectively. Thus, the size of the RGBD frames from all twelve Kinects is 17.58 million bytes, which can be sent to the high-performance computer through the gigabit switch at 7 fps. This frame rate can be increased by decreasing the size of RGBD frames from Kinect.
IV-B Precision of Extrinsic Calibration
To evaluate the precision of the proposed extrinsic calibration method, we use a motion capture system to obtain the ground truth relative poses between the sensors. The motion capture system requires at least three reflective markers to track the pose of a rigid body, such as the Kinect and the chessboard in our experiments. We attached four reflective markers to both the Kinect and the chessboard (see Fig. 4). We placed the four markers on the outer corners of the chessboard, such that the relative poses between the chessboard and the motion capture system and the Kinect color camera are known. The motion capture system can track the poses of the markers that are attached to the Kinect to determine the poses of the Kinect color camera.
For convenience, we only attached markers to one Kinect and placed this Kinect on twelve different positions in the camera rig to capture RGD frames. The precision of the extrinsic calibration is evaluated using twelve pairs of RGBD frames from the captured twelve RGBD frames.
While determining the keypoints correspondences for each pair of RGBD frames, we increase the distance threshold of the keypoints (denoted by $dist\_thresh$) from one and a half times the minimum distance to the maximum distance (ten times the minimum distance in average) to analyze the calibration error in rotation and translation with respect to the keypoints correspondences.
In TABLE I, the average residual error in rotation and translation is presented for the initial estimation (denoted by Ini.) and optimized estimation (denoted by Opti.) using pose graph optimization with loop closure. It can be seen that the average residual error is reduced when raising the distance threshold to consider more keypoint correspondences. The residual error in both the rotation and translation for the optimized estimation is generally less than the residual error for the initial estimation. When the distance threshold is set to be larger than four times the minimum distance, the residual error remains at the same level. The aligned point clouds before and after pose graph optimization with loop closure can also be observed in Fig. 5. It can be seen in the red circle on the bottom of Fig. 5 that the point clouds are misaligned, however, these point clouds align well in Fig. 5 with optimized poses with loop closure constraint.
The operating range of the Kinect v1 depth sensor is between 0.5 m to 5.0 m [23]. Using an output panoramic frame can reconstruct scenes within the circle with a radius of 5 meters. Thus the reconstruction of some indoor scenes becomes very efficient using the proposed panoramic 3D vision system. Fig. 6 presents the reconstruction result of a bedroom (3.3 m $\times$ 3 m) and a living room (9 m $\times$ 3.5 m) using only one output panoramic frame from the proposed system.
V CONCLUSION
In this letter, we present a panoramic 3D vision system from multiple RGBD sensors. A new methodology that relies on well-matched keypoints provided by a feature descriptor-based calibration pattern has been proposed to calibrate the extrinsic parameters of the RGBD sensors in the system. We have also designed a LAN-based distributed system that enables the system to provide panoramic RGBD frames in real time. The reconstruction of indoor scenes is efficiently and conveniently performed using this panoramic 3D vision system considering the time consumption and captured RGBD frames. All the experiments have validated the accuracy of the proposed calibration method and the efficiency in reconstruction.
Directions for future work include research on integrating the proposed panoramic 3D vision system with state-of-the art SLAM systems for localization and mapping at large scales. The proposed system provides a 360${}^{\circ}$ FoV, leading to better constraints for localization and can be potentially used to reduce localization and mapping errors. It is particularly important to investigate direct registration methods such as [24, 25], which do not depend on time-consuming keypoint detectors or descriptors for large scale SLAM, using this panoramic 3D vision system.
APPENDIX
The detailed derivation process of the derivative with respect to the camera pose of a error element is as follows,
$$\begin{array}[]{l}\frac{{\partial(\exp(\xi\hat{~{}})p)}}{{\partial\xi}}=%
\mathop{\lim}\limits_{\delta\xi\to 0}\frac{{\exp(\delta\xi\hat{~{}})\exp(\xi%
\hat{~{}})p-\exp(\xi\hat{~{}})p}}{{\delta\xi}}\\
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}%
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}%
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}\approx%
\mathop{\lim}\limits_{\delta\xi\to 0}\frac{{(I+\delta\xi\hat{~{}})\exp(\xi\hat%
{~{}})p-\exp(\xi\hat{~{}})p}}{{\delta\xi}}\\
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}%
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}%
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}=\mathop{\lim%
}\limits_{\delta\xi\to 0}\frac{{\delta\xi\hat{~{}}\exp(\xi\hat{~{}})p}}{{%
\delta\xi}}=\mathop{\lim}\limits_{\delta\xi\to 0}\frac{{-(\exp(\xi\hat{~{}})p)%
\hat{~{}}\delta\xi}}{{\delta\xi}}\\
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}%
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}%
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}=-(\exp(\xi%
\hat{~{}})p)\hat{~{}}\end{array}$$
(5)
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Spin-selective optical absorption of singly charged excitons in a quantum dot
Alexander Högele
Martin Kroner
Stefan Seidl
Khaled Karrai
Center for NanoScience, Department für Physik, Ludwig-Maximilians-Universität,
Geschwister-Scholl-Platz 1, 80539 München, Germany
Mete Atatüre
Jan Dreiser
Atac Imamoğlu
Institute of Quantum Electronics, ETH Hönggerberg HPT G12, CH-8093 Zürich, Switzerland
Richard J. Warburton
School of Engineering and Physical Sciences,
Heriot-Watt University, Edinburgh EH14 4AS, UK
Antonio Badolato
Brian D. Gerardot
Pierre M. Petroff
Materials Department, University of California, Santa Barbara, California 93106, USA
(October 14, 2004)
Abstract
We report high resolution laser absorption spectroscopy of a
single InGaAs/GaAs self-assembled quantum dot embedded in a
field-effect structure. We show experimentally that the interband
optical absorption to the lower Zeeman branch of the singly
charged exciton is strongly inhibited due to spin (Pauli) blockade
of the optical transition. At high magnetic fields the optical
absorption to the upper Zeeman branch dominates the absorption
spectrum. We find however that the spin blockade is not complete
and a 10% leakage remains at high magnetic fields. Applying a
gate voltage to empty the dot of its resident electron turns the
spin blockade off. This effect is observed at 1.5 K and up to
9 Tesla.
The coherence time of an excess electron spin strongly confined in
a quantum dot (QD) structure is expected to be orders of magnitude
longer than the typical timescales required for its coherent
manipulation Golovach ; Hanson ; Kroutvar ; Elzerman . Motivated
by this observation, several groups have proposed to use single QD
spins as quantum bits (qubits) Loss , and to manipulate,
couple and measure individual spins using either transport
Elzerman or optical techniques Imamoglu . In the case
of self-assembled InGaAs QDs with strong confinement along the
growth direction, the lowest energy optical transitions are those
arising from the excitation of a $J_{z}$=$+\frac{3}{2}$
$(J_{z}$=$-\frac{3}{2})$ valence electron to a
$S_{z}$=$+\frac{1}{2}$ $(S_{z}$=$-\frac{1}{2})$ conduction state.
If the QD already has an excess conduction electron, only one of
these optical transitions is allowed; the other is spin (Pauli)
blocked Warburton1 . In contrast, a neutral QD with
asymmetric confinement potential always has a pair of exciton
transitions. It has been suggested that Pauli blocking of
absorption or fluorescence can be used to implement high
efficiency all-optical single-spin measurements
Imamoglu ; Calarco and conditional spin dynamics
Calarco .
In this Letter, we report resonant absorption measurements on a
single QD charged with a single excess electron in the regime of
Pauli blocking. We observe that for high magnetic fields
(B$>$5 Tesla) where the electron is (with high probability) in the
lowest energy $S_{z}$=$+\frac{1}{2}$ state, the absorption of a
left-hand-circularly polarized ($\sigma^{-}$) laser is suppressed
by a factor of 10 as compared to that of a right-hand-circularly
polarized ($\sigma^{+}$) laser. In contrast, we observe that both
$\sigma^{+}$ and $\sigma^{-}$ transmission dips have equal
strength for a neutral QD, irrespective of the applied magnetic
field. Using the fact that the strength of $\sigma^{+}$ and
$\sigma^{-}$ absorption in a charged QD is a measure of the spin
polarization of the resident electron, we were able to determine
the electron g-factor by fitting the magnetic field dependence of
the transmission data. Our results represent a first step towards
all-optical coherent spin measurement.
The InGaAs dots investigated in this work were self-assembled
in the Stranski-Krastanow growth mode by molecular beam
epitaxy. The QDs are separated from a highly n-doped GaAs back contact by 25 nm of intrinsic GaAs which acts as a tunnel barrier. The electrons are prevented from tunneling to the gate
electrode, the metalized sample surface, by a 110 nm thick
GaAs/AlAs superlattice blocking barrier. The whole structure forms
a field-effect device. The electron occupation of the dots is controlled by
applying a gate voltage and monitored with the QD photoluminescence charging diagrams
Warburton2 . An individual QD is identified to be spectrally
separated by typically more than 5 meV from neighboring dots assuring that the observed features originate
unambiguously from a single QD. We used high resolution laser
spectroscopy and a low temperature confocal microscope to measure
the differential transmission through the QDs. This technique Alen ; PhysE ; PRL gives optical spectra as shown in
Fig. 1. The spectral resolution is much less than the QD
absorption width. A dip in the transmission is obtained when the resonantly scattered laser light is diverted away from the photodetector placed immediately behind the sample.
In the absence of a magnetic field, a negatively charged exciton
(X${}^{1-}$) strongly confined in a QD shows a single unpolarized
absorption peak in the optical interband spectrum, as demonstrated
in Fig. 1 with high resolution transmission spectroscopy.
In the case of X${}^{1-}$, electrons are forced to fill the lowest
energy $s$-shell such that the total electron spin is 0 in
accordance with the Pauli principle. As a consequence, the
exchange interaction between the confined electrons and the hole
is suppressed so that the optical spectrum is free of fine
structure features irrespective of the asymmetry of the confining
potential PRL ; Bayer1 . This is not the case for the neutral
exciton X${}^{0}$: the electron-hole exchange leads generally to a
splitting into two linearly polarized dipole-allowed optical
transitions PRL ; Gammon ; Bayer2 . For the dot investigated
here the measured neutral exciton splitting was 10 $\mu$eV.
In a high magnetic field and at low enough temperatures, the
ground state of a singly charged QD is that of a spin-polarized
electron in the lowest Zeeman level. As shown in Fig. 2a,
an interband photon can only be absorbed under the condition that
the photo-generated electron is of opposite spin. In principle, at
T = 0 K this translates into a single absorption line in the
optical spectrum irrespective of the applied magnetic field. In
spite of the applied magnetic field, the exciton absorption should
not show a Zeeman splitting, i.e. the optical transition at the
lower exciton Zeeman energy should be spin blocked due to the
Pauli principle independently of the photon polarization.
We have investigated Pauli blocking experimentally.
Fig. 2b shows transmission spectra in the presence of a
magnetic field. At T = 1.5 K and under magnetic field two
well-resolved transmission resonances are observed. Both lines
correspond to the two optical transitions shown in
Fig. 2a. The energy positions of the resonances are
plotted in Fig. 3a showing clearly a behavior quadratic
in field, consistent with the exciton diamagnetic shift
Schulhauser . The energy splitting, plotted in
Fig. 3b, is proportional to the applied magnetic field
consistent with a Zeeman splitting of the charged exciton. We have
conducted the transmission measurements as a function of the light
polarization. At finite magnetic fields the two dipole-allowed
transitions have right and left circular polarizations.
Fig. 2b, c and d show the corresponding spectra using
linearly polarized light. At low temperatures, the strength of the
low energy resonance decreases with increasing magnetic field
while the opposite is true for the higher energy resonance.
Fig. 4a shows the relative weight of the resonances as a
function of magnetic field. For each magnetic field the
polarization was optimized to maximize independently each of the
two resonances. At high magnetic field it can clearly be seen that
the absorption of the higher energy branch dominates. The low
energy branch is dramatically inhibited, saturating at high field
at a level of one tenth of the stronger peak, without ever
completely disappearing (Fig. 4b). The high magnetic
field limit confirms roughly the expected Pauli blocking behavior
in that one resonance clearly prevails.
A further confirmation of the Pauli blockade picture is obtained
by increasing the temperature to T = 12 K. In this case, the
spectra obtained at B = 4 T show two Zeeman-split absorptions with
comparable peak strengths (Fig.s 2c and 2d). The
interpretation is that the resident electron is thermally
activated and occupies both spin states with about equal
probability. Lowering the temperature to 1.5 K clearly favors the
higher Zeeman branch confirming that the spin polarization of the
resident electron is the source of the Pauli blocking.
At T = 1.5 K the degree of spin polarization depends on the
magnetic field. We have used a two-level Boltzmann statistical
distribution for the spin state occupation. Considering that the
relative strength of both resonance peaks is a measure of the spin
polarization of the resident electron, we fitted the evolution of
the data in magnetic field, as shown in Fig. 4a. As a
fit parameter we used a Landé factor for the resident electron
of $\mid$g${}_{e}$$\mid$=1.8$\pm$0.2. This value is about twice as
large as the one reported for InGaAs/GaAs self-assembled QDs in
Ref. Bayer2 . However, several other dots emitting around 1.3 eV close to the center of the inhomogeneous QD energy distribution exhibit an electron g-factor of 0.6 as measured at 4.2 K. The deviation of the g-factor from dot to dot is not surprising since the value strongly depends on the individual In content and strain distribution. To obtain an optimal fit, we needed to
introduce empirically a limit to the maximum achievable electron
spin polarization of about 90% in order to account for the fact
that the lower energy absorption branch does not vanish at high
fields, as shown in Fig. 4b. This result indicates that
the Pauli blockade of the optical transition is not complete as
ideally predicted. It was also observed for the dots with the electron g-factor of 0.6 consistent with the temperature and the Zeeman energy. At the present time, the precise origin for the
apparent limitation of the Pauli blocking is still unknown and
requires more detailed measurements and analysis.
In conclusion, using high-resolution transmission spectroscopy we
have confirmed that a singly negatively charged exciton confined
in a single self-assembled QD has a dominant absorption line at
the higher Zeeman energy branch as a result of the spin
polarization of the resident electron. Unexpectedly, at high
fields the absorption at the lower Zeeman branch, although
drastically inhibited as expected from Pauli blocking, is still
not completely extinguished.
We would like to thank A. O. Govorov for helpful discussions.
Financial support for this work was provided in Germany by the DFG
grant no. SFB 631, in Switzerland by NCCR Quantum Photonics grant
no. 6993.1 and in the UK by the EPSRC.
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Approach for modelling quantum-mechanical collapse
A. Yu. Ignatiev
[email protected]
Theoretical Physics Research Institute,
Melbourne 3163,
Australia
Abstract
A long-standing quantum-mechanical puzzle is whether the collapse of the wave function is a real physical process or simply an epiphenomenon. This puzzle lies at the heart of the measurement problem.
One way to choose between the alternatives is to assume that one or the other is correct and attempt to draw physical, observable consequences which then could be empirically verified or ruled out.
As a working hypothesis, we propose simple models of collapse as a real physical process for direct binary symmetric measurements made on one particle.
This allows one to construct irreversible unstable
Schrödinger equations capable of describing
continuously the process of collapse induced by the interaction of the quantum system with the measuring device. Due to unknown initial conditions the collapse outcome remains unpredictable so no contradictions with quantum mechanics arise.
Our theoretical framework predicts a finite time-scale of the collapse and links with experiment.
Sensitive probes of the
collapse dynamics could be done using Bose-Einstein condensates,
ultracold neutrons or ultrafast optics.
If confirmed, the formulation could be relevant to the transition from quantum fluctuations to classical inhomogeneities in early cosmology and to establishing the ultimate limits on the speed of quantum computation and information processing.
pacs: 03.65.Ta, 03.65.Sq, 03.75.Nt, 78.47.J-
I Introduction
Modern technology has turned the in-depth studies of quantum mechanics into a fast-growing research area.
The physical meaning and the evolution of quantum state are among the focal points in these studies l ; vN ; pbr .
The wave function collapse (also known as the reduction of the wave packet) is
an irreversible, unpredictable change of quantum state which occurs during the measurement and which cannot be described by the reversible Schrödinger
equation. The nature and physical mechanisms of collapse are among the most debatable issues of quantum mechanics, and our understanding of this mysterious process remains rather limited.
as the Many-Worlds interpretation of quantum mechanics.
Collapse has been discussed from numerous, often very different perspectives, starting from purely philosophical and ending with purely mathematical. Philosophical discussions focus on various interpretations of quantum mechanics, and point out that in some of them, collapse is not a physical process but rather a (trivial) mental process of updating the available information.
This approach is often illustrated with the example of picking balls from a box with one black and one white ball. Initially, the probability to pick the white ball is 1/2. But after the first picked ball turns out black, then the probability to pick up the white ball becomes 1. In this case, which is completely classical, even the term ‘collapse’ itself seems unnecessary. Indeed, before the advent of quantum mechanics, no-one was talking about collapses in the theory of probability.
Such a viewpoint emerged when the options for quantum experimenting were limited by the level of technology. Since then, completely new kinds of experiments became a reality, and it is now possible to consider the collapse problem afresh.
Besides, the quantum collapse without observers has become a vital issue in cosmology where a physical mechanism is needed for turning quantum fluctuations into classical which would then evolve into galaxies.
As a consequence, the modern approach consists in translating philosophical questions into physical. In this vein, known as “experimental metaphysics”, we do not postulate ‘absence of collapse’ or otherwise, but rather ask how we could find out if there is a collapse or not using ordinary physical methods and principles.
The problem of the collapse under measurement is intimately related (but not identical) to the problem of quantum-to-classical transition or the transition from possible to actual.
The theory of environmental decoherence has made impressive progress in this area over recent decades. Optimistically, a view was advocated by some authors that decoherence has solved the measurement problem and the concept of collapse has become superfluous.
However, this issue is far from being settled (see, e.g., adler and references therein).
The conventional quantum mechanics models the transition from quantum to classical as a black box.
The decoherence theory has shone some light on that box, but do we know what is inside? What happens at the final, most intriguing stage of the measurement process? Only 15 percent of experts believe that the decoherence program has solved the measurement problem SKZ .
For this reason, it seems warranted to consider first the case of ‘pure’ collapse, i.e., neglect the environmental decoherence altogether. This approach can be justified in the end by comparing the two characteristic time-scales: for collapse and for decoherence. If the former is much shorter than the latter, then decoherence indeed can be neglected. This case is also more interesting from a practical point of view as in quantum computing decoherence needs to be reduced as much as possible.
II Non-linear effective equations for collapse
The problem of collapse during the measurement is essentially a many body problem (or even an infinitely many degrees of freedom problem). For this reason, exact equations governing the collapse process are hard to obtain, and even if obtained, would be difficult to use.
In such cases, a common method to proceed is to eliminate the unwanted degrees of freedom by applying the phenomenological, effective theory approach. The essence of this approach is to ‘integrate out’ the unknown quantities. This means introducing new terms built out of the known quantities subject to certain symmetry requirements and multiplying them by unknown coefficients. This has become a standard practice of quantum field theory over the recent decades but is less appreciated in the context of quantum mechanics.
When introducing new terms, our strategy is coservative, and we will only rely on the most general physical criteria: instability, irreversibility, and symmetry breaking. The first two features have been advocated already by Niels Bohr, while the third one is among the most universal phenomena in contemporary physics.
We start with the familiar setting where collapse is supposed to occur: a quantum system with a wave function plus a measuring apparatus.
Following the spirit of the effective-theory, phenomenological type of approach,
we set ourselves the following task: Take the Schrödinger equation and add new terms which are made entirely out of the wave function and unknown coefficients. The new equations should be able to describe both ordinary evolution and the measurement process (i.e., the collapse of the wave function).
This may sound as an extraordinary proposal which is bound to fail. Indeed, it seems that nothing could be so different as the continuous, deterministic evolution and the sudden, unpredictable collapse. Further, it seems unavoidable that some random, stochastic sources, or an additional physical phenomena should necessarily be introduced to account for the collapse as was done, for instance, in Refs.grw ; p89 ; w11 ; pen ; dio .
gravity-induced collapse ideas of Penrose pen and Diósi dio .
Nevertheless, we require from the outset that no new stochastic sources or gravitation should be invoked in explaining the collapse 111To avoid confusion, we emphasize that the Continuous Spontaneous Localization (CSL) theory of Ghirardi-
Rimini-Weber and Pearle grw ; p89 ; w11 and the gravity-induced collapse models, for instance, those of Penrose pen and Diósi dio are in principle different from our proposal because they address an essentially different problem: collapse during a free evolution and not during a measurement. So it would be meaningless to ask which theory is better. More details are in Sec. V..
The main difficulty in this search is the fact that quantum and classical systems essentially speak different languages and it is hard to make them talk to each other.
The quantum language is based on operators while the classical one—on ordinary functions.
To get around this obstacle, we can try to rewrite quantum mechanics in a more classical form, and thus to obtain a zeroth approximation suitable for our effective theory.
The most convenient for us is the approach known as the complex quantum Hamilton-Jacobi (CQHJ) formulation lp ; john ; yang ; gdt ; poi ; chou which has attracted growing attention recently.
It is natural to expect that the new formulation could give us new insights into fundamental long-standing issues such as the mechanism of quantum collapse.
For brevity and simplicity, we focus first
on the simplest yet typical case of the wave function collapse. This case is realised in a one-dimensional Ôsymmetric binary measurementÕ of the momentum where only two outcomes are possible and the
initial wave function is symmetric with respect to these
outcomes.
It can be shown (see Refs.lp ; john ; yang ; gdt ; poi ; chou ; ign and Appendix) that
the Schrödinger equation in this case becomes
$$p_{t}=-\nabla H,$$
(1)
where
$$p(x,t)=\frac{\hbar}{i}\frac{\psi^{\prime}}{\psi},\;\;\;H=V+\frac{p^{2}}{2m}+%
\frac{\hat{p}p}{2m}.$$
(2)
The parity-symmetric wave function before the collapse is the 1:1 superposition of two plane waves with opposite directions:
$$\psi=\frac{1}{\sqrt{2}}(e^{ikx}+e^{-ikx}).$$
(3)
We need to construct the phenomenological equation that would collapse this wave function into either $\psi_{+}=e^{ikx}\;\;\;{\rm or}\;\;\;\psi_{-}=e^{-ikx}$.
Following Bohr, we assume that the collapse is caused by the interaction of the quantum system with a semi-classical measuring device. Usually, the interaction is described by the Hamiltonian (or potential) that does not depend on the quantum state. However, such interactions cannot cause the collapse and should be extended to include state-dependent interactions. The phenomenological state-dependent potentials (or their analogues) are familiar in many areas of quantum theory (for instance, equations of Ginzburg-Landau, Gross-Pitaevsky, quantum chemistry etc.).
A state-dependent interaction is not a fundamental
interaction but an approximate one that is obtained, e.g., as a result
of mean-field approximation which assumes a definite classical value for a mean field. So the proposed approach does not attempt to derive the definite outcomes from a fundamental theory.
The state-dependent potential $V_{S}$ should be added to the ordinary potential $V$ in Eq. (22) to obtain:
$$p_{t}=-\nabla(H+V_{S}).$$
(4)
The gradient of the potential $V_{S}$ is a more convenient quantity than the potential itself, and we introduce a special term for it. It is natural to refer to this gradient as some kind of generalized force; in the context of our problem, we will call it “the collapsing force”:
$$F_{c}=-\nabla V_{S}.$$
(5)
The fact that the potential $V_{S}$ should be state-dependent can be equivalently stated as requirement that the collapsing force $F_{c}$ should be $p$-dependent.
Due to the shortness of collapse, the interaction with the apparatus is expected to be dominant during the measurement process, so our Eq. (4) can be written in a simpler form
$$p_{t}=F_{c}.$$
(6)
To find the form of $F_{c}$, we invoke the general principles as formulated in the beginning:
(1) Symmetry;
(2) Irreversibility;
(3) Instability.
It is natural to start with the simplest kind of functions, such as low-degree polynomials (which can also be viewed as the first terms of a Taylor expansion):
$$F_{c}=c_{0}+c_{1}p+c_{2}p^{2}+c_{3}p^{3}+\dots$$
(7)
According to Eq. (6), the force must be a vector under parity symmetry ($p\rightarrow-p$) as expected.
Therefore, the even-power terms in this expansion, $c_{0}$ and $c_{2}p^{2}$, should be dropped as violating this symmetry.
The requirement that the measurement gives two outcomes, $p=+q$ or $p=-q$, translates into the condition that $F_{c}$ must vanish at $p=\pm q$:
$$F_{c}(+q)=F_{c}(-q)=0.$$
(8)
Therefore, we must have $c_{1}=-c_{3}q^{2}$, and introducing $g=-c_{3}$ we obtain
$$F_{c}=gp(q^{2}-p^{2}).$$
(9)
Consequently, $g$ is interpreted as the effective coupling constant controlling the strength of interaction between the system and apparatus.
In general, $g$ could be a complex number, but in order to satisfy the condition of irreversibility we assume that it is real.
Altogether, our general equation (6) takes on a ‘collapsible’ form
$$p_{t}=gp(q^{2}-p^{2}).$$
(10)
In addition to Eq. (5), the collapsing force can be also written in the ‘potential’ form in ‘momentum space’ with the Higgs-type ‘collapsing potential’:
$F_{c}=-\partial V_{c}/\partial p,\;\;V_{c}=(g/2)(q^{2}-p^{2})^{2}$ (Fig.1).
This illustrates the remarkable link between our model and the phenomena of symmetry breaking, phase transitions, and synergetics. The collapsing potential can be compared to the ‘mean field’ and the measured quantity—to the ‘order parameter’.
Solving Eq. (10) with the initial condition (3),
yields
$$\displaystyle p(x,t)=\frac{qp_{0}(x)B(t)}{\sqrt{p_{0}^{2}(x)[B^{2}(t)-1]+q^{2}%
}},$$
(11)
$$\displaystyle\;\;B(t)=\exp{(gq^{2}t)},\;\;\;p_{0}(x)=iq\tan{kx}.$$
(12)
Finally, using the definition (2) we can determine the continuous behavior of the wave function all the way through the measuring process—in other words, the process of the wave function collapse:
$$\psi(x,t)=\frac{B(t)\cos{kx}+\sqrt{1-B^{2}(t)\sin^{2}{kx}}}{B(t)+1}N(t),$$
(13)
where $N(t)$ is the normalization factor.
In particular, we obtain the following formula for the characteristic timescale $\tau_{c}$ of the collapse:
$\tau_{c}\sim 1/(gq^{2})$.
Asymptotically, when $B(t)\gg 1$, the wave function (13) can take one of the two forms (in each domain of length $\pi/2k$):
$$\psi_{+}=e^{ikx}\;\;\;{\rm or}\;\;\;\psi_{-}=e^{-ikx}.$$
(14)
This two-valuedness arises due to the two-valuedness of the square roots in Eqs. (11,13).
An interesting question now is: How does the wave function ‘know’ which way to collapse: $\psi_{+}$ or $\psi_{-}$?
The answer is that if the initial wave function has the exact form (3), then it does not know, i.e., the two-valued ambiguity in (14) cannot be resolved.
This is because the initial $p_{0}$ is purely imaginary. If, on the other hand, $p_{0}$ has a non-zero real part $Re\;p_{0}$, then the definite outcome arises: In case of positive real part, $Re\;p_{0}>0$, the asymptotic wave function is $\psi_{+}=e^{ikx}$; if the real part is negative, $Re\;p_{0}<0$, the asymptotic wave function is $\psi_{-}=e^{-ikx}$.
Thus the measurement outcome is completely determined by the sign of the real part of $p_{0}$.
It is natural to presume that $Re\;p_{0}$ can take up positive and negative values with equal probability as required by the Born rule.
We emphasize that only infinitesimal non-zero values of $Re\;p_{0}$ are required for the emergence of a definite outcome in the scheme based on the simple equation (10).
A question can be raised whether $Re\;p_{0}$ can be considered a hidden variable, the dependence on which of the outcome is a step function.
Using a step function to represent the outcome dependence on $Re\;p_{0}$ leads to the following problems:
If the dependence of the outcome on $Re\;p_{0}$ is represented by a step function, the domain of this function should be infinitesimally small. If finite values of $Re\;p_{0}$ are allowed to be included in that domain, then these values would be incompatible with the state described by the wave function (3). But if we consider other states compatible with finite values of $Re\;p_{0}$, then such a step function would not be a correct predictor of the outcome.
Therefore, the outcome dependence on $Re\;p_{0}$ has a more singular character than the step function. By the same argument, the outcome dependence on
$Re\;p_{0}$ cannot be represented by any function with finite domain. Also, it cannot be represented by a discrete binary variable $\lambda$ taking on values $\pm 1$ because in that case the dependence of this ‘stand alone’ variable on $Re\;p_{0}$ would be lost.
For these reasons we prefer not to call $Re\;p_{0}$ a hidden variable.
The real issue, though, is not so much in the name but in a possible contradiction with various no-go theorems which have been formulated in relation to theories with hidden variables of various sorts. These issues are discussed in Sec. V.F where it is shown that no contradictions arise.
III Interpretation
A few remarks are in order concerning the interpretation of our ‘collapsible equation’, Eq. (10).
(a) It is not supposed to be a fundamental equation describing exactly the collapse process in every detail. Rather, it is a phenomenological, effective approximation which is meant to capture the main features of the collapse: instability, irreversibility, and symmetry breaking, which could lead to observable consequences such as the finite time-scale of the collapse process.
In any case, the construction of more elaborate models of the wave function collapse seems to require more experimental data than is available at the moment.
(b) In contrast with the Bohm-Bub models of explicit collapse bb , Eq. (10) does not contain hidden variables.
There is a large number of works that replace the Schrödinger equation with a non-linear one (see, e.g., bbm ; wn ). In contrast, our collapsible equations complement the Schrödinger equation. They have a different mathematical structure and distinct observational consequences.
(c) The “superluminality” argument which is often invoked to limit the use of non-linear equations in quantum mechanics does not apply in our case for the reasons discussed in Sec. V.
IV Experimental consequences
As with any phenomenological equation, care is required in comparing its predictions with experiment. Although our derivation allows one to establish some necessary conditions of its validity, these conditions may not be sufficient, and only experiment can decide on this issue.
First of all, a certain balance between the characteristic times involved should be observed. Ideally, one would aim at the interaction-switching time much less than the collapse time, and the collapse time much less than the ‘energy’ time-scale $\hbar/\tau_{E}$ where $E$ is the characteristic energy scale involved in a particular experiment.
One scheme of an experiment could look as follows. The initial state is created, then the measurement interaction is switched on at the instant $t=0$, then the measurement result is read-off. In the conventional picture, the interval between the second and third events can be as small as possible (it is bounded from below by the resolution time of the detector). However, in our picture, the collapse time cannot be less than $\tau_{c}$. Therefore, if the time-scale $\tau_{c}$ exceeds the resolution time-scale, the theory can be tested.
To compare the sensitivities of different kinds of experiments, it is convenient to introduce the dimensionless collapse time $\kappa=\tau_{c}/\tau_{E}$.
Except for the pioneering work of Papaliolios pap back in 1967, no dedicated searches for the finite time of the collapse have been performed (to my knowledge). So we attempt to obtain rough estimates on the allowed values of $\tau_{c}$ and $\kappa$ from various other experimental results assuming that null effects were observed in them on a time scale of $\tau_{m}$ which is the shortest time scale probed in a specific experiment (Table I).
It is natural to start with the experiments gkz ; bhi that tested specific non-linear versions of the Schrödinger equation bbm ; wn ; mdo . However, they turn out not to be the most sensitive for our purposes.
The well-known observation of ‘quantum jumps’ nsd is another natural place to look for constraints, but again they are found to be rather weak.
Next, we turn to the ultrafast measurements using femtosecond pulses of laser light blr .
Here, the limits begin to improve.
Finally, an excellent source of constraints appears to be experiments with Bose-Einstein condensates.
The
observed dynamics
turned out surprising in many respects, and it would be interesting to see if the idea of a finite-time collapse could be helpful there.
This area is virtually unexplored, and there is not one, but many promising directions of attack. Consequently, without attempting to be exhaustive, we restrict ourselves to outlining just two general strategies.
One approach is to look for the longest characteristic times $\tau_{E}$, comparable or better than those achieved in the BEC experiments. For example, the distances between the energy levels of ultra-cold neutrons bouncing from a horizontal mirror in the earth gravity field are of the order of $10^{-12}$ eV (or peV) dur which corresponds to $\tau_{E}\sim 10^{-3}$ s. Because times, such as the time of flight, can be measured with much better accuracy, this type of experiment (suggested for other purposes) could be a useful starting point for a test proposal.
Alternative strategy would be to start from the experiments with the shortest possible times $\tau_{m}$, such as those involving ultrashort pulses. This is a rapidly evolving area progressing from the femtosecond scale to atto-, zepto-, and even yoctoseconds ($10^{-24}$ s).
We emphasize that if the theory is confirmed that would not mean at all that the conventional quantum mechanics is refuted. It would only mean that the approximation of the instantaneous collapse is too crude and should be replaced by a more realistic approximation of a finite time-collapse. As was mentioned in the introduction, the assumption of instantaneous collapse is just an approximation rather than a fundamental principle of quantum theory, and the validity of this approximation does not affect the validity of quantum theory as a whole.
Similarly, if our specific Eq. (10) is experimentally excluded, that would not mean that the whole approach is invalidated, because a variety of different equations could be built on the same conceptual basis which could lead to different experimental predictions.
V Discussion
In constructing the collapse-enabled Schrödinger equations, our method is based on three well-established principles:
1. Symmetry breaking. The binary symmetric measurements on one particle in the absence of decoherence are the fundamental prototype measurements that should be studied first, both theoretically and experimentally.
Furthermore, the collapsible equations for such measurements should be constructed phenomenologically using the minimal set of most general assumptions.
Specifically, we single out the following two conditions that are widely accepted as reliable, non-controversial characteristics of the collapse process:
2. Irreversibility. This is generally acknowledged as the most important, essential feature of the collapse.
3. Instability (or amplification). The key role of these principles in the theory of measurement was succinctly formulated by Bohr as follows:“…every atomic phenomenon is closed in the sense that its observation is based on registrations obtained by means of suitable amplification devices with irreversible functioning such as, for example, permanent marks on the photographic plate caused by the penetration of electrons into the emulsion.” (italics added—A.I.). In the earlier literature, the same principles were often described in the language of “uncontrollable perturbations” arising when the quantum system interacts with the measuring device.
We note that the requirement of finite time is not among our basic principles. Rather, it comes out as a natural consequence of them.
Our method reflects the fact that decades of concerted efforts towards clarifying the fundamental mechanisms of collapse and the measurement process have met with limited success. This means that the collapse problem is an extremely challenging as well as an highly interdisciplinary one. It involves several distinct areas of theoretical physics and modern mathematics. As a consequence, there is little hope of progress unless one uses the phenomenological approach first and ask questions about first principles later.
Wave function collapse can be regarded as a ‘macroscopic quantum phenomenon’. Therefore we can turn for inspiration to the successful theories of other macroscopic quantum phenomena such as superfluidity and superconductivity.
In particular, it would be interesting here to draw a parallel with the puzzle of superconductivity, which took almost 50 years to unravel and it is still at the forefront of physics research.
This ultimate success was based on the consistent use of the phenomenological, effective-theory type of approaches before embarking on the final quest for a microscopic theory. The same strategy could be worth trying in the collapse problem.
The analogy can be continued in another respect. The phenomenological theory of superconductivity of Ginzburg and Landau uses a non-linear equation for the wave function. This does not mean, however, that the superposition principle is violated. Likewise, our use of a non-linear collapsing equation does not necessarily mean the violation of that principle because it is meant to be understood as a description at the phenomenological level which is not necessarily fundamental.
Many alternative models can be constructed by choosing different representations, forms of the collapsing force and/or adding hidden variables to the fundamental general equations belonging to the family exemplified by Eqs. (4) and (10).
However, given the probable complexity of the collapse phenomenon, it would be desirable to focus first (both theoretically and experimentally) on the most fundamental, the “purest” form of collapse—the binary symmetric measurement.
With suitable modifications, our methods could be applied in other subfields of quantum theory where a non-potential generalization of the Schrödinger equation is required by the conditions of the problem. Examples include the theory of dissipative systems, motion under the forces of radiative friction, etc. More generally, one can expect that interesting connections with nanoscience, quantum gravity and quantum cosmology could also be possible.
V.1 Schematic of main finding
Quantum state collapse is represented as a kinetic process of irreversible symmetry breaking driven by the instability of the initial state (Fig. 1-3).
V.2 Physical ideas underlying our framework
Our approach and results may look deceptively simple and questions could
be raised as to what makes this possible.
The explanation is that all the hard work is done by combining six powerful insights
from several fronts
of theoretical physics, which have not been put together before:
-complex quantum Hamilton-Jacobi formulation of quantum mechanics
-principle of symmetry breaking
-the effective theory approach
-the idea of irreversibility
-the measured quantity as the order parameter
-Bohr’s insight that quantum phenomena occur due to irreversible amplifying interaction of a classical and quantum systems.
V.3 Where do we stand in terms of various interpretations?
Although our formulation is designed to be interpretation-neutral, it can also be seen as the logical development of the conventional Copenhagen interpretation.
As discussed in the main text, those who favour the statistical ensemble interpretation of quantum mechanics would probably be skeptical about our approach. In the spirit of “experimental metaphysics”, we believe that experimental tests may help to clarify or update one’s philosophical position.
The same can be said about the Many-Worlds interpretation. In fact, experimental searches for collapse dynamics are needed from the standpoint of any interpretation because the results of such experiments will help to decide which one is closer to Nature.
If these searches fail to produce evidence for collapse, this would be a serious argument in favour of the collapse-denying interpretations, including Many-Worlds and statistical. Such arguments are hard to achieve from just theoretical reasoning.
Also, it is well-known that in some situations the wave function is ‘more real’ than in others (superconductivity, Bose-Einstein condensation etc.). Consequently, collapse may be a more real phenomenon in those situations.
In any case, the wave function satisfying our collapsible equation should be interpreted as an ‘effective’ or ‘interpolating’ wave function rather than a ‘genuine’ wave function which ceases to exist once the system starts to interact with the measuring device, according to the standard (von Neumann’s) model of the measurement process.
V.4 Comparison with the approaches of Bohr and von Neumann
In any treatment of the measurement problem
the question arises: is the measuring apparatus to be treated as a quantum system or not? On this crucial point, the opinions of quantum theorists appear to be split.
One school of thought (Bohr, Heisenberg, Dirac, and Born) accepts Bohr’s idea that the measuring device should be described approximately as classical and not quantum-mechanical. Consequently, the wave function of the apparatus is not introduced and the Schrödinger equation for the combined set “system plus apparatus” is not discussed or solved.
The alternative school (von Neumann, Landau and Peierls, Pauli, and most modern textbooks) assumes that the measuring device should be treated on the same footing as any other quantum system. As a consequence, most modern treatments introduce the wave function of the apparatus as a matter of fact that does not require further justification.
At the moment, one of the fruitful if challenging issues in quantum mechanics is the question whether the superposition principle is universally valid or not. Obviously, this is just another way to ask the same question.
Again, this problem is attacked not by purely philosophical discussion, but in the spirit of ‘experimental metaphysics’. However, despite impressive progress in the area, it would be fair to say that the final solution is not yet on the horizon.
We do not introduce explicitly the wave function of the measuring device. This is in accord with Bohr’s ideas that the the measuring device is best
described approximately and classically rather than quantum-mechanically. In fact, Bohr appears to
have reservations regarding the crucial step of “quantizing the apparatus” (first made by von Neumann) as he did not discussed it in his published works.
From the modern perspective, this may be viewed as a flaw or even a serious defect of such an interpretation.
On the other hand, Bohr’s reservation
might probably be justified because to this day, the efforts spent in quantizing the apparatus have not been completely successful. Moreover, this problem is considered fundamental but intractable in the forseeable future.
For example,
it was pointed out bh that no observer can write down his own wave function just because the human brain capacity is not large enough for that. A similar type of argument goes for an inanimate device as well.
We, of course, do not pretend to solve this part of the problem. Instead, we go around it and concentrate on the behaviour of the quantum system rather than the measuring device. In this way, we act in the spirit of ‘effective’ theory, which is well-known and extremely fruitful in the area of quantum fields (see, e.g., wei ), but is less appreciated in the context of quantum mechanics.
The essence of the effective theory approach is to integrate out all unknown degrees of freedom and focus on what is essential and observable.
V.5 Relation to decoherence
Although the theory of decoherence made an important step towards solving the collapse problem, it has not solved it.
Our paper does not include decoherence explicitly, but this is not because it is irrelevant. The real reason is that the collapse puzzle is a difficult and unsolved task. The common approach of a theoretical physicist in such cases is to tackle first an idealized setting where decoherence is ignored.
Moreover, this simple setting could be much more realistic than it seems at first sight. Indeed, we could imagine physical conditions under which the effects of decoherence are greatly reduced, such as very low noise, low temperature and similar types of environment. In this ‘very clean’ environment we expect that decoherence will be greatly affected, but the collapse will not. Consequently, analysis of collapse in the absence of decoherence is not merely an academic exercise, but a viable physical model for specific environmental conditions.
V.6 The Einstein-Podolsky-Rosen set-up and no-go theorems
In this paper, we do not treat any EPR-related issues leaving them for further work. We feel that the phenomenon of collapse, if real, could be complicated and not necessarily universal for all types of systems and experiments. One may even wonder if the quantum collapse could be as complex as its gravitational cousin. In any case, it seems prudent to approach this difficult problem in a step-by-step manner starting with the simplest possible set-up: a measurement made on one non-relativistic particle.
As Braginsky and Khalili note in Ref. bh (p. 28): “The presence of two or more degrees of freedom can change the character of the measurement substantially”.
Perhaps, to emphasise the differences, it could be helpful to introduce special terms like, for example, “contact” and “non-contact” collapse, the latter describing the EPR-type situations.
We also note that under a different set of assumptions, an EPR-type experimental search for a finite-time collapse was recently proposed in Ref. MP .
Because of the fact that our simple model (10) treats only the case of one particle and two measurement outcomes, it avoids tensions with Bell’s inequalities bell as well as various other no-go theorems due to Gleason glea ; Kochen and Specker ks ; Conway and Kochen (or “the Free Will theorem”) ck ; Colbeck and Renner cr ; Pusey, Barrett and Rudolph pbr .
More specifically, both Gleason’s and Kochen-Specker’s theorems require that the dimensionality of Hilbert space be greater than two glea ; ks —which does not apply in our case (10).
The Free Will theorem, in both original and strengthened formulations, requires a two-particle state due to its TWIN axiom ck . Therefore, it does not clash with our model (10) either.
Colbeck and Renner ck extended Bell’s prohibition of local hidden variables by considering hidden variables that have both local and global components and ruling them out as well. Again, our model (10) is immune from this exclusion theorem because it (the theorem) requires at least a two-particle system, among other things cr .
Pusey, Barrett and Rudolph pbr have recently proved another no-go theorem excluding a certain class of theories with hidden parameters $\lambda$.
The following quote from Ref. pbr summarizes their work nicely: “Our main result is that for distinct quantum states
$|\psi_{0}\rangle$ and $|\psi_{1}\rangle$, if the distributions $\mu_{0}(\lambda)$ and $\mu_{1}(\lambda)$ overlap (more precisely: if $\Delta$, the intersection of their supports, has non-zero measure) then there is a contradiction with the predictions of quantum theory.”
It follows immediately that this result does not apply to our model (10) because for distinct quantum states the distributions of our infinitesimal parameters do not overlap.
V.7 Non-linear quantum mechanics and superluminality
It has been proposed by Gisin gis among others that non-linear modifications of quantum mechanics could lead to faster than light communication. However, several points have to be taken into account here:
1) Our collapsible Schrödinger equations are not of the type considered in Ref. gis . Our non-linearity arises only at the measurement stage, while equations studied in Ref. gis are non-linear always. For this reason, superluminal communication is not made possible by our equations and the objection does not apply.
2) For completeness, we also mention that in subsequent papers fss ; kent ; bgh it was pointed out that the argument of Ref.gis is not free from possible loopholes. However, due to the above point, we do not need to appeal to these loopholes in order to demonstrate that our equations are free from the superluminality objection.
V.8 Is consciousness needed for collapse?
In our approach, an observer or the consciousness is not required for collapse to occur (cf. tha ). Moreover, our experimental upper limits on the collapse time (Table 1) show that this process is too fast for human consciousness which typically operates on much longer time-scales.
V.9 Collapse and relativity
Relativistic generalisations of the collapse concept are notoriously difficult and are not discussed here. We would like to stress, however, that at the moment the proposed collapse mechanism does not appear to be incompatible with relativity. On the contrary, a relativistic generalisation of our collapsible equations seems a worthwhile direction to pursue.
V.10 de Broglie-Bohm formulation
Some readers may get an impression that we are using or advocating some version of the de Broglie-Bohm (dBB) interpretation of quantum mechanics hol .
We emphasise that this is not the case 222This phrase should not be construed as an implicit criticism of such interpretations.. What we do is just borrowing a useful form of rewriting the Schrödinger equation. This can be considered as a purely mathematical transformation that is completely neutral with regard to interpretational issues. The dBB interpretation involves much more than such a rewriting, e.g., particle trajectories. These extra features are not necessary in our approach.
V.11 Other types of measurements
Today we know many more types of measurements than in 1930s.
Examples include weak, indirect, non-demolition, continuous, interrupted and other varieties of measurements. We emphasise that all these are beyond the scope of the present paper.
Our focus here is only on the simplest, “canonical” kind of measurement, which was also historically one of the first examples where the measurement problem was born. Without a thorough understanding of this example, there is little chance of progress in the more intricate cases.
Moreover, it would be perhaps over-optimistic to expect that one and the same theory can describe all kinds of measurements. (It would be similar to expecting, for instance, that one and the same theory can describe both strong and gravitational interactions.)
V.12 Theories of dynamical collapse
On a first view, our framework can be seen as a competitor to the well-known theories that represent collapse as a continuous dynamical process bas . In fact, it is not. The main reason is that the two approaches set different goals from the start even if the term “collapse” happens to figure prominently in both of them.
Briefly, this term is used in two rather different senses so one wonders if two different terms would be more appropriate here.
Our collapse has been induced by the measurement (“M-collapse”).
In the theories of dynamical collapse, it is occurring spontaneously, during the free evolution (“S-collapse”). This is a considerable difference with far-reaching consequences.
The theories of dynamical collapse do not address the mechanism of M-collapse. Likewise, we do not discuss the S-collapse. We ask the question: what happens to the wave function during the measurement? The theories of dynamical collapse ask: Why do the wave functions not spread out so much that the classical picture becomes blurred (problem of macro-objectification)?
It might be possible that in the end, the answers to both questions could come from one and the same ultimate source. However, for all similarities between the two problems there seem to be as many differences, so it would be reasonable to treat them as separate issues for the time being.
Our model and the dynamical collapse theories are made for two different purposes; they have different mathematical structure and different observational consequences.
V.13 Meaning of collapsing force
Introduction of the collapsing force (Fig. 4) is the main phenomenological element of our program.
As usual in such cases, we do not attempt to derive it from first principles, but rather postulate it on the combined strength of a series of independent arguments:
—The irreversible collapse during a measurement can be viewed as sort of relaxation process. Then the appearance of a collapsing force is naturally understood as the emergence of a relaxation-driving dissipative force without which relaxation would be impossible.
—In the same vein, the kinetic mechanism of collapse invites a parallel with the classical kinetic theory. The collapsing force, then, plays a similar role to the collision term in the Boltzmann equation.
—The task of constructing the collapsible Schrödinger equation can be formulated as the problem of finding a smooth interpolation between the uncollapsed and collapsed wave functions. A new term that is needed for such an interpolation would be equivalent to a collapsing force.
—What arguments force us to assume that the collapse occurs instantaneously, given that we are not aware of any other instantaneous processes in Nature?
—If, on the other hand, the collapse is not instantaneous but takes a finite time, then it seems plausible for purely mathematical reasons that one or another kind of extra terms (‘generalised collapsing forces’) should be introduced into the CQHJ-Schrödinger equation to produce this finite time.
—Within the CQHJ approach, the Schrödinger equation acquires a form so similar to Newton’s second law that it almost ‘begs for’ introducing
such a new force and exploring its consequences.
V.14 Is quantum mechanics to be modified?
Do our equations require a modification of standard quantum mechanics? At first sight, the answer to this would be “Yes”, but we believe that such a conclusion would be premature and unwarranted.
As was mentioned in the main text, our goal was not to modify quantum mechanics, but to obtain effective equations for the collapse process.
The fact that our equations are non-linear, and the Schrödinger equation is linear is not enough, in our view, to claim that these equations violate quantum mechanics.
The reason is that the Schrödinger equation is exact, and our equation is approximate. But it is well-known that the exact and linear equation (Schrödinger) can lead to approximate non-linear ones when suitable approximations are made (equations of Hartree-Fock, Gross-Pitaevsky etc.).
Indeed, as emphasized by Fock foc , in quantum theory the distinction between “fundamental” and “effective” concepts becomes a matter of convention because approximations play the principal, not a secondary role. For example, from the perspective of quantum field theory, the Schrödinger equation itself becomes an approximate “effective” equation, rather than a “fundamental” law.
Finally, from purely experimental perspective, the ‘fundamental or effective’ question is hardly the most
pressing one.
VI Conclusions and outlook
To sum up, a new class of quantum-mechanical non-linear equations for the wave function collapse has been proposed
that combines conceptual simplicity and predictive power.
Also, we have reported a set of constraints and outlined future opportunities for the sensitive experimental tests.
It is exciting to realize that only two or three orders of magnitude separate us from deeper, dedicated probes of the wave function collapse. The results will help to clarify both foundational and practical issues in quantum mechanics such as the status of the Many-Worlds interpretation and the ultimate limits on the speed of quantum computation and information processing.
Acknowledgments. The author thanks M. Albrow, R. Dörner, L. Ignatieva, V. A. Kuzmin, J.-P. Magnot, M. E. Shaposhnikov, and F. Thaheld for taking time for reading, valuable comments and discussions.
*
Appendix A Complex quantum Hamilton-Jacobi equation
To make the article self-contained, we provide below a
derivation of the complex quantum Hamilton-Jacobi equation.
Let us start with the ordinary Schrödinger equation for a particle of mass $m$ in the potential $V({\bf r})$:
$$i\hbar\psi_{t}=-\frac{\hbar^{2}}{2m}\Delta\psi+V\psi,$$
(15)
and
introduce a new variable ${\bf p}({\bf r},t)$:
$${\bf p}=\frac{\hbar}{i}\frac{\nabla\psi}{\psi}.$$
(16)
Our goal is to obtain a closed dynamical equation for ${\bf p}$.
Differentiating Eq. (16) with respect to time, we find:
$${\bf p}_{t}=\frac{\hbar}{i}\frac{\nabla\psi_{t}}{\psi}-{\bf p}\frac{\psi_{t}}{%
\psi}.$$
(17)
It is convenient to start $\psi$-elimination from the second term of Eq. (17). For this purpose, we first rewrite $\Delta\psi$ as
$$\Delta\psi=\nabla\left(\frac{i}{\hbar}{\bf p}\psi\right)=\frac{i}{\hbar}\left(%
\psi\nabla\cdot{\bf p}+{\bf p}\nabla\psi\right).$$
(18)
Inserting this into the Schrödinger equation (15) and multiplying both sides by $i{\bf p}/(\hbar\psi)$, we obtain:
$$-{\bf p}\frac{\psi_{t}}{\psi}=\frac{1}{2m}{\bf p}\nabla\cdot{\bf p}+\frac{i}{2%
m\hbar}{\bf p}p^{2}+\frac{i}{\hbar}{\bf p}V.$$
(19)
To eliminate $\psi$ from the first term of Eq. (17), we substitute the right-hand side of Eq. (18) into the Schrödinger equation (15) instead of $\Delta\psi$, then take the gradient of both parts and divide both of them by $\psi$, which yields:
$$\displaystyle-i\hbar\frac{\nabla\psi_{t}}{\psi}=\frac{i\hbar}{2m}\nabla(\nabla%
\cdot{\bf p})-\frac{1}{2m}{\bf p}\nabla\cdot{\bf p}-$$
$$\displaystyle\frac{1}{2m}\nabla p^{2}-\frac{i}{2\hbar m}p^{2}{\bf p}-\nabla V-%
\frac{i}{\hbar}{\bf p}V.$$
(20)
We now add this formula, part-by-part, to Eq. (19) and then use Eq. (17) to finally obtain:
$${\bf p}_{t}=-\nabla V-\frac{1}{2m}\nabla p^{2}+\frac{i\hbar}{2m}\nabla(\nabla%
\cdot{\bf p}).$$
(21)
So the complete elimination of the wave function is indeed possible, and Eq. (21) reproduces the dynamical equation that
plays the role of the Schrödinger equation in the CQHJ approach.
A slight rearrangement gives it an elegant shape:
$$\displaystyle{\bf p}_{t}=-\nabla H,$$
(22)
$$\displaystyle{\hat{\bf p}}=-i\hbar\nabla,\;\;\;H=V+\frac{p^{2}}{2m}+\frac{{%
\hat{\bf p}}{\bf p}}{2m}.$$
(23)
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Pose for Action – Action for Pose
Umar Iqbal
Computer Vision Group, University of Bonn, Germany
{uiqbal, garbade, gall}@iai.uni-bonn.de
Martin Garbade
Computer Vision Group, University of Bonn, Germany
{uiqbal, garbade, gall}@iai.uni-bonn.de
Juergen Gall
Computer Vision Group, University of Bonn, Germany
{uiqbal, garbade, gall}@iai.uni-bonn.de
Abstract
In this work we propose to utilize information about human actions to improve pose estimation in monocular videos. To this end, we present a pictorial structure model that exploits high-level information about activities to incorporate higher-order part dependencies by modeling action specific appearance models and pose priors. However, instead of using an additional expensive action recognition framework, the action priors are efficiently estimated by our pose estimation framework. This is achieved by starting with a uniform action prior and updating the action prior during pose estimation. We also show that learning the right amount of appearance sharing among action classes improves the pose estimation. We demonstrate the effectiveness of the proposed method on two challenging datasets for pose estimation and action recognition with over 80,000 test images.111The models and source code are available at http://pages.iai.uni-bonn.de/iqbal_umar/action4pose/.
I Introduction
Human pose estimation from RGB images or videos is a challenging problem in computer vision, especially for realistic and unconstrained data taken from the Internet. Popular approaches for pose estimation [6, 18, 38, 5, 2, 29] adopt the pictorial structure (PS) model, which resembles the human skeleton and allows for efficient inference in case of tree structures [8, 7]. Even if they are trained discriminatively, PS models struggle to cope with the large variation of human pose and appearance.
This problem can be addressed by conditioning the PS model on additional observations from the image.
For instance, [18] detects poselets, which are examples of consistent appearance and body part configurations, and condition the PS model on these.
Instead of conditioning the PS model on predicted configurations of body parts from an image, we propose to condition the PS model on high-level information like activity.
Intuitively, the information about the activity of a person can provide a strong cue about the pose and vice versa the activity can be estimated from pose. There have been only few works [39, 42, 15] that couple action recognition and pose estimation to improve pose estimation. In [39], action class confidences are used to initialize an optimization scheme for estimating the parameters of a subject-specific 3D human model in indoor multi-view scenarios. In [42], a database of 3D poses is used to learn a cross-modality regression forest that predicts the 3D poses from a sequence of 2D poses, which are estimated by [38]. In addition, the action is detected and the 3D poses corresponding to the predicted action are used to refine the pose. However, both approaches cannot be applied to unconstrained monocular videos. While [39] requires a subject-specific model and several views, [42] requires 3D pose data for training. More recently, [15] proposed an approach to jointly estimate action classes and refine human poses. The approach decomposes the human poses estimated at each video frame into sub-parts and tracks these sub-parts across time according to the parameters learned for each action. The action class and joint locations corresponding to the best part-tracks are selected as estimates for the action class and poses.
The estimation of activities, however, comes at high computational cost since the videos are pre-processed by several approaches, one for pose estimation [16] and two for extracting action related features [32, 34].
In this work, we present a framework for pose estimation that infers and integrates activities with a very small computational overhead compared to an approach that estimates the pose only. This is achieved by an action conditioned pictorial structure (ACPS) model for 2D human pose estimation that incorporates priors over activities. The framework of the approach is illustrated in Figure 1. We first infer the poses for each frame with a uniform distribution over actions.
While the binaries of the ACPS are modeled by Gaussian mixture models, which depend on the prior distribution over the action classes, the unaries of the ACPS model are estimated by action conditioned regression forests. To this end, we modify the approach [5], which consists of two layers of random forests, on two counts. Firstly, we replace the first layer by a convolutional network and use convolutional channel features to train the second layer, which consists of regression forests. Secondly, we condition the regression forests on a distribution over actions and learn the sharing among action classes. In our experiments, we show that these modifications increase the pose estimation accuracy by more than 40% compared to [5]. After the poses are inferred with a uniform distribution over actions, we update the action prior and the ACPS model based on the inferred poses to obtain the final pose estimates. Since the update procedure is very efficient, we avoid the computational expensive overhead of [15].
We evaluate our approach on the challenging J-HMDB [13] and Penn-Action [44] datasets, which consist of videos collected from the Internet and contain large amount of scale and appearance variations. In our experiments, we provide a detailed analysis of the impact of conditioning unaries and binaries on a distribution over actions and the benefit of appearance sharing among action classes. We demonstrate the effectiveness of the proposed approach for pose estimation and action recognition on both datasets. Compared to [15], the pose estimation accuracy is improved by over 30%.
II Related Work
State-of-the-art approaches for pose estimation are mostly based on neural networks [30, 28, 17, 35, 20, 21, 10] or on the pictorial structure framework [38, 18, 5, 43].
Several approaches have been proposed to improve the accuracy of PS models for human pose estimation. For instance, joint dependencies can be modeled not only by the PS model, but also by a mid-level image representation such as poselets [18], exemplars [23] or data dependent probabilities learned by a neural network [1].
Pose estimation in videos can be improved by taking temporal information or motion cues into account [16, 2, 45, 10, 14, 12]. In [16] several pose hypotheses are generated for each video frame and a smooth configuration of poses over time is selected from all hypotheses. Instead of complete articulated pose, [22] and [2] track individual body parts and regularize the trajectories of the body parts through the location of neighboring parts. Similar in spirit, the approach in [43] jointly tracks symmetric body parts in order to better incorporate spatio-temporal constraints, and also to avoid double-counting. Optical flow information has also been used to enhance detected poses at each video frame by analyzing body motion in adjacent frames [9, 45].
Recent approaches for human pose estimation use different CNN architectures to directly obtain the heatmaps of body parts [28, 17, 35, 20, 21, 10]. [28, 35, 20] and [21] use fully convolutional neural network architectures, where [35] proposes a multi-staged architecture that sequentially refines the output in each stage. For pose estimation in videos, [17] combines the heatmaps of body parts from multiple frames with optical flow information to leverage the temporal information in videos. More recently, [10] proposes a convolutional recurrent neural network architecture that takes as input a set of video frames and sequentially estimates the body part locations in each frame, while also using the information of estimated body parts in previous frames.
As done in this work, a few works also combine both PS model and CNNs [1, 29]. In contrast to the approaches that use temporal information in videos for pose refinement, we utilize the detected poses in each video frame to extract high-level information about the activity and use it to refine the poses.
Action recognition based on 3D human poses has been investigated in many works [41]. With the progress in the area of 2D human pose estimation in recent years, 2D poses have also gained an increased attention for action recognition [37, 26, 6, 13, 19, 3]. However, utilizing action recognition to aid human pose estimation is not well studied, in particular not in the context of 2D human pose estimation. There are only a few works [39, 42, 31, 40, 15] that showed the benefit of it. The approaches in [39, 42] rely on strong assumption.
The approach [39] assumes that a person-specific 3D model is given and considers pose estimation in the context of multiple synchronized camera views. The approach [42] focuses on 3D pose estimation from monocular videos and assumes that 3D pose data is available for all actions.
The approach [31] adopts a mixture of PS models, and learns a model for each action class. For a given image, each model is weighted by the confidence scores of an additional action recognition system and the pose with the maximum weight is taken. A similar approach is adopted in [40] to model object-pose relations. These approaches, however, do not scale with the number of action classes since each model needs to be evaluated.
The closest to our work is the recent approach of [15] that jointly estimates the action classes and refines human poses. The approach first estimates human poses at each video frame and decomposes them into sub-parts. These sub-parts are then tracked across video frames based on action specific spatio-temporal constraints. Finally, the action labels and joint locations are inferred from the part tracks that maximize a defined objective function. While the approach shows promising results, it does not re-estimate the parts but only re-combines them over frames i. e., only the temporal constraints are influenced by an activity. Moreover, it relies on two additional activity recognition approaches based on optical flow and appearance features to obtain good action recognition accuracy that results in a very large computational overhead as compared to an approach that estimates activities using only the pose information. In this work, we show that additional action recognition approaches are not required, but predict the activities directly from a sequence of poses. In contrast to [15], we condition the pose model itself on activities and re-estimate the entire pose per frame.
III Overview
Our method exploits the fact that the information about the activity of a subject provides a cue about pose and appearance of the subject, and vice versa.
In this work we utilize the high-level information about a person’s activity to leverage the performance of pose estimation, where the activity information is obtained from previously inferred poses. To this end, we propose an action conditioned pictorial structure (PS) that incorporates action specific appearance and kinematic models. If we have only a uniform prior over the action classes, the model is a standard PS model, which we will briefly discuss in Section IV. Figure 1 depicts an overview of the proposed framework.
IV Pictorial Structure
We adopt the joint representation [5] of the PS model [8], where the vector $\mathbb{x}_{j}\in\mathcal{X}$ represents the 2D location of the $j^{th}$ joint in image $\mathbb{I}$, and $\mathcal{X}$ = $\{\mathbb{x}_{j}\}_{j\in\mathcal{J}}$ is the set of all body joints. The structure of a human body is represented by a kinematic tree with nodes of the tree being the joints ${j}$ and edges $\mathcal{E}$ being the kinematic constraints between a joint $j$ and its unique parent joint $p$ as illustrated in Figure 1. The pose configuration in a single image is then inferred by maximizing the following posterior distribution:
$$\mathrm{p}(\mathcal{X}|\mathbb{I})\propto\prod_{j\in\mathcal{J}}\phi_{j}(\mathbb{x}_{j}|\mathbb{I})\prod_{j,p\in\mathcal{E}}\psi_{jp}(\mathbb{x}_{j},\mathbb{x}_{p})$$
(1)
where the unary potentials $\phi_{j}(\mathbb{x}_{j}|\mathbb{I})$ represent the likelihood of the $j^{th}$ joint at location $\mathbb{x}_{j}$. The binary potentials $\psi_{jp}(\mathbb{x}_{j},\mathbb{x}_{p})$ define the deformation cost for the joint-parent pair $(j,p)$, and are often modeled by Gaussian distributions for an exact and efficient solution using a distance transform [8].
We describe the unary and binary terms in Section IV-A and Section IV-B, respectively. In Section V-A, we then discuss how these can be adapted to build an action conditioned PS model.
IV-A Unary Potentials
Random regression forests have been proven to be robust for the task of human pose estimation [24, 27, 5]. A regression forest $\mathcal{F}$ consists of a set of randomized regression trees, where each tree $T$ is composed of split and leaf nodes. Each split node represents a weak classifier which passes an input image patch $P$ to a subsequent left or right node until a leaf node $L_{T}$ is reached. As in [5], we use a separate regression forest for each body joint. Each tree is trained with a set of randomly sampled images from the training data. The patches around the annotated joint locations are considered as foreground and all others as background. Each patch consists of a joint label $c\in\{0,j\}$, a set of image features $F_{P}$, and its 2D offset $\mathbb{d}_{P}$ from the joint center.
During training, a splitting function is learned for each split node by randomly selecting and maximizing a goodness measure for regression or classification.
At the leaves the class probabilities $\mathrm{p}(c|L_{T})$ and the distribution of offset vectors $\mathrm{p}(\mathbb{d}|L_{T})$ are stored.
During testing, patches are densely extracted from the input image $\mathbb{I}$ and are passed through the trained trees.
Each patch centered at location $\mathbb{y}$ ends in a leaf node $L_{T}(P(\mathbb{y}))$ for each tree $T\in\mathcal{F}$.
The unary potentials $\phi_{j}$ for the joint $j$ at location $\mathbb{x}_{j}$ are then given by
$$\phi_{j}(\mathbb{x}_{j}|\mathbb{I})=\sum_{\mathbb{y}\in\Omega}\dfrac{1}{|\mathcal{F}|}\sum_{T\in\mathcal{F}}\Big{\{}\mathrm{p}(c=j|L_{T}(P(\mathbb{y})))\\
\cdot\mathrm{p}(\mathbb{x}-\mathbb{y}|L_{T}(P(\mathbb{y}))\Big{\}}.$$
(2)
In [5] a two layer approach is proposed. The first layer consists of classification forests that classify image patches according to the body parts using a combination of color features, HOG features, and the output of a skin color detector. The second layer consists of regression forests that predict the joint locations using the features of the first layer and the output of the first layer as features. For both layers, the split nodes compare feature values at different pixel locations within a patch of size $24\times 24$ pixels.
We propose to replace the first layer by a convolutional network and extract convolutional channel features (CCF) [36] from the intermediate layers of the network to train the regression forests of the second layer. In [36] several pre-trained network architectures have been evaluated for pedestrian detection using boosting as classifier. The study shows that the “conv3-3” layer of the VGG-16 net [25] trained on the ImageNet (ILSVRC-2012) dataset performs very well even without fine tuning, but it is indicated that the optimal layer depends on the task. Instead of pre-selecting a layer, we use regression forests to select the features based on the layers “conv2-2”, “conv3-3”, “conv4-3”, and “conv5-3”. An example of the CCF extracted from an image is shown in Figure 2. Since these layers are of lower dimensions than the original image, we upsample them using linear interpolation to make their dimensions equivalent to the input image. This results in a 1408 (128+256+512+512) dimensional feature representation for each pixel. As split nodes in the regression forests, we use axis-aligned split functions. For an efficient feature extraction at multiple image scales, we use patchwork as proposed in [11] to perform the forward pass of the convolutional network only once.
IV-B Binary Potentials
Binary potentials $\psi_{jp}(\mathbb{x}_{j},\mathbb{x}_{p})$ are modeled as a Gaussian mixture model for each joint $j$ with respect to its parent joint $p$ in the kinematic tree. As in [5], we obtain the relative offsets between child and parent joints from the training data and cluster them into $k=1,\dots,K$ clusters using k-means clustering. Each cluster $k$ takes the form of a weighted Gaussian distribution as
$$\psi_{jp}(\mathbb{x}_{j},\mathbb{x}_{p})=w_{jp}^{k}\cdot\\
\exp\left(-\dfrac{1}{2}\left(\mathbb{d}_{jp}-\mu_{jp}^{k}\right)^{T}\left(\Sigma_{jp}^{k}\right)^{-1}\left(\mathbb{d}_{jp}-\mu_{jp}^{k})\right)\right)$$
(3)
with mean $\mu_{jp}^{k}$ and covariance $\Sigma_{jp}^{k}$, where $\mathbb{d}_{jp}=(\mathbb{x}_{j}-\mathbb{x}_{p})$. The weights $w_{jp}^{k}$ are set according to the cluster frequency $\mathrm{p}(k|j,p)^{\alpha}$ with a normalization constant $\alpha=0.1$ [5].
For inference, we select the best cluster $k$ for each joint
by computing the max-marginals for the root node and backtrack the best pose configuration from the maximum of the max-marginals.
V Action Conditioned Pose Estimation
As illustrated in Figure 1, our goal is to estimate the pose $\mathcal{X}$ conditioned by the distribution $\mathrm{p_{{}_{A}}}$ for a set of action classes $a\in\mathcal{A}$. To this end, we introduce in Section V-A a pictorial structure model that is conditioned on $\mathrm{p_{{}_{A}}}$. Since we do not assume any prior knowledge of the action, we estimate the pose first with the uniform distribution $\mathrm{p_{{}_{A}}}(a)=1/|\mathcal{A}|,\forall a\in\mathcal{A}$. The estimated poses for $N$ frames are then used to estimate the probabilities of the action classes $\mathrm{p_{{}_{A}}}(a|\mathcal{X}_{n=1\dots N}),\forall a\in\mathcal{A}$ as described in Section V-B. Finally, the poses $\mathcal{X}_{n}$ are updated based on the distribution $\mathrm{p_{{}_{A}}}$.
V-A Action Conditioned Pictorial Structure
In order to integrate the distribution $\mathrm{p_{{}_{A}}}$ of the action classes obtained from the action classifier into (1), we make the unaries and binaries dependent on $\mathrm{p_{{}_{A}}}$:
$$\mathrm{p}(\mathcal{X}|\mathrm{p_{{}_{A}}},\mathbb{I})\propto\prod_{j\in\mathcal{J}}\phi_{j}(\mathbb{x}_{j}|\mathrm{p_{{}_{A}}},\mathbb{I})\cdot\prod_{j,p\in\mathcal{E}}\psi_{jp}(\mathbb{x}_{j},\mathbb{x}_{p}|\mathrm{p_{{}_{A}}}).$$
(4)
While the unary terms are discussed in Section V-A1, the binaries $\psi_{jp}(\mathbb{x}_{j},\mathbb{x}_{p}|\mathrm{p_{{}_{A}}})$ are represented by Gaussians as in (3). However, instead of computing mean and covariance from all training poses with equal weights, we weight each training pose based on its action class label and $\mathrm{p_{{}_{A}}}(a)$. In our experiments, we will also investigate the case where $\mathrm{p_{{}_{A}}}(a)$ is simplified to
$$\mathrm{p_{{}_{A}}}(a)=\begin{cases}1&\text{if}\quad a=\operatorname*{argmax}_{a^{\prime}}\mathrm{p_{{}_{A}}}(a^{\prime}|\mathcal{X}_{n=1\dots N})\\
0&\text{otherwise.}\end{cases}$$
(5)
V-A1 Conditional Joint Regressors
Figure 3 shows examples of patches of the wrist extracted from images of different action classes. We can see a large amount of appearance variation across different classes regardless of the fact that all patches belong to the same body joint. However, it can also be seen that within individual activities this variation is relatively small. We exploit this observation and propose action specific unary potentials for each joint $j$. To this end we adopt conditional regression forests [4, 27] that have been proven to be robust for facial landmark detection in [4] and 3D human pose estimation in [27].
While [4] trains a separate regression forest for each head pose and selects a specific regression forest conditioned on the output of a face pose detector, [27] proposes partially conditioned regression forests, where a forest is jointly trained for a set of discrete states of a human attribute like human orientation or height and the conditioning only happens at the leaf nodes. Since the continuous attributes are discretized, interpolation between the discrete states is achieved by sharing the votes.
In this work we resort to partially conditional forests due to its significantly reduced training time and memory requirements.
During training we augment each patch $P$ with its action class label $a$. Instead of $\mathrm{p}(c|L_{T})$ and $\mathrm{p}(\mathbb{d}|L_{T})$, the leaf nodes model the conditional probabilities $\mathrm{p}(c|a,L_{T})$ and $\mathrm{p}(\mathbb{d}|a,L_{T})$. Given the distribution over action classes $\mathrm{p_{{}_{A}}}$, we obtain the conditional unary potentials:
$$\displaystyle\phi_{j}(\mathbb{x}_{j}|\mathrm{p_{{}_{A}}},\mathbb{I})$$
$$\displaystyle=\sum_{\mathbb{y}\in\Omega}\dfrac{1}{|\mathcal{F}|}\sum_{T\in\mathcal{F}}\sum_{a\in\mathcal{A}}\Big{\{}\mathrm{p_{{}_{A}}}(a)$$
$$\displaystyle\cdot\mathrm{p}(c=j|a,L_{T}(P(\mathbb{y}))).\mathrm{p}(\mathbb{x}-\mathbb{y}|a,L_{T}(P(\mathbb{y}))\Big{\}}$$
$$\displaystyle=\sum_{a\in\mathcal{A}}\phi_{j}(\mathbb{x}_{j}|a,\mathbb{I})\mathrm{p_{{}_{A}}}(a).$$
(6)
Since the terms $\phi_{j}(\mathbb{x}_{j}|a,\mathbb{I})$ need to be computed only once for an image $\mathbb{I}$, $\phi_{j}(\mathbb{x}_{j}|\mathrm{p_{{}_{A}}},\mathbb{I})$ can be efficiently computed after an update of $\mathrm{p_{{}_{A}}}$.
V-A2 Appearance Sharing Across Actions
Different actions sometimes share body pose configurations and appearance of parts as shown in Figure 3. We therefore propose to learn the sharing among action classes within a conditional regression forest. To this end, we replace the term $\phi_{j}(\mathbb{x}_{j}|a,\mathbb{I})$ in (V-A1) by a weighted combination of the action classes:
$$\phi^{\textit{sharing}}_{j}(\mathbb{x}_{j}|a,\mathbb{I})=\sum_{a^{\prime}\in\mathcal{A}}\gamma_{a}(a^{\prime})\phi_{j}(\mathbb{x}_{j}|a^{\prime},\mathbb{I})$$
(7)
where the weights $\gamma_{a}(a^{\prime})$ represent the amount of sharing between action class $a$ and $a^{\prime}$. To learn the weights $\gamma_{a}$ for each class $a\in\mathcal{A}$, we apply the trained conditional regression forests to a set of validation images scaled to a constant body size and maximize the response of (7) at the true joint locations and minimize it at non-joint locations. Formally, this can be stated as
$$\gamma_{a}=\operatorname*{argmax}_{\gamma}\sum_{n_{a}}\sum_{j}\Bigg{\{}\sum_{a^{\prime}\in\mathcal{A}}\gamma(a^{\prime})\phi^{*}_{j}\left(\mathbb{x}^{\textit{gt}}_{j,n_{a}}|{a^{\prime}},\mathbb{I}_{n_{a}}\right)\\
-\max_{\mathbb{x}\in\mathbb{X}_{j,n_{a}}^{\textit{neg}}}\left(\sum_{a^{\prime}\in\mathcal{A}}\gamma(a^{\prime})\phi^{*}_{j}\left(\mathbb{x}|{a^{\prime}},\mathbb{I}_{n_{a}}\right)\right)\Bigg{\}}-\lambda\|\gamma\|^{2}$$
(8)
subject to $\sum_{a^{\prime}\in\mathcal{A}}\gamma(a^{\prime})=1$ and $\gamma(a^{\prime})\geq 0$. $\mathbb{I}_{n_{a}}$ denotes the $n^{th}$ scaled validation image of action class $a$, $\mathbb{x}^{\textit{gt}}_{j,n_{a}}$ is the annotated joint position for joint $j$ in image $\mathbb{I}_{n_{a}}$, and $\mathbb{X}_{j,n_{a}}^{\textit{neg}}$ is a set of image locations which are more than $5$ pixels away from $\mathbb{x}^{\textit{gt}}_{j,n_{a}}$.
The set of negative samples is obtained by computing $\phi^{*}_{j}(\mathbb{x}|{a^{\prime}},\mathbb{I}_{n_{a}})$ and taking the $10$ strongest modes, which do not correspond to $\mathbb{x}^{\textit{gt}}_{j,n_{a}}$, for each image. For optimization, we use the smoothed unaries
$$\phi_{j}^{*}(\mathbb{x}|a,\mathbb{I})=\sum_{\mathbb{y}\in\Omega}\exp\left(-\dfrac{{\|\mathbb{x}-\mathbb{y}\|}^{2}}{\sigma^{2}}\right)\phi_{j}(\mathbb{y}|a,\mathbb{I})$$
(9)
with $\sigma=3$ and replace $\max$ by the softmax function to make (8) differentiable. The last term in (8) is a regularizer that prefers sharing, i. e., $\|\gamma\|^{2}$ attains its minimum value for uniform weights. In our experiments, we use $\lambda=0.4$ as weight for the regularizer.
We optimize the objective function by constrained local optimization using uniform weights for initialization $\gamma(a^{\prime})=1/|\mathcal{A}|$. In our experiments, we observed that similar weights are obtained when the optimization is initialized by $\gamma(a)=1$ and $\gamma(a^{\prime})=0$ for $a^{\prime}\neq a$, indicating that the results are not sensitive to the initialization.
In (8), we learn the weights $\gamma_{a}$ for each action class but we could also optimize for each joint independently. In our experiments, however, we observed that this resulted in over-fitting.
V-B Action Classification
For pose-based action recognition, we use the bag-of-word approach proposed in [13].
From the estimated joint positions $\mathcal{X}_{n=1\dots N}$, a set of features called NTraj+ is computed that encodes spatial and directional joint information. Additionally, differences between successive frames are computed to encode the dynamics of the joint movements. Since we use a body model with 13 joints, we compute the locations of missing joints (neck and belly) in order to obtain the same $15$ joints as in [13]. We approximate the neck position as the mean of the face and the center of shoulder joints. The belly position is approximated by the mean of the shoulder and hip joints.
For each of the $3,223$ descriptor types, a codebook is generated by running k-means $8$ times on all training samples and choosing the codebook with minimum compactness. These codebooks are used to extract a histogram for each descriptor type and video.
For classification, we use an SVM classifier in a multi-channel setup. To this end, for each descriptor type $t$, we compute a distance matrix $D_{t}$ that contains the $\chi^{2}$-distance between the histograms $(h_{i}^{t},h_{j}^{t})$ of all video pairs $(v_{i},v_{j})$. We then obtain the kernel matrix that we use for the multi-class classification as follows
$$K(v_{i},v_{j})=\exp\left(-\frac{1}{L}\sum^{L}_{t=1}\frac{D_{t}(h_{i}^{t},h_{j}^{t})}{\mu^{t}}\right)$$
(10)
where $L$ is the number of descriptor types and $\mu^{t}$ is the mean of the distance matrix $D_{t}$. For classification we use a one-vs-all approach with $C=100$ for the SVM.
VI Experiments
In order to evaluate the proposed method, we follow the same protocol as proposed in [15]. In particular, we evaluate the proposed method on two challenging datasets, namely sub-J-HMDB [13]
and the Penn-Action dataset [44]. Both datasets provide annotated 2D poses and activity labels for each video. They consist of videos collected from the Internet and contain large amount of scale and appearance variations, low resolution frames, occlusions and foreshortened body poses. This makes them very challenging for human pose estimation.
While sub-J-HMDB [13] comprises videos from 12 action categories with fully visible persons, the Penn-Action dataset comprises videos from 15 action categories with a large amount of body part truncations. As in [15], we discard the activity class “playing guitar” since it does not contain any fully visible person.
For testing on sub-J-HMDB, we follow the 3-fold cross validation protocol proposed by [13]. On average for each split, this includes $229$ videos for training and $87$ videos for testing with $8,124$ and $3,076$ frames, respectively.
The Penn-Action dataset consists of $1,212$ videos for training and $1,020$ for testing with $85,325$ and $74,308$ frames, respectively.
To evaluate the performance of pose estimation, we use the APK (Average Precision of Keypoints) metric [38, 15].
VI-A Implementation Details
For the Penn-Action dataset, we split the training images half and half into a training set and a validation set. Since the dataset sub-J-HMDB is smaller, we create a validation set by mirroring the training images. The training images are scaled such that the mean upper body size is $40$ pixels. Each forest consists of $20$ trees, where $10$ trees are trained on the training and $10$ on the validation set, with a maximum depth of $20$ and a minimum of $20$ patches per leaf. We train each tree with $50,000$ positive and $50,000$ negative patches extracted from $5,000$ randomly selected images and generate $40,000$ split functions at each node.
For the binary potentials, we use $k=24$ mixtures per part.
For learning the appearance sharing among action classes (Section V-A2) and training the action classifier (Section V-B), we use the $10$ trees trained on the training set and apply them to the validation set. The action classifier and the sharing are then learned on the validation set.
For pose estimation, we create an image pyramid and perform inference at each scale independently. We then select the final pose from the scale with the highest posterior (4). In our experiments, we use $4$ scales with scale factor $0.8$.
The evaluation of 260 trees (20 trees for each of the 13 joints) including feature extraction takes roughly 15 seconds on average.222Measured on a 3.4GHz Intel processor using only one core with NVidia GeForce GTX 780 GPU. The image size for all videos in sub-J-HMDB is $320\times 240$ pixels. Inference with the PS model for all 4 scales takes around 1 second. The action recognition with feature computation takes only 0.18 seconds per image and it does not increase the time for pose estimation substantially.
VI-B Pose Estimation
We first evaluate the impact of the convolutional channel features (CCF) for pose estimation on split-1 of sub-J-HMDB. The results in Table I show that the CCF outperform the combination of color features, HOG features, and the output of a skin color detector, which is used in [5].
In Table I(a) we evaluate the proposed ACPS model under different settings on split-1 of sub-J-HMDB when using CCF features for joint regressors. We start with the first step of our framework where neither the unaries nor the binaries depend on the action classes. This is equivalent to the standard PS model described in Section IV, which achieves an average joint estimation accuracy of $51.5\%$. Given the estimated poses, the pose-based action recognition approach described in Section V-B achieves an action recognition accuracy of $66.3\%$ for split-1.
Having estimated the action priors $\mathrm{p_{{}_{A}}}$, we first evaluate action conditioned binary potentials while keeping the unary potentials as in the standard PS model. As described in Section V-A, we can use in our model the probabilities $\mathrm{p_{{}_{A}}}$ or replace them by the distribution (5), which considers only the classified action class. The first setting is denoted by “Cond. ($\mathrm{p_{{}_{A}}}$)” and the second by “Cond. (5)”. It can be seen that the conditional binaries based on (5) already outperform the baseline by improving the accuracy from $51.5\%$ to $53.8\%$. However, taking the priors from all classes slightly decreases the performance. This shows that conditioning the binary potentials on the most probable class is a better choice than using priors from all classes.
Secondly, we analyze how action conditioned unary potentials affect pose estimation. For the unaries, we have the same options “Cond. ($\mathrm{p_{{}_{A}}}$)” and “Cond. (5)” as for the binaries. In addition, we can use appearance sharing as described in Section V-A2, which is denoted by “AS”. For all three binaries, the conditional unaries based on (5) decrease the performance.
Since the conditional unaries based on (5) are specifically designed for each action class, they do not generalize well in case of a misclassified action class..
However, adding appearance sharing to the conditional unaries boost the performance for both conditioned on (5) and $\mathrm{p_{{}_{A}}}$. Adding appearance sharing outperforms all other unaries without appearance sharing, i. e., conditional unaries, independent unaries and the unaries conditioned on $\mathrm{p_{{}_{A}}}$. For all unaries, the binaries conditioned on (5) outperform the other binaries.
This shows that appearance sharing and binaries conditioned on the most probable class performs best, which gives an improvement of the baseline from $51.5\%$ to $55.3\%$.
In Table I(b), we also evaluate the proposed ACPS when using the weaker features from [5]. Although the accuracies as compared to CCF features are lower, the benefit of the proposed method remains the same. For the rest of this paper, we will use CCF for all our experiments.
In Table III we compare the proposed action conditioned PS model with other state-of-the-art approaches on all three splits of sub-J-HMDB. In particular, we provide a comparison with [5, 38, 15, 16, 2, 1]. The accuracies for the approaches [38, 15, 16, 2] are taken from [15] where the APK threshold $0.2$ is used. We also evaluated the convolutional network based approach [1] using the publicly available source code trained on sub-J-HMDB. Our approach outperfroms the other methods by a margin, and notably improves wrist localization by more than 5% as compared to the baseline.
Table IV compares the proposed ACPS with the state-of-the-art on the Penn-Action dataset. The accuracies for the approaches [38, 15, 16] are taken from [15]. We can see that the proposed method improves the baseline from 75.5% to 81.1%, while improving the elbow and wrist localization accuracy by more than 7% and 10%, respectively. The proposed method also significantly outperforms other approaches Only the approach [10] achieves a higher accuracy than our method. [10], however, uses a better multi-staged CNN architecture as baseline compared to our network for computing CCF features. Since the gain of ACPS compared to our baseline even increases when better features are used as shown in Table I(a) & Table I(b), we expect at least a similar performance gain when we use the baseline architecture from [10] for ACPS.
VI-C Action Recognition
In Table VI, we compare the action recognition accuracy obtained by our approach with state-of-the-approaches for action recognition. On sub-J-HMDB, the obtained accuracy using only pose as feature is comparable to the other approaches. Only the recent work [3] which combines pose, CNN, and motion features achieves a better action recognition accuracy. However, if we combine our pose-based action recognition with Fisher vector encoding of improved dense trajectories [33] using late fusion, we outperform other methods that also combine pose and appearance. The results are similar on the Penn-Action dataset.
In Table V, we report the effect of different action recognition approaches on pose estimation. We report the pose estimation accuracy for split-1 of sub-J-HMDB when the action classes are not inferred by our framework, but estimated using improved dense trajectories with Fisher vector encoding (IDT-FV) [33] or the fusion of our pose-based method and IDT-FV. Although the action recognition rate is higher when pose and IDT-FV are combined, the pose estimation accuracy is not improved. If the action classes are not predicted but are provided (GT),
the accuracy improves slightly for sub-J-HMDB and
from 64.8% to 68.1% for the Penn-Action dataset. We also experimented with several iterations in our framework, but the improvements compared to the achieved accuracy of $51.6\%$ were not more than 0.1% on all three splits of sub-J-HMDB.
VII Conclusion
In this paper, we have demonstrated that action recognition can be efficiently utilized to improve human pose estimation on realistic data. To this end, we presented a pictorial structure model that incorporates high-level activity information
by conditioning the unaries and binaries on a prior distribution over action labels. Although the action priors can be estimated by an accurate, but expensive action recognition system, we have shown that the action priors can also be efficiently estimated during pose estimation without substantially increasing the computational time of the pose estimation. In our experiments, we thoroughly analyzed various combinations of unaries and binaries and showed that learning the right amount of appearance sharing among action classes improves the pose estimation accuracy. While we expect further improvements by using a more sophisticated CNN architecture as baseline and by including a temporal model, the proposed method has already shown its effectiveness on two challenging datasets for pose estimation and action recognition.
Acknowledgements: The work was partially supported by the ERC Starting Grant ARCA (677650).
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A Clear View of a Cloudy Brown Dwarf Companion from High-Resolution Spectroscopy
Jerry W. Xuan
Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA
Jason Wang
51 Pegasi b Fellow
Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA
Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA) and Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208, USA
Jean-Baptiste Ruffio
Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA
Heather Knutson
Division of Geological & Planetary Sciences, California Institute of Technology, Pasadena, CA 91125, USA
Dimitri Mawet
Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA
Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr.,Pasadena, CA 91109, USA
Paul Mollière
Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany
Jared Kolecki
Department of Astronomy, The Ohio State University, 100 W 18th Ave, Columbus, OH 43210 USA
Arthur Vigan
Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France
Sagnick Mukherjee
Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA
Nicole Wallack
Division of Geological & Planetary Sciences, California Institute of Technology, Pasadena, CA 91125, USA
Ji Wang
Department of Astronomy, The Ohio State University, 100 W 18th Ave, Columbus, OH 43210 USA
Ashley Baker
Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA
51 Pegasi b Fellow
Randall Bartos
Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr.,Pasadena, CA 91109, USA
Geoffrey A. Blake
Division of Geological & Planetary Sciences, California Institute of Technology, Pasadena, CA 91125, USA
Charlotte Z. Bond
UK Astronomy Technology Centre, Royal Observatory, Edinburgh EH9 3HJ, United Kingdom
Marta Bryan
Department of Astronomy, University of California at Berkeley, CA 94720, USA
51 Pegasi b Fellow
Benjamin Calvin
Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA
Department of Physics & Astronomy, 430 Portola Plaza, University of California, Los Angeles, CA 90095, USA
Sylvain Cetre
W. M. Keck Observatory, 65-1120 Mamalahoa Hwy, Kamuela, HI, USA
Mark Chun
Institute for Astronomy, University of Hawaii at Hilo, 640 N Aohoku Pl, Hilo, HI 96720, USA
Jacques-Robert Delorme
W. M. Keck Observatory, 65-1120 Mamalahoa Hwy, Kamuela, HI, USA
Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA
Greg Doppmann
W. M. Keck Observatory, 65-1120 Mamalahoa Hwy, Kamuela, HI, USA
Daniel Echeverri
Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA
Luke Finnerty
Department of Physics & Astronomy, 430 Portola Plaza, University of California, Los Angeles, CA 90095, USA
Michael P. Fitzgerald
Department of Physics & Astronomy, 430 Portola Plaza, University of California, Los Angeles, CA 90095, USA
Katelyn Horstman
Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA
Julie Inglis
Division of Geological & Planetary Sciences, California Institute of Technology, Pasadena, CA 91125, USA
Nemanja Jovanovic
Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA
Ronald López
Department of Physics & Astronomy, 430 Portola Plaza, University of California, Los Angeles, CA 90095, USA
Emily C. Martin
Department of Astronomy & Astrophysics, University of California, Santa Cruz, CA95064, USA
Evan Morris
Department of Astronomy & Astrophysics, University of California, Santa Cruz, CA95064, USA
Jacklyn Pezzato
Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA
Sam Ragland
W. M. Keck Observatory, 65-1120 Mamalahoa Hwy, Kamuela, HI, USA
Bin Ren
Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA
Garreth Ruane
Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA
Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr.,Pasadena, CA 91109, USA
Ben Sappey
Center for Astrophysics and Space Sciences, University of California, San Diego, La Jolla, CA 92093
Tobias Schofield
Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA
Andrew Skemer
Department of Astronomy & Astrophysics, University of California, Santa Cruz, CA95064, USA
Taylor Venenciano
Physics and Astronomy Department, Pomona College, 333 N. College Way, Claremont, CA 91711, USA
J. Kent Wallace
Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr.,Pasadena, CA 91109, USA
Peter Wizinowich
W. M. Keck Observatory, 65-1120 Mamalahoa Hwy, Kamuela, HI, USA
Abstract
Direct imaging studies have mainly used low-resolution spectroscopy ($R\sim 20-100$) to study the atmospheres of giant exoplanets and brown dwarf companions, but the presence of clouds has often led to degeneracies in the retrieved atmospheric abundances (e.g. C/O, metallicity). This precludes clear insights into the formation mechanisms of these companions. The Keck Planet Imager and Characterizer (KPIC) uses adaptive optics and single-mode fibers to transport light into NIRSPEC ($R\sim 35,000$ in $K$ band), and aims to address these challenges with high-resolution spectroscopy. Using an atmospheric retrieval framework based on petitRADTRANS, we analyze KPIC high-resolution spectrum ($2.29-2.49~{}\mu$m) and archival low-resolution spectrum ($1-2.2~{}\mu$m) of the benchmark brown dwarf HD 4747 B ($m=67.2\pm 1.8~{}M_{\rm{Jup}}$, $a=10.0\pm 0.2$ au, $T_{\rm eff}\approx 1400$ K). We find that our measured C/O and metallicity for the companion from the KPIC high-resolution spectrum agree with that of its host star within $1-2\sigma$. The retrieved parameters from the $K$ band high-resolution spectrum are also independent of our choice of cloud model. In contrast, the retrieved parameters from the low-resolution spectrum are highly sensitive to our chosen cloud model. Finally, we detect CO, H${}_{2}$O, and CH${}_{4}$ (volume mixing ratio of log(CH${}_{4}$)=$-4.82\pm 0.23$) in this L/T transition companion with the KPIC data. The relative molecular abundances allow us to constrain the degree of chemical disequilibrium in the atmosphere of HD 4747 B, and infer a vertical diffusion coefficient that is at the upper limit predicted from mixing length theory.
††facilities: Keck (KPIC)††software: petitRADTRANS (Mollière et al., 2019), dynesty (Speagle, 2020), PyAstronomy (https://github.com/sczesla/PyAstronomy)
1 Introduction
The Keck Planet Imager and Characterizer (KPIC) is a new suite of instrument upgrades at Keck II, including a single-mode fiber injection unit (Mawet et al., 2017; Delorme et al., 2021) that feeds light into the upgraded NIRSPEC (Martin et al., 2018; López et al., 2020), enabling high-resolution spectroscopy (HRS111We will use HRS to abbreviate both high-resolution spectroscopy (the technique) and high-resolution spectra (the data) in this paper. The same is true for LRS: low-resolution spectra or spectroscopy.) at $R\sim 35,000$ in $K$ band. By using single-mode fibers to inject light from planets and brown dwarfs at high-contrast, KPIC provides suppression of the stellar point-spread function at the fiber input and a stable line spread function that is independent of incoming wavefront aberrations (Mawet et al., 2017; Wang et al., 2021b). By observing at high-resolution, we can further distinguish between star and planet light from their spectral differences (Wang et al., 2017; Mawet et al., 2017). Recently, Wang et al. (2021b) published the KPIC detections of HR 8799 c, d, e, demonstrating the ability of KPIC to detect molecular lines and measure the rotational line broadening of planets at high contrast ($\Delta K\approx 11$) and small separations ($\approx 0.4$”) from their host star.
The atmospheric composition of a substellar object holds a wealth of information about its formation, accretion, and evolutionary history, as well as fundamental physical processes that shape its atmosphere. It is therefore important to assess how well KPIC and other fiber-fed, high-resolution spectrographs (e.g. Subaru/REACH Kotani et al. 2020 and VLT/HiRISE Otten et al. 2021) can measure the atmospheric compositions of directly imaged planets and brown dwarfs. Specifically, previous studies of gas giant planet formation have highlighted the carbon-to-oxygen ratio (C/O) and metallicity (e.g. [C/H]) of the atmosphere as informative probes of formation history (e.g. Öberg et al., 2011; Madhusudhan, 2012; Piso et al., 2015). To first order, a companion with a C/O and metallicity similar to that of its host star is consistent with formation via gravitational instability in a protostellar disk or fragmentation in a molecular cloud, akin to binary star formation (Bate et al., 2002). On the other hand, differences between the companion and stellar C/O are suggestive of core accretion (Pollack et al., 1996) as the likely formation mechanism, and in that scenario, could be used to constrain where the companion formed in the disk relative to ice lines of major C- and O-bearing molecules (e.g. H${}_{2}$O, CO${}_{2}$, and CO). This picture can be complicated by a variety of effects such as the relative amount of solids incorporated into the planet’s atmosphere (e.g. Madhusudhan et al., 2014; Öberg & Bergin, 2016; Mordasini et al., 2016; Gravity Collaboration et al., 2020; Pelletier et al., 2021).
So far, atmospheric characterization of directly imaged companions has mostly relied on low-resolution spectroscopy (LRS) with resolving powers of $R\approx 20-100$. LRS is sensitive to continuum emission originating from the deepest observable layer of the atmosphere and modified by opacity sources further up. Many of these companions have temperatures warm enough for silicate clouds to condense in their atmospheres (Marley & Robinson, 2015), and there is much evidence that cloud opacity plays an important role in the LRS of directly imaged companions and brown dwarfs with L or L/T transition spectral types (e.g. Skemer et al., 2014; Burningham et al., 2017; Gravity Collaboration et al., 2020). However, due to our limited knowledge of cloud physics, a reliable assessment of atmospheric abundances from LRS could be fraught with degeneracies between clouds, the pressure-temperature profile, and chemical abundances (e.g. Burningham et al., 2017). In addition, the retrieval results can also be highly sensitive to systematics in different data sets that are combined to obtain a wider wavelength coverage (Wang et al., 2020a). More encouragingly, Mollière et al. (2020) report atmospheric abundances that are relatively robust to clouds and model choices, though Burningham et al. (2021) show that issues such as an unphysically small radius could persist despite improvements in cloud modeling and extensive wavelength coverage ($1-15\mu$m).
Recently, Wang et al. (2022) presented the first atmospheric free retrievals at high-resolution for a directly imaged companion. They studied the L-type brown dwarf HR 7672 B ($T_{\rm eff}\approx 1800$ K) using KPIC HRS and near-infrared photometry, and measured carbon and oxygen abundances that are consistent within $<1.5\sigma$ to that of its host star. In this paper, we present a detailed atmospheric study of HD 4747 B using both KPIC HRS ($K$ band) and archival low-resolution spectra (LRS) from 1-2.2 $\mu$m that we re-extract in a uniform manner. While the KPIC HRS resolves individual molecular lines and conains direct information about a companion’s atmospheric abundances, LRS provides spectral shape and luminosity measurements, which has the potential to complement the HRS.
Compared to HR 7672 B, HD 4747 B is a colder L/T transition object ($T_{\rm eff}\approx 1400$ K) with strong evidence for clouds and a similar color to directly imaged planets such as HR 8799 c,d,e (Crepp et al., 2018; Peretti et al., 2019). Like HR 7672 B, the wealth of prior knowledge available for HD 4747 B makes it a valuable benchmark object to test whether we can make robust inferences with spectroscopic data. First, we are able to precisely measure the dynamical mass of HD 4747 B (§ 2.2). Mass is a fundamental quantity that is poorly constrained for most directly imaged companions (Bowler, 2016). Furthermore, given its high mass, HD 4747 B is expected to have formed via direct gravitational collapse in the same cloud or disk as its host star, which means that we can assume chemical homogeneity: the brown dwarf and primary star should share the same chemical composition. Finally, with the companion mass, observed luminosity, and stellar age, we can independently estimate the brown dwarf’s radius from evolutionary models.
In this paper, we use the open-source radiative transfer code petitRADTRANS (Mollière et al., 2019, 2020) to fit the HRS and LRS for HD 4747 B in a retrieval framework. The main goals of our study are to measure the atmospheric composition of this brown dwarf companion using both the HRS and LRS, and to present a detailed characterization of its atmosphere, including constraints on clouds, chemical equilibrium or disequilibrium, and the detection of CH${}_{4}$. In this process, we also explore the relative advantages and disadvantages of HRS versus LRS.
This paper is organized as follows: in § 2, we summarize the system properties including our mass measurement for HD 4747 B. Our spectroscopic data and data reduction procedure is described in § 3. We then discuss our spectral analysis framework in § 4. We present individual and joint retrievals of the HRS and LRS in § 5, § 6 and § 7, respectively. We summarize the lessons learned in § 8, and conclude in § 9.
2 System properties
2.1 Host star
In this section, we summarize relevant properties of the host star. HD 4747 is a main-sequence, solar-type star located $\approx 19$ parsec away based on its Gaia eDR3 parallax (Brown et al., 2021). Chromospheric emission in the Ca II H&K lines are visible in the stellar spectrum (log$R_{\rm HK}=-4.72\pm 0.02$), which Peretti et al. (2019) used to derive an age of $2.3\pm 1.4$ Gyr from the age-log$R_{\rm HK}$ calibration of Mamajek & Hillenbrand (2008). This agrees with the gyro-chronological age estimate of $3.3^{+2.3}_{-1.9}$ Gyr from Crepp et al. (2018). These studies also converged on $T_{\rm eff}$ around 5300–5400 K, and a surface gravity log($g$) of 4.5–4.65. Of particular relevance to this study are the C/O ratio and metallicity of the host star, since we expect these to be roughly similar to those of the brown dwarf. HD 4747 is found to have a sub-solar metallicity, with [Fe/H]=$-0.23\pm 0.05$ from Peretti et al. (2019) and [Fe/H]=$-0.22\pm 0.04$ from Crepp et al. (2018). Previous studies including Brewer et al. (2016) and Peretti et al. (2019) also measured the elemental abundances for the star, but either did not take into account non-local thermodynamic equilibrium (LTE) effects on their oxygen abundances (Amarsi et al., 2019) or do not quote error bars. We instead carry out a new analysis using the method described in Kolecki et al. (2021) to derive the abundances for different elements, and correct the results to account for 3D non-LTE effects (Amarsi et al., 2019) on the results. For this analysis, we used an archival spectrum from FEROS (Kaufer et al., 1997) which covers 350-920 nm at $R=48,000$. Using this spectrum, we measure the equivalent widths of absorption lines and compare them to model stellar atmospheres in an iterative approach using the MOOG code (Sneden, 1973). From our derived carbon and oxygen abundances, we find C/O=$0.48\pm 0.08$. The iron abundance is [Fe/H]=$-0.30\pm 0.5$, in agreement with previous studies.
Since Fe condenses out for temperatures below $\approx 1800$ K (Marley & Robinson, 2015), it is not a relevant gaseous absorber in the photosphere of HD 4747 B. Therefore, the more useful metrics for comparison are C and O. From our analysis above, we find [C/H]$=-0.08\pm 0.06$ and [O/H]$=-0.02\pm 0.04$ for the host star. [C/H] is defined as $\log_{10}({N_{\rm C}/N_{\rm H}})_{\rm star}$ - $\log_{10}({N_{\rm C}/N_{\rm H}})_{\rm sun}$, where $N_{\rm C}$ and $N_{\rm H}$ are the number fraction of C and H respectively. [O/H] is defined similarly. We adopt Asplund et al. (2009) as our solar reference in order to be consistent with petitRADTRANS, which we use to model the atmosphere of HD 4747 B.
2.2 Orbit and Dynamical mass
The orbit and mass of HD 4747 B have been measured by several studies using relative astrometry from Keck/NIRC2, host star radial velocities (RV) from Keck/HIRES, and Gaia-Hipparcos absolute astrometry (Brandt et al., 2019; Xuan & Wyatt, 2020). Here, we take advantage of 23 yr of RV observations published in Rosenthal et al. (2021) and the improved precision of the Hipparcos-Gaia Catalog of Accelerations (HGCA) (Brandt, 2021) based on Gaia eDR3 (Brown et al., 2021) to update the orbit and mass of HD 4747 B. HD 4747 B shows significant proper motion anomalies (PMa) in both the Gaia and Hipparcos epochs, with S/N of $77.2$ and $9.1$, and the position angle and amplitude of the PMa is consistent with being induced by the brown dwarf companion. For the relative astrometry, we use data points tabulated in Brandt et al. (2019), except for the two GPI epochs measured by Crepp et al. (2018), which we replaced with our new measurements from § 3.2. We choose not to use the companion RV as measured by KPIC for this fit, because it does not appreciably improve our already well-constrained orbital solution.
To fit the relative astrometry, radial velocity, and absolute astrometry from Gaia and Hipparcos together, we use the orvara package (Brandt et al., 2021b), which is designed to jointly fit these types of data and takes into account the Gaia and Hipparcos astrometry at the epoch astrometry level using htof (Brandt et al., 2021a). We use the priors listed in Table 4 of Brandt et al. (2021b) for the fitted parameters. The posterior is sampled using the parallel-tempering MCMC sampler (Vousden et al., 2016), a fork of emcee (Foreman-Mackey et al., 2013) over 50000 steps with 10 temperatures and 100 walkers. The fits converged as determined by visual inspection of the chains, and we discarded the first ten percent as burn-in. In orvara, the system parallax and other linear parameters are analytically marginalized out to speed up the fits.
The resulting orbit and mass measurements are tabulated in Table 1, while the model fits are shown in Appendix A. We find a companion mass of $m=67.2\pm 1.8~{}M_{\rm{Jup}}$, which is consistent with previous values, but more precise. We checked the orvara results with a second fit where we model the Gaia and Hipparcos astrometry using the methodology in Xuan & Wyatt (2020). This gives $m=67.1\pm 2.0~{}M_{\rm{Jup}}$, consistent with the orvara result. Furthermore, the companion mass and orbital parameters we find are also consistent with results from orbit fits that only use RV and imaging data (no Gaia-Hipparcos astrometry) from Peretti et al. (2019) and Crepp et al. (2018). We adopt the companion mass from our orvara fit for the spectral analysis in this paper.
3 Spectroscopic data
3.1 High-resolution spectroscopy
3.1.1 KPIC Observations
We observed HD 4747 B on UT 2020 September 28 with Keck/NIRSPEC. The data were collected using the first version of the KPIC fiber injection unit (FIU) (Delorme et al., 2021). The FIU is located downstream of the Keck II adaptive optics system and is used to inject light from a selected object into one of the single mode fibers connected to NIRSPEC. We obtain spectrum in K band, which is broken up into nine echelle orders from 1.94-2.49 $\mu$m. The observing strategy is identical to that of Wang et al. (2021b). In short, we placed the companion on the fiber with the highest throughput and acquired six exposures of 600 seconds each, for a total integration time of 1 hour. The relative astrometry of the companion was computed using whereistheplanet.com (Wang et al., 2021a), based on data in Peretti et al. (2019). For calibration purposes, we acquired a pair of 60 second exposures of the host star before observing the companion, and a pair of 60 second exposures of a telluric standard star (HIP 6960) after the companion exposures so as to share nearly the same airmass. Using exposures on the host star, we calculated an end-to-end throughput from the top of the atmosphere to the detector of $1.8-2.0\%$ during the observations.
3.1.2 Data reduction
To extract the spectra from the raw data, we follow the procedure outlined in Wang et al. (2021b), which the KPIC team has implemented in a public Python pipeline.222https://github.com/kpicteam/kpic_pipeline The images for all objects were reduced in the same way. First, we removed the thermal background from the images using combined instrument background frames taken during daytime. As shown in Wang et al. (2021b), the thermal background of our data is dominated by the warm optics rather than the sky background. We also remove persistent bad pixels identified using the background frames. Then, we use data from the telluric standard star to fit the trace of each column in the four fibers and nine spectral orders, which give us the position and standard deviation of the PSF (in spatial direction) at each column. The trace positions and widths were additionally smoothed using a cubic spline in order to mitigate random noise. We adopt the trace locations and widths as the line spread function (LSF) positions and widths in the dispersion dimension.
For every frame, we then extracted the 1D spectra in each column of each order. To remove residual background light, we subtracted the median of pixels that are at least 5 pixels away from every pixel in each column. Finally, we used optimal extraction to sum the flux using weights defined by the 1D Gaussian LSF profiles calculated from spectra of the telluric star.
The extracted spectra have a median signal-to-noise ratio (S/N) of $\approx 8$ per pixel element, which has a typical width of $0.2~{}\textup{\AA}$, and consists of a mixture of light from the brown dwarf companion and stellar speckles. The S/N of KPIC is optimized for wavelengths around $2.3~{}\mu$m, where CO has a series of strong absorption lines. For our analysis, we use three spectral orders from 2.29-2.49 $\mu$m, which contain the strongest absorption lines from the companion and have relatively few telluric absorption lines. Note that the three spectral orders have gaps in between them, so we have data over $\approx 0.13~{}\mu$m (instead of $0.2~{}\mu$m; see Fig. 3).
As a preliminary analysis, we cross-correlate our KPIC spectra with single-molecule templates assuming $T_{\rm eff}=1400$ K and log($g$)=5.5 from the Sonora model grid (Marley et al., 2021). In short, we estimate the maximum likelihood value for both the single-molecule companion flux and speckle flux in the data as a function of RV (radial velocity) shift using the method described in Wang et al. (2021b), which is based on Ruffio (2019). We find that H${}_{2}$O and CO are detected with S/N of 8.5 and 13.5 respectively (Fig. 1). CH${}_{4}$ is not detected with statistical confidence in this crude analysis, but we present evidence for a weak CH${}_{4}$ detection in § 5.4.
3.2 Low-resolution spectroscopy
3.2.1 Gemini Planet Imager IFS
The Gemini Planet Imager (GPI) observed HD 4747 B on UT 2015 December 24 and 25, in the $K_{1}$ (1.90-2.19 $\mu$m, $R=66$) and $H$ (1.50-1.80 $\mu$m, $R=46.5$) bands, respectively, and the data were published in Crepp et al. (2018). After doing some fits to the published spectrum, we found that the average flux levels of the $K_{1}$ and $H$ bands are inconsistent, and the error bars appear to be significantly over-estimated.
We therefore re-extracted the GPI spectrum using the pyKLIP package (Wang et al., 2015), which models a stellar point spread function (PSF) with Karhunen-Loève Image Processing (KLIP, also known as Principal Component Analysis) following the framework in Soummer et al. (2012) and Pueyo (2016). We tested various model choices to minimize the residuals after stellar PSF subtraction while preserving the companion signal. A key parameter we tuned was the number of Karhunen-Loève (KL) modes. KL modes represent an orthogonal basis for patterns in the images that are used to model the stellar PSF. We chose 5 and 12 KL modes to subtract the stellar PSF in the $H$ and $K_{1}$ band data, respectively. After subtracting the stellar PSF, we first extracted the companion’s relative astrometry in terms of separation and position angle, which are reported in Appendix B (Table 5). Then, we extracted the flux at the companion’s determined location as a function of wavelength, which gave us the raw spectrum. Note that rather than using spectral differential imaging (SDI) to subtract the stellar PSF, we only used angular differential imaging (ADI). For a bright companion like HD 4747 B, ADI is more than sufficient to properly remove the PSF of the star given sufficient parallactic angle rotation.
To flux-calibrate the raw spectrum, we used the satellite spot flux ratios333https://www.gemini.edu/instrumentation/gpi/capability to find the companion-to-star flux ratio. To obtain the observed flux density of the companion, we empirically determined the flux scaling factor $R^{2}/d^{2}$ by fitting a PHOENIX model (Husser et al., 2013) of the star ($T_{\rm eff}=5400$, log($g$)=4.5, and [Fe/H] = -0.5) using the star’s 2MASS J, H, K (Cutri et al., 2003) and the Gaia G band magnitudes (Riello et al., 2021). The zeropoint fluxes and filter transmission of the photometric bands are downloaded from the SVO Filter Service444http://svo2.cab.inta-csic.es/theory/fps/ and the Gaia website.555https://www.cosmos.esa.int/web/gaia/edr3-passbands To obtain measurement uncertainties, we injected 20 fake companions at the same separation and equally spaced position angles in the data, and repeated the same spectral extraction process. We avoided using the fake injections that were within $20\degree$ of the real companion to avoid biasing the fluxes. We inflated the uncertainties on the extracted spectra by $2.5\%$ to account for errors in the stellar flux calibration. The value of $2.5\%$ is estimated by comparing our empirically computed flux scaling factor with the value of $R^{2}/d^{2}$ of the star (using the radius from Peretti et al. 2019 and the Gaia parallax).
3.2.2 SPHERE IFS
HD 4747 B was observed on UT 2016 December 12 and 2017 September 28 with the Spectro-Polarimetric High-contrast Exoplanet Research (SPHERE; Beuzit et al. 2019). The SPHERE Integral Field Spectrograph (IFS) (Claudi et al., 2008) collects data in the $YH$ band from 0.95-1.6 $\mu$m ($R=29$). The extracted spectra was published in Peretti et al. (2019), but is not available. We therefore reduced the raw data using the SPHERE pipeline (Vigan, 2020), and performed a similar post-processing procedure with pyKLIP as described above for the GPI spectra. The only difference is that we needed to use ADI+SDI to perform PSF-subtraction for the SPHERE IFS data, which did not have enough parallactic angle rotation (only $\approx 0.2\degree$). For the SPHERE IFS data, flux calibration is based on unocculted observations of the host star. We chose to use the 2017 data for our analysis since it was taken under much better observing conditions and yields slightly higher spectral S/N than the 2016 data, despite shorter integration times.
3.2.3 Results and comparison with previous LRS
Our newly extracted GPI and SPHERE spectra are plotted in black circles and squares respectively in Fig. 2, and available in Appendix B (Table 4). The absolute flux scaling of our GPI spectrum agrees well with the published spectrum in gray from Crepp et al. (2018), but the uncertainties are much smaller. The shape of our new SPHERE spectrum also agrees well with that in Peretti et al. (2019). From the PSF-subtracted images, the brown dwarf companion is detected with a median S/N per wavelength bin of $\approx 61$ and $\approx 26$ in the GPI $H$ and $K_{1}$ bands, and $\approx 20$ in the SPHERE data. When comparing the extracted spectrum to that of field brown dwarfs from the SPEX library (Burgasser, 2014) in Fig. 2, we find that the newly extracted GPI spectrum is in better agreement compared to the previously published spectrum. As in Crepp et al. (2018) and Peretti et al. (2019), we find a spectral type near the L/T transition (the best matching spectra were from a L9.5 and T0 dwarf). The SPHERE IFS spectrum increases our wavelength coverage by a factor of $\sim 2$, which we find is important for constraining model atmosphere parameters in our fits to the LRS.
4 Spectral analysis
4.1 Forward modeling the KPIC high-resolution spectrum
Here, we briefly describe the framework to forward model and fit the HRS from KPIC, which follows Wang et al. (2021b). When a companion of interest is aligned with one of the KPIC fibers, the companion light and a fraction of light from the host star’s speckle field are injected into the fiber. At the projected separation of HD 4747 B ($\approx 0.6$″), we find the speckles are roughly the same brightness as our companion ($K_{s}\approx 14.4$ from Crepp et al. 2016). Furthermore, the light is transmitted through Earth’s atmosphere and modulated by the instrument optics. Thus, we build two forward models (one for the companion, one for speckles) and jointly fit them as a linear combination. Below we detail how we generate each of the model components.
The companion spectral templates are generated with petitRADTRANS. We shift the templates in wavelength space to fit for the radial velocity. Then, we rotationally broaden the templates by a projected rotation rate $v\sin{i}$ using the fastRotBroad function in PyAstronomy (Czesla et al., 2019), and convolve the templates with the instrumental LSF. The effect of limb darkening is included in petitRADTRANS by integrating intensities along multiple angles between the ray and atmospheric normal.
Next, we multiply the companion model by the telluric response function, which characterizes the atmospheric transmission as a function of wavelength and includes telluric absorption lines. The telluric model is calculated by dividing the spectrum of the standard star (HIP 6960) by a PHOENIX stellar model with matching properties ($T_{\rm eff}=9200$ and log($g$)=4.0).
To model the speckle contribution to the data, we use on-axis observations of the host star taken before the companion exposures. These observations are reduced in the same way as the companion spectra, but have much higher S/N. Unlike the companion models, the host star observations are already modulated by telluric transmission.
The last step is to remove continuum variations. The KPIC spectra are not flux-calibrated and contain a smoothly varying continuum due to stellar speckles and wavelength-dependent atmospheric refraction. Therefore, we apply high-pass filtering with a median filter of 100 pixels ($\sim 0.002~{}\mu$m) on both the data and models to subtract the continuum following Wang et al. (2021b). To determine the optimal filter size, we carried out a series of injection-recovery tests, and found that $\sim 100$ pixels is best at recovering weak molecular signals for our data set. Larger filter sizes (e.g. 200 pixels or more) do not remove the continuum sufficiently, and smaller filter sizes (50 pixels or less) tend to be overly aggressive at removing weak molecular signals.
Finally, we flux-normalize both the companion and stellar models and multiply them by different flux scaling factors, which are fitted parameters. The flux scales are in units of counts as measured by the NIRSPEC detector. After scaling, the companion and speckle models are added and the same high-pass filter is applied on the final model before fitting it to the data.
4.2 Atmospheric retrieval setup
We implement a ‘retrieval’ framework based on petitRADTRANS to model the data, which means that we freely retrieve the chemical abundances, vertical temperature structure, and cloud properties from the data. Previous studies have used retrievals to model HRS of self-luminous exoplanets and brown dwarfs (e.g. Burningham et al., 2017; Mollière et al., 2020), and show that it can be a powerful alternative to fitting self-consistent grid models, which solve for the abundances and temperature profiles with physical assumptions such as chemical equilibrium. The retrieval approach allows more flexibility to fit the data and can potentially provide much more detailed information about the atmospheric properties, with the caveat that it is important to check for physical plausibility of the models since retrievals need not be self-consistent.
In our main set of retrievals, we fit for the chemical abundances in terms of C/O and atmospheric metallicity [C/H]666We denote the atmospheric metallicity as [C/H] because we are only sensitive to C- and O-bearing molecules in this brown dwarf’s atmosphere. along with a quench pressure (where the chemical timescale of a certain reaction is equal to the mixing timescale) to allow for disequilibrium chemistry, the temperature profile (§ 4.2.1), the cloud structure (§ 4.2.2), and other parameters such as the radius. We denote these quenched chemistry retrievals to distinguish from free retrievals where the abundances of each gas species is fit independently. Each component of the model is described in the subsections below.
We use the correlated-k and line-by-line opacity sampling methods in petitRADTRANS for the low-resolution and high-resolution retrievals respectively. For high-resolution, we include opacities for CO, H${}_{2}$O, CH${}_{4}$, NH${}_{3}$, and CO${}_{2}$, and for low-resolution we additionally include Na and K. This is because the alkali lines have wings which affect the $\sim 1~{}\mu$m portion of the LRS, while their opacities are negligible over the portion of $K$ band covered by our HRS. We repeated our baseline HRS retrieval with Na and K included and found that the addition of these two species did not influence the results or improve the fit.
Because the native high-resolution opacities are at $R=10^{6}$, much higher than the resolution of our HRS resolution ($R\approx 35,000$), we down-sampled the opacity tables by a factor of six in order to speed up the retrievals (by roughly the same factor) and reduce the corresponding computational cost. We checked that the maximum deviation in synthetic spectra obtained by using the down-sampled opacities relative to the full-resolution opacity model is $<5\%$ of the minimum HRS error bars. In addition, we repeated our fiducial HRS retrieval with the native opacities ($R=10^{6}$) and found that it yielded the same results. We re-binned the correlated-k opacities to $R=200$ for our fits to the LRS, which has a maximum resolution of $66$. We also repeated our fiducial LRS retrieval at the native $R=1000$ opacities and found the results are fully consistent.
4.2.1 Temperature structure and chemistry
We retrieve the pressure-temperature (PT) profile of the brown dwarf between $P=10^{-4}-10^{3}$ bars, which sets the vertical extent of the atmosphere. We use the P-T profile parametrization from Mollière et al. (2020) which has six free parameters. The spatial coordinate is an optical depth $\tau=\delta P^{\alpha}$, where $\delta$ and $\alpha$ are the first two parameters. The atmosphere then consists of a high altitude region (top of atmosphere to $\tau=0.1$) fitted with three temperature points equi-distant in log pressure, a middle radiative region ($\tau=0.1$ to radiative-convective boundary) which uses the Eddington approximation with $T_{0}$ as the ‘internal temperature’, and a lower region (radiative-convective boundary to bottom of atmosphere), which is set to follow the moist adiabatic temperature gradient once the atmosphere becomes unstable to convection (Mollière et al., 2020). We ignore stellar irradiation as a source of heat because the total incident energy on HD 4747 B at periastron ($\approx 2.7$ au) is approximately four orders of magnitude less than its luminosity, which is dominated by the brown dwarf’s internal energy.
In our quenched chemistry retrievals, the C/O, [C/H], and P-T profile determine the equilibrium chemical abundances (mass fractions of molecules) as a function of pressure, by interpolating the chemical equilibrium table from Mollière et al. (2020). The opacities we include in the models are listed in § 4.2. In petitRADTRANS, the abundances of all metals except oxygen are assumed to scale together such that [C/H] = [Si/H] = [N/H], etc. Then, C/O and [C/H] are combined to set the oxygen abundance (Mollière et al., 2020). We use Asplund et al. (2009) as our reference for the solar metallicity in these models.
Finally, we include a quench pressure $P_{\rm quench}$ which fixes the abundances of H${}_{2}$O, CO, and CH${}_{4}$ where $P<P_{\rm quench}$ using the equilibrium values found at $P_{\rm quench}$ (Zahnle & Marley, 2014; Mollière et al., 2020). The inclusion of $P_{\rm quench}$ allows for the possibility of disequilibrium chemistry, which occurs where the atmospheric mixing timescale is shorter than the chemical reaction timescale. We only include a quench pressure for the net reaction between H${}_{2}$O, CO, and CH${}_{4}$ because these molecules are the only ones detectable in our KPIC HRS (see § 5.4 for the CH${}_{4}$ detection), and chemical kinetics modeling indicates that the abundances of these three molecules are closely linked to each other by a series of reactions (e.g. Moses et al., 2013). In summary, our quenched chemistry retrievals use C/O, [C/H] and $P_{\rm quench}$ to set the abundances of each gas species for a given P-T profile.
4.2.2 Clouds
Crepp et al. (2018) and Peretti et al. (2019) analyzed LRS for HD 4747 B and found evidence for a cloudy atmosphere. We summarize their results in Table 2 along with our new measurements. In this study, we consider both clear and cloudy models in order to explore the sensitivity of our retrieved abundances to the assumed cloud properties. For our cloudy model, we use the EddySed model from Ackerman & Marley (2001) as implemented in petitRADTRANS (Mollière et al., 2020). In this model, the cloud particles both absorb and scatter the outgoing photons from the atmosphere according to measured optical properties (Mollière et al., 2019). The cloud particles can be either crystalline or amorphous, and the opacities of the clouds are computed assuming either homogeneous and spherical particles, modeled with Mie theory, or irregularly-shaped cloud particles, modeled with the Distribution of Hollow Spheres (DHS) (Min et al., 2005; Mollière et al., 2019).
For HD 4747 B, we consider models with two different cloud species (MgSiO${}_{3}$ and Fe) and properties (amorphous or crystalline particles). We choose to focus on MgSiO${}_{3}$ and Fe for several reasons. First, the condensation curves of these two species intersect the thermal profile of a $T_{\rm eff}=1400$ K, log($g$)=5.5 object from the Sonora atmospheric model (Marley et al., 2021) at $\sim 10-50$ bars. While the Sonora model is cloudless, it provides a rough estimate of which cloud species are relevant. Second, recent theoretical work has shown that MgSiO${}_{3}$ is expected to be the most important cloud species for substellar objects with $T_{\rm eff}>950$ K due to its low nucleation energy barriers and the relatively high elemental abundances of Mg, Si, and O (Gao et al., 2020). Finally, studies using mid-IR spectroscopy from Spitzer have found direct evidence for a MgSiO${}_{3}$ absorption feature at $\sim 10~{}\mu$m in field brown dwarfs (Cushing et al., 2006; Luna & Morley, 2021), and specifically amorphous MgSiO${}_{3}$ (Burningham et al., 2021). Although MgSiO${}_{3}$ and Fe clouds do not have distinct features in the near-IR, they still impact the near-IR spectrum by contributing a wavelength-dependent opacity. Our baseline model uses amorphous MgSiO${}_{3}$ modeled with Mie theory (abbreviated MgSiO${}_{3}$, ‘am’) for the clouds. In addition, we also consider models with MgSiO${}_{3}$, ‘cd’, which assumes crystalline cloud particles modeled with DHS, as well as models with two cloud species (MgSiO${}_{3}$ + Fe) for the LRS.
4.2.3 Methane opacities
Given that HD 4747 B is located near the L/T transition for brown dwarfs, we might expect to observe methane in its atmosphere. Previous $L$ band studies have detected methane in field brown dwarfs with spectral types as early as mid-L, or up to $T_{\rm eff}\approx 1800$ K (Noll et al., 2000; Johnston et al., 2019). In this study, we adopted the HITEMP CH${}_{4}$ line list from Hargreaves et al. (2020), which we convert into opacities following the petitRADTRANS documentation. When cross-correlating a model generated with the HITEMP CH${}_{4}$ opacities with a late T dwarf, we obtained a CCF S/N of $\approx 15$, in comparison to $\approx 5$ when cross-correlating with a model generated from the default CH${}_{4}$ opacities from ExoMol (Yurchenko & Tennyson, 2014) in petitRADTRANS.
4.2.4 Additional fit parameters
petitRADTRANS computes the flux density as emitted at the surface of the object. For the LRS, we scale the model by the distance and companion radius, where the radius is another free parameter, and the distance is taken from the Gaia eDR3 parallax (Brown et al., 2021). For the HRS, we also fit the companion’s radial velocity and $v\sin{i}$, as well as an error multiple term to account for any underestimation in the data uncertainties.
Due to imperfect starlight subtraction in the spectral extraction process, we found that our LRS likely still contains correlated noise from the wavelength-dependence of speckles, as has been noted by several previous studies on high-contrast companions (e.g. De Rosa et al., 2016; Samland et al., 2017; Currie et al., 2018; Wang et al., 2020b, 2021c). This is evident in the residual frames, where we can see speckles at 5-20% of the companion intensity in the PSF-subtracted images. We therefore adopt a Gaussian process with a squared exponential kernel to empirically estimate the correlated noise in the GPI H, K and SPHERE YJH bands when fitting models to the data. Following Wang et al. (2020b), we assume that our extracted error bars contain a fraction $f_{\rm amp}$ of correlated noise, and $1-f_{\rm amp}$ of white noise, and fit for $f_{\rm amp}$ and the scale of correlation $l$. This adds $2\times 3=6$ additional parameters to the retrievals.
As an alternative model, we also tried fitting the LRS with error inflation terms and flux scaling factors for the SPHERE and GPI spectra along the lines of Mollière et al. (2020), but found that our results were very sensitive to our choice of prior for the flux scaling factor. We conclude that our GP model is better suited to account for correlated noise from speckles, and use it in all LRS fits presented in this work.
4.3 Priors
We adopt uniform or log-uniform priors for all model parameters except for the mass, for which we use a Gaussian prior of $67.2\pm 1.8~{}M_{\rm{Jup}}$ from the dynamical mass measurement (§ 2.2). For the parametric P-T profile parameters, we exclude profiles that contain temperature inversions, as the heat budgets of widely separated companions are dominated by their internal luminosities. For the companion’s radius, we use a uniform prior between 0.6 - 1.2 $R_{\rm{Jup}}$. When including a quench pressure, we use a log-uniform prior from $10^{-4}$ - $10^{3}$, which is the full pressure range of our models. The priors for all retrieval parameters are tabulated in Appendix C.
4.3.1 Model fitting with nested sampling
We use nested sampling as implemented by dynesty (Speagle, 2020) to find the posterior distributions for the model parameters. Specifically, we use 200 live points and adopt the stopping criterion that the estimated contribution of the remaining prior volume to the total evidence is less than 1%. We repeated a few retrievals using 1000 live points and found the evidence remains roughly the same, implying the fits have converged when using 200 live points.
One advantage of adopting nested sampling is that we can use the Bayesian evidence from each fit to calculate the Bayes factor $B$, which assesses the relative probability of model $M_{2}$ compared to $M_{1}$. We will use the Bayes factor to compare different models throughout this paper to determine whether a given $M_{2}$ is justified over $M_{1}$. In Table 2, we take a baseline model (MgSiO${}_{3}$, am) to be $M_{1}$ and compare other models to it. Based on Jeffreys (1983), a model with 100 times lower $B$ than the model with the highest $B$ can be ‘decisively’ rejected. $B$ of $\lesssim 10$ is considered weak evidence for preferring one model over the other. We first run retrievals with only the HRS (§ 5), only the LRS (§ 6), as well as joint retrievals with both HRS and LRS (§ 7).
5 High-resolution retrievals (KPIC)
5.1 Overview
From our HRS retrievals of HD 4747 B, we find that both clear and cloudy models yield consistent results for the atmospheric parameters (abundances, temperature structure, quenching) and bulk properties (radius, radial velocity, spin). A few selected parameters are plotted in Fig. 4 and tabulated in Table 2. The insensitivity of the HRS retrieval results to clouds, a major finding of this paper, is discussed in § 5.2. In Fig. 3, we plot the data, a best fit model, and residuals for the baseline HRS retrieval. We report values from this retrieval as the final results of this paper, with selected parameters shown in the first row of Table 2 and joint posterior distributions in Appendix C. We also plot the contribution from the planet and star separately at their best-fit flux levels. We compute the auto-correlation function of the residuals and find that there is no evidence for correlated noise or strong systematics. Unless otherwise specified, we quote results from the baseline EddySed cloud model (MgSiO${}_{3}$, am). See Appendix C for the posterior distributions of other parameters from our baseline model.
To make sure that we are fitting the correct signal, we check the RV and flux level of the companion. From our orbital posteriors for HD 4747 B, the expected RV shift on the night of our HRS observation is $15.0\pm 0.1$ km/s in the Earth’s reference frame, which is a combination of the system barycenter velocity, the Earth’s relative velocity with respect to HD 4747, and the companion’s orbital velocity. The fitted RV of $15.0\pm 0.4$ km/s agrees perfectly with this value (see Fig. 4). In addition, the companion flux level in the spectral orders from 2.29 to 2.49 $\mu$m is $85\pm 10$ counts, comparable to the speckle flux levels in these orders. Taking into account the difference in wavelengths and the difference in integration time (600 s for the companion, 60 s for the on-axis star), we estimate that our measured companion flux corresponds to $\Delta K_{s}=8.3\pm 0.3$ mag, which is within $3\sigma$ of the photometric $\Delta K_{s}=9.05\pm 0.14$ mag reported by Crepp et al. (2018). The agreement between these contrast values are reasonably good given the time-varying throughput of KPIC (Delorme et al., 2021), and the fact that we subtract out the continuum with high-pass filtering, effects which complicate a direct flux comparison.
Fig. 4 also shows the projected spin rate $v\sin{i}=13.2^{+1.4}_{-1.5}$ km/s, which is comparable to the rotation rates observed for field brown dwarfs with similar spectral types (e.g. Konopacky et al., 2012). We also plot the retrieved quench pressure $P_{\rm quench}$ in Fig. 4, which indicates that the chemical reaction timescale becomes longer than the vertical mixing timescale at pressures lower than $P_{\rm quench}$. Thus, disequilibrium chemistry is clearly affecting the atmosphere (see § 5.3 for details).
We compute $T_{\rm eff}$ by sampling from our posterior to generate low-resolution models over a large wavelength range (0.5 to 30 $\mu$m) and calculating the integrated flux. We then solve for $T_{\rm eff}$ using the Stefan-Boltzmann law. When computing $T_{\rm eff}$, we include opacities from Na and K, which are important sources of opacity near visible wavelengths. As shown in Table 2, the retrieved radius and $T_{\rm eff}$ from HRS have broad distributions, which reflect the relatively weak luminosity constraints from the HRS (log($L_{\rm bol}/L_{\odot}$)=$-4.33^{+0.23}_{-0.25}$). This is because the HRS is not flux-calibrated and we remove the continuum in our fits. Comparing to values of radius and $T_{\rm eff}$ from previous work based on LRS (Crepp et al., 2018; Peretti et al., 2019), our retrieved values from the HRS retrievals are consistent at the $1-2\sigma$ level (see Table 2). We discuss the constraints on these parameters from the LRS in § 6.2.
We compare our retrieved [C/H], [O/H], and C/O with that of the host star (see § 2.1) in Fig. 5. Our retrieved C abundance agrees well with the host star value, while the O abundance is lower by about $1\sigma$. This results in our retrieved C/O for the companion being higher by about $2\sigma$ compared to the stellar value. Here and elsewhere in the paper, we compute the ‘$\sigma$ difference’ between two measurements by dividing the difference in the two median values by the quadrature sum of the uncertainties from both measurements. We discuss the implications of our measured abundances for HD 4747 B in § 8.4.
5.2 Why are our KPIC HRS insensitive to clouds in HD 4747 B?
Clouds represent a significant source of uncertainty in many published models of substellar atmospheres (e.g. Burningham et al., 2017; Wang et al., 2020a). However, we find that the retrieved parameters from our KPIC HRS are insensitive to the choice of cloud model for HD 4747 B. As shown in Fig 4, the posteriors for radius, RV, $v\sin{i}$, C/O, [C/H], and quench pressure are nearly identical across the various models. The same is true for other parameters.
Table 2 shows that the different cloud models are indistinguishable for the KPIC HRS; the clear model fits as well as the cloudy models, with $B=0.61$, which does not pass the threshold of $B=10/0.1$ to be considered statistically favored/disfavored. This indicates that the data can be fitted adequately without clouds; indeed the cloud parameters for the EddySed models span their respective prior ranges almost uniformly as shown in Appendix C. As we will discuss in § 6, the LRS show that the atmosphere of HD 4747 B is cloudy. This implies that cloud opacity must be minimal at the pressures probed by our HRS.
To understand this, we plot in Fig. 6 the retrieved P-T profiles (black and blue lines), cloud condensation curves (dashed lines), and emission contribution functions. The left and right panels show results from the HRS and LRS retrievals, respectively. The emission contribution function for HRS shows that we are sensitive to pressures ranging from a few bars, where the continuum forms, up to $\approx 10^{-2}$ bars in the cores of individuals lines. Note that the contribution functions use the wavelength axes on the top. In the EddySed model, the cloud base is set at the intersection of the P-T profile and a given cloud condensation curve (dashed lines). For MgSiO${}_{3}$, this corresponds to a pressure of $\approx 10-20$ bars when using our HRS-retrieved P-T profile. As the cloud mass fraction drops exponentially above the cloud base in the EddySed model (controlled by $K_{\rm zz}$ and $f_{\rm sed}$), we find that the cloud opacity decreases to negligible levels by the time we reach pressures of a few bars where the continuum forms. For this reason, we do not consider models with Fe clouds in our HRS retrievals, since the Fe cloud base forms even deeper than that of MgSiO${}_{3}$.
Therefore, our KPIC HRS are insensitive to clouds because we cover both a relatively small wavelength range (2.29-2.49 $\mu$m) and a range where molecular opacities from H${}_{2}$O, CO, and CH${}_{4}$ are significant. The small wavelength range means that the cloud opacity is effectively constant in wavelength. The strong molecular opacity in HRS allows us to resolve many individual absorption lines and obtain good constraints on the atmospheric composition for molecules present in this region of the spectrum. The opacity of these molecules decrease at shorter wavelength due to decreasing excitation cross sections, so the continuum shifts to higher pressures (deeper down) at shorter wavelengths. This effect is visible in the LRS contribution function, where close to $1~{}\micron$, the emission originates from roughly the same pressure as the MgSiO${}_{3}$ cloud base, making the $y$ and $J$ bands particularly sensitive to clouds (see Fig. 10).
Could the KPIC HRS be affected by clouds at lower pressures (higher altitudes) than predicted by the EddySed model? Several studies have found that including clouds at lower pressures than predicted by EddySed produces better fits to mid-IR spectra of isolated brown dwarfs (e.g. Burningham et al., 2021; Luna & Morley, 2021). As shown in Fig. 6, our HRS P-T profiles show a nearly isothermal region between about 0.1-1 bars, which could suggest a degeneracy with clouds (Burningham et al., 2017). To check whether the P-T parameterization affects our results, we run a retrieval with a fixed P-T, namely the self-consistent profile over-plotted in gray. We find that all posteriors from this fixed P-T fit overlap within $1\sigma$ with those from our baseline retrieval. Thus, we conclude that the isothermal part of the P-T we retrieve is not biasing our conclusions. To further examine the possibility of clouds at lower pressures, we also run an opaque cloud model with infinite opacity below a retrieved pressure, and a gray cloud model that adds a constant cloud opacity at each pressure layer. When fitting the HRS with these more flexible cloud models, we also find consistent results with the baseline model. In the second model, the gray opacity is bounded to lie below $\sim 0.03~{}g/\rm{cm}^{-3}$, and the pressure of the infinitely opaque cloud is required to be deeper than $\sim 1$ bar. Therefore, even with these less constraining cloud parameterizations, we find that our HRS still prefers solutions with minimal cloud opacity.
5.3 Disequilibrium chemistry with deep quenching pressure
In our retrievals, we include a simple model for disequilibrium chemistry using the quench pressure prescription in petitRADTRANS, which is motivated by Zahnle & Marley (2014). Specifically, the abundances of CH${}_{4}$, CO, and H${}_{2}$O are held constant at atmospheric pressures lower than the retrieved $P_{\rm quench}$ parameter. We find that when including quenching, the goodness of fit increases drastically compared to fits with full chemical equilibrium. For example, between two clear retrievals with and without quenching, we find that $B\approx 380$ in favor of the quenched retrieval. From the Bayes factor interpretation of Benneke & Seager (2013), this represents a detection of quenching at $\approx 3.9\sigma$ significance. The quench pressure retrieved is also highly consistent between retrievals with different cloud models, with 1 and 2$\sigma$ intervals of $50-260$ and $14-836$ bars (Fig. 4). In this section, we explore reasons why the data prefer disequilibrium chemistry in the atmosphere of HD 4747 B. The physical implications of our retrieved quench pressure, including an estimate of the vertical diffusion coefficient ($K_{\rm zz}$), are discussed in § 8.2.
To understand why the data prefer a deep quench pressure, we plot lines of constant log(CO/CH${}_{4}$) volume mixing ratios (VMR) along with the best-fit P-T profile from our baseline HRS retrieval in Fig. 7. We calculate CO/CH${}_{4}$ from this quenched chemistry retrieval by finding the abundances of each molecule in the chemical grid, iterating over our posterior distribution of C/O, [C/H], and P-T profile. We find that CO/CH${}_{4}$=$13.6^{+5.8}_{-4.6}$. If the atmosphere was in chemical equilibrium, we repeat our calculation and find that we would expect CO/CH${}_{4}$=$1.35^{+0.21}_{-0.17}$, which is ten times smaller than our retrieved value in the quenched chemistry model. Thus, the relative under-abundance of CH${}_{4}$ relative to CO in our HRS leads our models to prefer a deep quench pressure. The value of CO/CH${}_{4}$ also determines our retrieved the quench pressure, whose $2\sigma$ interval is indicated by the thick blue region in Fig. 7. Because the P-T profile nearly overlaps the curve of CO/CH${}_{4}$ = 10 at $\sim 20$ bars and deeper, a broad range of quench pressures deeper than $\sim 20$ bars are consistent with the data.
As another way to visualize the detection of disequilibrium chemistry, we plot the molecular abundances in VMR as a function of pressure in Fig. 8. The solid lines show the VMRs for the HRS quenched chemistry retrieval, while dashed lines show the VMRs for the same model with quenching turned off manually. By comparing the solid and dashed lines, we see that at the pressures probed by our observations, the relative abundances of CO, CH${}_{4}$, and H${}_{2}$O differ by several orders of magnitude between the quenched model and expectations from chemical equilibrium.
5.4 Detection of methane in the HRS
In this section, we take a closer look at the relatively weak methane absorption signal in our HRS, which leads us to prefer quenched models where the CO/CH4 ratio is a factor of ten higher than predicted in models assuming chemical equilibrium. We confirm the presence of detectable levels of methane in the HRS by running a pair of free retrievals, one with only H${}_{2}$O and CO, and one with H${}_{2}$O, CO, and CH${}_{4}$. The results of these retrievals are listed in Table 3. In these free retrievals, we fit the abundances of each absorbing species independently and assume a constant abundance as a function of pressure. Although we also considered models that included NH${}_{3}$ and CO${}_{2}$, we only obtained upper limits on their abundances, and therefore excluded them from our fits in this section. Finally, given the insensitivity of the HRS to clouds, we carry out these tests with the clear model to save computation time.
We find that the data strongly prefer the model with CH${}_{4}$, with a Bayes factor of 84 ($3.4\sigma$ significance; Benneke & Seager 2013). As shown in Table 3, we obtain log(CH${}_{4}$)$=-4.82\pm 0.23$ from the free retrieval, and the CH${}_{4}$ posterior in Fig. 9 shows no strong covariance with the abundances of either CO or H${}_{2}$O. If the atmosphere was in chemical equilibrium, we would expect a CH${}_{4}$ VMR that is ten times higher than what we retrieve, according to the same calculation described in § 5.3. We note that the abundances from the free retrieval with CH${}_{4}$ also agree well with the corresponding VMRs from our quenched chemistry retrievals. This is not surprising given the deep quench pressure we retrieve, which makes the molecules abundances constant in the regions where our HRS is sensitive (see Fig. 8).
We separately visualize the CH${}_{4}$ detection in cross-correlation space by carrying out an analysis similar to that described in Zhang et al. (2021). First, we make a ‘pure CH${}_{4}$ template’ from the best-fit companion model with CH${}_{4}$, H${}_{2}$O, and CO by manually setting the abundances of H${}_{2}$O and CO to zero. If the model without CH${}_{4}$ is fitting poorly due to its inability to fit CH${}_{4}$ lines in the data, we would expect the residuals of this model, which we denote $R$ = (data - model without CH${}_{4}$), to contain CH${}_{4}$ lines. Therefore, we cross-correlate $R$ with the pure CH${}_{4}$ template, plotted as the blue CCF in Fig. 9. In addition, we plot the CCF of $R^{\prime}=$ (data - model with CH${}_{4}$) with the pure CH${}_{4}$ template in red for comparison. The blue CCF shows a peak at 0 km/s (solid gray line), where we expect a real signal to be since the models have been shifted by the best fit companion RV. If the residuals were dominated by telluric CH${}_{4}$ for example, the CCF peak would appear at the dotted gray line (negative of the RV, or -15 km/s). Thus, even though the height of the CH${}_{4}$ peak in the blue CCF is small compared to the surrounding structure, the fact that it is located at the companion RV is evidence of a real signal from CH${}_{4}$.
In our CCF framework, the y-axis is the estimated flux level (in counts) of the companion signal from a least-squares minimization. As shown in Fig. 9, we find a flux level of $\approx 50$ counts for CH${}_{4}$, which is an estimate of the companion flux in the residuals. Importantly, this value is consistent with the flux value found when we repeat the same CCF analysis with H${}_{2}$O (i.e. comparing a model with only CO and CH${}_{4}$ and the baseline model of CO, H${}_{2}$O, and CH${}_{4}$). For a molecule such as NH${}_{3}$, which we see no evidence of in the KPIC HRS, the flux value from the CCF becomes unbounded as the least-squares routine used for computing the CCF fails to converge.
Finally, we check for cross-talk between H${}_{2}$O and CH${}_{4}$ by cross-correlating $R$ with the pure water template and detect no CCF peaks. Furthermore, we note that in a retrieval with only CO and CH${}_{4}$ (no H${}_{2}$O), the retrieved CH${}_{4}$ abundance is consistent with the value from the full model including CO, H${}_{2}$O and CH${}_{4}$.
We therefore conclude that the data strongly favor the presence of detectable levels of methane in the HRS, with an abundance significantly lower than that predicted by equilibrium chemistry models. The detection of methane at log(CH${}_{4}$) = ${-4.82\pm 0.23}$ demonstrates the ability of KPIC to retrieve species that are more than an order of magnitude lower in VMR than the dominant molecular constituents in the data, in only 1 hour of integration time.
6 Low-resolution retrievals (GPI + SPHERE)
6.1 Overview
In this section, we present the results from our fits to the LRS and compare our retrieved parameters to those from the HRS fits. We fit the LRS using the same models as before. These include one clear model and four different implementations of the EddySed cloud model where we vary our assumptions about the unknown cloud properties. The cloudy models consist of two MgSiO${}_{3}$ retrievals with am and cd particles (explained in § 4.2.2), and two retrievals with MgSiO${}_{3}$ and Fe clouds (again, am and cd). In Fig. 10, we plot the data, best-fit cloudy and clear models, the residuals, and the GP models of the residuals. The posteriors for a few key parameters from these retrievals are plotted in Fig. 11 and tabulated in Table 2. See Appendix C for the posterior distributions of other parameters in the baseline model.
When comparing the clear and cloudy models in Fig. 10, we see that the data shortward of $\sim 1.2~{}\mu$m is poorly fit by models without clouds. This causes the clear model to have $B\approx 7\times 10^{-26}$; it is overwhelmingly ruled out compared to the baseline EddySed model. In addition, when we plot the models over a larger wavelength range in Fig. 12, we find that the cloudy models agree with the NIRC2 L band photometry from Crepp et al. (2016), while the clear model over-predicts the $L$ band flux by $\approx 2\sigma$. We did not include these photometric points in our retrievals.
Fig. 10 shows that the SPHERE J band data from $\approx 1.2-1.35~{}\mu$m is not well fit by even the cloudy model, which could either be caused by model mismatch or speckle contamination that artificially raises the flux. The GP model finds that $\sim 60\%$ of the SPHERE error bars and $\sim 90\%$ of the GPI error bars are from correlated noise, with correlation length scales of $\sim 6$ and $\sim 2$ wavelength channels, respectively. This confirms our initial intuition that the noise in the SPHERE and GPI images is likely dominated by correlated speckle noise based on visual inspection of the images. For the SPHERE data set, we estimate that the retrieved length scale is roughly equal to the number of steps that a speckle would move across the PSF for our brown dwarf’s separation; indeed, we see speckles moving across the companion PSF in the reduced images. Overall, the SPHERE spectrum is less reliable than that from GPI because only 4 exposures are available, compared to the $\sim 40$ exposures from GPI.
Finally, the P-T profile retrieved from our baseline LRS retrieval shows a bi-modal distribution (see Fig. 6). The degeneracy seen here may be related to issues with the LRS (see § 6.3).
6.2 Comparison with prior knowledge
Because the LRS is flux-calibrated, we can check whether our retrieved radii and effective temperatures are physical and consistent with prior knowledge for this benchmark companion. Using the known age and mass of HD 4747 B ($3\pm 2$ Gyr and $m=67.2\pm 1.8$ $M_{\rm{Jup}}$), we interpolate the COND evolutionary models (Baraffe et al., 2003) to find a model-predicted radius of $0.8^{+0.07}_{-0.03}R_{\rm{Jup}}$, and a predicted $T_{\rm eff}=1450^{+350}_{-180}$ K. Peretti et al. (2019) compared the SPHERE spectrum of HD 4747B to those of field brown dwarfs to derive a more tightly constrained $T_{\rm eff}=1350\pm 50$ K (see Table 2), which we adopt in the subsequent discussion.
We calculate the effective temperatures of the models in our retrievals by integrating the flux over 0.5-30 $\mu$m. For the baseline EddySed model, we find $T_{\rm eff}=1473^{+17}_{-20}$ K, and a radius of $0.70^{+0.05}_{-0.03}~{}R_{\rm{Jup}}$. Compared to prior expectations however, the radius retrieved is too small by $\approx 1.5\sigma$ while $T_{\rm eff}$ is too high by $\approx 2\sigma$. From substellar evolutionary models, the minimum possible radius of a brown dwarf should be $\approx 0.74R_{\rm{Jup}}$, which is imposed by electron degeneracy pressure (Chabrier et al., 2009). We find that $T_{\rm eff}$ and radius are correlated in the LRS retrievals, as shown in Fig. 11, which is expected as different combinations of these two parameters can produce the same total luminosity. However, our total luminosity agrees well with the luminosity predicted by evolutionary models.
Several previous retrieval studies have also found smaller than expected radius for L dwarfs, which may be attributed to the presence of heterogeneous surface features, such as patchy clouds, that are not captured in current 1-D retrieval frameworks (e.g. Kitzmann et al., 2020; Gonzales et al., 2020; Burningham et al., 2021). On the other hand, Gonzales et al. (2021) retrieved a radius consistent with evolutionary models for a seemingly cloudless L dwarf. Whether the radii from evolutionary models are correct is an assumption that is now being tested by a growing sample of transiting brown dwarfs from TESS (e.g. Carmichael et al., 2020).
In our retrievals with both MgSiO${}_{3}$ and Fe clouds, we retrieve slightly larger radii that are more consistent with the evolutionary model prediction. This could indicate that a single cloud model (MgSiO${}_{3}$) may be inadequate in attenuating the flux from the deep atmosphere. However, models with two cloud species do not improve the fit significantly ($B$ = 1.5-3 compared to the baseline model with MgSiO${}_{3}$ only). Furthermore, the MgSiO3, cd model actually has the largest retrieved radius, but our data cannot distinguish between crystalline and amorphous particles. We conclude that our retrieved radius is sensitive to aspects of the cloud models that are poorly constrained by the existing data for this object.
6.3 LRS at longer wavelengths could improve abundance and cloud constraints
While the LRS can provide tighter constraints on the cloud parameters and radius compared to the HRS, we find that many retrieved parameters, including the atmospheric abundances, are very sensitive to model choices. In Fig. 11, we overplot the posteriors distributions of a few parameters from our four EddySed models. The retrieved C/O and [C/H] have large uncertainties and can disagree at the $3\sigma$ level between models. The values also span a significant portion of the parameter space ($>$1 dex in metallicity), and show much stronger covariance compared those measured from the HRS (see Fig. 4). However, all cloudy models fit the LRS well, with Bayes factors within a factor of $\sim 3$ (see Table 2), so we cannot distinguish between them.
We note that Mollière et al. (2020) were able to obtain much better constraints on the composition of HR 8799e, which also has a cloudy atmosphere, using LRS data sets from 0.95-2.5 $\mu$m. Their LRS had SNR between 4-11 per wavelength bin, much lower than the SNR of our LRS (between 20-60 per wavelength bin). Unlike Mollière et al. (2020), however, our study does not have LRS in the second half of $K$ band ($2.2-2.5~{}\mu$m), which contains a strong CO bandhead as well as significant H${}_{2}$O and CH${}_{4}$ opacities. When we compute the CO abundances from our LRS retrievals, we find that they are not well constrained, with 1$\sigma$ intervals that are $\gtrsim 3$ wider than the CO constraint from HRS. In Fig. 12, we plot random draws of our baseline model color-coded by metallicity out to 5 $\mu$m. As shown, the models diverge quickly in the $2.2-2.5~{}\mu$m range. The fact that we miss this crucial wavelength region could explain why Mollière et al. (2020) obtain more robust constraints on atmospheric abundances, and P-T profiles that agree better with self-consistent models than we do, despite using data with a lower SNR.
Fig. 12 also shows a clear gradient in metallicity beyond 2.5 $\mu$m. In some of our cloudy LRS retrievals, we see a covariance between metallicity and cloud mass fraction, where lower metallicities correspond to higher cloud mass fractions, as well as larger, more physically consistent radii (see Fig. 11). The degeneracy between metallicity and cloud mass fraction might arise because both molecular opacities and clouds contribute opacity, and our data has insufficient wavelength coverage to probe more regions where the gas and cloud opacities are sufficiently different. From the LRS retrievals, we consistently find a factor of $\sim 2-3$ more CH${}_{4}$ and H${}_{2}$O than observed in the HRS, implying that the LRS retrievals could be compensating for our imperfect cloud models by increasing the gas opacities.
Using a more flexible cloud model might alleviate some of these issues. For example, Burningham et al. (2021) retrieved the 1-15 $\mu$m LRS of a field L dwarf and found the data preferred silicate clouds much higher up than the predicted cloud base locations from condensation curves. In addition, their retrieved cloud particles also have smaller sizes (sub-micron) than predicted by the EddySed model (a few microns). Similarly, Luna & Morley (2021) found that sub-micron cloud particles at lower pressures than predicted by EddySed are required to fit the mid-IR silicate feature ($\approx 8-10~{}\mu$m) of many L dwarfs. They found that the microphysical cloud model CARMA (Turco et al., 1979; Toon et al., 1988; Gao et al., 2018) allows them to fit their data much better and even place constraints on which cloud species are producing the observed absorption feature.
Both the above-mentioned studies benefited from data at $\sim 10~{}\mu$m that significantly help with constraining cloud properties. Thus, to obtain better abundance measurements with LRS, it is not only important to obtain full coverage in the near-IR (which we lack), but also to acquire data in the mid-IR. JWST can obtain low- and medium-resolution spectroscopy of brown dwarfs spanning the near- to mid-IR wavelengths using the NIRSpec and MIRI instruments. Future ground-based instruments such as SCALES at Keck (Stelter et al., 2020) will also provide LRS in the mid-IR.
7 Joint retrievals
In this section, we describe the results of joint retrievals to both the HRS and LRS for HD 4747 B. In practice, we set up two radiative transfer routines with petitRADTRANS using line-by-line (for HRS) and correlated-k (for LRS) opacity sampling respectively. The HRS and LRS models share the same atmospheric parameters and priors, but each has some unique parameters (e.g. RV and $v\sin{i}$ for HRS, GP kernel parameters for LRS). Within one nested sampling retrieval, we add the log likelihoods from the HRS and LRS components at each step of sampling to get the total log likelihood. We consider both clear and cloudy EddySed models for our joint retrievals.
Because the LRS prefer clouds, the cloudy model (MgSiO${}_{3}$, am) is overwhelmingly preferred in our joint retrieval, with a Bayes factor in excess of $10^{34}$ compared to the clear model. From the cloudy model, we retrieve C/O$=0.70\pm 0.03$ and [C/H]$=0.34\pm 0.07$. The retrieved uncertainties on these parameters are lower than in the HRS-only retrieval (which had C/O$=0.66\pm 0.04$ and [C/H]$=-0.10^{+0.18}_{-0.15}$). In addition, the C/O from our joint fit is consistent with the C/O from our HRS fit. This is not surprising, because the HRS places tight constraints on the relative line depths (and hence the relative abundance ratios) of CH${}_{4}$, H${}_{2}$O, and CO. However, the joint fit pushes the metallicity to higher values, which corresponds to increased gas abundances as shown in Fig. 13. The joint fit results translate to a $>4\sigma$ discrepancy in [C/H] between HD 4747 A&B, while there is no discrepancy if we take the results from the HRS fit. This implies that the joint fit might be compensating for inadequacies in modeling clouds by increasing the gas opacities, as discussed in § 6.3 for the LRS-only case. We ran additional joint retrievals where we varied the cloud parameters (e.g. adding Fe clouds) and found similar results.
If we compare the log likelihoods of the HRS part of the joint fit to that from the HRS-only fit, we find that the HRS is fit less well by $\approx e^{10}$ (which translates to $\approx 4\sigma$) in the joint fit, implying a trade-off between fitting the LRS and HRS. We can qualitatively compare the LRS S/N per wavelength bin to the CCF S/N of the HRS, which approximates the total constraining power of the HRS. When including all molecules in our model, we find a CCF S/N of $\approx 15$ for the HRS. For the LRS, the average S/N per wavelength bin is $\approx 20$ for the SPHERE $YJH$ data and $\approx 60/30$ between the GPI $H/K_{1}$ data. This explains why the joint fits prioritize fitting the LRS at the expense of the HRS.
As discussed in § 6, the LRS are very model-sensitive and additionally contaminated by correlated noise. For this reason, we adopt the HRS-only results as the best estimate of HD 4747 B’s atmospheric properties in this paper (see first row of Table 2). We leave it to future work, preferably aided by longer wavelength coverage in LRS, to achieve a more satisfactory joint retrieval.
8 Discussion
8.1 Next steps for high-resolution spectroscopy
Our KPIC HRS provide a better handle on the atmospheric abundances of HD 4747 B, and are less sensitive to model choices than our LRS. In fact, our $K$ band HRS are essentially agnostic to clouds in the brown dwarf’s atmosphere; all retrieved parameters are consistent independent of our chosen cloud model (Fig. 4). As discussed in § 5.2, this is because our HRS covers a wavelength region ($2.29-2.49~{}\mu$m) of high molecular opacity, and probes emission across a wide range of atmospheric pressures where cloud opacity is negligible (up to $10^{-2}$ bars in line cores). While clouds affect the continuum near $1~{}\mu$m in the LRS, they have little effect on the line depths across the wavelength range of our HRS. The relative lines depths are sensitive to relative molecular abundances, which directly constrains C/O. These results advocate for using HRS to measure atmospheric abundances.
In the future, it is important to explore whether these findings hold true for other substellar objects. In upcoming papers, we will present KPIC HRS retrievals of brown dwarf companions and giant planets spanning a range of effective temperatures and surface gravities. Ultimately, it would also be useful to constrain cloud properties with HRS. For transmission spectroscopy, Gandhi et al. (2020) found that their simulated near-IR HRS for warm Neptunes are more sensitive to molecular abundances than LRS for the same reasons highlighted in this study. While both clouds and metallicity affect the line depths in HRS, Gandhi et al. (2020) show that increasing the wavelength coverage (e.g. going from 0.9-1.7 $\mu$m to 0.9-2.5 $\mu$m) helps distinguish between clouds and metallicity and provide better constraints on both. Thus, if we wish to obtain constraints on clouds and abundances at the same time, it would be important to extend our current HRS to a broader range of wavelengths. KPIC Phase II will allow us to obtain $L$ band data ($\approx 3.4-4.1~{}\mu$m) to complement existing $K$ band data (Delorme et al., 2021), and future upgrades could benefit from including $H$ and $J$ bands as well.
In this study, we have assumed that the atmosphere of HD 4747 B is globally uniform. However, it would be important to examine the impact of 3-D effects, including non-uniform cloud coverage. Past studies with photometry or LRS show that many brown dwarfs exhibit clear rotational variability signals (e.g. Apai et al., 2013; Zhou et al., 2018; Biller et al., 2018; Manjavacas et al., 2021; Vos et al., 2022), which appear to be caused by inhomogeneities in their cloud properties. Therefore, time-resolved observations are important to understanding clouds and 3-D effects.
With HRS, we can use the time-varying line depth and shape to map the 2-D brightness distributions of these objects (e.g. Crossfield et al., 2014). In this paper, we used 1 hour of KPIC data for HD 4747 B. Given our measured $v\sin{i}$ and assuming a radius of $0.8R_{\rm{Jup}}$, we would expect a 5 or 7 hour rotation period if $i$ is equal to the orbital inclination or $i=90\degree$. Thus, it may be possible to sample a full rotation period within a single observing night, with the caveat that measurements of the true rotation period remain difficult for high-contrast companions (Biller et al., 2021).
8.2 Methane and the presence of disequilibrium chemistry
Our HRS retrievals indicate that the ratio of CO/CH4 (VMR) is $\approx 10$ times higher than expected by equilibrium chemistry (see § 5.3). To gain more physical intuition, we convert the quench pressure from our HRS retrievals to an estimate of the vertical diffusion coefficient, $K_{\rm zz}$. To do this, we match the chemical timescale of the CO-CH${}_{4}$ reaction from Zahnle & Marley (2014) with the mixing timescale $\tau_{\rm mix}=L^{2}/K_{\rm zz}$. While the length scale $L$ is typically taken to be the pressure scale height $H$ for lack of a better estimate, Smith (1998) show that this assumption is not valid across several reactions in the atmospheres of Jupiter and Neptune. In fact, Smith (1998) find that $L\approx 0.1H$, which changes the inferred $K_{\rm zz}$ by two orders of magnitude. Similarly, Ackerman & Marley (2001) also note that the mixing length is generally shorter than the pressure scale height $H$ in stable atmospheric regions. Due to the uncertainty in $L$, we adopt $L=\alpha H$, where $\alpha$ is a scaling factor, and $H=\frac{k_{B}T}{\mu mg}$ ($\mu$: mean molecular weight, $g$: surface gravity, $T$: the local temperature). For each value of quench pressure from our posteriors, we compute the necessary quantities to derive a posterior for $K_{\rm zz}$. For instance, if $\alpha=0.1$, we find $K_{\rm zz}=5\times 10^{8}-1\times 10^{12}~{}\rm{cm^{2}s^{-1}}$ ($1\sigma$). On the other hand, if $\alpha=1$, we obtain $K_{\rm zz}=5\times 10^{10}-1\times 10^{14}~{}\rm{cm^{2}s^{-1}}$.
There have been few quantitative measurements of $K_{\rm zz}$ for substellar companions. Miles et al. (2020) used M-band LRS to constrain the CO abundance and estimate the vertical diffusion coefficient, $K_{\rm zz}$, for seven field brown dwarfs. However, their objects have $T_{\rm eff}$ between 250-750 K, much colder than HD 4747 B. In terms of objects with $T_{\rm eff}\gtrsim 1000~{}$K, Barman et al. (2015) reported a detection of CH${}_{4}$ in HR 8799b ($T_{\rm eff}\sim 1000~{}$K) with Keck/OSIRIS data, which they used to estimate $K_{\rm zz}$ between $10^{6}-10^{8}~{}\rm{cm^{2}s^{-1}}$. However, this CH${}_{4}$ detection was not confirmed by an independent study (Petit dit de la Roche et al., 2018), and recently Ruffio et al. (2021) concluded that future higher-resolution follow up is needed to resolve the discrepant CH${}_{4}$ signal strengths found by different analyzes. Ruffio et al. (2021) point out that if the CH${}_{4}$ abundance was over-estimated by Barman et al. (2015), that would imply a larger $K_{\rm zz}$. Using LRS, Mollière et al. (2020) found a well-constrained quench pressure for HR 8799e ($T_{\rm eff}\sim 1100~{}$K) from petitRADTRANS retrievals, which could similarly be converted to a $K_{\rm zz}$ constraint. In summary, our finding HD 4747 B, which is $\sim 300-400~{}$K hotter than HR 8799b&e and much older (a few Gyr from § 2.1) than most directly imaged planets, represents an important new data point because hotter objects are expected to be closer to equilibrium, making chemical disequilibrium processes harder to detect (e.g. Moses et al., 2013).
Zahnle & Marley (2014) provide an upper limit for $K_{\rm zz}$ from mixing length theory (Gierasch & Conrath, 1985) assuming full convection. For HD 4747 B, their Equation 4 translates to an upper limit of $\approx 10^{9}~{}\rm{cm^{2}s^{-1}}$. Depending on $\alpha$, our retrieved $K_{\rm zz}$ either exceeds this upper limit by $\gtrsim 2\sigma$ (if $L=H$), or is very close to this limit (if $L=0.1H$). Together, this suggests that convection is driving the vertical mixing in HD 4747 B, and that the mixing efficiency is likely close to its predicted maximum. We check whether our inferred $K_{\rm zz}$ makes sense by comparing them to those predicted by self-consistent atmospheric models with disequilibrium chemistry from Karalidi et al. (2021) and Mukherjee et al. (2022, in prep). For an object with properties similar to HD 4747 B, our measured CH${}_{4}$ VMR is consistent with $K_{\rm zz}\sim 10^{8}-10^{12}$ in these models (with the assumption that $L=H$). These values of $K_{\rm zz}$ are roughly consistent with our estimate based on $P_{\rm quench}$, and also near the upper limit from Zahnle & Marley (2014). On the modeling front, it would be valuable to carry out 3-D hydrodynamical simulations (e.g. Zhang & Showman, 2018; Tan & Showman, 2021) of brown dwarf interiors to independently estimate $K_{\rm zz}$ (Tan, 2022) and compare the results to that inferred by our data. Such simulations could also reveal what physical processes might cause a discrepancy between mixing length theory and our observations.
8.3 Dynamical versus Spectroscopic Mass Constraints
For a majority of substellar companions observed by direct imaging, there are no dynamical mass constraints. To assess whether our mass prior plays an important role in the results, we repeat our HRS and LRS retrievals with the baseline cloud model but use uniform priors in mass from 10 to 100 $M_{\rm{Jup}}$ (‘free-mass’). For the HRS free-mass retrieval, we find that all parameters change by less than $1\sigma$ compared to the mass-prior retrieval. The mass itself shows a broad distribution (33-76 $M_{\rm{Jup}}$ at $1\sigma$) that encompasses the dynamical mass. Because our KPIC HRS are not flux calibrated, the radius is not well constrained. In this case, we get large uncertainties in the spectroscopic mass because mass is inferred from the retrieved surface gravity, which depends on the poorly-constrained radius.
Our LRS free-mass retrieval also yields posteriors for all parameters that are consistent between 1-2 $\sigma$ with the mass-prior retrieval. Furthermore, the mass retrieved by the LRS is $59^{+7}_{-8}~{}M_{\rm{Jup}}$, which agrees within about $1\sigma$ with the dynamical mass. This provides confidence that reasonable mass constraints can be placed on substellar objects from LRS. The radius retrieved is $0.77\pm 0.03~{}R_{\rm{Jup}}$, consistent with evolutionary model predictions and close to the radius from the mass-prior retrieval, suggesting the two retrievals find a similar surface gravity.
8.4 Atmospheric abundances of HD 4747 AB
We retrieve [C/H] and [O/H] values that are $1\sigma$ consistent with those of the host star, as discussed in § 5.1. Both companion and the star are mildly sub-solar in terms of their metal content. However, our retrieved C/O=$0.66\pm 0.04$ is higher by approximately $2\sigma$ than the stellar C/O=$0.48\pm 0.08$.
The question is whether the marginal discrepancy in C/O is from astrophysical or systematic reasons.
For example, Wang et al. (2022) carried out retrieval experiments on simulated HRS ($2.2-2.35~{}\mu$m, $R=35,000$) and found that their formal error bars are likely under-estimated due to systematic errors at the $\sim 0.15$ level in C/O. Using KPIC HRS from 2.23-2.33 $\mu$m, they found $\approx 1-1.5\sigma$ discrepancies between the [C/H] and [O/H] abundances of HR 7672 A and B, another benchmark brown dwarf system. On an earlier study of benchmark brown dwarfs, Line et al. (2015) quoted $1\sigma$ uncertainties of $0.2-0.3$ in their brown dwarf C/O (much larger than our formal C/O uncertainty of $0.04$), and concluded that a $2\sigma$ agreement between the stellar and companion C/O is sufficiently good given the caveats. It is also possible that the uncertainties on stellar abundances are under-estimated given non-LTE effects (Line et al., 2015).
Another factor that might contribute to the $2\sigma$ discrepancy in C/O is uncertainties in the chemistry of condensates. The chemical model of petitRADTRANS we use accounts for the equilibrium condensation of various species and reports the global (rather than gas phase) C and O abundances (Mollière et al., 2019). In particular, species such as MgSiO${}_{3}$ and Mg${}_{2}$SiO${}_{4}$ contain 3 or 4 oxygen atoms per molecule, and are expected to hold a significant portion of O (Line et al., 2015). From our HRS retrievals, we find that $\approx 18\%$ of O is condensed into solids such as MgSiO${}_{3}$. In order to decrease the global C/O of the brown dwarf by $\approx 0.1$ (therefore making the companion and stellar C/O agree at the $1\sigma$ level), we require a $\sim 20\%$ increase in the net O abundance. Keeping everything else unchanged, this means the MgSiO${}_{3}$ mass fraction, which is predicted by the chemical model to be $\sim 2\times 10^{-3}$ in our retrievals, needs to be doubled to $\sim 4\times 10^{-3}$. From the LRS retrievals, the cloud base MgSiO3 fraction can be as high as $10^{-2}$. Therefore, a factor of $\sim 2$ uncertainty in the abundance of MgSiO${}_{3}$ could make our C/O consistent at the $1\sigma$ level with the stellar value.
Given these caveats, we conclude that the $2\sigma$ difference between our retrieved C/O for HD 4747 B and the stellar value is not significant, and HD 4747 AB are consistent with being chemically homogeneous. Chemical homogeneity is expected by models where brown dwarf companions form via gravitational fragmentation in molecular clouds (e.g. Padoan & Nordlund, 2004) or massive protostellar disks (e.g. Stamatellos et al., 2007). Simulations suggest that brown dwarfs typically form as part of unstable, high-order multiple systems, which undergo chaotic interactions that reduce the multiplicity over time (e.g. Bate et al., 2002; Thies et al., 2010; Bate, 2012). With a semi-major axis of 10 au, HD 4747 B is unlikely to have been directly affected by such encounters, but its relatively high orbital eccentricity ($\approx 0.73$) could encode such a dynamically ‘hot’ past.
9 Conclusions
Using high-resolution spectra ($R\sim 35,000$) obtained by Keck/KPIC, we retrieve [C/H]=$-0.10^{+0.18}_{-0.15}$, [O/H]=$-0.18^{+0.18}_{-0.15}$, and C/O=$0.66\pm 0.04$ for the benchmark brown dwarf companion HD 4747 B (formal error bars). The C and O abundances are consistent with the stellar values to $\lesssim 1\sigma$, while the C/O ratio is consistent at the $2\sigma$ level, as expected for a binary-star like formation scenario. This shows that we can measure the atmospheric abundances for high contrast substellar companions to the $20\%$ level with KPIC and our current modeling framework, which Wang et al. (2022) also show for another benchmark brown dwarf. We outline some other key findings from our study below.
We measure precise abundances from the KPIC HRS ($2.29-2.49~{}\mu$m), which are insensitive to our choice of cloud model. Our abundance measurements suggest that HD 4747 B has a CO/CH${}_{4}$ ratio that is 10 times higher than predicted by equilibrium chemistry, corresponding to a quench pressure of $50-260$ bars ($1\sigma$). This translates to a high vertical diffusion coefficient $K_{\rm zz}$ which depends on the assumed length scale $L$. However, even if $L$ is ten times smaller than the pressure scale height, we get $K_{\rm zz}=5\times 10^{8}-1\times 10^{12}~{}\rm{cm^{2}s^{-1}}$, which implies a mixing strength that is at or above the upper limit predicted by mixing length theory.
The composition retrieved from our LRS (1-2.2 $\mu$m) are both sensitive to model choices, and can be biased by the presence of speckles. For this reason, HRS provides a more reliable picture of the atmospheric composition in the current data sets, although the LRS could be improved with additional observation at longer wavelengths including the $L$ and $M$ bands. Despite these challenges, the current LRS does provide a spectroscopic mass estimate that is $1\sigma$ consistent with the dynamical mass for the brown dwarf.
Although our joint retrieval results are likely biased by the limited LRS wavelength coverage, joint analyzes of LRS and HRS remain a promising avenue to constrain cloud properties and abundances simultaneously and provide a more complete picture of substellar atmospheres. When extended wavelength coverage is available, it would also be important to consider possible 3-D effects, including patchy clouds. These might be constrained by obtaining multiple spectra sampling a rotation period. Additional modeling work on condensation, chemistry, and vertical mixing rates are also important to inform future observational results.
We thank the referee for helpful comments that improved the manuscript. J.X. thanks Michael Zhang for advice in computing the CH${}_{4}$ opacities, Jonathan Fortney for discussions on chemical disequilibrium, and Konstantin Batygin for discussions on brown dwarf formation.
We wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Maunakea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain. This research has benefitted from the SpeX Prism Library maintained by Adam Burgasser at http://www.browndwarfs.org/spexprism. Funding for KPIC has been provided by the California Institute of Technology, the Jet Propulsion Laboratory, the Heising-Simons Foundation, the Simons Foundation, and the United States National Science Foundation Grant No. AST-1611623. AV acknowledges funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant agreements No. 757561 (HiRISE). HAK acknowledges support from the President’s and Director’s Research & Development Fund Program, which is jointly funded by the Jet Propulsion Laboratory and the California Institute of Technology under a contract with the National Aeronautics and Space Administration. The computations presented here were conducted in the Resnick High Performance Center, a facility supported by Resnick Sustainability Institute at the California Institute of Technology.
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Appendix A Orbit fits for HD 4747 B
Our orbit fit for the HD 4747 system is shown in Fig. 14.
Appendix B Extracted Low-resolution spectrum and GPI astrometry
Our extracted spectrum for HD 4747 B based on observations with GPI (Crepp et al., 2018) and SPHERE (Peretti et al., 2019) are given in Table 4. Our relative astrometry measurements based on the GPI data are listed in Table 5.
\startlongtable
Appendix C Priors and posteriors for retrieval parameters
Here we list the priors on our retrieved parameters and include joint posterior distributions of selected parameters from our baseline HRS and LRS retrievals. |
Curves in characteristic $2$ with non-trivial
$2$-torsion
Wouter Castryck, Marco Streng, Damiano Testa
Abstract.
Cais, Ellenberg and Zureick-Brown recently observed that over finite fields of characteristic two,
almost all smooth plane projective curves of a given odd degree admit a non-trivial rational $2$-torsion point
on their Jacobian. We extend this observation to curves given
by Laurent polynomials with a fixed Newton polygon, provided that the polygon
satisfies a certain combinatorial property. This includes many classical families, such
as hyperelliptic curves of odd genus and $C_{a,b}$ curves. In the hyperelliptic case, we give an alternative
proof using an explicit description of the $2$-torsion subgroup.
1. Introduction
The starting point of this note is a recent theorem by Cais, Ellenberg and Zureick-Brown [CEZB, Thm. 4.2], asserting
that over a finite field $k$ of characteristic $2$, almost all smooth plane projective curves of a given odd degree $d\geq 3$ have a non-trivial $k$-rational $2$-torsion point on their Jacobian. Here, ‘almost all’ means that the corresponding proportion converges to $1$ as $\#k$ and/or $d$ tend to infinity.
The underlying observation is that such curves admit
•
a ‘geometric’ $k$-rational half-canonical divisor $\Theta_{\text{geom}}$: the canonical class of a smooth plane projective curve of degree $d$ equals $(d-3)H$, where $H$ is the class of hyperplane sections; if $d$ is odd then $\frac{1}{2}(d-3)H$ is half-canonical,
•
an ‘arithmetic’ $k$-rational half-canonical divisor $\Theta_{\text{arith}}$ (whose class is sometimes called the canonical theta characteristic), related to the fact that over a perfect field of characteristic $2$, the derivative of a Laurent series is always a square [Mum, p. 191].
The difference $\Theta_{\text{geom}}-\Theta_{\text{arith}}$ maps to a $k$-rational $2$-torsion point on the Jacobian. The proof of [CEZB, Thm. 4.2] then amounts to showing that, quite remarkably, this point is generically non-trivial.
There exist many classical families of curves admitting such a ‘geometric’ half-canonical divisor. Examples include hyperelliptic curves of odd genus $g$, whose canonical class is given by $(g-1)g^{1}_{2}$ (where $g^{1}_{2}$ denotes the hyperelliptic pencil), and smooth projective curves in $\mathbf{P}_{k}^{1}\times\mathbf{P}_{k}^{1}$ of even bidegree $(a,b)$ (both $a$ and $b$ even, that is), where the canonical class reads $(a-2)R_{1}+(b-2)R_{2}$ (here $R_{1},R_{2}$ are the two rulings of $\mathbf{P}_{k}^{1}\times\mathbf{P}_{k}^{1}$). The families mentioned so far are parameterized by sufficiently generic polynomials that are supported on the polygons
$d$$d$$0$smooth plane curvesof degree $d$$2g+2$$2$$0$hyperelliptic curvesof genus $g$$a$$b$$0$curves in $\mathbf{P}_{k}^{1}\times\mathbf{P}_{k}^{1}$of bidegree $(a,b)$,
respectively. The following lemma, which is an easy consequence of the theory of toric surfaces (see Section 2), gives a purely combinatorial reason for the existence of a half-canonical divisor in these cases.
Lemma 1.
Let $k$ be a perfect field and let $\Delta$ be a two-dimensional lattice polygon. For each edge $\tau\subset\Delta$, let $a_{\tau}X+b_{\tau}Y=c_{\tau}$ be its supporting line,
where $\gcd(a_{\tau},b_{\tau})=1$. Suppose that the system of congruences
(1)
$$\left\{\ a_{\tau}X+b_{\tau}Y\equiv c_{\tau}+1\pmod{2}\ \right\}_{\tau\text{ %
edge of }\Delta}$$
admits a solution in $\mathbf{Z}^{2}$. Then any sufficiently general Laurent polynomial $f\in k[x^{\pm 1},y^{\pm 1}]$ that is supported on $\Delta$ defines a curve carrying a $k$-rational half-canonical divisor
on its non-singular complete model.
In the proof of Lemma 1 below, where we describe this half-canonical divisor explicitly, we will be more precise on the meaning of ‘sufficiently general’.
Here again, when specializing to characteristic $2$, there is, in addition, an arithmetic $k$-rational half-canonical divisor. So it is natural to wonder whether the proof of [CEZB, Thm. 4.2] still applies in these cases. We will show that it usually does.
Theorem 2.
Let $\Delta$ be a lattice polygon satisfying the conditions of Lemma 1, where in addition we assume that $\Delta$ is not unimodularly equivalent to
$1$$3$$(3,1)$$k$$1$$0$for some $k\geq 1$or$1$$0$for some $0\leq k<\ell\geq 3$with $k$ even and $\ell$ odd.$(\ell,1)$$(k,2)$
Then there exists a non-empty Zariski open subset $S_{\Delta}/\mathbf{F}_{2}$ of the space of
Laurent polynomials that are supported on $\Delta$ having the following property.
For every perfect field $k$ of characteristic $2$ and every $f\in S_{\Delta}(k)$,
the Jacobian of the non-singular complete model of the
curve defined by $f$ has a non-trivial $k$-rational $2$-torsion point.
Right before the proof of Theorem 2 we will define the set $S_{\Delta}$ explicitly. As a consequence, if $k$
is a finite field of characteristic $2$, then the proportion of Laurent polynomials
that are supported on $\Delta$, which define a curve whose Jacobian has a
non-trivial $k$-rational $2$-torsion point, tends to $1$ as $\#k\rightarrow\infty$.
(See the end of
Section 3, where we also discuss asymptotics for
increasing dilations of $\Delta$, i.e. the analogue of $d\rightarrow\infty$ in the smooth plane curve case.)
This observation seems new even for hyperelliptic curves of odd genus.111In
view of the asymptotic consequences discussed in Section 3, this observation shows that
[CFHS, Principle 3] can fail for $g>2$.
In this case we can give an alternative proof yielding sufficient conditions having a more arithmetic flavor (such as being ordinary); see Section 4. Another interesting class of examples is given by the polygons
$a$$b$$0$
where $a$ and $b$ are not both even. The case $a=b$ corresponds to the smooth plane curves of odd degree considered in [CEZB]. The case $\gcd(a,b)=1$ corresponds to so-called $C_{a,b}$ curves. The
case $b=2$, $a=2g+1$ (a subcase of the latter) corresponds
to hyperelliptic curves having a prescribed $k$-rational Weierstrass point $P$. Note that in this case $g^{1}_{2}\sim 2P$, so there is indeed always a $k$-rational half-canonical divisor, regardless of the parity of $g$. Finally, the case $b=3$, $a\geq 4$ corresponds to trigonal curves having maximal Maroni invariant (that is trigonal curves for which the series $(h^{0}(ng^{1}_{3}))_{n\in\mathbf{Z}_{\geq 0}}$ starts increasing by steps of $3$ as late as the Riemann-Roch theorem allows it to do); if $a=6$, these are exactly the genus-$4$ curves having a unique $g^{1}_{3}$.
Remark 3.
This explains why Denef and Vercauteren had to allow a factor $2$ while generating
cryptographic hyperelliptic and $C_{a,b}$ curves in characteristic $2$; see Sections 6 of [DV1, DV2].
2. Half-canonical divisors from toric geometry
Let $k$ be a perfect field and let $f=\sum_{(i,j)\in\mathbf{Z}^{2}}c_{i,j}x^{i}y^{j}\,\in k[x^{\pm 1},y^{\pm 1}]$ be an absolutely
irreducible Laurent polynomial. Let
$$\Delta(f)=\textup{conv}\left\{\,\left.(i,j)\in\mathbf{Z}^{2}\,\right|\,c_{i,j}%
\neq 0\,\right\}$$
be the Newton polygon of $f$, which we assume to be two-dimensional. We say that $f$ is non-degenerate with respect to its Newton polygon if for every face $\tau\subset\Delta(f)$ (vertex, edge, or $\Delta(f)$ itself)
the system
$$f_{\tau}=\frac{\partial f_{\tau}}{\partial x}=\frac{\partial f_{\tau}}{%
\partial y}=0\qquad\text{with }f_{\tau}=\sum_{(i,j)\in\tau\cap\mathbf{Z}^{2}}c%
_{i,j}x^{i}y^{j}$$
has no solutions over an algebraic closure of $k$.
For a given Newton polygon $\Delta=\Delta(f)$, the condition of non-degeneracy is generically
satisfied, in the sense that it is characterized by the non-vanishing of
$$\rho_{\Delta}:=\text{Res}_{\Delta}\left(f,x\frac{\partial f}{\partial x},y%
\frac{\partial f}{\partial y}\right)\,\in\mathbf{Z}[c_{i,j}|(i,j)\in\Delta\cap%
\mathbf{Z}^{2}]$$
(where $\text{Res}_{\Delta}$ is the sparse resultant; $\rho_{\Delta}$ does not vanish identically in any characteristic [CV, §2]).
Let $C_{f}$ be the curve in $\mathbf{T}_{k}^{2}=\text{Spec}\,k[x^{\pm 1},y^{\pm 1}]$ cut out by $f$ and let $\text{Tor}_{k}(\Delta(f))$ be the toric surface corresponding to $\Delta(f)$; it is a compactification of $\mathbf{T}_{k}^{2}$ to which the self-action of $\mathbf{T}_{k}^{2}$ extends algebraically. There is a natural dimension-preserving bijection between the orbits of this extended action and the faces of $\Delta(f)$; for each face $\tau$, write $O(\tau)$ for the corresponding orbit. Then geometrically, the condition of non-degeneracy means that $C_{f}$ is a non-singular curve in $\mathbf{T}_{k}^{2}=O(\Delta(f))$ compactifying to a curve $C^{\prime}_{f}$ in $\text{Tor}_{k}(\Delta(f))$ that does not contain any of the zero-dimensional $O(\tau)$’s and that intersects the one-dimensional $O(\tau)$’s transversally.
$\Delta(f)$$\tau_{1}$$\tau_{2}$$\tau_{3}$$\tau_{4}$$\mathbf{R}^{2}$$\mathbf{T}^{2}$$C_{f}^{\prime}$$O(\tau_{2})$$O(\tau_{4})$$O(\tau_{1})$$O(\tau_{3})$$\text{Tor}_{k}(\Delta(f))$
In particular, since $\text{Tor}_{k}(\Delta(f))$ is normal, non-degeneracy implies that $C^{\prime}_{f}$ is a non-singular complete model of $C_{f}$. See [CDV, §2] for a more detailed description of this construction.
Example 4.
Assume that $\Delta(f)=\textup{conv}\{(0,0),(d,0),(0,d)\}$, i.e. $f$ is a degree-$d$ polynomial with non-zero coefficients at $1,x^{d},y^{d}$.
In this case $\text{Tor}_{k}(\Delta(f))$ is just the projective plane, and the toric orbits are $\mathbf{T}_{k}^{2}=O(\Delta(f))$,
the three coordinate points $(1:0:0)$, $(0:1:0)$ and $(0:0:1)$
(which are the orbits of the form $O(\text{vertex})$),
and the three coordinate axes from which the coordinate points are removed (these are the orbits of the form $O(\text{edge})$).
Then $f$ is non-degenerate with respect to its Newton polygon if and only if $C_{f}$ compactifies to a non-singular projective plane curve that
is non-tangent to any of the coordinate axes, and that does not contain any of the coordinate points.
Example 5.
Let $g\geq 2$ be an integer, and consider $f=y^{2}+h_{1}(x)y+h_{0}(x)$, where $\deg h_{1}\leq g+1$, $\deg h_{0}=2g+2$, and $h_{0}(0)\neq 0$. Then
$\Delta(f)=\textup{conv}\{(0,0),(2g+2,0),(0,2)\}$, and $\text{Tor}_{k}(\Delta(f))$ is the weighted projective plane $\mathbf{P}_{k}(1:g+1:1)$. Here again,
$f$ is non-degenerate with respect to its Newton polygon if and only if $C^{\prime}_{f}$ is a non-singular curve that is non-tangent to the coordinate axes
and does not contain any coordinate points. In this case $C^{\prime}_{f}$ is a hyperelliptic
curve of genus $g$ (cf. Remark 8).
Now for each edge $\tau\subset\Delta(f)$ let
$\nu_{\tau}\in\mathbf{Z}^{2}$ be the inward pointing primitive normal vector to $\tau$, let $p_{\tau}$ be any element of $\tau\cap\mathbf{Z}^{2}$, and let $D_{\tau}$ be the $k$-rational divisor on $C^{\prime}_{f}$ cut out by $O(\tau)$. If $f$ is non-degenerate with respect to its Newton polygon, then one can show
(2)
$$\text{div}\frac{dx}{xy\frac{\partial f}{\partial y}}=\sum_{\tau\text{ edge}}(-%
\langle\nu_{\tau},p_{\tau}\rangle-1)D_{\tau}.$$
Here $\langle\cdot,\cdot\rangle$ is the standard inner product on $\mathbf{R}^{2}$.
See [CDV, Cor. 3] for an elementary but elaborate proof of (2).
It is possible to give a more conceptual proof using adjunction theory.
Remark 6.
Using the theory of sparse resultants, one can show that $\partial f/\partial y$ does
not vanish identically,
so that the left-hand side of (2) makes sense.
Note also that $df=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy=0$, so
we could as well have written
(3)
$$\text{div}\frac{dy}{xy\frac{\partial f}{\partial x}}.$$
Proof of Lemma 1.
Assume that $f$ is such that $\Delta(f)=\Delta$ and that it is non-degenerate with respect to its Newton polygon (which, as mentioned above, is a non-empty Zariski open condition).
Let $(i_{0},j_{0})\in\mathbf{Z}^{2}$ be a solution to
the given system of congruences. Then the translated polygon $(-i_{0},-j_{0})+\Delta$ is such that all corresponding $\langle\nu_{\tau},p_{\tau}\rangle$’s are odd. So by applying the above to $x^{-i_{0}}y^{-j_{0}}f$, we find that
$$\sum_{\tau\text{ edge}}\frac{-\langle\nu_{\tau},p_{\tau}\rangle-1}{2}D_{\tau}$$
is a $k$-rational half-canonical divisor on $C^{\prime}_{x^{-i_{0}}y^{-j_{0}}f}=C^{\prime}_{f}$.
∎
Remark 7.
Let $\Delta=\textup{conv}\{(0,0),(d,0),(0,d)\}$ with $d$ odd. Then the condition of non-degeneracy restricts our attention to smooth plane curves of degree $d$ that do not contain the coordinate points and that intersect the coordinate axes transversally. But of course any smooth plane curve of degree $d$ carries a $k$-rational half-canonical divisor. This shows that the non-degeneracy condition, even though it is generically satisfied,
is sometimes a bit stronger than needed.222The reader might want to note that there always exists an automorphism of $\mathbf{P}_{k}^{2}$ that puts our smooth plane curve in a non-degenerate position (at least if $\#k$ is sufficiently large). But for
more general instances of $\Delta$, the automorphism group of $\text{Tor}_{k}(\Delta)$ may be much smaller (e.g. the only automorphisms may be the ones coming from the $\mathbf{T}_{k}^{2}$-action), in which case it
might be impossible to resolve tangency to the one-dimensional toric orbits.
In general, the according weaker condition reads that $C_{f}$ compactifies to a non-singular curve in $\text{Tor}_{k}(\Delta)$.
Here we have to revisit Remark 6, however:
there do exist instances of absolutely irreducible Laurent polynomials $f\in k[x^{\pm 1},y^{\pm 1}]$ for which $C_{f}$ compactifies to a non-singular curve in $\text{Tor}_{k}(\Delta(f))$, yet for
which $\partial f/\partial y$ does vanish identically (example: $f=1+x^{2}y^{2}+x^{3}y^{2}$).
For these instances of $f$
the left-hand side of (2) does not make sense. But in that
case $\partial f/\partial x$ does not vanish identically (otherwise $C_{f}$ would have singularities), and one can prove that (2) holds with the left-hand side replaced by (3).
Remark 8.
We mention two other well-known features of non-degenerate Laurent polynomials, that can be seen as consequences to (2); see for instance [CV] and the references therein:
•
the geometric genus of $C_{f}$ equals $\#\left(\Delta(f)^{\circ}\cap\mathbf{Z}^{2}\right)$, where $\Delta(f)^{\circ}$ denotes the
interior of $\Delta(f)$, and
•
in case $\#\left(\Delta(f)^{\circ}\cap\mathbf{Z}^{2}\right)\geq 2$, $C_{f}$ is hyperelliptic if and only if $\Delta(f)^{\circ}\cap\mathbf{Z}^{2}$ is contained in a line.
3. Proof of the main result
Lemma 9.
Let $\Delta$ be a two-dimensional lattice polygon and
suppose as in Lemma 1 that (1) admits a solution in $\mathbf{Z}^{2}$. If $\Delta$ is not among the polygons excluded in the hypothesis of Theorem 2, then there is a solution of (1) contained in $\Delta\cap\mathbf{Z}^{2}$.
Proof.
Let us first classify all two-dimensional lattice polygons $\Delta$ for which the reduction-modulo-$2$ map $\pi_{\Delta}\colon\Delta\cap\mathbf{Z}^{2}\rightarrow\left(\mathbf{Z}/(2)%
\right)^{2}$ is not surjective.
If the interior lattice points of $\Delta$ lie on a line,
then surjectivity fails if and only if $\Delta$ is among
$(0,0)$$(0,1)$$(k,0)$,(for some $k\geq 1$)(a)$(0,0)$$(1,2)$$(2,0)$,(b)$(0,1)$$(1,2)$$(2,1)$,$(1,0)$,(c)$(0,1)$$(k,2)$$(\ell,1)$$(0,0)$($0\leq k\leq\ell\geq 3$, $k$ even)(d)
(up to unimodular equivalence). This assertion follows from Koelman’s classification; see [Koe, Ch. 4] or [Cas, Thm. 10]. As a consequence, any two-dimensional $\Delta$ for which $\pi_{\Delta}$ is not surjective must contain one of the polygons (a-d) as an ‘onion skin’, i.e. as a lattice polygon obtained by subsequently taking the convex hull of the interior lattice points. But using the criterion from [HS, Lem. 9-11] one sees that the only polygons in
(a-d) that can appear as
convex hull of the interior lattice points
of a strictly bigger lattice polygon $\Gamma$ are the polygons (a) with $k=1$ or $k=2$, the polygon (b) and the polygon (c). The only corresponding instance of $\Gamma$ for which $\pi_{\Gamma}$ is not surjective is
$(0,-1)$$(-1,2)$$(2,0)$(e)
which, again by [HS, Lem. 9-11], is an onion skin only of itself.
This ends the classification.
Now let $\Delta$ be a two-dimensional lattice polygon and suppose that (1) admits a solution in $\mathbf{Z}^{2}$. If
$\pi_{\Delta}$ is surjective, then it clearly also admits a solution in $\Delta\cap\mathbf{Z}^{2}$. So we may assume that
$\Delta$ is among (a-e). Then the lemma follows by noting that cases (b), (c) and (d) with $\ell$ even admit the solution $(1,1)\in\Delta\cap\mathbf{Z}^{2}$, and that cases (a), (e) and (d) with $\ell$ odd were excluded in the énoncé.
∎
Remark 10.
Because of Remark 8, the excluded polygons correspond to certain classes of smooth plane quartics, rational curves, and hyperelliptic curves, respectively.
We can now define the variety $S_{\Delta}$ mentioned in the statement of Theorem 2. Namely, we will prove the existence of a
non-trivial $k$-rational $2$-torsion point under the assumption that
•
$\Delta(f)=\Delta$ and $f$ is non-degenerate with respect to its Newton polygon (i.e. the genericity assumption from Lemma 1), and
•
for at least one solution $(i_{0},j_{0})\in\Delta\cap\mathbf{Z}^{2}$ to the system of congruences (1), the corresponding coefficient $c_{i_{0},j_{0}}$ is non-zero.
So we can let $S_{\Delta}$ be defined by $c_{i_{0},j_{0}}\rho_{\Delta}\neq 0$.
Remark 11.
Here again, one can weaken the non-degeneracy condition as described in Remark 7. When that stronger version is applied
to $\Delta=\textup{conv}\{(0,0),(d,0),(0,d)\}$ with $d$ odd, one exactly recovers [CEZB, Thm. 4.2].
Proof of Theorem 2.
By replacing $f$ with $x^{-i_{0}}y^{-j_{0}}f$
if needed, we assume that $(0,0)\in\Delta$ is a solution to the system of congruences (1) and that the constant term of $f$ is non-zero.
As explained in [Mum, p. 191], $C_{f}^{\prime}$ comes equipped with a $k$-rational divisor $\Theta_{\text{arith}}$ such that $2\Theta_{\text{arith}}=\text{div}\,dx$.
(Recall that the derivative of a Laurent series over $k$ is always a square, so the order of $dx$ at a point of $C_{f}^{\prime}$ is indeed even.)
On the other hand, Lemma 1 and its proof provide us with a $k$-rational divisor $\Theta_{\text{geom}}$ such that
$$2\Theta_{\text{geom}}=\text{div}\,\frac{dx}{xy\frac{\partial f}{\partial y}}.$$
In order to prove that $\Theta_{\text{geom}}\not\sim\Theta_{\text{arith}}$ (and hence that $\text{Jac}(C^{\prime}_{f})$ has a non-trivial $k$-rational $2$-torsion point), we
need to show that
$$xy\frac{\partial f}{\partial y}$$
is a non-square when considered as an element of the function field $k(C_{f})$. If it were a square, then there would exist Laurent polynomials $\alpha,G,H$ such that
(4)
$$H^{2}xy\frac{\partial f}{\partial y}+\alpha f=G^{2}\qquad\text{in $k[x^{\pm 1}%
,y^{\pm 1}]$},$$
where $f\nmid H$.
Taking derivatives with respect to $y$ yields
$$(\alpha+H^{2}x)\frac{\partial f}{\partial y}=\frac{\partial\alpha}{\partial y}f,$$
which together with (4) results in
$$\left((\alpha+H^{2}x)\alpha+H^{2}xy\frac{\partial\alpha}{\partial y}\right)f=(%
\alpha+H^{2}x)G^{2}.$$
Since $f$ is irreducible, it follows that $f\mid(\alpha+H^{2}x)$ or $f\mid G^{2}$. Using (4) and $f\nmid H$, the latter implies that $f\mid\frac{\partial f}{\partial y}$, which is
a contradiction (by the theory of sparse resultants, see Remark 6; one can alternatively repeat the argument using (3) if wanted).
So we know that $f\mid(\alpha+H^{2}x)$. Along with (4) we conclude that there exists a Laurent polynomial $\beta\in k[x^{\pm 1},y^{\pm 1}]$ such that
$$H^{2}x\left(y\frac{\partial f}{\partial y}+f\right)+\beta f^{2}=G^{2}.$$
Taking derivatives with respect to $x$ yields
$$H^{2}\left(f+x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}+xy%
\frac{\partial^{2}f}{\partial x\partial y}\right)+\frac{\partial\beta}{%
\partial x}f^{2}=0.$$
Since $f$ has a non-zero constant term, the large factor between brackets is non-zero. On the other hand, since $f\nmid H$, it must be a multiple of $f^{2}$.
Note that $\Delta(f^{2})=2\Delta(f)$, while $\Delta(f+\cdots+xy\partial^{2}f/(\partial x\partial y))\subset\Delta(f)$. This is a contradiction.
∎
We end this section by discussing some asymptotic consequences to Theorem 2.
Growing field size.
Let $\Delta$ be a two-dimensional
lattice polygon satisfying the conditions of Theorem 2.
Let $k$ be a finite field of characteristic $2$. Because
non-degeneracy is characterized by the non-vanishing of $\rho_{\Delta}$, the
proportion of Laurent polynomials $f\in k[x^{\pm 1},y^{\pm 1}]$ that are
non-degenerate with respect to their Newton polygon $\Delta(f)=\Delta$ (amongst all
Laurent polynomials that are supported on $\Delta$) converges to $1$
as $\#k\rightarrow\infty$. Then Theorem 2 implies:
$$\lim_{\#k\rightarrow\infty}\text{Prob}\left(\,\text{Jac}(C_{f}^{\prime})(k)[2]%
\neq 0\,\left|\,\text{$f\in k[x^{\pm 1},y^{\pm 1}]$ is non-degenerate
with \phantom{x}}\right.\right.$$
$$\left.\left.\text{\phantom{$\in k[x^{\pm 1},y^{\pm 1}]$} respect to its Newton%
polygon $\Delta(f)=\Delta$}\right.\right)=1.$$
As soon as $\#(\Delta^{\circ}\cap\mathbf{Z}^{2})\geq 2$ this is deviating statistical behavior: in view
of Katz-Sarnak-Chebotarev-type density theorems [KS, Theorem 9.7.13], for a general smooth proper family of
genus $g$ curves, one expects that the
probability of having a non-trivial
rational $2$-torsion point on the Jacobian
approaches the chance that a random matrix in $\text{GL}_{g}(\mathbf{F}_{2})$ satisfies
$\det(M-\text{Id})=0$, which is
$$-\sum_{r=1}^{g}\prod_{j=1}^{r}\frac{1}{1-2^{j}}$$
by [CFHS, Thm. 6]. For $g=1,2,3,4,\dots$, these probabilities are $1,\frac{2}{3},\frac{5}{7},\frac{32}{45},\dots$ (converging to about $0.71121$).
In the table below we denote by $\square_{i}$ the square $[0,i]^{2}$ (for $i=2,3,4$),
by $H_{g}$ the hyperelliptic polygon $\textup{conv}\{(0,0),(2g+2,0),(0,2)\}$ (for $g=7,8$), and
by $E$ the exceptional polygon $\textup{conv}\{(1,0),(3,1),(0,3)\}$ from the statement of Theorem 2.
Each entry
corresponds to a sample of $10^{4}$ uniformly randomly chosen
Laurent polynomials $f\in k[x^{\pm 1},y^{\pm 1}]$ that are supported on $\square_{2},\square_{3},\dots$
The table presents the proportion of $f$’s for which $\text{Jac}(C_{f}^{\prime})$ has a non-trivial $k$-rational $2$-torsion point,
among those $f$’s that are non-degenerate with
respect to their Newton polygon $\Delta(f)=\square_{2},\square_{3},\dots$ An entry ‘-’
means that we did not do the computation because it would take too long. The computations were
carried out using Magma [BCP].
$$k\phantom{i}$$
$$\begin{array}[]{c}\square_{2}\\
\text{\footnotesize{$(g=1)$}}\\
\end{array}$$
$$\begin{array}[]{c}\square_{3}\\
\text{\footnotesize{$(g=4)$}}\\
\end{array}$$
$$\begin{array}[]{c}\square_{4}\\
\text{\footnotesize{$(g=9)$}}\\
\end{array}$$
$$\begin{array}[]{c}H_{7}\\
\text{\footnotesize{$(g=7)$}}\\
\end{array}$$
$$\begin{array}[]{c}H_{8}\\
\text{\footnotesize{$(g=8)$}}\\
\end{array}$$
$$\begin{array}[]{c}E\\
\text{\footnotesize{$(g=3)$}}\\
\end{array}$$
$$\mathbb{F}_{2}\phantom{i}$$
$$0/0$$
0.370
0.958
0.995
0.670
0.143
$$\mathbb{F}_{4}\phantom{i}$$
0.750
0.621
1.000
1.000
0.795
0.449
$$\mathbb{F}_{8}\phantom{i}$$
0.884
0.654
1.000
1.000
0.852
0.591
$$\mathbb{F}_{16}\phantom{i}$$
0.940
0.697
1.000
1.000
0.872
0.661
$$\mathbb{F}_{32}\phantom{i}$$
0.968
0.704
1.000
1.000
0.877
0.696
$$\mathbb{F}_{64}\phantom{i}$$
0.986
-
1.000
1.000
0.880
0.694
$$\mathbb{F}_{128}\phantom{i}$$
0.992
-
1.000
1.000
0.889
0.708
$$\mathbb{F}_{256}\phantom{i}$$
0.996
-
1.000
1.000
0.888
-
$$\begin{array}[]{r}\text{\footnotesize{asymptotic}}\\
\text{\footnotesize{prediction}}\\
\end{array}$$
$$1$$
$$\frac{32}{45}\approx 0.711$$
$$1$$
$$1$$
$$\frac{8}{9}\approx 0.889$$
$$\frac{5}{7}\approx 0.714$$
Note that the conditions of Theorem 2 are
satisfied for $\square_{2}$, $\square_{4}$ and $H_{7}$. So here we proved that the proportion converges to $1$.
In the case of $H_{8}$, by the material in Section 4
(see Corollary 16) we know that the proportion converges
to $\frac{8}{9}$. In the other two cases $\square_{4}$ and $E$ we have no clue, so our best guess is that these follow
the $\text{GL}_{g}(\mathbf{F}_{2})$-model.
Growing polygon.
Let $k$ be a finite field of characteristic $2$. If $\Delta$ is a two-dimensional lattice polygon satisfying the conditions
of Lemma 1, then the same holds for each odd Minkowski multiple $(2n+1)\Delta$.
It seems reasonable to assume that the proportion of Laurent polynomials $f\in k[x^{\pm 1},y^{\pm 1}]$ that are
non-degenerate with respect to their Newton polygon $\Delta(f)=(2n+1)\Delta$ (amongst all
Laurent polynomials that are supported on $(2n+1)\Delta$) converges to
a certain strictly positive constant.
If $\text{Tor}_{k}(\Delta)$ is smooth then
this is certainly true for the larger proportion of Laurent polynomials $f$ satisfying
the weaker condition from Remark 7, namely that
$C_{f}$ compactifies to a non-singular curve in $\text{Tor}_{k}((2n+1)\Delta)$. Indeed,
using [Poo2, Thm. 1.1] one
can show that this proportion converges to
$$Z_{\text{Tor}_{k}(\Delta)}((\#k)^{-3})^{-1}=(1-(\#k)^{-1})(1-(\#k)^{-2})^{r(%
\Delta)-2}(1-(\#k)^{-3})$$
as $n\rightarrow\infty$; here $Z_{\text{Tor}_{k}(\Delta)}$ is the Hasse-Weil Zeta function of $\text{Tor}_{k}(\Delta)$, and $r(\Delta)$ is the number of vertices of $\Delta$.
On the other hand, the number of solutions to (1) inside $(2n+1)\Delta\cap\mathbf{Z}^{2}$ tends to infinity. So
the assumption would allow
one to conclude:
$$\lim_{n\rightarrow\infty}\text{Prob}\left(\,\text{Jac}(C_{f}^{\prime})(k)[2]%
\neq 0\,\left|\,\text{$f\in k[x^{\pm 1},y^{\pm 1}]$ is non-degenerate
with \phantom{x}}\right.\right.$$
$$\left.\left.\text{\phantom{$\in k[x^{\pm 1},y^{\pm 1}]$} respect to its Newton%
polygon $\Delta(f)=(2n+1)\Delta$}\right.\right)=1.$$
This is again deviating statistical behavior: in view of Cohen-Lenstra type heuristics, one naively expects a probability of about
$$1-\prod_{j=1}^{\infty}(1-2^{-j})\approx 0.71121;$$
see [CEZB] for some additional comments.
When applied to $\Delta=\textup{conv}\{(0,0),(1,0),(0,1)\}$ one recovers the claim made before [CEZB, Thm. 4.2]
(taking into account Remark 7).
4. Hyperelliptic curves of odd genus
Let $C$ be a hyperelliptic curve
of genus $g$
over a perfect field $k$.
Then $C$ has a smooth weighted projective plane model
(5)
$$C\colon\quad Y^{2}+H(X,Z)Y=F(X,Z),$$
where $H$ and $F$ in $k[X,Z]$ are homogeneous
of degrees $g+1$ and $2g+2$ respectively.
The Newton polygon of (the defining polynomial of) the corresponding
affine model $y^{2}+H(x,1)y-F(x,1)=0$
is contained in
a triangle
with vertices $(0,0)$, $(2g+2,0)$ and $(0,2)$,
and is generically equal to this triangle.
In particular, Theorem 2 implies
that if the characteristic of $k$ is $2$
and $C$ is sufficiently general of odd genus, then its Jacobian
has a non-trivial $k$-rational $2$-torsion point.
The purpose of this section is to show
that we can replace ‘sufficiently general’ by ‘ordinary’
in this case (Corollary 14), and to
give explicit examples of the rational half-canonical divisors
from the proof of Theorem 2.
Theorem 12.
Let $C/k$ be a hyperelliptic curve over a perfect
field $k$ of characteristic $2$
given by a smooth model (5).
The Jacobian of $C$ has no
rational point of order $2$
if and only if $H(X,Z)$
is a power of an irreducible odd-degree
polynomial in $k[X,Z]$.
Corollary 13.
Let $C/k$ be a hyperelliptic curve of odd $2$-rank
over a perfect field $k$ of characteristic $2$.
Then the Jacobian of $C$ has a $k$-rational point
of order $2$.
Corollary 14.
Let $C/k$ be an ordinary hyperelliptic curve of odd genus
over a perfect field $k$ of characteristic $2$.
Then the Jacobian of $C$ has a $k$-rational point
of order $2$.
Corollary 15.
Let $C/k$ be a hyperelliptic curve of genus $2^{m}-1$ over a perfect field
$k$ of characteristic $2$, for some integer $m\geq 2$.
If the Jacobian of $C$ has no $k$-rational point of order $2$,
then it has $2$-rank zero, but it is not supersingular.
Finally, for integers $g,r\geq 1$, let $c_{g,r}$ be the proportion of
equations (5) over $\mathbf{F}_{2^{r}}$
that define a curve of genus $g$ whose
Jacobian has at least one rational point of order $2$.
Corollary 16.
The limit $\lim_{r\rightarrow\infty}c_{g,r}$ exists and
we have
$$\lim_{r\rightarrow\infty}c_{g,r}=\begin{cases}1&\text{if $g$ is odd,}\\
g/(g+1)&\text{if $g$ is even}.\end{cases}$$
Proof of Theorem 12.
All we need to do is describe the two-torsion of the Jacobian
$\text{Jac}(C)$ of $C$. Since we were not able to find a ready-to-use statement in the literature,
we give a stand-alone treatment, even though what follows
is undoubtedly known to several experts in the field; for instance, it
is implicitly contained in [EP]. Let $\overline{k}$ be an algebraic closure
of $k$. Note that $C$ has a unique point $Q_{(a:b)}=(a:\sqrt{F(a,b)}:b)\in C(\overline{k})$
for every root $(a:b)\in\mathbf{P}^{1}_{\overline{k}}$ of $H=H(X,Z)$.
This gives $n$ points, where $n\in\{1,\dots,g+1\}$ is the number
of distinct roots of $H$.
Let $D$ be the divisor of zeroes of a vertical line,
so $D$ is effective of degree $2$.
All such divisors $D$ are linearly equivalent,
and are linearly equivalent to $2Q_{(a:b)}$ for each $(a:b)$.
In particular, if we let
$$A=\ker\left(\xymatrix{\displaystyle\bigoplus_{(a:b)}(\mathbf{Z}/2\mathbf{Z})%
\ar^{\sum}[r]&(\mathbf{Z}/2\mathbf{Z})}\right),$$
then we have a homomorphism
$$\displaystyle A$$
$$\displaystyle\quad\longrightarrow\quad\text{Jac}(C)(\overline{k})[2]$$
$$\displaystyle(c_{(a:b)}\ \mbox{mod}\ 2)_{(a:b)}$$
$$\displaystyle\quad\longmapsto\quad(\sum_{(a:b)}c_{(a:b)}Q_{(a:b)})-(\frac{1}{2%
}\sum_{(a:b)}c_{(a:b)})D.$$
In fact, this map is an isomorphism. Indeed, it is injective
because if the divisor of a function is invariant under the hyperelliptic involution, then so
is the function itself, i.e. it is contained in $\overline{k}(x)$. But at the points $Q_{(a:b)}$
such functions can only admit poles or zeroes having an even order.
Surjectivity follows from the fact that $\text{Jac}(C)(\overline{k})[2]$ is generated by divisors that are supported on
the Weierstrass locus of $C$. This can be seen using Cantor’s algorithm
[Kob, Appendix.§6-7], for the application of which one needs to
transform the curve to a so-called imaginary model; this is always possible over $\overline{k}$.
Alternatively,
surjectivity follows from the injectivity and the fact that $\#\text{Jac}(C)(\overline{k})[2]=2^{n-1}$ by
[EP, Thm. 1.3].
Then in particular, the rational $2$-torsion subgroup
$\text{Jac}(C)(k)[2]$
is isomorphic to the subgroup of elements of $A$
that are invariant under $\mathrm{Gal}(\overline{k}/k)$,
that is, to
$$A_{k}=\ker\left(\bigoplus_{P\mid H}(\mathbf{Z}/2\mathbf{Z})\rightarrow(\mathbf%
{Z}/2\mathbf{Z}):(c_{P})_{P}\mapsto\sum_{P}c_{P}\deg(P)\right)$$
where the sum is taken over the irreducible factors $P$ of $H$.
The only way for $A_{k}$ to be trivial is for $H$
to be the power of an irreducible factor $P$
of odd degree.
∎
Proof of Corollary 13.
Let $n$ be the degree of the radical $R$ of $H$.
The $2$-rank of $C$ equals $n-1$
(as in the proof of Theorem 12; see e.g. [EP, Thm. 1.3]).
So if the $2$-rank is odd, then $R$ has even degree, which
implies that $H$ is not a power of an odd-degree polynomial.
In particular, Theorem 12
implies that $C$ has a non-trivial $k$-rational $2$-torsion point.
∎
Proof of Corollary 14.
This is a special case of Corollary 13
since in characteristic $2$, the $2$-rank of an ordinary abelian variety equals
its dimension.
∎
Proof of Corollary 15.
If there is no rational point of order $2$, then $H$ is a power of
a polynomial of odd degree dividing $\deg(H)=g+1=2^{m}$.
In other words, it is a power
of a linear polynomial and hence the $2$ rank of $C$ is zero.
There are no supersingular hyperelliptic
curves of genus $2^{m}-1$ in characteristic $2$ by [SZ, Thm. 1.2].
∎
Proof of Corollary 16.
As $r\rightarrow\infty$, the proportion of equations (5)
for which $H$ is not separable becomes negligible.
By Theorem 12 it therefore
suffices to prove the corresponding limit for the proportion
of degree $g+1$ polynomials that are not irreducible of odd degree.
If $g$ is odd then this proportion is clearly $1$. If $g$ is even
then this
is the same as the proportion of reducible polynomials of degree $g+1$,
which converges to $1-(g+1)^{-1}$.
∎
Remark 17.
In Corollary 16, instead of working with the proportion of equations (5),
we can work with the corresponding proportion of $\mathbf{F}_{2^{r}}$-isomorphism classes of
hyperelliptic curves of genus $g$. This is because
the subset of equations (5) that define a hyperelliptic curve of genus $g$ whose
only non-trivial geometric automorphism
is the hyperelliptic involution (inside the affine space of all
equations of this form) is non-empty [Poo1], open, and defined over $\mathbf{F}_{2}$ (being invariant under
the $\text{Gal}(\overline{\mathbf{F}}_{2},\mathbf{F}_{2})$-action). See also [Zhu].
Theorem 2 proves Corollary 14
for sufficiently general curves.
We finish by showing that the $2$-torsion points
from both proofs are equal.
The proof of Theorem 2 provides
$\Theta_{\mathrm{arith}}$ and $\Theta_{\mathrm{geom}}$
with $2\Theta_{\mathrm{arith}}\sim 2\Theta_{\mathrm{geom}}$,
hence the class of
$T=\Theta_{\mathrm{arith}}-\Theta_{\mathrm{geom}}$
is two-torsion.
We have $2\Theta_{\mathrm{arith}}=\text{div}\,dx$.
To compute $2\Theta_{\mathrm{geom}}$, we need to take
an appropriate model as in the proof of
Lemma 1.
The bivariate polynomial $y^{2}+H(x,1)y+F(x,1)$
gives an affine model of our hyperelliptic curve $C$,
and if $g$ is odd, then
the system from Lemma 1 admits the solution $(1,1)$.
By the proof of that lemma, we should then look at the
toric model $C^{\prime}_{f}$ where
$$f=x^{-1}(y+H(x,1)+y^{-1}F(x,1)).$$
Then $\Theta_{\mathrm{geom}}$
is given by $2\Theta_{\mathrm{geom}}=\text{div}\frac{1}{xy\frac{\partial f}{\partial y}}dx$,
so
we compute
$$\frac{\partial f}{\partial y}=x^{-1}(1+y^{-2}F(x,1))=x^{-1}y^{-1}\ H(x,1).$$
We find
$$T=\Theta_{\mathrm{arith}}-\Theta_{\mathrm{geom}}=\frac{1}{2}\ \text{div}\,xy%
\frac{\partial f}{\partial y}=\frac{1}{2}\text{div}\,H(x,1),$$
where $\text{div}\,H(x,1)$ is twice the sum of all points
$P_{(a:b)}$ as $(a:b)$ ranges over the roots of $H(X,Z)$ in $\mathbf{P}^{1}_{\overline{k}}$ (with multiplicity),
minus $(g+1)$ times the divisor $D$ of degree $2$ at infinity.
So both proofs give the same $2$-torsion point in cases
where they both apply.
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the Jacobian of a genus $2$ curve is prime, Proc. London Math. Soc. 104(6), pp. 1235-1270 (2012)
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J. Denef, F. Vercauteren, Computing zeta functions of $C_{a,b}$ curves using Monsky-Washnitzer cohomology, Fin. Fields App. 12(1), pp. 78-102 (2006)
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A. Elkin, R. Pries, Ekedahl-Oort strata of hyperelliptic curves in characteristic $2$, to appear in Algebra & Number Theory
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C. Haase, J. Schicho, Lattice polygons and the number $2i+7$, American Mathematical Monthly 116(2), pp. 151-165 (2009)
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N. Katz, P. Sarnak, Random matrices, Frobenius eigenvalues, and monodromy,
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B. Poonen, Varieties without extra automorphisms. II. Hyperelliptic curves,
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E-mail: [email protected].
Address: Departement Wiskunde, KU Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium.
E-mail: [email protected].
Address: Mathematisch Instituut, Universiteit Leiden, Postbus 9512, 2300 RA Leiden, The Netherlands.
E-mail: [email protected].
Address: Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom. |
Higher Kazhdan projections, $\ell_{2}$-Betti numbers and Baum-Connes conjectures
Kang Li
Piotr W. Nowak
Sanaz Pooya
Abstract
We introduce higher-dimensional analogs of Kazhdan projections in matrix algebras over group $C^{*}$-algebras
and Roe algebras. These projections are constructed in the framework of cohomology with coefficients in unitary representations and
in certain cases give rise to non-trivial $K$-theory classes.
We apply the higher Kazhdan projections to establish a relation between
$\ell_{2}$-Betti numbers of a group and surjectivity of different Baum-Connes type assembly maps.
Kazhdan projections are certain idempotents whose existence in the maximal group $C^{*}$-algebra $C^{*}_{\max}(G)$ of a group $G$ characterizes Kazhdan’s property $(T)$
for $G$. They have found an important application in higher index theory, as the non-zero $K$-theory class represented by a Kazhdan projection in $K_{0}(C^{*}_{\max}(G))$ obstructs the Dirac-dual Dirac method of proving the Baum-Connes conjecture.
This idea has been used effectively in the coarse setting [higson, higson-lafforgue-skandalis], where Kazhdan projections were used to construct
counterexamples to the coarse Baum-Connes conjecture and to the Baum-Connes conjecture with coefficients.
However, Kazhdan projections related to property $(T)$ are not applicable to the classical Baum-Connes conjecture, as the image of a Kazhdan projection
in the reduced group $C^{*}$-algebra vanishes for any infinite group.
The goal of this work is to introduce
higher-dimensional analogs of Kazhdan projections with the motivation of applying them
to the $K$-theory of group $C^{*}$-algebra and to higher index theory.
These higher Kazhdan projections are defined in the context of higher cohomology with coefficients in unitary representations
and they are elements of matrix algebras over group $C^{*}$-algebras. Their defining feature is that their
image in certain unitary representations is the orthogonal projection onto the harmonic $n$-cochains.
We show that the higher Kazhdan projections exist in many natural cases and that they are non-zero.
We then investigate the classes represented by higher Kazhdan projection in the $K$-theory of group $C^{*}$-algebras.
This is done by establishing a connection between the properties of the introduced projections and $\ell_{2}$-Betti numbers of the
group. In particular, we can conclude that the higher Kazhdan projections over the reduced group $C^{*}$-algebra can be non-zero
and can give rise to non-zero $K$-theory
classes in $K_{0}(C^{*}_{r}(G))$.
To present the connections between the existence of higher Kazhdan projections and $\ell_{2}$-invariants recall that given
a finitely generated group $G$ the subring $\Lambda^{G}\subseteq\mathbb{Q}$ is generated
from $\mathbb{Z}$ by adjoining inverses of all orders of finite subgroups, as in [luck, Theorem 0.3].
The cohomological Laplacian in degree $n$ can be represented by a matrix with coefficients in the group ring $\mathbb{R}G$, denoted $\Delta_{n}$.
In the case of the regular representation $\lambda$ and the reduced group $C^{*}$-algebra $C^{*}_{r}(G)$
the operator $\Delta_{n}\in\mathbb{M}_{k}(C^{*}_{r}(G))$ is the cohomological Laplacian in degree $n$ in the $\ell_{2}$-cohomology of $G$.
We will say that $\Delta_{n}$ has a spectral gap in a chosen completion of $\mathbb{M}_{n}(\mathbb{C}G)$ if its spectrum is contained in $\{0\}\cup[\epsilon,\infty)$ for some $\epsilon>0$.
Recall that a group $G$ is of type $F_{n}$ if it has an Eilenberg-MacLane space
with a finite $n$-skeleton. Denote by $k_{n}$ the number of simplices in the chosen model of the Eilenberg-MacLane space.
Let $\beta_{(2)}^{n}(G)$ denote the $n$-th $\ell_{2}$-Betti number of $G$.
The following is a consequence of the existence of spectral gap for the Laplacian and [luck].
Proposition 1.
Let $G$ be of type $F_{n+1}$. Assume $\Delta_{n}\in\mathbb{M}_{k_{n}}(C^{*}_{r}(G))$ has a spectral gap.
If the Baum-Connes assembly map $K_{0}^{G}(\underline{E}G)\to K_{0}(C^{*}_{r}(G))$
is surjective
then
$$\beta_{(2)}^{n}(G)\in\Lambda^{G}.$$
In particular, if $G$ is torsion-free then $\beta_{(2)}^{n}(G)\in\mathbb{Z}.$
The above theorem illustrates a possible strategy
for finding counterexamples to the Baum-Connes conjecture: a group $G$ of type $F_{n+1}$ such that
$\beta_{(2)}^{n}(G)\notin\Lambda_{G}$ (in particular, if $\beta_{(2)}^{n}(G)$ is irrational) with 0 isolated in the spectrum of $\Delta_{n}$ will not satisfy the
Baum-Connes conjecture; more precisely, the Baum-Connes assembly map for $G$ will not be surjective.
We refer to [gomez-aparicio-etal, valette]
for an overview of the Baum-Connes conjecture. It is worth noting that such a counterexample, with bounded orders of finite subgroups, would also not satisfy the Atiyah conjecture, see e.g. [luck-book].
Our primary application is the construction of classes in the $K$-theory of the
Roe algebra of a box space of a residually finite group $G$ and their relation to the Lück approximation theorem.
Let $\{N_{i}\}$ be a family of finite index normal subgroups of a residually finite group $G$, satisfying $\bigcap N_{i}=\{e\}$.
Let $\lambda_{i}$ denote the associated quasi-regular representation of $G$ on $\ell_{2}(G/N_{i})$ and consider the family $\mathcal{N}=\{\lambda,\lambda_{1},\lambda_{2},\dots\}$ of unitary representations of $G$.
The $C^{*}$-algebra $C^{*}_{\mathcal{N}}(G)$ is the completion of the group ring $\mathbb{C}G$
with respect to the norm induced by the family $\mathcal{N}$ (see
Section 1.1 for a precise definition). We will denote
by $\beta^{n}(G)=\dim_{\mathbb{C}}H^{n}(G,\mathbb{C})$ the standard $n$-th Betti number of $G$.
Theorem 2.
Let $G$ be an exact, residually finite group of type $F_{n+1}$ and let $\{N_{i}\}$ and $\mathcal{N}$ be as above.
Assume that $\Delta_{n}\in\mathbb{M}_{k_{n}}(C^{*}_{\mathcal{N}}(G))$ has a spectral gap
and that the coarse Baum-Connes assembly map
$KX_{0}(Y)\to K_{0}(C^{*}(Y))$
for the box space
$Y=\coprod G/N_{i}$ of $G$
is surjective. Then
$$\beta_{(2)}^{n}(N_{i})=\beta^{n}(N_{i})$$
for all but finitely many $i$.
Note that the conclusion of the above theorem in fact is a strenghtening of Lück’s approximation theorem [luck-gafa].
Indeed, the expression in the formula in the above theorem can be rewritten as
$$[G:N_{i}]\left(\beta_{(2)}^{n}(G)-\dfrac{\beta^{n}(N_{i})}{[G:N_{i}]}\right)=0,$$
Theorem 2 thus
forces a much stronger equality of the involved Betti numbers.
On the other hand, as shown in [luck-approx-survey, Theorem 5.1] there are examples where the speed of convergence
of $\beta^{n}(N_{i})/[G:N_{i}]$ to $\beta_{(2)}^{n}(G)$ can be as slow as needed.
Theorem 2 in particular provides new strategies for contradicting the coarse Baum-Connes conjecture.
It is an important open question wether there exist such counterexamples that do not contain expander graphs.
However, it is also natural to conjecture that Theorem 2 could also lead to new counterexamples
to the Baum-Connes conjecture with coefficients
by embedding a counterexample to the coarse Baum-Connes conjecture, obtained through Theorem 2,
isometrically into a finitely generated group, as in [osajda].
Such constructions could lead to new counterexample to the Baum-Connes conjecture with coefficients, by applying arguments similar to the
ones in [higson-lafforgue-skandalis].
We remark that algebraic conditions implying the existence of gaps in the spectrum of the operators arising from the
cochain complexes with coefficients in unitary representations
were recently provided in
[bader-nowak]. Those conditions involve writing the elements $(\Delta_{n}^{+}-\lambda\Delta_{n}^{+})\Delta_{n}^{+}$
and $(\Delta_{n}^{-}-\lambda\Delta_{n}^{-})\Delta_{n}^{-}$, where $\Delta_{n}^{+}=d_{n}^{*}d_{n}$ and $\Delta_{n}^{-}=d_{n-1}d_{n-1}^{*}$ are
matrices over $\mathbb{R}G$ and the two summands of the Laplacian, as sums of squares in $\mathbb{M}_{n}(\mathbb{R}G)$.
Such conditions can be verified using computational methods.
Acknowledgments
We are grateful to Dawid Kielak, Roman Sauer, Thomas Schick and Alain Valette for helpful comments.
This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 677120-INDEX).
Contents
1 Higher Kazhdan projections
1.1 Kazhdan projections related to property $(T)$
1.2 Kazhdan projections in higher degree
1.3 $K$-theory classes - a special case
1.4 The Baum-Connes assembly map
1.5 Examples
1.5.1 Free groups
1.5.2 Kähler groups
1.5.3 Lattices in $\operatorname{PGL}_{n+1}(\mathbb{Q}_{p})$
2 The coarse Baum-Connes conjecture
2.1 Roe algebras, box spaces and Laplacians
2.1.1 Roe algebras
2.1.2 Laplacians on box spaces
2.2 The lifting map
2.3 Surjectivity of the coarse Baum-Connes assembly map
3 Final remarks
1 Higher Kazhdan projections
Recall that a group has type $F_{n}$
if it admits an
Eilenberg-MacLane space $K(G,1)$ with a finite $n$-skeleton. The condition $F_{1}$ is equivalent to $G$ being finitely generated, and $F_{2}$ is equivalent
to
$G$ having a finite presentation.
1.1 Kazhdan projections related to property $(T)$
We begin with revisiting the classical notion of Kazhdan projections in the context of property $(T)$.
Let $G$ be a finitely generated group with $S=S^{-1}$ a fixed generating set. The real (respectively, complex) group ring of $G$ will be denoted
by $\mathbb{R}G$ (respectively, $\mathbb{C}G$).
Consider a family $\mathcal{F}$ of unitary representations and let the associated group $C^{*}$-algebra be the completion
$$C^{*}_{\mathcal{F}}(G)=\overline{\mathbb{C}G}^{\|\cdot\|_{\mathcal{F}}},$$
where
$$\|f\|_{\mathcal{F}}=\sup_{\pi\in\mathcal{F}}\|\pi(f)\|.$$
We will always assume that $\mathcal{F}$ is a faithful family (i.e. for every $0\neq f\in\mathbb{C}G$ we have $\pi(f)\neq 0$ for some $\pi\in\mathcal{F}$) to ensure $\|\cdot\|_{\mathcal{F}}$ is a norm.
In particular, we obtain the maximal group $C^{*}$-algebra $C^{*}_{\max}(G)$ if $\mathcal{F}$ is the family of all unitary representations,
and the reduced group $C^{*}$-algebra $C^{*}_{r}(G)$ if $\mathcal{F}=\{\lambda\}$, where $\lambda$ is the left regular representation.
The following is a classical characterization of property $(T)$ for $G$ due to Akemann and Walter [akemann-walter]. We will use this characterization
as a definition and refer to [bekka-etal] for a comprehensive overview of property $(T)$.
Theorem 3 (Akemann-Walter [akemann-walter]).
The group $G$ has property $(T)$ if and only if there exists a projection $p_{0}=p_{0}^{*}=p_{0}^{2}\in C^{*}_{\max}(G)$
with the property that for every unitary representation $\pi$ of $G$
the image $\pi(p_{0})$ is the orthogonal projection onto the subspace of invariant vectors of $\pi$.
See e.g. [higson-roe-book, Section 3.7] and [drutu-nowak] for a broader discussion of Kazhdan projections.
We will now interpret Kazhdan projections in the setting of group cohomology.
Let $\pi$ be a unitary representation of $G$ on a Hilbert space $\mathcal{H}$.
Denote by $d_{0}$ the $\#S\times 1$ matrix
$$\left[\begin{array}[]{c}1-s_{1}\\
\vdots\\
1-s_{n}\end{array}\right],$$
where $s_{i}$ runs through the elements of $S$, with coefficients in $\mathbb{R}G$
and the Laplacian $\Delta_{0}\in\mathbb{R}G$ is
$$\Delta_{0}=d_{0}^{*}d_{0}=2\left(\#S-\sum_{s\in S}s\right)\ \ \ \ \in\mathbb{R%
}G.$$
For any unitary representation $\pi$ of $G$ on a Hilbert space $\mathcal{H}$ we have
$$\ker\pi(\Delta_{0})=\ker\pi(d_{0})=\mathcal{H}^{\pi},$$
where $\mathcal{H}^{\pi}\subseteq\mathcal{H}$ denotes the closed subspace of invariant vectors of $\pi$.
Note also that the same space can be interpreted as reduced111Recall that in a setting where the cochain spaces in a cochain complex
$C^{n}\ _{\overrightarrow{\ \ d_{n}\ \ }}C^{n+1}$ are equipped with a Banach space structure the corresponding reduced cohomology is defined as the quotient $\overline{H}^{n}=\ker d_{n}/\overline{\operatorname{im}d_{n-1}}$, where $\overline{\operatorname{im}d_{n-1}}$ is the closure of the image of the codifferential $d_{n-1}$. Cohomology is said to be reduced in degree $n$ if $\overline{H}^{n}=H^{n}$.
cohomology in degree 0:
$$\ker\pi(\Delta_{0})=\mathcal{H}^{\pi}\simeq\overline{H}^{0}(G,\pi).$$
The image $\pi(p_{0})$ of the Kazhdan projection is the orthogonal projection
$$C^{0}(G,\pi)\,_{\overrightarrow{\ \ \ \ \ \pi(p_{0})\ \ \ \ \ }}\,\ker\pi(%
\Delta_{0}).$$
The Kazhdan projection $p_{0}$ exists in $C^{*}_{\mathcal{F}}(G)$ if and only if $\pi(\Delta_{0})$ has a uniform spectral gap for all $\pi\in\mathcal{F}$.
The projection $p_{0}$ is non-zero if at least one $\pi\in\mathcal{F}$ has a non-zero invariant vector.
1.2 Kazhdan projections in higher degree
We will now discuss a generalization of Kazhdan projections in the setting of higher group cohomology.
Let $G$ be a group of type $F_{n+1}$ with a chosen model $X$ of $K(G,1)$ with finite $n+1$-skeleton.
As in [bader-nowak] we can consider the cochain complex for cohomology of $G$ with coefficients in $\pi$, where
$$C^{n}(G,\pi)=\,\left\{f:X^{(n)}\to\mathcal{H}\right\},$$
By $\mathbb{M}_{m\times m^{\prime}}(\mathbb{R}G)=\mathbb{M}_{m\times m^{\prime}}%
\otimes\mathbb{R}G$ we denote the space of
$m\times m^{\prime}$ matrices with coefficients in the group ring $\mathbb{R}G$.
When $m=m^{\prime}$ the resulting algebra is denoted by $\mathbb{M}_{m}(\mathbb{R}G)$.
The codifferentials can be represented by elements
$$d_{n}\in\mathbb{M}_{k_{n+1}\times k_{n}}(\mathbb{R}G),$$
where $k_{i}$ denotes the number of $i$-simplices in $X$,
in the sense that for every unitary representation $\pi$ the codifferential is given by the operator
$$\pi(d_{n}):C^{n}(G,\pi)\to C^{n+1}(G,\pi),$$
where $\pi$ is applied to an element of $\mathbb{M}_{m\times m^{\prime}}(C^{*}_{\mathcal{F}}(G))$ entry-wise;
see e.g. [bader-nowak-deformations, bader-nowak].
The Laplacian element in degree $n$ is defined to be
$$\Delta_{n}=d_{n}^{*}d_{n}+d_{n-1}d_{n-1}^{*}\in\mathbb{M}_{k_{n}}(\mathbb{R}G),$$
and we denote the two summands by
$$\Delta_{n}^{+}=d_{n}^{*}d_{n},\ \ \ \ \ \Delta_{n}^{-}=d_{n-1}d_{n-1}^{*},$$
as in [bader-nowak].
In particular, $\Delta_{n}$, $\Delta_{n}^{+}$, and $\Delta_{n}^{-}$ are elements of any completion of $\mathbb{M}_{k_{n}}(\mathbb{R}G)$ such as
$\mathbb{M}_{k_{n}}(C^{*}_{\max}(G))$ or $\mathbb{M}_{k_{n}}(C^{*}_{r}(G)).$
For a unitary representation $\pi$ of $G$ we can now consider the operator $\pi(\Delta_{n})$. The kernel of $\pi(\Delta_{n})$
is the space of harmonic $n$-cochains for $\pi$, and we have the standard Hodge-de Rham isomorphism
$$\ker\pi(\Delta_{n})\simeq\overline{H}^{n}(G,\pi),$$
between that kernel and the reduced cohomology of $G$ with coefficients in $\pi$ (see e.g. [bader-nowak, Section 3]).
Similarly, the kernel of the projection $\pi(\Delta_{n}^{+})$ are the $n$-cocycles for $\pi$, and the kernel of $\pi(\Delta_{n}^{-})$ are the
$n$-cycles.
Definition 4 (Higher Kazhdan projections).
A Kazhdan projection in degree $n$ is a projection $p_{n}=p_{n}^{*}=p_{n}^{2}\in\mathbb{M}_{k_{n}}(C^{*}_{\mathcal{F}}(G))$
such that
for every unitary representation $\pi\in\mathcal{F}$ the projection $\pi(p_{n})$ is the orthogonal projection $C^{n}(G,\pi)\to\ker\pi(\Delta_{n})$.
A partial Kazhdan projection in degree $n$ is a projection $p_{n}^{+},p_{n}^{-}\in\mathbb{M}_{k_{n}}(C^{*}_{\mathcal{F}}(G))$, such
that for every unitary representation $\pi\in\mathcal{F}$ the projection $\pi(p_{n}^{+})$ (respectively, $\pi(p_{n}^{-})$) is the
orthogonal projection onto the kernel of $\pi(d_{n})$ (respectively, onto the kernel of $\pi(d_{n-1}^{*})$).
In degree 0 and when $\mathcal{F}$ is the family of all unitary representations the projection $p_{0}\in C^{*}_{\max}(G)$ is the classical Kazhdan projection.
Clearly, $p_{n}$ is non-zero if and only if for at least one $\pi\in\mathcal{F}$ we have $\pi(p_{n})\neq 0$.
From now on we will shorten the subscript $k_{n}$, denoting the number of $n$-simplices in the chosen $K(G,1)$, to $k$, as the dimension will
be clear from the context.
Similarly as in the case of property $(T)$, higher Kazhdan projections exist in the presence of a spectral gap.
Proposition 5.
Assume that $\Delta_{n}\in\mathbb{M}_{k}(C^{*}_{\mathcal{F}}(G))$ (respectively, $\Delta_{n}^{+}$, $\Delta_{n}^{-}$) has a spectral gap.
Then the Kazdhan projection $p_{n}$ (respectively, the partial Kazhdan projections $p_{n}^{+}$, $p_{n}^{-}$) exist in the $C^{*}$-algebra
$\mathbb{M}_{k_{n}}(C^{*}_{\mathcal{F}}(G))$.
Proof.
By assumption, the spectrum of $\Delta_{n}$ satisfies
$$\sigma(\pi(\Delta_{n}))\subseteq\{0\}\cup[\epsilon,\infty)$$
for every unitary representation $\pi\in\mathcal{F}$ and some $\epsilon>0$.
In particular, $\sigma(\Delta_{n})\subseteq\{0\}\cup[\epsilon,\infty)$
when $\Delta_{n}$ is viewed as an element of $\mathbb{M}_{k}(C^{*}_{\mathcal{F}}(G))$.
Due to the presence of the spectral gap we can apply continuous functional calculus and conclude that the limit of the heat semigroup
$$\lim_{t\to\infty}e^{-t\Delta_{n}}$$
is the spectral projection $p_{n}\ \in\mathbb{M}_{k}(C^{*}_{\mathcal{F}}(G))$.
By construction, $p_{n}$ has the required properties.
The proof for the partial projections is analogous.
∎
Note that the fact that the spectrum of $\Delta_{n}\in\mathbb{M}_{k}(C^{*}_{\mathcal{F}}(G))$ has a gap implies that both $\Delta_{n}^{+}$ and $\Delta_{n}^{-}$
also have spectral gaps and in particular all three higher Kazhdan projections exist in
$\mathbb{M}_{k}(C^{*}_{\mathcal{F}}(G))$.
We have the following decomposition for higher Kazhdan projections.
Lemma 6.
Assume that $\Delta_{n}\in\mathbb{M}_{k}(C^{*}_{\mathcal{F}}(G))$ has a spectral gap. Then
$$p_{n}=p_{n}^{+}p_{n}^{-}.$$
Proof.
It suffices to notice that $\Delta_{n}^{+}$ and $\Delta_{n}^{-}$ commute in $\mathbb{M}_{k}(C^{*}_{\mathcal{F}}(G))$ and satisfy
$$\Delta_{n}^{+}\Delta_{n}^{-}=\Delta_{n}^{-}\Delta_{n}^{+}=0,$$
using the property that $d_{i}d_{i-1}=0$ for every $i\in\mathbb{N}$. Consequently, the corresponding spectral projections commute
and their product is the projection onto the intersections of their kernels, which is precisely the kernel of $\Delta_{n}$.
∎
1.3 $K$-theory classes - a special case
In this section we consider the case when $\mathcal{F}=\{\lambda\}$, where $\lambda$ is the left regular
representation of $G$ on $\ell_{2}(G)$ and $C^{*}_{\mathcal{F}}(G)=C^{*}_{r}(G)$ is the reduced group $C^{*}$-algebra
of $G$. We refer to [higson-roe-book, valette] for background on $K$-theory and assembly maps.
The projection $p_{n}\in\mathbb{M}_{k}(C^{*}_{r}(G))$ exists if $\lambda(\Delta_{n})$ has a spectral gap at $0$.
In this case the kernel of the Laplacian $\lambda(\Delta_{n})$ is isomorphic to the reduced $\ell_{2}$-cohomology of $G$:
$$\ker\lambda(\Delta_{n})\simeq\ell_{2}\overline{H}^{n}(G)\simeq\ker\lambda(d_{n%
})\big{/}\,\overline{\operatorname{im}\lambda(d_{n-1})},$$
and it is non-trivial if and only the $n$-th $\ell_{2}$-Betti number $\beta_{(2)}^{n}(G)$ is non-zero.
The projection $p_{n}\in\mathbb{M}_{k}(C^{*}_{r}(G))$ gives rise to a $K$-theory class $[p_{n}]\in K_{0}(C^{*}_{r}(G))$.
We are interested in the properties of the classes defined by the higher Kazhdan projections $[p_{n}]\in K_{0}(C^{*}_{r}(G))$.
Recall that the canonical trace $\tau_{G}$ on $C^{*}_{r}(G)$ is given by
$$\tau_{G}(\alpha)=\langle\alpha\delta_{e},\delta_{e}\rangle,$$
where $\delta_{e}\in\ell_{2}(G)$ is the Dirac delta at the identity element $e\in G$. For an element $\alpha\in\mathbb{C}G$ we have $\tau(\alpha)=\alpha(e)$.
The induced trace on $\mathbb{M}_{k}(C^{*}_{r}(G))$ is defined as
$$\tau_{k,G}\left((a_{ij})\right)=\sum_{i=1}^{k}\tau_{G}(a_{ii}),$$
and there is an induced map
$$\tau_{*}:K_{0}(C^{*}_{r}(G))\to\mathbb{R}$$
defined by $\tau_{*}([p])=\tau_{k,G}(p)$,
for a class represented by a projection $p\in\mathbb{M}_{k}(C^{*}_{r}(G))$.
Proposition 7.
Assume that $\Delta_{n}$ has a spectral gap in $\mathbb{M}_{k}(C^{*}_{r}(G))$.
Then
$$\tau_{*}([p_{n}])=\beta_{(2)}^{n}(G).$$
In particular, if $\beta_{(2)}^{n}(G)\neq 0$ then
$[p_{n}]\neq 0$ in $K_{0}(C^{*}_{r}(G))$.
Proof.
By definition, the $n$-th $\ell_{2}$ Betti number of $G$ is the von Neumann dimension (over the group von Neumann algebra $L(G)$)
of the
orthogonal projection onto the kernel of the Laplacian. Under our assumptions the projection
is now an element of the smaller algebra $\mathbb{M}_{k}(C^{*}_{r}(G))\subseteq\mathbb{M}_{k}(L(G))$ and
$$\beta_{(2)}^{n}(G)=\tau_{k,G}(p_{n}).$$
Therefore, the induced trace on the group $K_{0}(C^{*}_{r}(G))$ satisfies $\tau_{*}([p_{n}])=\beta_{(2)}^{n}(G)$,
as claimed
∎
A similar argument shows non-vanishing of the classes represented by the partial projections.
Proposition 8.
Assume that $0$ is isolated in the spectrum of $\Delta_{n}^{+}$ (respectively, $\Delta_{n}^{-}$) in $\mathbb{M}_{k}(C^{*}_{r}(G))$.
If $\beta_{(2)}^{n}(G)\neq 0$ then
$[p_{n}^{+}]\neq 0$ (respectively, $[p_{n}^{-}]\neq 0$) in $K_{0}(C^{*}_{r}(G))$.
Proof.
By assumption $p_{n}^{+}$ exists in $\mathbb{M}_{k}(C^{*}_{r}(G))$ and $p_{n}$ exists in $\mathbb{M}_{k}(L(G))$ and we will compare the
values of the trace over the latter algebra.
Since $\tau_{k,G}(p_{n})\neq 0$ and
$$\operatorname{im}(p_{n})\subseteq\operatorname{im}(p_{n}^{+})$$
by monotonicity of the von Neumann dimension over $L(G)$
we have
$$\tau_{k,G}(p_{n}^{+})\geq\tau_{k,G}(p_{n})>0.$$
Consequently $\tau_{*}([p_{n}^{+}])\neq 0$ in $K_{0}(C^{*}_{r}(G))$.
∎
1.4 The Baum-Connes assembly map
The Baum-Connes assembly map is the map
$$\mu_{i}:K_{i}^{G}(\underline{E}G)\to K_{i}(C^{*}_{r}(G)),$$
$i=0,1$.
The trace conjecture of Baum and Connes and the modified trace conjecture by Lück [luck] predict the range of the composition
$\tau_{*}\circ\mu_{0}$.
In particular, the modified trace conjecture, formulated and proved by Lück [luck],
states that $\tau_{*}\circ\mu_{i}$ takes values in a subring $\Lambda^{G}\subseteq\mathbb{Q}$,
generated from $\mathbb{Z}$ by adjoining the inverses of cardinalities of finite subgroups.
Theorem 9 (Lück [luck, Theorem 0.3]).
If the Baum-Connes assembly map is surjective then the composition $\tau_{*}\circ\mu_{0}$ takes values in the ring
$\Lambda^{G}\subseteq\mathbb{Q}$.
We can now state the relation between the Baum-Connes conjecture and $\ell_{2}$-Betti numbers.
Proposition 1.
Let $G$ be of type $F_{n+1}$. Assume that $0$ is isolated in the spectrum of $\Delta_{n}\in\mathbb{M}_{k}(C^{*}_{r}(G))$.
If the Baum-Connes assembly map $K_{0}^{G}(\underline{E}G)\to K_{0}(C^{*}_{r}(G))$
is surjective
then
$$\beta_{(2)}^{n}(G)\in\Lambda^{G}.$$
In particular, if $G$ is torsion-free then $\beta_{(2)}^{n}(G)\in\mathbb{Z}.$
Remark 10 (The Euler class).
Consider an infinite group $G$ such that $K(G,1)$ can be chosen to be a finite complex.
Assume that $\lambda(\Delta_{n})$ has a spectral gap in $\mathbb{M}_{k}(C^{*}_{r}(G))$ for every $n\geq 0$
(in particular, $G$ is non-amenable).
Then we can define the Euler class
$$\Xi(G)=\sum_{i\geq 0}(-1)^{i}[p_{i}]$$
in the $K$-theory $K_{0}(C^{*}_{r}(G))$.
We have that
$$\tau_{*}(\Xi(G))=\chi_{(2)}(G)$$
is the $\ell_{2}$-Euler characteristic of the chosen $K(G,1)$.
Atiyah’s $L_{2}$-index theorem associates the $L_{2}$-index, analogous to the Fredholm index but defined via the trace on the von Neumann algebra,
to a lift $\widetilde{D}$ of an elliptic operator $D$ on a compact manifold $M$ to the universal cover of $M$.
Loosely speaking, in the presence of the spectral gap the $L_{2}$-indices of certain operators appearing in the Atiyah $L_{2}$-index theorem
are in fact traces of their Baum-Connes indices.
1.5 Examples
In case of an infinite group $G$ the projection $\lambda(p_{0})\in C^{*}_{r}(G)$ is always 0 since the kernel of $\lambda(\Delta_{0})$
consists precisely of constant functions in $\ell_{2}(G)$.
We will now discuss certain cases in which higher Kazhdan projections exist.
We will use the following fact, which is a reformulation of [bader-nowak, Proposition 16, 2) and 3)]
Lemma 11.
Let $\pi$ be a unitary representation of $G$.
The Laplacian $\pi(\Delta_{n})$ has a spectral gap around $0$ if and only if the cohomology groups $H^{n}(G,\pi)$ and $H^{n+1}(G,\pi)$ are both reduced.
Sketch of proof.
Using the notation of [bader-nowak], given a cochain complex
$$\dots\to C^{n-1}\to C^{n}\to C^{n+1}\to\dots,$$
where the $C^{n}$ have Hilbert space structures and the codifferentials $d_{n}:C^{n}\to C^{n+1}$ are bounded operators,
we have $\Delta_{n}=d_{n}^{*}d_{n}+d_{n-1}d_{n-1}^{*}$.
We have an orthogonal decomposition
$$C^{n}=C_{n}^{+}\oplus\ker\Delta_{n}\oplus C_{n}^{-},$$
where $C_{n}^{-}=\ker d_{n}\cap\ker(d_{n-1}^{*})^{\perp}$ and $C_{n}^{+}=\ker d_{n-1}^{*}\cap(\ker d_{n}^{*})^{\perp}$.
Then the restritictions of $d_{n}^{*}d_{n}$ and of $d_{n-1}d_{n-1}^{*}$ are invertible on $C_{n}^{+}$ and $C_{n}^{-}$, respectively if and only if
the cohomology groups $H^{n}$ and $H^{n+1}$ are reduced [bader-nowak]. Clearly, the first condition is equivalent to the fact that $0$ is isolated in
the spectrum of the Laplacian $\Delta_{n}$.
∎
1.5.1 Free groups
Consider the free group $F_{n}$ on $n\geq 2$ generators. The standard Eilenberg-MacLane space $K(F_{n},1)$ is the wedge $\bigvee_{n}S^{1}$ of $n$ circles,
whose universal cover is the tree, the Cayley graph of $F_{n}$. Then for $F_{n}$ the projection $p_{1}\in\mathbb{M}_{k}(C^{*}_{r}(F_{n}))$ exists and gives rise to a non-zero class in
$K$-theory $C^{*}_{r}(F_{n})$.
Indeed, since for $n\geq 2$, the free group $F_{n}$ is non-amenable we have that the $\ell_{2}$-cohomology
is reduced in degree 1 (i.e., the range of the codifferential into the 1-cochains is closed).
Since $K(F_{n},1)$ is 1-dimensional, the cohomology in degree 2 is reduced as there are no 2-cochains.
We thus have $d_{1}=0$, $p_{1}^{+}=\operatorname{I}$ and $p_{1}=p_{1}^{-}$.
These facts together with lemma
11 imply that the cohomological Laplacian $\Delta_{1}$ in degree 1 has a spectral gap.
The $\ell_{2}$-Betti number of a free group $F_{n}$ on $n$ generators is
$$\beta_{(2)}^{1}(F_{n})=n-1.$$
By the previous discussion,
$$[p_{1}]\neq 0\ \ \ \text{ in }\ \ \ K_{0}(C^{*}_{r}(F_{n})).$$
Moreover, we can identify precisely the class of $p_{1}$ in the $K$-theory group.
Indeed, $K_{0}(C^{*}_{r}(F_{n}))$ is isomorphic to $\mathbb{Z}$ and generated by $1\in C^{*}_{r}(F_{n})$. Because of this the value of the trace
determines the class of $p_{1}$ to be
$$[p_{1}]=n-1\ \ \ \ \in\ \ \ \ \mathbb{Z}\simeq K_{0}(C^{*}_{r}(F_{n})).$$
The corresponding Euler class (see Remark 10) in $K_{0}(C^{*}_{r}(F_{n}))$ is given by
$$\Xi(F_{n})=[p_{0}]-[p_{1}]=-[p_{1}].$$
More generally, the argument used in the above example together with [kahlerbook, Corollary 4.8] gives
the following.
Corollary 12.
Let $G$ be a finitely presented group with infinitely many ends such that $H^{2}(G,\ell_{2}(G))$ is reduced. Then the projection
$p_{1}(G)$ exists in $\mathbb{M}_{k_{n}}(C^{*}_{r}(G))$ and
$$[p_{1}]\neq 0\ \ \ \text{ in }\ \ \ K_{0}(C^{*}_{r}(G)).$$
1.5.2 Kähler groups
As shown in [efremov], [gromov-shubin, Remark 2], see also [luck-book, Theorem 2.68] or [luck-survey, Section 8],
the property that the Laplacian has a spectral gap in the analytic setting (i.e. $L_{2}$-cohomology defined in terms differential forms)
is equivalent to the existence of the spectral gap in the combinatorial setting.
This is most often phrased in terms of Novikov-Shubin invariants associated to spectral density functions: spectral gap is equivalent to
the Novikov-Shubin invariant being infinite, which is the same as the property that the spectral density function is constant in the neighborhood
of 0. Then it is shown that the combinatorial and analytic spectral density functions are dilation equivalent, and in particular, equal in a neighborhood of 0.
Consider now a closed Kähler hyperbolic manifold $M$ and denote $G=\pi_{1}(M)$.
As shown by Gromov [gromov-Kahler], see also [luck-book, Section 11.2.3], the Laplacian acting on $L_{2}$-differential forms on the universal cover
$\widetilde{M}$ has a spectral gap in every dimension. Gromov also showed that the $\ell_{2}$-cohomology of $M$
vanishes except for the middle dimension.
Consequently, if $G$ is a fundamental group of such a Kähler manifold then the assumptions of
Proposition 7
are satisfied and
the projection $p_{n}$ exists in $\mathbb{M}_{k_{n}}(C^{*}_{r}(G))$ in every dimension.
Moreover, the Euler class
$$\Xi(G)=\sum_{i\geq 0}(-1)^{i}[p_{i}]\ \ \ \ \in K_{0}(C^{*}_{r}(G))$$
exist and its trace is the $\ell_{2}$-Euler characteristic of $G$.
1.5.3 Lattices in $\operatorname{PGL}_{n+1}(\mathbb{Q}_{p})$
As shown in [bader-nowak, Proposition 19], given $n\geq 2$, a sufficiently large prime $p$ and a lattice $\Gamma\subseteq\operatorname{PGL}_{n+1}(\mathbb{Q}_{p})$ we have that
$H^{n}(\Gamma,\pi)$ is reduced for every unitary representation $\pi$. This fact is equivalent to $\pi(\Delta_{n}^{-})$ having a spectral gap for every unitary
representation $\pi$ of $\Gamma$ and it is easy to check that such a spectral gap must be then uniform; i.e., $\Delta_{n}^{+}$ has a spectral
gap in $\mathbb{M}_{k}(C^{*}_{\max}(\Gamma))$.
At the same time there exists a finite index subgroup $\Gamma^{\prime}\subseteq\Gamma$
such that $H^{n}(\Gamma^{\prime},\mathbb{C})\simeq H^{n}(\Gamma,\ell_{2}(\Gamma/\Gamma^{%
\prime}))\neq 0$.
As a consequence we obtain the following.
Proposition 13.
Let $\Gamma\subseteq\operatorname{PGL}_{n+1}(\mathbb{Q}_{p})$ be a lattice. Then the partial Kazhdan projection $p_{n}^{+}$
exists in $\mathbb{M}_{k}(C^{*}_{\max}(\Gamma))$ and is non-zero.
2 The coarse Baum-Connes conjecture
We will now prove our main results and describe the $K$-theory classes induced by the higher Kazhdan projections in Roe algebras and their
applications to the coarse Baum-Connes conjecture of a box space
of a residually finite group. The arguments we will use were introduced in [higson, higson-lafforgue-skandalis] and developed further in
[willett-yu1]. We also refer to [willett-yu-book, Chapter 13] for more details.
2.1 Roe algebras, box spaces and Laplacians
2.1.1 Roe algebras
The Roe algebra $C^{*}(X)$ of a discrete, bounded geometry metric space $X$ is the completion in $B(\ell_{2}(X;\mathcal{H}_{0}))$
of the $*$-algebra $\mathbb{C}[X]$
of finite propagation operators, which are locally compact; i.e. for a discrete space the
matrix coefficients are compact operators $T_{x,y}\in\mathcal{K}(\mathcal{H}_{0})$ on a fixed, infinite-dimensional Hilbert space $\mathcal{H}_{0}$.
If $G$ acts on $X$ then the equivariant Roe algebra is defined to be the closure of the subalgebra
$\mathbb{C}[X]^{G}\subseteq\mathbb{C}[X]$ of equivariant finite propagation operators, i.e. satisfying
$T_{x,y}=T_{gx,gy}$ for any $g\in G$ and $x,y\in X$.
The uniform Roe algebra $C^{*}_{u}(X)$ of $X$ is the completion in $B(\ell_{2}(X))$ of the $*$-algebra $\mathbb{C}_{u}[X]$ of
finite propagation operators. The equivariant uniform Roe algebra $\mathbb{C}_{u}^{*}(X)^{G}$ is defined as before by considering $G$-invariant operators in
$\mathbb{C}_{u}[X]$ and taking the closure inside $C^{*}_{u}(X)$.
See Definitions 3.2 and 3.6 in [willett-yu1] for a detailed description of the Roe algebra and the equivariant Roe algebra.
2.1.2 Laplacians on box spaces
Consider a finitely generated residually finite group $G$.
Let $\{N_{i}\}$ be a family of finite index normal subgroups of $G$ satisfying $\bigcap N_{i}=\{e\}$. Consider the space
$$Y=\coprod G/N_{i},$$
viewed as a box space with $d(G/N_{i},G/N_{j})\geq 2^{i+j}$.
Let $\lambda_{i}$ be the quasi-regular representation of $G$ on $\ell_{2}\left(G/N_{i}\right)$ given by pulling back the regular representation
of $G/N_{i}$ on $\ell_{2}(G/N_{i})$ to $G$ via the quotient map $G\to G/N_{i}$ and denote by $\mathcal{N}$ the family
of representations
$$\mathcal{N}=\{\lambda,\lambda_{1},\lambda_{2},\dots\}.$$
The standing assumption in this section will be that the cohomological $n$-Laplacian $\Delta_{n}$ has a spectral gap in
$\mathbb{M}_{k}(C^{*}_{\mathcal{N}}(G))$.
The cohomological Laplacian element $\Delta_{n}\in\mathbb{M}_{k}(\mathbb{C}G)$ in degree $n$ maps to $\lambda_{i}(\Delta_{n})$ in
$\mathbb{M}_{k}(\lambda_{i}(C^{*}_{\mathcal{N}}(G)))$ and we will denote
$$\Delta_{n}^{i}=\lambda_{i}(\Delta_{n}).$$
Define the Laplace element of the box space to be
$$D_{n}=\oplus_{i}\ \Delta_{n}^{i}\in\bigoplus\mathbb{M}_{k}(\mathbb{C}_{u}[G/N_%
{i}])\subseteq\mathbb{M}_{n}(\mathbb{C}_{u}[Y]).$$
Note that $D_{n}$ has a spectral gap in $C^{*}(Y)$ if and only if all the $\Delta_{n}^{i}$ have a spectral gap in $C^{*}_{\mathcal{N}}(G)$.
2.2 The lifting map
From now on we will assume the group $G$ satisfies the operator norm localization property.
As shown by Sako [sako] the operator norm localization property is equivalent to exactness for bounded geometry
metric spaces, in particular for finitely generated groups, so the assumptions of Theorem 2 guarantee this property
for $G$.
Note that we will only be using a special case of the setup described in [willett-yu1], as our
box space consists of a family of finite quotients of a single group $G$, which naturally serves as a covering space for all of the finite quotients.
This setting is the same as the one in [higson], however we do follow the detailed version of the arguments as in [willett-yu1].
Willett and Yu in [willett-yu1, Lemma 3.8] define a lifting map, a $*$-homomorphism
$$\phi:\mathbb{C}[Y]\to\dfrac{\prod_{i}\mathbb{C}[G]^{N_{i}}}{\bigoplus_{i}%
\mathbb{C}[G]^{N_{i}}},$$
into the algebra of sequences of elements of the $N_{i}$-invariant subspaces of $\mathbb{C}[G]$ modulo the relation of
being equal at all but finitely many entries.
We have
$$\mathbb{M}_{k}\left(\dfrac{\prod_{i}\mathbb{C}[G]^{N_{i}}}{\bigoplus_{i}%
\mathbb{C}[G]^{N_{i}}}\right)=\dfrac{\prod_{i}\mathbb{M}_{k}\left(\mathbb{C}[G%
]^{N_{i}}\right)}{\bigoplus_{i}\mathbb{M}_{k}\left(\mathbb{C}[G]^{N_{i}}\right)}$$
and for every $k\in\mathbb{N}$ the map $\phi$ induces a $*$-homomorphism
$$\phi^{(k)}:\mathbb{M}_{k}(\mathbb{C}[Y])\to\mathbb{M}_{k}\left(\dfrac{\prod%
\mathbb{C}[G]^{N_{i}}}{\bigoplus\mathbb{C}[G]^{N_{i}}}\right),$$
by applying $\phi$ entrywise.
We recall that in [willett-yu1, Lemma 3.12] it was shown that if $G$ has the operator norm localization property then the map
$\phi$ extends to a map
$$\phi:C^{*}(Y)\to\dfrac{\prod_{i}C^{*}(G)^{N_{i}}}{\bigoplus_{i}C^{*}(G)^{N_{i}%
}},$$
and similarly induces the corresponding map $\phi^{(n)}$ on the $k\times k$ matrices over these algebras.
Here, $\dfrac{\prod_{i}C^{*}(G)^{N_{i}}}{\bigoplus_{i}C^{*}(G)^{N_{i}}}$ is the algebra of sequences of elements of $C^{*}(G)^{N_{i}}$ modulo the relation of
being asymptotically equal.
The same formula as the one for $\phi$ in [willett-yu1, Lemma 3.8] gives a uniform version of the lifting map,
$$\phi_{u}:\mathbb{C}_{u}[Y]\to\dfrac{\prod_{i}\mathbb{C}_{u}[G]^{N_{i}}}{%
\bigoplus_{i}\mathbb{C}_{u}[G]^{N_{i}}},$$
which is in the same way a $*$-homomorphism.
If $G$ has the operator norm localization property (see e.g. [chen-etal, sako]) then it also has the uniform version of that property, as proved by Sako [sako].
This implies that $\phi_{u}$ extends to a $*$-homomorphism
$$\phi_{u}:C^{*}_{u}(Y)\to\dfrac{\prod_{i}C^{*}_{u}(G)^{N_{i}}}{\bigoplus_{i}C^{%
*}_{u}(G)^{N_{i}}}.$$
See 13.3.11 and 13.3.12 in [willett-yu-book] for the detailed arguments, which apply verbatim here.
Similarly, $\phi_{u}^{(n)}$ in both cases induces a map on matrices over the respective algebras.
2.3 Surjectivity of the coarse Baum-Connes assembly map
We will now focus on the lift of the Laplacian $D_{n}$.
In order to do this we will use an argument based on Lemma 5.6 in [willett-yu1].
Lemma 14.
Let $G$ be an exact group and assume that $\Delta_{n}$ has a spectral gap in $C^{*}_{\mathcal{N}}(G)$.
Denote by
$$P_{n}=\bigoplus\lambda_{i}(p_{n})\ \ \ \ \in\mathbb{M}_{k}(C_{u}^{*}(Y))$$
the spectral projection associated to $D_{n}\in\mathbb{M}_{k}(C_{u}^{*}(Y))$.
Then
$$\phi_{u}^{(k)}(P_{n})=\prod_{i=1}^{\infty}\lambda(p_{n}),$$
where $\lambda(p_{n})$ is the projection onto the harmonic $n$-cochains in the $\ell_{2}$-cohomology of $G$.
Proof.
The projection $P_{n}$ is the spectral projection of $D_{n}$.
The Laplacian element $D_{n}$ defined above lifts to an element
$$\phi_{u}^{(k)}(D_{n})\in\mathbb{M}_{k}\left(\dfrac{\prod_{i=1}^{\infty}\mathbb%
{C}_{u}[G]^{N_{i}}}{\bigoplus_{i=1}^{\infty}\mathbb{C}_{u}[G]^{N_{i}}}\right),$$
whose spectrum is contained in $\{0\}\cup[\epsilon,\infty)$ for some $\varepsilon>0$. Indeed, since the same
is true for $D_{n}\in C^{*}_{u}(Y)$, by assumption, and spectral gaps are preserved by homomorphisms of unital $C^{*}$-algebras.
It is easy to see that from the definition of $\phi$ in the proof of [willett-yu1, Lemma 3.8] that the lift of $D_{n}$
is represented by the constant sequence
$$\phi_{u}^{(k)}(D_{n})=\prod_{i=1}^{\infty}\Delta_{n},$$
in the algebra
$$\dfrac{\prod_{i=1}^{\infty}\mathbb{M}_{k}(\mathbb{C}G)}{\bigoplus_{i=1}^{%
\infty}\mathbb{M}_{k}(\mathbb{C}G)}\subseteq\mathbb{M}_{k}\left(\dfrac{\prod_{%
i=1}^{\infty}\mathbb{C}_{u}[G]^{N_{i}}}{\bigoplus_{i=1}^{\infty}\mathbb{C}_{u}%
[G]^{N_{i}}}\right).$$
Indeed, this follows from the fact that the size of the support of $\Delta_{n}^{i}$ and the formula for the lift $\phi_{(u)}$ are both independent
of the particular quotient $G/N_{i}$ for $i$ sufficiently large.
Consider now the spectral projection $\widetilde{P}_{n}$ associated to $\phi_{u}^{(k)}(D_{n})$. Then
$$\widetilde{P}_{n}=\lim_{t\to\infty}e^{-t\phi_{u}^{(k)}(D_{n})}=\lim_{t\to%
\infty}\phi_{u}^{(k)}\left(e^{-tD_{n}}\right)=\phi_{u}^{(k)}(P_{n}).$$
Thus the associated spectral projection is of the form $\prod_{i=1}^{\infty}\lambda(p_{n})$, where
$$\lambda(p_{n})\in\mathbb{M}_{k}(C^{*}_{r}(G))$$
is the
projection onto the harmonic $n$-cochains in the $\ell_{2}$-cochains of $G$, as claimed.
∎
We now define the higher Kazhdan projection of the box space $Y$ to be $P_{n}\otimes q\in\mathbb{M}_{n}(C^{*}(Y))$, where $q$ is any rank one projection
on $\mathcal{H}_{0}$. We consider the associated $K$-theory class $[P_{n}\otimes q]\in K_{0}(C^{*}(Y))$. We will show that under certain conditions
this class is non-zero and does not lie in the image of the coarse Baum-Connes conjecture for $Y$.
The first thing to notice is that the lift of the projection $P_{n}\otimes q$ can be described explicitly as an element of $C^{*}(G)^{G}$.
Lemma 15.
Let $q\in\mathcal{K}(\mathcal{H}_{0})$. Then
$$\phi(P_{n}\otimes q)=\phi_{u}(P_{n})\otimes q.$$
Proof.
Let $\alpha_{i}$ be a sequence of finite propagation operators such that
$P_{n}=\lim_{i}\alpha_{i}$.
Then
$$\phi(\alpha_{i}\otimes q)=\phi_{u}(\alpha_{i})\otimes q,$$
by the definition of $\phi$ ([willett-yu1, Lemma 3.8]).
Passing to the limit and using continuity of $\phi$ we obtain the claim.
∎
Following [higson, willett-yu1] define the map $d_{*}:K_{0}(C^{*}(Y))\to\dfrac{\prod\mathbb{Z}}{\bigoplus\mathbb{Z}}$ by
taking the map
$$d:C^{*}(Y)\to\dfrac{\prod\mathcal{K}(\ell_{2}(G/N_{i};\mathcal{H}_{0}))}{%
\bigoplus\mathcal{K}(\ell_{2}(G/N_{i};\mathcal{H}_{0}))},$$
defined by
$$A\mapsto\prod_{i=1}^{\infty}Q_{i}AQ_{i},$$
where $Q_{i}:\ell_{2}(Y)\to\ell_{2}(G/N_{i})$ is the projection on the $i$-th box $G/N_{i}$,
and considering the induced map on $K$-theory
$$d_{*}:K_{0}(C^{*}(Y))\to K_{0}\left(\dfrac{\prod\mathcal{K}(\ell_{2}(G/N_{i};%
\mathcal{H}_{0}))}{\bigoplus\mathcal{K}(\ell_{2}(G/N_{i};\mathcal{H}_{0}))}%
\right)\simeq\dfrac{\prod\mathbb{Z}}{\bigoplus\mathbb{Z}}.$$
The above map $d_{*}$ can recognize when our $K$-theory class is non-zero. The map $d_{*}$ applied to
the projection $P_{n}$ gives the sequence of dimensions of images of $\lambda_{i}(p_{n})$.
On the other hand, consider the trace $\tau_{i}$ on $C^{*}(G)^{N_{i}}\simeq C^{*}_{r}(N_{i})\otimes\mathcal{K}$
by considering the tensor product of the standard trace on $C^{*}_{r}(N_{i})$ with the canonical trace on $\mathcal{K}$.
These $\tau_{i}$ induce the map
$$T:K_{0}\left(\dfrac{\prod_{i}C^{*}(G)^{N_{i}}}{\bigoplus_{i}C^{*}(G)^{N_{i}}}%
\right)\to\dfrac{\prod\mathbb{R}}{\bigoplus\mathbb{R}}.$$
The following diagram was analyzed in [willett-yu1].
{tikzcd}
& K_0( ∏iC*(G)Ni⊕iC*(G)Ni )\arrow[rd, ”T”]&
KX_0(Y) \arrow[ru, ”~μ_c”] \arrow[rd, ”μ_c”] &&∏R⊕R
&K_0(C^*(Y)) \arrow[uu, ”ϕ_*” ] \arrow[ru, ”d_*”]
The next lemma was formulated in [willett-yu1] for projections in $C^{*}(X)$, however the proof applies equally to projections in
$\mathbb{M}_{n}(C^{*}(X))$.
Lemma 16 (Lemma 6.5 in [willett-yu1]).
If $p$ is a projection in $\mathbb{M}_{n}(C^{*}(Y))$ such that the class $[p]\in K_{0}(C^{*}(Y))$ is in the image of the coarse assembly map
$\mu_{0}:KX_{0}(Y)\to K_{0}(C^{*}(Y))$, then
$$d_{*}([p])=T(\phi_{*}([p]))\ \ \ \ \ \ \text{ in }\dfrac{\prod\mathbb{R}}{%
\bigoplus\mathbb{R}}.$$
These maps take values in $\prod\mathbb{R}/\oplus\mathbb{R}$ viewed as an object in the category of abelian groups; that is, the equality of the two traces is an equality
of coordinates for all but finitely many $i$ (see Section 13.3 in [willett-yu-book]).
The next statement allows to show that when it is non-compact, the projection $P_{n}$ gives rise to a non-zero $K$-theory class in
$K_{0}(C^{*}(Y))$.
Proposition 17.
Assume that $H^{n}(G,\ell_{2}(G/N_{i}))\neq 0$ for infinitely many $i\in\mathbb{N}$.
Then $d_{*}[P_{n}]\neq 0$.
Proof.
We have
$$\displaystyle d_{*}(P_{n})$$
$$\displaystyle=\prod_{i=1}^{\infty}\dim\lambda_{i}(p_{n})=\prod_{i=1}^{\infty}%
\dim H^{n}(G;\lambda_{i}).$$
Indeed, since
$$\mathbb{M}_{n}(C^{*}(Y))=C^{*}(\mathbb{Z}_{n}\times Y),$$
the map
$$d^{(n)}:\mathbb{M}_{n}(C^{*}(X))\to\dfrac{\prod\mathbb{M}_{n}(\mathcal{K}(\ell%
_{2}(G/N_{i};\mathcal{H}_{0})))}{\bigoplus\mathbb{M}_{n}(\mathcal{K}(\ell_{2}(%
G/N_{i};\mathcal{H}_{0})))}.$$
can be rewritten as
$$d:C^{*}(X\times\mathbb{Z}_{n})\to\dfrac{\prod\mathcal{K}(\ell_{2}(G/N_{i}%
\times\mathbb{Z}_{n};\mathcal{H}_{0}))}{\bigoplus\mathcal{K}(\ell_{2}(G/N_{i}%
\times\mathbb{Z}_{n};\mathcal{H}_{0}))}.$$
By the assumption on cohomology of $G$ with coefficients in $\lambda_{i}$, the projection $p$ is non-compact in $C^{*}(Y\times\mathbb{Z}_{n})$
and as shown in the proof of Theorem 6.1 on page 1407 in [willett-yu1], we have $d_{*}[p]\neq 0$.
∎
The next lemma shows that the trace of the lift is naturally related to $\ell_{2}$-Betti numbers.
Lemma 18.
$T(\phi_{*}([P_{n}]))=\prod_{i=1}^{\infty}[G:N_{i}]\beta_{(2)}^{n}(G)=\prod_{i=%
1}^{\infty}\,\beta_{(2)}^{n}(N_{i})$.
Proof.
For a finite index subgroup $N\subseteq G$ the trace $\tau_{N}$ on $C^{*}(G)^{N}$ is defined as the tensor product of the canonical trace $\operatorname{tr}_{N}$ on $C^{*}_{r}(N)$
with the the canonical (unbounded trace) $\operatorname{Tr}$ on the compact operators, via the isomorphism
$$\psi_{N}:{C^{*}(G)^{N}\ }_{\overrightarrow{\ \ \ \simeq\ \ \ \ }}\ C^{*}_{r}(N%
)\otimes\mathcal{K}.$$
The isomorphism $\psi_{N}$ is defined by considering a fundamental domain $D\subset G$ for $N\subseteq G$
and for $A\in C^{*}(G)^{N}$ identifying
$$A\mapsto\sum_{g\in N}u_{g}\otimes A^{(g)},$$
where $A^{(g)}\in\mathcal{K}(\ell_{2}(D,\mathcal{H}_{0}))$ is defined by the formula
$$A^{(g)}_{x,y}=A_{x,gy},$$
for $x,y\in D$.
With this identification we observe that for an element $A\in C^{*}(G)^{G}$ of the form $A=\alpha\otimes q$, where $\alpha\in C^{*}_{r}(G)$
and $q$ is a rank 1 projection on $\mathcal{H}_{0}$, we have
$$\displaystyle\tau_{N}(A)$$
$$\displaystyle=\tau_{N}\left(\sum_{g\in N}u_{g}\otimes A^{(g)}\right)$$
$$\displaystyle=\sum_{g\in N}\operatorname{tr}(u_{g})\operatorname{Tr}(A^{(g)})$$
$$\displaystyle=\operatorname{Tr}(A^{(e)})$$
In our case $A^{(e)}$ is a $D\times D$ matrix defined by restricting $A$ to $D$. Therefore
$$\operatorname{Tr}(A^{(e)})=\sum_{x\in D}\alpha_{x,x}^{(e)}\cdot\operatorname{%
rank}(q)=[G:N]\alpha_{e,e}^{(e)}.$$
The same relation passes to traces of matrices over the respective $C^{*}$-algebras.
In the case of the projection $P_{n}$ these formulas yield
$$\tau_{N}(P_{n})=[G:N]\beta_{(2)}^{n}(G),$$
as claimed.
∎
Before we summarize this discussion we will observe one more fact that will allow to relate our results to Lück’s approximation theorem.
Recall that the Betti number of a group $G$ is the number $\beta^{n}(G)=\dim_{\mathbb{C}}H^{n}(G,\mathbb{C})$.
Lemma 19.
For a finite index subgroup $N\subseteq G$ we have
$$\dim_{\mathbb{C}}H^{n}(G,\ell_{2}(G/N))=\beta^{n}(N).$$
Proof.
Since $N$ is of finite index in $G$ we have
$$\operatorname{CoInd}_{N}^{G}\mathbb{C}=\operatorname{Ind}_{N}^{G}\mathbb{C}=%
\ell_{2}(G/N),$$
see e.g. [brown-book, Proposition 5.9].
Applying Shapiro’s lemma [brown-book, Proposition 6.2] we obtain
$$H^{n}(G,\ell_{2}(G/N))\simeq H^{n}(G,\operatorname{CoInd}_{N}^{G}\mathbb{C})%
\simeq H^{n}(N,\mathbb{C}).$$
∎
We are now in the position to formulate the main theorem of this section.
Theorem 2.
Let $G$ be an exact, residually finite group of type $F_{n+1}$ and let $\{N_{i}\}$ and $\mathcal{N}$ be as above.
Assume that $\Delta_{n}\in\mathbb{M}_{k_{n}}(C^{*}_{\mathcal{N}}(G))$ has a spectral gap
and that the coarse Baum-Connes assembly map
$KX_{0}(Y)\to K_{0}(C^{*}(Y))$
for the box space
$Y=\coprod G/N_{i}$ of $G$
is surjective. Then
$$\beta_{(2)}^{n}(N_{i})=\beta^{n}(N_{i})$$
for all but finitely many $i$.
Proof.
The claim follows from the explicit computation of the values of both traces in
proposition 17 and
lemmas 18 and 19.
∎
Note that the Theorem 2
provides a strenghtening of Lück’s approximation theorem [luck-gafa] in the case described by the above theorem.
Indeed, we can rewrite the conclusion of Theorem 2 as vanishing of
$$[G:N_{i}]\left(\beta_{(2)}^{n}(G)-\dfrac{\beta^{n}(N_{i})}{[G:N_{i}]}\right)=0$$
for all but finitely many $i$.
Compare this with [luck-approx-survey, Theorem 5.1], where examples with slow speed of convergence have been constructed.
The speed of convergence of Betti numbers of finite quotients to the $\ell_{2}$-Betti number of a residually finite group
was also studied in [clair-whyte], however the techniques used there are different. In our case both the assumptions and the conclusions are stronger.
3 Final remarks
Question 20.
Is there a spectral gap characterization of the existence of higher Kazhdan projections in $C^{*}_{\mathcal{N}}(G)$?
As mentioned earlier, Kazhdan’s property $(T)$ for $G$ is characterized by either the vanishing of first cohomology of $G$ with every unitary coefficients,
or equivalently, by the fact that the first cohomology of $G$ with any unitary coefficients is always reduced.
Related higher-dimensional generalizations of property $(T)$ were discussed in [bader-nowak-deformations], see also [dechiffre-etal].
As pointed out in [bader-nowak], the generalizations of these two conditions to higher degrees
are not equivalent.
The existence of higher Kazhdan projections is related to the property that cohomology is reduced and to the existence of gaps in the spectrum
of the Laplacian rather than to vanishing of cohomology. Indeed, in our reasoning it is crucial that
cohomology does not vanish.
It would be interesting to determine if the existence of higher Kazhdan projections can be viewed as a higher-dimensional rigidity property.
Remark 21 (Higher Kazhdan projections and $K$-amenability).
Clearly, if for a particular $G$ we have at the same time that $\beta_{(2)}^{n}(G)=0$ and
$\pi(\Delta_{n})$ has a nontrivial kernel and a spectral gap for some $\pi\neq\lambda$,
then $G$ cannot be amenable.
Consider the map
$$K_{*}(C^{*}_{\max}(G))\to K_{*}(C^{*}_{r}(G)).$$
If $[p_{n}]\neq 0$ in the former, but $[p_{n}^{r}]=0$ in the latter then the map above cannot be an isomorphism.
In other words, if $\beta_{(2)}^{n}(G)=0$ and we could ensure condition that the class of $[p_{n}]$ is not 0, then the group
$G$ would not be $K$-amenable, as this last condition forces the above map in $K$-theory to be an isomorphism.
Remark 22 (Ghost projections).
We can extend the notion of a ghost operator to matrix algebras over the Roe algebra by defining an element $T\in\mathbb{M}_{n}(C^{*}(Y))$ to
be a ghost if $T_{i,j}\in C^{*}(Y)$ is a ghost for every $1\leq i,j\leq n$. It can be shown that the kernel of the lifting map $\phi^{(n)}$ consists precisely of
ghost operators. See [willett-yu-book, Corollary 13.3.14]. This observation provides a new cohomological tool to construct ghost projections
in the case when an $\ell_{2}$-Betti number of $G$ vanishes. The problem of constructing new examples
of ghost projection was posed by Willet and Yu [willett-yu1].
References
K. Li, P. W. Nowak and S. Pooya, Institute of Mathematics of the Polish Academy of Sciences, Śniadeckich 8, 00-656 Warszawa, Poland
[email protected], [email protected], [email protected] |
Metal-poor hypervelocity star candidates from the Sloan Digital Sky Survey
Yinbi Li11affiliation: Key Laboratory of Optical Astronomy, National
Astronomical Observatories, Chinese Academy of Sciences, Beijing
100012, China. 22affiliation: Graduate University of
Chinese Academy of Sciences, 19A Yuquan Road, Beijing 100049, China , Ali Luo11affiliation: Key Laboratory of Optical Astronomy, National
Astronomical Observatories, Chinese Academy of Sciences, Beijing
100012, China. , Gang Zhao11affiliation: Key Laboratory of Optical Astronomy, National
Astronomical Observatories, Chinese Academy of Sciences, Beijing
100012, China. ,
Youjun Lu11affiliation: Key Laboratory of Optical Astronomy, National
Astronomical Observatories, Chinese Academy of Sciences, Beijing
100012, China. , Juanjuan Ren11affiliation: Key Laboratory of Optical Astronomy, National
Astronomical Observatories, Chinese Academy of Sciences, Beijing
100012, China. 22affiliation: Graduate University of
Chinese Academy of Sciences, 19A Yuquan Road, Beijing 100049, China , Fang Zuo11affiliation: Key Laboratory of Optical Astronomy, National
Astronomical Observatories, Chinese Academy of Sciences, Beijing
100012, China.
[email protected]
Abstract
Hypervelocity stars are believed to be ejected out from the Galactic center through dynamical
interactions of (binary) stars with the central massive black hole(s). In this letter, we report
13 metal-poor F-type hypervelocity star candidates selected from 370,000 stars of the
data release 7 of the Sloan Digital Sky Survey. With a detailed analysis of the kinematics of
these stars, we find that seven of them were likely ejected from the Galactic center (GC)
or the Galactic disk, four neither originated from the GC nor the Galactic disk, and the
other two were possibly ejected from either the Galactic disk or other regions. Those candidates
which unlikely originated from the GC or the Galactic disk, may be explained by other mechanisms, like the
tidal disruption of the Milky Way’s dwarf galaxies in the Galactic potential, or the gravitational
interactions with a massive black hole at the center of M31 or M32.
galaxies: kinematics and dynamics — Galaxy:
structure — stars: kinematics and dynamics — stars: late-type — galaxies: individual
(M31, M32 and Leo A)
1 INTRODUCTION
The hypervelocity stars (HVSs), recently discovered in the Galactic halo (Brown et al., 2005; Hirsch et al., 2005; Edelmann et al., 2005),
are moving so fast that they may escape from the Galaxy. A natural explanation of these HVSs is that
they are ejected out from the Galactic center (GC) by interactions of stars with the massive black
hole (MBH) or the hypothetical binary MBHs as predicted by Hills (1988) and Yu & Tremaine (2003). The ejection mechanisms of HVSs can
be divided into three categories: tidal breakup of binary stars in the vicinity of a single MBH
(Hills, 1988; Yu & Tremaine, 2003; Bromley et al., 2006), and the binary stars are probably injected into the vicinity of the MBH from
the young stellar disk in the GC (e.g., Lu et al., 2010; Zhang et al., 2010) or from the Galactic bulge (Perets, 2009a, b);
single star encounters with a binary MBH (Yu & Tremaine, 2003; Sesana et al., 2006, 2007; Merritt, 2006);
or single star encounters with a cluster of stellar mass black holes around the MBH (O’Leary & Loeb, 2008).
More than 16 HVSs have been reported in the literature (Brown et al., 2009)(See also: Tillich et al., 2009; Kollmeier et al., 2009; Tillich et al., 2010),
most of them are 3–4 M${}_{\sun}$ late B-type stars. Assuming a Salpeter initial mass function (IMF),
the expected solar mass HVSs are about 10 times more abundant than 3–4 M${}_{\sun}$ HVSs (Brown et al., 2009). Kollmeier et al. (2009)
systematically searched for such low mass HVSs in about 290,000 spectra of the Sloan Digital Sky Survey (SDSS), however,
they found only 6 metal-poor stars that can be possibly taken as HVS candidates, which might suggest that the IMF of the parent
population of these HVSs is top heavy. A top heavy IMF of the HVS parent population is possibly consistent with the disk
origination (Bartko et al., 2010; Lu et al., 2010; Kollmeier et al., 2010). In order to distinguish the ejection mechanisms of HVSs and put constraints
on the origin of the parent population of HVSs, it is quite necessary to search for the low-mass HVSs.
In this letter, we aim to find F/G type low mass HVS candidates from the data release 7 (DR7) of SDSS, which provides a
large catalog of stars with precise multi-color photometry and medium resolution ($R\sim 1800$)spectra (York et al., 2000).
We report the finding of 13 old metal-poor F-type HVS candidates, and also discuss their possible origins by detailed
analysis of their kinematics. The letter is organized as follows. In Section 2, we introduce the searching process
of our low-mass HVS candidates. Section 3 discusses their possible
ejection mechanisms through kinematics. Finally, a brief conclusion is given in Section 4.
2 TARGET SELECTION
We analyze over 370,000 stars in the SDSS DR7 with their five-band photometry $ugriz$ and spectra, which are flux- and wavelength-calibrated in $3800-9200$ Å and are reduced by the automated SEGUE Stellar Parameter Pipeline (Lee et al., 2008) to produce reliable heliocentric radial velocities RV${}_{\sun}$ and atmospheric parameters.
2.1 High Velocity Objects
In order to find out the F/G type HVS candidates, we first select the high-velocity
objects from the Galactic radial velocity distribution of F/G type main sequence samples
with the following methods.
1.
Select F/G type main sequence samples with the five photometric criteria and log(g) $>$ 3.0 in
section 2.3.1 of Ivezić et al. (2008): [130775].
2.
Find objects at the high velocity end, which significantly deviate from the best fit Gaussian distribution
of line-of-sight velocities in the Galactic rest frame $V_{\rm rf}$: [369].
The numbers in brackets indicate the number of stars left after each selection step. The $V_{\rm rf}$
distribution of our 130775 samples, shown in the top panel of Figure 1, can be well fitted by
the two Gaussian distribution (red solid curve). The two Gaussian components have mean velocities
(and scatters) of 0 km s${}^{-1}$ (92 km s${}^{-1}$) and 124 km s${}^{-1}$ (55 km s${}^{-1}$) respectively.
Approximately 82% of the samples belong to the low velocity component, and 18% of them belong
to the high velocity component (two dashed green curves). These two components probably correspond to
the Galactic halo population (0 $\pm$ 100 km s${}^{-1}$) and thick disk population
(90 $\pm$ 45 km s${}^{-1}$; Williams et al., 2011) respectively. The normalized residual of the observation
from this two Gaussian distribution is shown in the bottom panel of the Figure 1, and at its
high velocity tail, high-significance deviations are found. So, we choose the high velocity objects with
$V_{\rm rf}>309$ km s${}^{-1}$ and $V_{\rm rf}<-285$ km s${}^{-1}$, which have normalized
residuals larger than 1 and are denoted by two red arrows.
2.2 HVS Candidates
To determine which high velocity objects are unbound to the Galaxy, we first obtain the phase space coordinates for them. The Galactic cartesian coordinate system adopted here is centered on the GC: the X axis points from the Sun to the GC with the Sun at $x$ = $-$8 kpc; the Y axis points in the direction of Galactic rotation; the Z axis points towards the Northern Galactic Pole. Assuming the motion of the local standard of rest (LSR) is 220 km s${}^{-1}$, and the velocity of the Sun with respect to the LSR is (11.1 km s${}^{-1}$,
12.24 km s${}^{-1}$, 7.25 km s${}^{-1}$, Schönrich et al., 2010), the phase-space coordinates to the GC for these high velocity objects can be derived by equatorial coordinates ($ra$, $dec$), galactic coordinates ($l$, $b$), $RV_{\sun}$, two components of the proper motion $\mu_{\alpha}\cos(\delta)$, $\mu_{\delta}$, and the heliocentric distances ($D_{\sun}$). Here, $D_{\sun}$ is equal to $\frac{1}{100}\times 10^{0.2~{}(r-M_{r})}$, where r is the deredden r-band apparent magnitude given by the DR7 and $M_{r}$ is the r-band absolute magnitude calculated by the photometric parallax relation of Ivezić et al. (2008).
Then, we adopt two different Galactic potential models, i.e., a spherically
symmetric model (Xue et al., 2008, hereafter Xue08) and a triaxial model (Gnedin et al., 2005, hereafter Gnedin05) to estimate the
escape velocities ($V_{esc}$). There are totally 13 objects (mpHVS1–mpHVS13, “mp” means metal-poor) as listed in Table 1, which are unbound to the Galaxy in the Xue08 model, but only three (mpHVS1–mpHVS3) are unbound in the Gnedin05 model. The rest stars in the 369 high velocity objects are all bound to the Galaxy in both models. We cross-check the $V_{esc}$ for these 13 unbound candidates using other Galactic potential models in the literature, e.g., a spherically symmetric model from Kenyon et al. (2008), two axisymmetric models from Koposov et al. (2010) and Paczynski (1990) with the same definition of unbound stars as in Kenyon et al. (2008), and find that the $V_{esc}$ from these divergent models are between the values obtained from the Gnedin05 and Xue08 models, and mpHVS1, mpHVS2 and mpHVS3 are unbound to the Galaxy in all models. Note here, a small fraction of the bound stars at the 2-$\sigma$ level could be actually unbound due to the velocity measurement error, but the possibilities are certainly much less than those of unbound candidates. The basic information of these 13 candidates are listed in Table 1– 3. Table 1 summarizes their basic parameters, i.e., the name and its notation, $\mu_{\alpha}\cos(\delta)$ and $\mu_{\delta}$, $RV_{\sun}$, $V_{\rm rf}$, g-band apparent magnitude g, and stellar atmospheric parameters $T_{\rm eff}$, log($g$), and [Fe/H]. Table 2 lists their 6D phase-space coordinates and heliocentric distances $D_{\sun}$. Except mpHVS13, the estimates of $D_{\sun}$ and velocities of other candidates are all valid, because their spectroscopic metallicities [Fe/H] are in the valid range of that given by Ivezić et al. (2008). Table 3 compares their Galactic total velocities $V_{\rm G}$ with $V_{\rm esc}$ of the Gnedin05 and Xue08 models at their Galactocentric distances $D_{\rm G}$, from which we can see whether they are unbound in a certain potential model. In addition, Kollmeier et al. (2009) found 6 analogous metal-poor HVS candidates, so we also check whether they are in our high velocity objects. Consequently, only SDSS J074557.31+181246.7 is selected out and belongs to our bound objects, while the other five candidates are not in our high velocity objects mainly because they are inconsistent with the selection criterion 1 in Section 2.1.
3 POSSIBLE ORIGINS OF 13 mpHVS CANDIDATES
To investigate the origins of these 13 mpHVS candidates, it is essential to study their kinematics. By varying the 3D velocities and positions within their measurement errors, we calculate 10000 tracing-back trajectories and the 1 $\sigma$ level intersection places with the Galactic disk for each of mpHVS1–3 with the Gnedin05 and Xue08 models, and those for each of mpHVS4–13 with the Xue08 model, because they are bound in the Gnedin05 model, which are shown in Figure 2. From the left panel of Figure 2, we can see:
(1) the trajectories of mpHVS1 and mpHVS2 do not intersect with the Galactic disk.
(2) the trajectories of mpHVS3 intersect with the Galactic disk about 12 and 15 Myr ago for the Gnedin05 and Xue08 models respectively, and the 1 $\sigma$ intersection places are respectively shown by red and green ellipses.
In the right panel:
(1) the trajectories of mpHVS4 and mpHVS13 do not intersect with the Galactic disk.
(2) the trajectories of mpHVS5, mpHVS7, mpHVS8, mpHVS10, mpHVS11 and mpHVS12 do intersect with the Galactic disk about 10 to 20 Myr ago, and the 1 $\sigma$ intersection regions are depicted by 6 ellipses with different colors.
(3) for mpHVS6 and mpHVS9, most trajectories do intersect with the Galactic disk, but most of the intersection regions are far away from the GC even to hundreds of kpc, so we just describe part of the trajectories for the two candidates, which are calculated with their present 6D phase-space coordinates, and do not show their intersection regions in the right panel.
The trajectories of mpHVS3 are consistent with being ejected out from the GC or the Galactic disk, while those of mpHVS5, mpHVS7, mpHVS8, mpHVS10, mpHVS11 and mpHVS12 are consistent with originating from the Galactic disk (see Figure 2). Therefore, a supernova explosion in a massive tight binary system (Blaauw, 1961) or dynamical interactions in dense stellar clusters (Poveda et al., 1967; Leonard, 1991; Gvaramadze et al., 2009; Gvaramadze & Gualandris, 2011) are possible originating mechanisms for these seven candidates, and dynamical interactions between (binary) stars and MBH(s) in the GC are also possible ejection mechanisms for mpHVS3. For mpHVS1, mpHVS2, mpHVS4 and mpHVS13, their trajectories are incompatible with the GC origin or the Galactic disk origin mechanisms (see Figure 2). However, their origins may be interpreted as following.
Sherwin et al. (2008) proposed that there are a large number of low mass ($\approx$ 1 M${}_{\sun}$) HVSs ejected by the MBH in the center of M31, and a fraction of them would be moving towards the Milky Way on the solar system side, with large Galactocentric radial approach velocities ($\sim-500$ km s${}^{-1}$) exceeding the local escape velocity. The Galactocentric radial velocities of mpHVS1, mpHVS2, mpHVS4 and mpHVS13 are $-476\pm$85 km s${}^{-1}$, $-597\pm$177 km s${}^{-1}$, $413\pm$96 km s${}^{-1}$ and $265\pm$126 km s${}^{-1}$ respectively, which means the approaching velocities of mpHVS1 and mpHVS2 are consistent well with the predicted $-500$ km s${}^{-1}$, but it seems unlikely that mpHVS4 and mpHVS13 originated from the central MBH of M31. The mpHVS1 and mpHVS2 are low mass stars ($\approx$ 1 M${}_{\sun}$) and on the side of the Sun (see Fig. 2), considering of these factors, they seem to have been ejected out from the central MBH of M31. The ejection rate of HVSs from M32 is even higher than that from M31 (Lu et al., 2007), and thus M32 is also a possible origin of the two mpHVS candidates.
Tidal disruptions of dwarf galaxies in the Milky Way can also produce high velocity stars as proposed by Abadi et al. (2009). And they suggested that the current detected HVSs surrounding the constellation Leo ($l\approx$ 230${}^{\circ}$, $b\approx$ 60${}^{\circ}$) may be a stream of stars from a dwarf galaxy that was recently tidally disrupted (but see a different explanation of this in Lu et al. (2010)), and Teyssier et al. (2009) further proposed that such tidal disruption process of the massive satellites or satellites on eccentric orbits can also generate a population of old isolated high speed “escaped” (unbound) or “wandering” (bound) stars 111The “wandering” stars may possibly be related to the bound stars in the 369 high velocity objects.. If mpHVS1, mpHVS2, mpHVS4 and mpHVS13 are stars in tidal stripping streams, they may originate from other star streams, as their loci ($l$: 58${}^{\circ}$$\sim$69${}^{\circ}$; $b$: -45${}^{\circ}$$\sim$33${}^{\circ}$) are far away from the Leo area. If they are the isolated old “escaped” tidal stripping stars, we can roughly estimate their stripping times since they only pass though our galaxy once and never come back. Their estimated tidal stripping occurred about $10^{8}$ yr ago, assuming the dwarfs lie at the virial radius. Such a short stripping time suggests that their parent dwarf galaxies or streams may be still detectable along their trajectories. Therefore, further investigations are needed to find such dwarf galaxies, but the current comparatively large data errors impede such studies.
From the results of numerical experiments of mpHVS6 and mpHVS9, they are possibly ejected out from either the Galactic disk by corresponding mechanisms described above, or other places. The Galactocentric radial velocities of mpHVS6 and mpHVS9 are $364\pm$223 km s${}^{-1}$ and $-273\pm$262 km s${}^{-1}$ respectively, so they are unlikely to be ejected by the central MBH of M31. The tidal interactions between satellite galaxies and our galaxy may be able to explain their origins, but they were surely not ejected out from star streams in the direction of constellation Leo because their loci ($l$:56${}^{\circ}$$\sim$185${}^{\circ}$; $b$: 26${}^{\circ}$$\sim$31${}^{\circ}$) are also far away from that region. More exact conclusions about the origins of these two candidates mainly depend on more accurate data measurements in the future.
Note here, the large velocities of these mpHVS candidates could partly be due to orbital motions,
if they were in compact binaries. Below, we give a simple estimation of the effect of binary orbital velocities
on the observed heliocentric radial velocities and the Galactic total velocities. If each of these mpHVS candidates
were in binaries, its companion could be a low mass main-sequence star, a neutron star or a black hole.
a.
a low mass ($\leq$ 1 M${}_{\sun}$) main-sequence companion: Rastegaev (2010)
calculated the distribution of orbital periods for 60 detected metal-poor F,G and early K type binaries, and found the
minimum orbital period was larger than 10 days, and correspondingly the minimum semi-major axis $\geq$ 0.1 AU.
If the mass and semi-major axis of the companion are 1 M${}_{\sun}$ and 0.1 AU respectively, the effects due to binary orbital
motion could be most significant. The average effect of the projected
orbital velocities on the line-of-sight is thus about 27 km s${}^{-1}$, and the maximum is about 67 km s${}^{-1}$,
which introduce only approximate 3 $\sim$ 7 km s${}^{-1}$ and 9 $\sim$ 31 km s${}^{-1}$ effect to the 3D velocities
of our candidates. In addition, if these candidates are really in such binaries, their heliocentric distances and
Galactic total velocities would be larger than our estimations, and the escape velocities would
be correspondingly smaller, so these binaries should more likely be able to escape from our Galaxy.
b.
a neutron star companion: The maximum theoretical mass of a neutron star is $\sim$ 3 M${}_{\sun}$ (Kiziltan et al., 2010),
and if its semi-major is 0.1 AU, the average line-of-sight projected orbital velocity is about 57 km s${}^{-1}$, and the
maximum value is about 141 km s${}^{-1}$. In this cases, approximate 11 $\sim$ 24 km s${}^{-1}$ and 49 $\sim$ 93 km s${}^{-1}$
effects on the total space velocities of these mpHVS candidates are introduced.
c.
a black hole companion: If the mass and semi-major axis of a black hole are 10 M${}_{\sun}$ and 0.1 AU respectively, the average
line-of-sight projected orbital velocity is about 115 km s${}^{-1}$, and the maximum velocity is about 284 km s${}^{-1}$,
which introduce approximate 35 $\sim$ 69 km s${}^{-1}$ and 150 $\sim$ 228 km s${}^{-1}$ effects on the total space velocities
of these candidates. In this cases, the binary effect appears significant, so multiple observations are important to investigate whether these HVS candidates are in binaries with black hole companions.
To close the discussion, we note here that only mpHVS1 is unbound with high statistical significance, although all 13 HVS candidates are unbound in some Galactic potential models. The proper motion estimates adopted in our analysis (see Table 1), given by the SDSS, may suffer some uncertainties as pointed out by Dong et al. (2011), which may effect our estimations on the 3D velocities of those mpHVS candidates. Applying the model of Dong et al. (2011) (see equation 6), we re-estimate statistically the true proper motions of those 13 HVS candidates at 3-$\sigma$ significance level. Considering of this correction, we find that the highest velocity candidate, mpHVS1, is still unbound to the Galaxy in both potential models though with a lower 3D velocity (877 km s${}^{-1}$). However, mpHVS4–13 of the other 12 candidates are bound to the Galaxy if adopting the Xue08 potential or mpHVS2–13 are bound if adopting the Gnedin05 potential. So, those mpHVS candidates, except mpHVS1, could be bound halo stars. If they were, it is possible to use them to set constraints on the Galactic potential (W.Brown 2011, private communication).
4 CONCLUSION
In this letter, we report 13 metal-poor F-type HVS candidates from over
370,000 stars in the SDSS DR7. Through the kinematic analysis, we find that seven of them are likely to have been
ejected from the GC or the Galactic disk. Two of them were possibly ejected out from either the Galactic disk
or other places. Meanwhile, the other four candidates were unlikely to originate from the GC or the
Galactic disk. Those candidates impossibly ejected from the GC or the Galactic disk may be explained by other
mechanisms, i.e., being ejected from the tidal break of the Milky Way’s dwarf galaxies (Abadi et al., 2009; Teyssier et al., 2009),
or from the center of M31 or M32 by the interactions of stars with the MBH (Sherwin et al., 2008). In order to understand
the origins of these mpHVS candidates well and solve the binary problem, second-epoch observations in future are needed.
We thank Yingchun Wei for useful discussions, Jeffrey A. Munn for a comment on the proper
motion provided by the SDSS, Željko Ivezić for a comment on evaluating the absolute
magnitude. This work makes use of data products from the SDSS, which is managed by the Astrophysical
Research Consortium for the Participating Institution. The work was funded by the National Science
Foundation of China (NSFC) under grant Nos. 10821061, 10973021, 10973017 and 11103030.
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Some generalizations of the Aluthge transform of operators and their consequences
Mojtaba Bakherad${}^{1}$ and Khalid Shebrawi${}^{2}$
${}^{1}$Department of Mathematics, Faculty of Mathematics, University of Sistan and Baluchestan, Zahedan, I.R.Iran.
[email protected]; [email protected]
${}^{2}$Department of Mathematics, Al-Balqa’ Applied University, Salt, Jordan.
[email protected]; [email protected]
Abstract.
Let $A=U|A|$ be the polar decomposition of $A$. The
Aluthge transform of the operator $A$, denoted by $\tilde{A}$, is defined as
$\tilde{A}=|A|^{\frac{1}{2}}U|A|^{\frac{1}{2}}$.
In this paper, first we generalize the definition of Aluthge transform for
non-negative continuous functions $f,g$ such that $f(x)g(x)=x\,\,(x\geq 0)$. Then, by using of this definition, we get some numerical radius inequalities. Among other inequalities, it is shown that if $A$ is bounded linear operator on a complex Hilbert space ${\mathscr{H}}$, then
$$h\left(w(A)\right)\leq\frac{1}{4}\left\|h\left(g^{2}\left(\left|A\right|\right%
)\right)+h\left(f^{2}\left(\left|A\right|\right)\right)\right\|+\frac{1}{2}h%
\left(w\left(\tilde{A}_{f,g}\right)\right),$$
where $f,g$ are non-negative continuous functions such that $f(x)g(x)=x\,\,(x\geq 0)$, $h$ is a non-negative non-decreasing convex function on $[0,\infty)$ and $\tilde{A}_{f,g}=f(|A|)Ug(|A|)$.
Key words and phrases:Aluthge transform, Numerical radius, Operator matrices, Polar decomposition.
2010 Mathematics Subject Classification: Primary 47A12, Secondary 47A63, 47A30
1. Introduction
Let ${\mathbb{B}}(\mathscr{H})$ denote the $C^{*}$-algebra of all bounded linear operators on a complex Hilbert space ${\mathscr{H}}$ with an inner product $\langle\cdot,\cdot\rangle$ and the corresponding norm $\|\cdot\|$. In the case when ${\rm dim}{\mathscr{H}}=n$, we identify ${\mathbb{B}}({\mathscr{H}})$ with the matrix algebra $\mathbb{M}_{n}$ of all $n\times n$ matrices with entries in
the complex field.
For an operator $A\in{\mathbb{B}}(\mathscr{H})$, let $A=U|A|$ (U is a partial
isometry with $\textrm{ker}U=\textrm{rng}|A|^{\bot}$) be the polar decomposition of $A$. The
Aluthge transform of the operator $A$, denoted by $\tilde{A}$, is defined as
$\tilde{A}=|A|^{\frac{1}{2}}U|A|^{\frac{1}{2}}.$
In [12], Okubo introduced a more general notion called $t$-Aluthge transform which has later
been studied also in detail. This is defined for any $0<t\leq 1$ by
$\tilde{A}_{t}=|A|^{t}U|A|^{1-t}$. Clearly, for $t=\frac{1}{2}$ we obtain the usual Aluthge transform. As for the case $t=1$, the operator $\tilde{A}_{1}=|A|U$ is called the Duggal transform of $A\in{\mathbb{B}}(\mathscr{H})$. For $A\in{\mathbb{B}}(\mathscr{H})$, we generalize the Aluthge transform of the operator $A$ to the form
$$\displaystyle\tilde{A}_{f,g}=f(|A|)Ug(|A|),$$
in which $f,g$ are non-negative continuous functions such that $f(x)g(x)=x\,\,(x\geq 0).$ The numerical radius of $A\in{\mathbb{B}}({\mathscr{H}})$ is defined by
$$\displaystyle w(A):=\sup\{|\langle Ax,x\rangle|:x\in{\mathscr{H}},\|x\|=1\}.$$
It is well known that $w(\,\cdot\,)$ defines a norm on ${\mathbb{B}}({\mathscr{H}})$, which is equivalent to the usual operator norm $\|\cdot\|$. In fact, for any $A\in{\mathbb{B}}({\mathscr{H}})$,
$\frac{1}{2}\|A\|\leq w(A)\leq\|A\|$;
see [6]. Let $r(\cdot)$ denote to the spectral radius. It is well known that for every operator
$A\in{\mathbb{B}}({\mathscr{H}})$, we have $r(A)\leq w(A)$. An important inequality for $\omega(A)$ is the power inequality stating that $\omega(A^{n})\leq\omega(A)^{n}\,\,(n=1,2,\cdots)$. The quantity $w(A)$ is useful in studying perturbation, convergence and approximation problems as well as integrative method, etc. For more information see [3, 7, 8, 9] and references therein.
Let $A,B,C,D\in{\mathbb{B}}({\mathscr{H}})$. The operator matrices $\left[\begin{array}[]{cc}A&0\\
0&D\end{array}\right]$ and $\left[\begin{array}[]{cc}0&B\\
C&0\end{array}\right]$ are called the diagonal and off-diagonal parts of the operator matrix $\left[\begin{array}[]{cc}A&B\\
C&D\end{array}\right]$, respectively.
In [11], It has been shown that if $A$ is an operator in ${\mathbb{B}}({\mathscr{H}})$, then
$$\displaystyle w(A)\leq\frac{1}{2}\left(\|A\|+\|A^{2}\|^{\frac{1}{2}}\right).$$
(1.1)
Several refinements and generalizations of inequality (1.1) have been given; see [1, 4, 14, 15]. Yamazaki [15] showed that for $A\in{\mathbb{B}}({\mathscr{H}})$ and $t\in[0,1]$ we have
$$\displaystyle w(A)\leq\frac{1}{2}\left(\|A\|+w(\tilde{A}_{t})\right).$$
(1.2)
Davidson and Power [5] proved that if $A$ and $B$ are positive operators in ${\mathbb{B}}({\mathscr{H}})$, then
$$\displaystyle\|A+B\|\leq\max\{\|A\|,\|B\|\}+\|AB\|^{\frac{1}{2}}.$$
(1.3)
Inequality (1.3) has been generalized in [2, 13]. In [13], the author extended this inequality to the form
$$\displaystyle\|A+B^{*}\|\leq\max\{\|A\|,\|B\|\}+\frac{1}{2}\left(\left\||A|^{t%
}|B^{*}|^{1-t}\right\|+\left\||A^{*}|^{1-t}|B|^{t}\right\|\right),$$
(1.4)
in which $A,B\in{\mathbb{B}}({\mathscr{H}})$ and $t\in[0,1]$.
In this paper, by applying the generalized Aluthge transform of operators, we establish some inequalities involving the numerical radius. In particular, we extend inequality (1.2) and (1.4) for two non-negative continuous functions. We also show some upper bounds for the numerical radius of $2\times 2$ operators matrices.
2. main results
To prove our numerical radius inequalities, we need several known lemmas.
Lemma 2.1.
[1, Theorem 2.2]
Let $X,Y,S,T\in{\mathbb{B}}({\mathscr{H}})$. Then
$$\displaystyle r(XY+ST)\leq\frac{1}{2}\left(w(YX)+w(TS)\right)+\frac{1}{2}\sqrt%
{\left(w(YX)-w(TS)\right)^{2}+4\|YS\|\|TX\|}.$$
Lemma 2.2.
[15, 11]
Let $A\in{\mathbb{B}}({\mathscr{H}})$. Then
$(a)\,\,w(A)=\underset{\theta\in\mathbb{R}}{\max}\left\|\textrm{Re}\left(e^{i%
\theta}A\right)\right\|.$
$(b)\,\,w\left(\left[\begin{array}[]{cc}0&A\\
0&0\end{array}\right]\right)=\frac{1}{2}\|A\|.$
Polarization identity: For all $x,y\in{\mathscr{H}}$, we have
$$\left\langle x,y\right\rangle=\frac{1}{4}\sum_{k=0}^{3}\left\|x+i^{k}y\right\|%
^{2}i^{k}.$$
Now, we are ready to present our first result. The following theorem shows a generalization of inequality (1.2).
Theorem 2.3.
Let $A\in{\mathbb{B}}({\mathscr{H}})$ and $f,g$ be two non-negative
continuous functions on $[0,\infty)$ such that $f(x)g(x)=x\,\,(x\geq 0)$.
Then, for all non-negative non-decreasing convex function $h$ on $[0,\infty)$, we have
$$h\left(w(A)\right)\leq\frac{1}{4}\left\|h\left(g^{2}\left(\left|A\right|\right%
)\right)+h\left(f^{2}\left(\left|A\right|\right)\right)\right\|+\frac{1}{2}h%
\left(w\left(\tilde{A}_{f,g}\right)\right).$$
Proof.
Let $x$ be any unit vector. Then
$$\displaystyle\textrm{Re}\left\langle e^{i\theta}Ax,x\right\rangle$$
$$\displaystyle=$$
$$\displaystyle\textrm{Re}\left\langle e^{i\theta}U\left|A\right|x,x\right\rangle$$
$$\displaystyle=$$
$$\displaystyle\textrm{Re}\left\langle e^{i\theta}Ug\left(\left|A\right|\right)f%
\left(\left|A\right|\right)x,x\right\rangle$$
$$\displaystyle=$$
$$\displaystyle\textrm{Re}\left\langle e^{i\theta}f\left(\left|A\right|\right)x,%
g\left(\left|A\right|\right)U^{\ast}x\right\rangle$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{4}\left\|\left(e^{i\theta}f\left(\left|A\right|\right)+g%
\left(\left|A\right|\right)U^{\ast}\right)x\right\|^{2}-\frac{1}{4}\left\|%
\left(e^{i\theta}f\left(\left|A\right|\right)-g\left(\left|A\right|\right)U^{%
\ast}\right)x\right\|^{2}$$
$$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\text{\ (by
polarization identity)}$$
$$\displaystyle\leq$$
$$\displaystyle\frac{1}{4}\left\|\left(e^{i\theta}f\left(\left|A\right|\right)+g%
\left(\left|A\right|\right)U^{\ast}\right)x\right\|^{2}$$
$$\displaystyle\leq$$
$$\displaystyle\frac{1}{4}\left\|\left(e^{i\theta}f\left(\left|A\right|\right)+g%
\left(\left|A\right|\right)U^{\ast}\right)\right\|^{2}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{4}\left\|\left(e^{i\theta}f\left(\left|A\right|\right)+g%
\left(\left|A\right|\right)U^{\ast}\right)\left(e^{-i\theta}f\left(\left|A%
\right|\right)+Ug\left(\left|A\right|\right)\right)\right\|$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{4}\left\|g^{2}\left(\left|A\right|\right)+f^{2}\left(%
\left|A\right|\right)+e^{i\theta}\tilde{A}_{f,g}+e^{-i\theta}\left(\tilde{A}_{%
f,g}\right)^{\ast}\right\|$$
$$\displaystyle\leq$$
$$\displaystyle\frac{1}{4}\left\|g^{2}\left(\left|A\right|\right)+f^{2}\left(%
\left|A\right|\right)\right\|+\frac{1}{4}\left\|e^{i\theta}\tilde{A}_{f,g}+e^{%
-i\theta}\left(\tilde{A}_{f,g}\right)^{\ast}\right\|$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{4}\left\|g^{2}\left(\left|A\right|\right)+f^{2}\left(%
\left|A\right|\right)\right\|+\frac{1}{2}\left\|\textrm{Re}\left(e^{i\theta}%
\tilde{A}_{f,g}\right)\right\|$$
$$\displaystyle\leq$$
$$\displaystyle\frac{1}{4}\left\|g^{2}\left(\left|A\right|\right)+f^{2}\left(%
\left|A\right|\right)\right\|+\frac{1}{2}w\left(\tilde{A}_{f,g}\right).$$
Now, taking the supremum over all unit vector $x\in{\mathscr{H}}$ and applying Lemma 2.2 in the above inequality produces
$$\displaystyle w\left(A\right)\leq\frac{1}{4}\left\|g^{2}\left(\left|A\right|%
\right)+f^{2}\left(\left|A\right|\right)\right\|+\frac{1}{2}w\left(\tilde{A}_{%
f,g}\right).$$
Therefore,
$$\displaystyle h\left(w\left(A\right)\right)$$
$$\displaystyle\leq$$
$$\displaystyle h\left(\frac{1}{4}\left\|g^{2}\left(\left|A\right|\right)+f^{2}%
\left(\left|A\right|\right)\right\|+\frac{1}{2}w\left(\tilde{A}_{f,g}\right)\right)$$
$$\displaystyle=$$
$$\displaystyle h\left(\frac{1}{2}\left\|\frac{g^{2}\left(\left|A\right|\right)+%
f^{2}\left(\left|A\right|\right)}{2}\right\|+\frac{1}{2}w\left(\tilde{A}_{f,g}%
\right)\right)$$
$$\displaystyle\leq$$
$$\displaystyle\frac{1}{2}h\left(\left\|\frac{g^{2}\left(\left|A\right|\right)+f%
^{2}\left(\left|A\right|\right)}{2}\right\|\right)+\frac{1}{2}h\left(w\left(%
\tilde{A}_{f,g}\right)\right)$$
$$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad(\textrm{by the convexity of %
}\,h)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}\left\|h\left(\frac{g^{2}\left(\left|A\right|\right)+f%
^{2}\left(\left|A\right|\right)}{2}\right)\right\|+\frac{1}{2}h\left(w\left(%
\tilde{A}_{f,g}\right)\right)$$
$$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad(\textrm{by the functional %
calculus})$$
$$\displaystyle\leq$$
$$\displaystyle\frac{1}{4}\left\|h\left(g^{2}\left(\left|A\right|\right)\right)+%
h\left(f^{2}\left(\left|A\right|\right)\right)\right\|+\frac{1}{2}h\left(w%
\left(\tilde{A}_{f,g}\right)\right)$$
$$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad(\textrm{by the convexity of %
}\,h).$$
∎
Theorem 2.3 includes some special cases as follows.
Corollary 2.4.
Let $A\in{\mathbb{B}}({\mathscr{H}})$. Then, for all non-negative non-decreasing convex
function $h$ on $[0,\infty)$ and all $t\in\left[0,1\right]$, we have
$$h\left(w(A)\right)\leq\frac{1}{4}\left\|h\left(\left|A\right|^{2t}\right)+h%
\left(\left|A\right|^{2\left(1-t\right)}\right)\right\|+\frac{1}{2}h\left(w%
\left(\tilde{A}_{t}\right)\right).$$
(2.1)
Corollary 2.5.
Let $A\in{\mathbb{B}}({\mathscr{H}})$. Then, for all $t\in\left[0,1\right]$ and $r\geq 1$, we have
$$w^{r}(A)\leq\frac{1}{4}\|\left|A\right|^{2tr}+\left|A\right|^{2\left(1-t\right%
)r}\|+\frac{1}{2}w^{r}\left(\tilde{A}_{t}\right).$$
In particular,
$$w^{r}(A)\leq\frac{1}{2}\left(\|A\|^{r}+w^{r}\left(\tilde{A}\right)\right).$$
Proof.
The first inequality follows from
inequality (2.1) for the function $h\left(x\right)=x^{r}\,\,(r\geq 1)$. For the particular case, it is enough to put $t=\frac{1}{2}.$
∎
Theorem 2.3 gives the next result for the off-diagonal operator matrix $\left[\begin{array}[]{cc}0&A\\
B&0\end{array}\right]$.
Theorem 2.6.
Let $A,B\in{\mathbb{B}}({\mathscr{H}})$, $f,g$ be two non-negative
continuous functions on $[0,\infty)$ such that $f(x)g(x)=x\,\,(x\geq 0)$ and $r\geq 1$. Then
$$\displaystyle w^{r}\left(\left[\begin{array}[]{cc}0&A\\
B&0\end{array}\right]\right)$$
$$\displaystyle\leq$$
$$\displaystyle\frac{1}{4}\max\left(\left\|g^{2r}\left(\left|A\right|\right)+f^{%
2r}\left(\left|A\right|\right)\right\|,\left\|g^{2r}\left(\left|B\right|\right%
)+f^{2r}\left(\left|B\right|\right)\right\|\right)$$
$$\displaystyle+\frac{1}{4}\big{(}\|f(|B|)g(|A^{\ast}|)\|^{r}+\|f(|A|)g(|B^{\ast%
}|)\|^{r}\big{)}.$$
Proof.
Let $A=U|A|$ and $B=V|B|$ be the polar decompositions of $A$ and $B$, respectively and let $T=\left[\begin{array}[]{cc}0&A\\
B&0\end{array}\right]$. It follows from the polar the composition of $T=\left[\begin{array}[]{cc}0&U\\
V&0\end{array}\right]\left[\begin{array}[]{cc}|B|&0\\
0&|A|\end{array}\right]$ that
$$\displaystyle\tilde{T}_{f,g}$$
$$\displaystyle=$$
$$\displaystyle f(|T|)\left[\begin{array}[]{cc}0&U\\
V&0\end{array}\right]g(|T|)$$
$$\displaystyle=$$
$$\displaystyle\left[\begin{array}[]{cc}f(|B|)&0\\
0&f(|A|)\end{array}\right]\left[\begin{array}[]{cc}0&U\\
V&0\end{array}\right]\left[\begin{array}[]{cc}g(|B|)&0\\
0&g(|A|)\end{array}\right]$$
$$\displaystyle=$$
$$\displaystyle\left[\begin{array}[]{cc}0&f(|B|)Ug(|A|)\\
f(|A|)Vg(|B|)&0\end{array}\right].$$
Using $|A^{\ast}|^{2}=AA^{\ast}=U|A|^{2}U^{\ast}$ and $|B^{\ast}|^{2}=BB^{\ast}=V|B|^{2}V^{\ast}$ we have $g(|A|)=U^{\ast}g(|A^{\ast}|)U$ and $g(|B|)=V^{\ast}g(|B^{\ast}|)V$ for every non-negative continuous function $g$ on $[0,\infty)$.
Therefore,
$$\displaystyle w\left(\tilde{T}_{f,g}\right)$$
$$\displaystyle=$$
$$\displaystyle w\left(\left[\begin{array}[]{cc}0&f(|B|)Ug(|A|)\\
f(|A|)Vg(|B|)&0\end{array}\right]\right)$$
(2.2)
$$\displaystyle\leq$$
$$\displaystyle w\left(\left[\begin{array}[]{cc}0&f(|B|)Ug(|A|)\\
0&0\end{array}\right]\right)+w\left(\left[\begin{array}[]{cc}0&0\\
f(|A|)Vg(|B|)&0\end{array}\right]\right)$$
$$\displaystyle=$$
$$\displaystyle w\left(\left[\begin{array}[]{cc}0&f(|B|)Ug(|A|)\\
0&0\end{array}\right]\right)+w\left(U^{*}\left[\begin{array}[]{cc}0&f(|A|)Vg(|%
B|)\\
0&0\end{array}\right]U\right)$$
$$\displaystyle=$$
$$\displaystyle w\left(\left[\begin{array}[]{cc}0&f(|B|)Ug(|A|)\\
0&0\end{array}\right]\right)+w\left(\left[\begin{array}[]{cc}0&f(|A|)Vg(|B|)\\
0&0\end{array}\right]\right)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}\|f(|B|)Ug(|A|)\|+\frac{1}{2}\|f(|A|)Vg(|B|)\|$$
$$\displaystyle\qquad\qquad\qquad\qquad\qquad\textrm{(by Lemma \ref{kit1}(b))}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}\|f(|B|)UU^{\ast}g(|A^{\ast}|)U\|+\frac{1}{2}\|f(|A|)%
VV^{\ast}g(|B^{\ast}|)V\|$$
$$\displaystyle\leq$$
$$\displaystyle\frac{1}{2}\|f(|B|)g(|A^{\ast}|)\|+\frac{1}{2}\|f(|A|)g(|B^{\ast}%
|)\|,$$
where $U=\left[\begin{array}[]{cc}0&I\\
I&0\end{array}\right]$ is unitary. Applying Theorem 2.3 and inequality (2.2), we have
$$\displaystyle w^{r}\left(T\right)$$
$$\displaystyle\leq$$
$$\displaystyle\frac{1}{4}\left\|g^{2r}\left(\left|T\right|\right)+f^{2r}\left(%
\left|T\right|\right)\right\|+\frac{1}{2}\left(w^{r}\left(\tilde{T}_{f,g}%
\right)\right)$$
$$\displaystyle\leq$$
$$\displaystyle\frac{1}{4}\max\left(\left\|g^{2r}\left(\left|A\right|\right)+f^{%
2r}\left(\left|A\right|\right)\right\|,\left\|g^{2r}\left(\left|B\right|\right%
)+f^{2r}\left(\left|B\right|\right)\right\|\right)$$
$$\displaystyle+\frac{1}{2}\left[\frac{1}{2}\left(\|f(|B|)g(|A^{\ast}|)\|+\|f(|A%
|)g(|B^{\ast}|)\|\right)\right]^{r}$$
$$\displaystyle\leq$$
$$\displaystyle\frac{1}{4}\max\left(\left\|g^{2r}\left(\left|A\right|\right)+f^{%
2r}\left(\left|A\right|\right)\right\|,\left\|g^{2r}\left(\left|B\right|\right%
)+f^{2r}\left(\left|B\right|\right)\right\|\right)$$
$$\displaystyle+\frac{1}{4}\|f(|B|)g(|A^{\ast}|)\|^{r}+\frac{1}{4}\|f(|A|)g(|B^{%
\ast}|)\|^{r}$$
$$\displaystyle\qquad\qquad(\textrm{by the convexity}\,h(x)=x^{r}).$$
∎
Corollary 2.7.
Let $A,B\in{\mathbb{B}}({\mathscr{H}})$. Then,
for all $t\in\left[0,1\right]$ and $r\geq 1$, we have
$$\displaystyle w^{\frac{r}{2}}\left(AB\right)$$
$$\displaystyle\leq$$
$$\displaystyle\frac{1}{4}\max\left(\left\|\left|A\right|^{2tr}+\left|A\right|^{%
2\left(1-t\right)r}\right\|,\left\|\left|B\right|^{2tr}+\left|B\right|^{2\left%
(1-t\right)r}\right\|\right)$$
$$\displaystyle+\frac{1}{4}\left(\left\||A|^{t}\left|B^{\ast}\right|^{1-t}\right%
\|^{r}+\left\|\left|B\right|^{t}|A^{\ast}|^{1-t}\right\|^{r}\right).$$
Proof.
Applying the power inequality of the numerical radius, we have
$$\displaystyle w^{\frac{r}{2}}\left(AB\right)$$
$$\displaystyle\leq$$
$$\displaystyle\max\left(w^{\frac{r}{2}}\left(AB\right),w^{\frac{r}{2}}\left(BA%
\right)\right)$$
$$\displaystyle=$$
$$\displaystyle w^{\frac{r}{2}}\left(\left[\begin{array}[]{cc}AB&0\\
0&BA\end{array}\right]\right)$$
$$\displaystyle=$$
$$\displaystyle w^{\frac{r}{2}}\left(\left[\begin{array}[]{cc}0&A\\
B&0\end{array}\right]^{2}\right)$$
$$\displaystyle\leq$$
$$\displaystyle w^{r}\left(\left[\begin{array}[]{cc}0&A\\
B&0\end{array}\right]\right)$$
$$\displaystyle\leq$$
$$\displaystyle\frac{1}{4}\max\left(\left\|\left|A\right|^{2tr}+\left|A\right|^{%
2\left(1-t\right)r}\right\|,\left\|\left|B\right|^{2tr}+\left|B\right|^{2\left%
(1-t\right)r}\right\|\right)$$
$$\displaystyle+\frac{1}{4}\left(\left\||A|^{t}\left|B^{\ast}\right|^{1-t}\right%
\|^{r}+\left\|\left|B\right|^{t}|A^{\ast}|^{1-t}\right\|^{r}\right)$$
$$\displaystyle\qquad\qquad\qquad\qquad\textrm{(by Theorem \ref{man-kal})}.$$
∎
Corollary 2.8.
Let $A,B\in{\mathbb{B}}({\mathscr{H}})$ be positive operators. Then,
for all $t\in\left[0,1\right]$ and $r\geq 1$, we have
$$\displaystyle\left\|A^{\frac{1}{2}}B^{\frac{1}{2}}\right\|^{r}$$
$$\displaystyle\leq$$
$$\displaystyle\frac{1}{4}\max\left(\left\|A^{tr}+A^{\left(1-t\right)r}\right\|,%
\left\|B^{tr}+B^{2\left(1-t\right)r}\right\|\right)$$
$$\displaystyle+\frac{1}{4}\left(\left\|A^{t}B^{1-t}\right\|^{r}+\left\|B^{t}A^{%
1-t}\right\|^{r}\right).$$
Proof.
Since the spectral radius of any operator is dominated by its numerical
radius, then $r^{\frac{1}{2}}\left(AB\right)\leq w^{\frac{1}{2}}\left(AB\right).$ Applying a commutativity property of the spectral radius, we get
$$\displaystyle r^{\frac{r}{2}}\left(AB\right)$$
$$\displaystyle=$$
$$\displaystyle r^{\frac{r}{2}}\left(A^{\frac{1}{2}}A^{\frac{1}{2}}B^{\frac{1}{2%
}}B^{\frac{1}{2}}\right)$$
(2.3)
$$\displaystyle=$$
$$\displaystyle r^{\frac{r}{2}}\left(A^{\frac{1}{2}}B^{\frac{1}{2}}B^{\frac{1}{2%
}}A^{\frac{1}{2}}\right)$$
$$\displaystyle=$$
$$\displaystyle r^{\frac{r}{2}}\left(A^{\frac{1}{2}}B^{\frac{1}{2}}\left(A^{%
\frac{1}{2}}B^{\frac{1}{2}}\right)^{\ast}\right)$$
$$\displaystyle=$$
$$\displaystyle\left\|A^{\frac{1}{2}}B^{\frac{1}{2}}\left(A^{\frac{1}{2}}B^{%
\frac{1}{2}}\right)^{\ast}\right\|^{\frac{r}{2}}$$
$$\displaystyle=$$
$$\displaystyle\left\|A^{\frac{1}{2}}B^{\frac{1}{2}}\right\|^{r}.$$
Now, the result follows from Corollary 2.7.
∎
An important special case of Theorem 2.6, which refines inequality (1.4) can be stated as follows.
Corollary 2.9.
Let $A,B\in{\mathbb{B}}({\mathscr{H}})$ and $r\geq 1$. Then
$$\displaystyle\left\|A+B\right\|^{r}$$
$$\displaystyle\leq$$
$$\displaystyle\frac{1}{2^{2-r}}\max\left(\left\|\left|A\right|^{2tr}+\left|A%
\right|^{2\left(1-t\right)r}\right\|,\left\|\left|B^{*}\right|^{2tr}+\left|B^{%
*}\right|^{2\left(1-t\right)r}\right\|\right)$$
$$\displaystyle+\frac{1}{2^{2-r}}\left(\left\||A|^{t}\left|B\right|^{1-t}\right%
\|^{r}+\left\|\left|B^{*}\right|^{t}|A^{\ast}|^{1-t}\right\|^{r}\right).$$
In particular, if $A$ and $B$ are normal, then
$$\displaystyle\left\|A+B\right\|^{r}\leq\frac{1}{2^{1-r}}\max\left(\left\|A%
\right\|^{r},\left\|B\right\|^{r}\right)+\frac{1}{2^{1-r}}\left\|AB\right\|^{%
\frac{r}{2}}.$$
Proof.
Applying Lemma 2.2 and Theorem 2.3, we have
$$\displaystyle\left\|A+B^{\ast}\right\|^{r}$$
$$\displaystyle=$$
$$\displaystyle\left\|T+T^{\ast}\right\|^{r}$$
$$\displaystyle\leq$$
$$\displaystyle 2^{r}\underset{\theta\in\mathbb{R}}{\max}\left\|\textrm{Re}\left%
(e^{i\theta}T\right)\right\|^{r}$$
$$\displaystyle=$$
$$\displaystyle 2^{r}w^{r}\left(T\right)$$
$$\displaystyle\leq$$
$$\displaystyle\frac{2^{r}}{4}\max\left(\left\|\left|A\right|^{2tr}+\left|A%
\right|^{2\left(1-t\right)r}\right\|,\left\|\left|B\right|^{2tr}+\left|B\right%
|^{2\left(1-t\right)r}\right\|\right)$$
$$\displaystyle+\frac{2^{r}}{4}\left(\left\||A|^{t}\left|B^{\ast}\right|^{1-t}%
\right\|^{r}+\left\|\left|B\right|^{t}|A^{\ast}|^{1-t}\right\|^{r}\right)$$
$$\displaystyle\textrm{(by Theorem \ref{man-kal})},$$
where $T=\left[\begin{array}[]{cc}0&A\\
B&0\end{array}\right].$ Now, the desired result follows by replacing $B$ by $B^{\ast}$.
For the particular case, since $A$ and $B$ are normal,
then $\left|B^{\ast}\right|=$ $\left|B\right|$ and $|A^{\ast}|=|A|.$ Applying equality $($2.3$)$ for the operators $\left|A\right|^{\frac{1}{2}}$ and $\left|B\right|^{\frac{1}{2}}$, we have
$$\displaystyle\left\|\left|A\right|^{\frac{1}{2}}\left|B\right|^{\frac{1}{2}}%
\right\|^{r}$$
$$\displaystyle=$$
$$\displaystyle r^{\frac{r}{2}}\left(\left|A\right|\left|B\right|\right)$$
$$\displaystyle\leq$$
$$\displaystyle\left\|\left|A\right|\left|B\right|\right\|^{\frac{r}{2}}$$
$$\displaystyle=$$
$$\displaystyle\left\|U^{\ast}AB^{\ast}V\right\|^{\frac{r}{2}}$$
$$\displaystyle=$$
$$\displaystyle\left\|AB^{\ast}\right\|^{\frac{r}{2}},$$
where $A=U|A|$ and $B=V\left|B\right|$ are the
polar decompositions of the operators $A$ and $B.$ This completes the proof
of the corollary.
∎
In the next result, we show another generalization of inequality (1.2).
Theorem 2.10.
Let $A\in{\mathbb{B}}({\mathscr{H}})$ and $f,g,h$ be non-negative non-decreasing continuous functions on $[0,\infty)$ such that $f(x)g(x)=x\,\,(x\geq 0)$. Then
$$\displaystyle h\left(w(A)\right)\leq\frac{1}{2}\Big{(}h\left(w\left(\tilde{A}_%
{f,g}\right)\right)+\left\|h(|A|)\right\|\Big{)}.$$
Proof.
Let $A=U|A|$ be the polar decomposition of A.
Then for every $\theta\in\mathbb{R}$, we have
$$\displaystyle\|\textrm{Re}\left(e^{i\theta}A\right)\|$$
$$\displaystyle=$$
$$\displaystyle r\left(\textrm{Re}\left(e^{i\theta}A\right)\right)$$
(2.4)
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}r\left(e^{i\theta}A+e^{-i\theta}A^{\ast}\right)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}r\left(e^{i\theta}U|A|+e^{-i\theta}|A|U^{\ast}\right)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}r\left(e^{i\theta}Ug(|A|)f(|A|)+e^{-i\theta}f(|A|)g(|A%
|)U^{\ast}\right).$$
Now, if we put $X=e^{i\theta}Ug(|A|)$, $Y=f(|A|)$, $S=e^{-i\theta}f(|A|)$ and $T=g(|A|)U^{\ast}$ in Lemma 2.1, then we get
$$\displaystyle r\big{(}e^{i\theta}Ug(|A|)f(|A|)+e^{-i\theta}f(|A|)g(|A|)U^{\ast%
}\big{)}$$
(2.5)
$$\displaystyle\leq$$
$$\displaystyle\frac{1}{2}\Big{(}w(f(|A|)Ug(|A|))+w(g(|A|)U^{\ast}f(|A|))\Big{)}$$
$$\displaystyle+\frac{1}{2}\sqrt{4\|e^{-i\theta}f(|A|)g(|A|)\|\|g(|A|)U^{\ast}e^%
{i\theta}Uf(|A|)\|}$$
$$\displaystyle\qquad\qquad\qquad\qquad\qquad\textrm{(by Lemma \ref{kit1})}$$
$$\displaystyle\leq$$
$$\displaystyle w(f(|A|)Ug(|A|))+\sqrt{\|f(|A|)\|\|f(|A|)\|\|g(|A|)\|\|g(|A|)\|}$$
$$\displaystyle=$$
$$\displaystyle w(f(|A|)Ug(|A|))+\sqrt{f(\|A\|)g(\|A\|)g(\|A\|)f(\|A\|)}$$
$$\displaystyle\qquad\qquad\qquad\qquad\qquad\textrm{(by the functional calculus)}$$
$$\displaystyle=$$
$$\displaystyle w(f(|A|)Ug(|A|))+\sqrt{\|A\|\|A\|}$$
$$\displaystyle=$$
$$\displaystyle w\left(\tilde{A}_{f,g}\right)+\|A\|.$$
Using inequalities (2.4), (2.5) and Lemma 2.2 we get
$$\displaystyle\omega(A)=\underset{\theta\in\mathbb{R}}{\max}\left\|\textrm{Re}%
\left(e^{i\theta}A\right)\right\|\leq\frac{1}{2}\Big{(}w\left(\tilde{A}_{f,g}%
\right)+\|A\|\Big{)}.$$
Hence
$$\displaystyle h\left(w(A)\right)$$
$$\displaystyle\leq$$
$$\displaystyle h\left(\frac{1}{2}\left[w\left(\tilde{A}_{f,g}\right)+\left\|A%
\right\|\right]\right)$$
$$\displaystyle\qquad\qquad\qquad(\textrm{by the monotonicity of}\,\,h)$$
$$\displaystyle\leq$$
$$\displaystyle\frac{1}{2}h\left(w\left(\tilde{A}_{f,g}\right)\right)+\frac{1}{2%
}h\left(\left\|A\right\|\right)$$
$$\displaystyle\qquad\qquad\qquad(\textrm{by the convexity of}\,\,h)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}h\left(w\left(\tilde{A}_{f,g}\right)\right)+\frac{1}{2%
}\left\|h(|A|)\right\|,$$
as required.
∎
Another proof for Theorem 2.3:
We can obtain Theorem 2.3 from Theorem 2.10.
To see this, first note that by the hypotheses of Theorem 2.3 we have
$$\displaystyle h(|A|)$$
$$\displaystyle=$$
$$\displaystyle h(g(|A|)f(|A|))$$
(2.6)
$$\displaystyle\leq$$
$$\displaystyle h\left(\frac{g^{2}(|A|)+f^{2}(|A|)}{2}\right)\qquad\textrm{(by %
the arithmetic-geometric inequality)}$$
$$\displaystyle\leq$$
$$\displaystyle\frac{1}{2}\left(h\left(g^{2}(|A|)\right)+h\left(f^{2}(|A|)\right%
)\right)\qquad\textrm{(by the convexity of\,}h).$$
Hence, using Theorem 2.10 and inequality (2.6) we get
$$\displaystyle h\left(w(A)\right)$$
$$\displaystyle\leq$$
$$\displaystyle\frac{1}{2}\Big{[}h\left(w\left(\tilde{A}_{f,g}\right)\right)+%
\left\|h(|A|)\right\|\Big{]}$$
$$\displaystyle\leq$$
$$\displaystyle\frac{1}{2}\Big{[}h\left(w\left(\tilde{A}_{f,g}\right)\right)+%
\frac{1}{2}\left\|h\left(g^{2}(|A|)\right)+h\left(f^{2}(|A|)\right)\right\|%
\Big{]}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}h\left(w\left(\tilde{A}_{f,g}\right)\right)+\frac{1}{4%
}\left\|h\left(g^{2}(|A|)\right)+h\left(f^{2}(|A|)\right)\right\|.$$
Remark 2.11.
For the special case $f(x)=x^{t}$ and $g=x^{1-t}\,\,(t\in[0,1])$, we obtain the inequality (1.2)
$$\displaystyle w(A)\leq\frac{1}{2}\left(w\left(\tilde{A}_{t}\right)+\|A\|\right),$$
where $A\in{\mathbb{B}}({\mathscr{H}})$ and $t\in[0,1]$.
Using Theorem 2.10, we get the following result.
Corollary 2.12.
Let $A,B\in{\mathbb{B}}({\mathscr{H}})$ and $f,g$ be two non-negative non-decreasing continuous functions such that $f(x)g(x)=x\,\,(x\geq 0)$. Then
$$\displaystyle 2w^{r}\left(\left[\begin{array}[]{cc}0&A\\
B&0\end{array}\right]\right)\leq\max\{\|A\|^{r},\|B\|^{r}\}+\frac{1}{2}\big{(}%
\left\|f(|B|)g(|A^{\ast}|)\right\|^{r}+\left\|f(|A|)g(|B^{\ast}|)\right\|^{r}%
\big{)},$$
where $r\geq 1$.
Proof.
Using Theorem 2.10 and inequality (2.2), we have
$$\displaystyle 2w^{r}\left(\left[\begin{array}[]{cc}0&A\\
B&0\end{array}\right]\right)$$
$$\displaystyle\leq\left\|\left[\begin{array}[]{cc}0&A\\
B&0\end{array}\right]\right\|^{r}+w^{r}\left(\tilde{T}_{f,g}\right)$$
$$\displaystyle=\max\{\|A\|^{r},\|B\|^{r}\}+\left(\frac{1}{2}\big{[}\left\|f(|B|%
)g(|A^{\ast}|)\right\|+\|f(|A|)g(|B^{\ast}|)\|\big{]}\right)^{r}$$
$$\displaystyle\leq\max\{\|A\|^{r},\|B\|^{r}\}+\frac{1}{2}\big{(}\left\|f(|B|)g(%
|A^{\ast}|)\right\|^{r}+\|f(|A|)g(|B^{\ast}|)\|^{r}\big{)}$$
and the proof is complete.
∎
Corollary 2.13.
Let $A,B\in{\mathbb{B}}({\mathscr{H}})$ and $f,g$ be two non-negative non-decreasing continuous functions on $[0,\infty)$ such that $f(x)g(x)=x\,\,(x\geq 0)$. Then
$$\displaystyle\|A+B\|\leq\max\{\|A\|,\|B\|\}+\frac{1}{2}\Big{(}\big{\|}f(|B|)g(%
|A|)\big{\|}+\big{\|}f(|A^{\ast}|)g(|B^{\ast}|)\big{\|}\Big{)}.$$
Proof.
Let $T=\left[\begin{array}[]{cc}0&A\\
B&0\end{array}\right]$. Then
$$\displaystyle\left\|A+B^{{}^{\ast}}\right\|$$
$$\displaystyle=$$
$$\displaystyle\left\|T+T^{{}^{\ast}}\right\|$$
$$\displaystyle\leq$$
$$\displaystyle 2\underset{\theta\in\mathbb{R}}{\max}\left\|\textrm{Re}\left(e^{%
i\theta}T\right)\right\|$$
$$\displaystyle=$$
$$\displaystyle w(T)\qquad\qquad\qquad(\textrm{by Lemma \ref{1}})$$
$$\displaystyle\leq$$
$$\displaystyle\max\{\|A\|,\|B\|\}+\frac{1}{2}\Big{(}\big{\|}f(|B|)g(|A^{\ast}|)%
\big{\|}+\big{\|}f(|A|)g(|B^{\ast}|)\big{\|}\Big{)}$$
$$\displaystyle\qquad\qquad\qquad\qquad(\textrm{by Theorem \ref{main1}}).$$
If we replace $B$ by $B^{\ast}$, then we get the desired result.
∎
In the last results, we present some upper bounds for operator matrices. For this purpose, we need the following lemma.
Lemma 2.14.
[10, Theorem 1]
Let $A\in{\mathbb{B}}({\mathscr{H}})$ and $x,y\in{\mathscr{H}}$ be any vectors. If $f$, $g$ are non-negative continuous functions on $[0,\infty)$ which are satisfying the relation $f(x)g(x)=x\,(x\geq 0)$, then
$$\displaystyle|\left\langle Ax,y\right\rangle|^{2}\leq\left\langle f^{2}(|A|)x,%
x\right\rangle\,\left\langle g^{2}(|A^{*}|)y,y\right\rangle.$$
Theorem 2.15.
Let
$A,B,C,D\in{\mathbb{B}}({\mathscr{H}})$ and $f_{i},g_{i}\,\,(1\leq i\leq 4)$ be non-negative continues functions such that $f_{i}(x)g_{i}(x)=x\,\,(1\leq i\leq 4)$ for all $x\in[0,\infty)$. Then
$$\displaystyle\omega\left(\left[\begin{array}[]{cc}A&B\\
C&D\end{array}\right]\right)$$
$$\displaystyle\leq\max\left\{\left\|f_{1}^{2}(|A|)+g_{2}^{2}(|B^{*}|)+f_{3}^{2}%
(|C|)\right\|^{\frac{1}{2}},\left\|g_{4}(|D^{*}|)\right\|\right\}$$
$$\displaystyle\,\,+\max\left\{\left\|g_{1}(|A^{*}|)\right\|,\left\|f_{2}^{2}(|B%
|)+g_{3}^{2}(|C^{*}|)+f_{4}^{2}(|D|)\right\|^{\frac{1}{2}}\right\}.$$
Proof.
Let $T=\left[\begin{array}[]{cc}A&B\\
C&D\end{array}\right]$ and $\mathbf{x}=\left[\begin{array}[]{c}x_{1}\\
x_{2}\end{array}\right]$ be a unit vector (i.e., $\|x_{1}\|^{2}+\|x_{2}\|^{2}=1$). Then
$$\displaystyle\left|\left\langle T\mathbf{x},\mathbf{x}\right\rangle\right|$$
$$\displaystyle=$$
$$\displaystyle\left|\left\langle\left[\begin{array}[]{cc}A&B\\
C&D\end{array}\right]\left[\begin{array}[]{c}x_{1}\\
x_{2}\end{array}\right],\left[\begin{array}[]{c}x_{1}\\
x_{2}\end{array}\right]\right\rangle\right|$$
$$\displaystyle=$$
$$\displaystyle\left|\left\langle\left[\begin{array}[]{c}Ax_{1}+Bx_{2}\\
Cx_{1}+Dx_{2}\end{array}\right],\left[\begin{array}[]{c}x_{1}\\
x_{2}\end{array}\right]\right\rangle\right|$$
$$\displaystyle=$$
$$\displaystyle\left|\left\langle Ax_{1},x_{1}\right\rangle+\left\langle Bx_{2},%
x_{1}\right\rangle+\left\langle Cx_{1},x_{2}\right\rangle+\left\langle Dx_{2},%
x_{2}\right\rangle\right|$$
$$\displaystyle\leq$$
$$\displaystyle\left|\left\langle Ax_{1},x_{1}\right\rangle\right|+\left|\left%
\langle Bx_{2},x_{1}\right\rangle\right|+\left|\left\langle Cx_{1},x_{2}\right%
\rangle\right|+\left|\left\langle Dx_{2},x_{2}\right\rangle\right|$$
$$\displaystyle\left\langle f_{1}^{2}(|A|)x_{1},x_{1}\right\rangle^{\frac{1}{2}}%
\left\langle g_{1}^{2}(|A^{*}|)x_{1},x_{1}\right\rangle^{\frac{1}{2}}+\left%
\langle f_{2}^{2}(|B|)x_{2},x_{2}\right\rangle^{\frac{1}{2}}\left\langle g_{2}%
^{2}(|B^{*}|)x_{1},x_{1}\right\rangle^{\frac{1}{2}}$$
$$\displaystyle+\left\langle f_{3}^{2}(|C|)x_{1},x_{1}\right\rangle^{\frac{1}{2}%
}\left\langle g_{3}^{2}(|C^{*}|)x_{2},x_{2}\right\rangle^{\frac{1}{2}}+\left%
\langle f_{4}^{2}(|D|)x_{2}x_{2}\right\rangle^{\frac{1}{2}}\left\langle g_{4}^%
{2}(|D^{*}|)x_{2},x_{2}\right\rangle^{\frac{1}{2}}$$
$$\displaystyle\leq$$
$$\displaystyle\left(\left\langle f_{1}^{2}(|A|)x_{1},x_{1}\right\rangle+\left%
\langle g_{2}^{2}(|B^{*}|)x_{1},x_{1}\right\rangle+\left\langle f_{3}^{2}(|C|)%
x_{1},x_{1}\right\rangle+\left\langle g_{4}^{2}(|D^{*}|)x_{2},x_{2}\right%
\rangle\right)^{\frac{1}{2}}$$
$$\displaystyle\qquad\qquad\qquad\qquad\qquad\textrm{(\normalsize{by the Cauchy-%
Schwarz inequality})}$$
$$\displaystyle+\left(\left\langle g_{1}^{2}(|A^{*}|)x_{1},x_{1}\right\rangle+%
\left\langle f_{2}^{2}(|B|)x_{2},x_{2}\right\rangle+\left\langle g_{3}^{2}(|C^%
{*}|)x_{2},x_{2}\right\rangle+\left\langle f_{4}^{2}(|D|)x_{1},x_{1}\right%
\rangle\right)^{\frac{1}{2}}$$
$$\displaystyle=$$
$$\displaystyle\left(\left\langle\left(f_{1}^{2}(|A|)+g_{2}^{2}(|B^{*}|)+f_{3}^{%
2}(|C|)\right)x_{1},x_{1}\right\rangle+\left\langle g_{2}^{2}(|B^{*}|)x_{2},x_%
{2}\right\rangle\right)^{\frac{1}{2}}$$
$$\displaystyle+\left(\left\langle\left(f_{2}^{2}(|B|)+g_{3}^{2}(|C^{*}|)+f_{4}^%
{2}(|D|)\right)x_{2},x_{2}\right\rangle+\left\langle g_{1}^{2}(|A^{*}|)x_{1},x%
_{1}\right\rangle\right)^{\frac{1}{2}}$$
$$\displaystyle\leq$$
$$\displaystyle\left(\left\|f_{1}^{2}(|A|)+g_{2}^{2}(|B^{*}|)+f_{3}^{2}(|C|)%
\right\|\|x_{1}\|^{2}+\left\|g_{4}^{2}(|D^{*}|)\right\|\|x_{2}\|^{2}\right)^{%
\frac{1}{2}}$$
$$\displaystyle+\left(\left\|f_{2}^{2}(|B|)+g_{3}^{2}(|C^{*}|)+f_{4}^{2}(|D|)%
\right\|\|x_{2}\|^{2}+\left\|g_{1}^{2}(|A^{*}|)\right\|\|x_{1}\|^{2}\right)^{%
\frac{1}{2}}.$$
Let
$$\displaystyle\alpha=\left\|f_{1}^{2}(|A|)+g_{2}^{2}(|B^{*}|)+f_{3}^{2}(|C|)%
\right\|,\quad\beta=\left\|g_{4}^{2}(|D^{*}|)\right\|,$$
$$\displaystyle\mu=\left\|f_{2}^{2}(|B|)+g_{3}^{2}(|C^{*}|)+f_{4}^{2}(|D|)\right%
\|\quad\textrm{and}\,\,\lambda=\left\|g_{1}^{2}(|A^{*}|)\right\|.$$
It follows from
$$\underset{\|x_{1}\|^{2}+\|x_{2}\|^{2}=1}{\max}\left(\alpha\|x_{1}\|^{2}+\beta%
\|x_{2}\|^{2}\right)=\underset{\theta\in[0,2\pi]}{\max}(\alpha\sin^{2}\theta+%
\beta\cos^{2}\theta)=\max\{\alpha,\beta\}$$
and
$$\underset{\|x_{1}\|^{2}+\|x_{2}\|^{2}=1}{\max}\left(\lambda\|x_{1}\|^{2}+\mu\|%
x_{2}\|^{2}\right)=\underset{\theta\in[0,2\pi]}{\max}(\lambda\sin^{2}\theta+%
\mu\cos^{2}\theta)=\max\{\lambda,\mu\}$$
that
$$\displaystyle\left|\left\langle T\mathbf{x},\mathbf{x}\right\rangle\right|$$
$$\displaystyle\leq$$
$$\displaystyle\left(\left\|f_{1}^{2}(|A|)+g_{2}^{2}(|B^{*}|)+f_{3}^{2}(|C|)%
\right\|\|x_{1}\|^{2}+\left\|g_{4}^{2}(|D^{*}|)\right\|\|x_{2}\|^{2}\right)^{%
\frac{1}{2}}$$
$$\displaystyle+\left(\left\|f_{2}^{2}(|B|)+g_{3}^{2}(|C^{*}|)+f_{4}^{2}(|D|)%
\right\|\|x_{2}\|^{2}+\left\|g_{1}^{2}(|A^{*}|)\right\|\|x_{1}\|^{2}\right)^{%
\frac{1}{2}}$$
$$\displaystyle\leq$$
$$\displaystyle\max\left\{\left\|f_{1}^{2}(|A|)+g_{2}^{2}(|B^{*}|)+f_{3}^{2}(|C|%
)\right\|^{\frac{1}{2}},\left\|g_{4}^{2}(|D^{*}|)\right\|^{\frac{1}{2}}\right\}$$
$$\displaystyle+\max\left\{\left\|g_{1}^{2}(|A^{*}|)\right\|^{\frac{1}{2}},\left%
\|f_{2}^{2}(|B|)+g_{3}^{2}(|C^{*}|)+f_{4}^{2}(|D|)\right\|^{\frac{1}{2}}\right\}$$
$$\displaystyle=$$
$$\displaystyle\max\left\{\left\|f_{1}^{2}(|A|)+g_{2}^{2}(|B^{*}|)+f_{3}^{2}(|C|%
)\right\|^{\frac{1}{2}},\left\|g_{4}(|D^{*}|)\right\|\right\}$$
$$\displaystyle+\max\left\{\left\|g_{1}(|A^{*}|)\right\|,\left\|f_{2}^{2}(|B|)+g%
_{3}^{2}(|C^{*}|)+f_{4}^{2}(|D|)\right\|^{\frac{1}{2}}\right\}.$$
Taking the supremum over all unit vectors $\mathbf{x}$ we get the desired result.
∎
Corollary 2.16.
Let $A,B,C,D\in{\mathbb{B}}({\mathscr{H}})$. Then
$$\displaystyle\omega\left(\left[\begin{array}[]{cc}A&B\\
C&D\end{array}\right]\right)$$
$$\displaystyle\leq$$
$$\displaystyle\max\left\{\left\||A|^{2\alpha}+|B^{*}|^{2\gamma}+|C|^{2\mu}%
\right\|^{\frac{1}{2}},\left\||D^{*}|^{\omega}\right\|\right\}$$
$$\displaystyle+\max\left\{\left\||A^{*}|^{\beta}\right\|,\left\||B|^{2\zeta}+|C%
^{*}|^{2\nu}+|D|^{\kappa}\right\|\right\},$$
where $\alpha+\beta=\gamma+\zeta=\mu+\nu=\omega+\kappa=1$. In particular,
$$\displaystyle\omega\left(\left[\begin{array}[]{cc}A&B\\
0&0\end{array}\right]\right)\leq\max\left\{\left\||A^{*}|^{\beta}\right\|,%
\left\||B|^{\zeta}\right\|\right\}+\left\||A|^{2\alpha}+|B^{*}|^{2\gamma}%
\right\|^{\frac{1}{2}},$$
in which $\alpha+\beta=\gamma+\zeta=1$.
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Abstract
The leading contributions to the electron (or muon) electric dipole
moment due to CP violation in the charged Higgs sector are at the
two–loop level.
A careful model-independent analysis of the heavy fermion
contribution is provided.
We also consider some specific scenarios to
demonstrate how charged Higgs sector CP violation can naturally give
rise to large electric dipole moments.
Numerical results show that the
electron electric dipole moment in such models can lie at the
experimentally accessible level.
March 1997
hep-ph/9703435
NHCU-HEP-97-5
UICHEP-TH/97-3
Electron Electric Dipole Moment from
CP Violation in
the Charged Higgs Sector
David Bowser-Chao${}^{(1)}$,
Darwin Chang${}^{(2,3)}$,
and
Wai-Yee Keung${}^{(1)}$
${}^{(1)}$Physics Department, University of Illinois at Chicago,
IL 60607-7059, USA
${}^{(2)}$Physics Department,
National Tsing-Hua University, Hsinchu 30043, Taiwan, R.O.C.
${}^{(3)}$Institute of Physics, Academia Sinica, Taipei, R.O.C.
PACS numbers: 13.40.Fn, 11.30.Er, 12.15.Cc, 14.80.Dq
Introduction
Experiment has established that neither parity (P) nor charge
conjugation (C) are unbroken symmetries. Kaon
physics show that CP also fails to be an exact symmetry. The
CPT theorem then implies that time-reversal (T) is broken as
well, leading to expectation of a T-odd electric dipole moment (EDM)
for one or more of the elementary particles.
The Standard Model (SM) of electroweak interactions explains CP
violation in the $K-{\bar{K}}$ system as the result of a single complex
phase in the Cabibbo-Kobayashi-Maskawa (CKM) matrix. It also predicts an
electron EDM $d_{e}$ of about $8\times 10^{-41}{\rm e}\cdot{\rm cm}$
[1]
and a muon EDM of about $2\times 10^{-38}{\rm e}\cdot{\rm cm}$, while the neutron EDM (calculated from the up and down quark
EDMs) is estimated [2] to be less than $10^{-31}{\rm e}\cdot{\rm cm}$,
The experimental limits (given at 95% C.L.) are several orders of
magnitude above these predictions, with the limit on the electron EDM
$|d_{e}|<6.2\times 10^{-27}{\rm e}\cdot{\rm cm}$ [3]. The limit
on $d_{\mu}$ is even further removed from the SM prediction, with $|d_{\mu}|<1.1\times 10^{-18}{\rm e}\cdot{\rm cm}$ [4],
although there is a proposal to measure
the muon EDM down to $10^{-24}$ e$\cdot$cm [5].
The neutron EDM limit is $|d_{n}|<11\times 10^{-26}{\rm e}\cdot{\rm cm}$ [6]. Clearly, measurement of a non-zero electron, muon, or
neutron EDM close to current or proposed limits would point to physics
beyond the Standard Model.
New sources of CP violation can come from complex couplings or vacuum
expectation values (VEV) associated with the Higgs boson sector. A
significant EDM for elementary fermions can be
generated if CP violation is mediated by neutral Higgs-boson exchange
[7, 8, 9, 10, 11]. Dominant contributions
come from one-loop or two-loop diagrams. The one-loop terms are
proportional to $(m/v)^{3}$ (with one factor of $m/v$ due to an internal
mass insertion), while the two-loop terms are
proportional[9] to $m/v$, with $m$ being the fermion mass
and $v=246$ GeV the vacuum expectation value of the SM Higgs field.
The one-loop contributions are thus strongly suppressed relative to the
two-loop terms, by a factor of $(m/v)^{2}$.
Exhaustive studies[10, 12] have been carried out at the
two loop level on the electron EDM generated by CP violation in the neutral
Higgs-boson sector.
The contributions containing a heavy fermion or gauge boson loop
were considered[10]
and, within an $SU(2)\times U(1)$ theory, are more or less
model-independent. Additional, relatively more model-dependent,
contributions involving a physical charged
Higgs-boson loop (with a CP violating charged-Higgs-neutral-Higgs
coupling) have been considered in Ref.[12]. On the other hand,
the corresponding contribution to $d_{e}$ due solely
to CP violation in charged Higgs sector has not been studied in
the literature, even though this category of CP violating mechanism has been
emphasized for other phenomenological effects such as
top quark decay[13], the neutron electric dipole
moment[14, 15], and $\Gamma(b\to s\gamma)$[16].
With charged Higgs sector CP violation involving fermions, the one-loop
contribution is
suppressed as in the neutral Higgs case but suffers an additional
factor of $m_{\nu}/m_{e}$, where $m_{\nu}$ is the mass of the electron
neutrino.
If no right-handed neutrino exists, or the neutrino is
massless, the two-loop diagrams are unequivocally the leading
contribution. It is also important to note that the recent measurement
of the decay rate of $b\to s\gamma$ by the CLEO collaboration
[17] stringently constrains[18] the mass of the $H^{\pm}$
only in the 2HDM II. The constraint is easily evaded in 2HDM
III[16, 19, 20] and other extensions.
In this letter, we make a model-independent study of the two–loop
contribution to $d_{e}$ and $d_{\mu}$
from charged Higgs exchange between fermions.
Useful formulas are given. We
discuss specific models to see how this type of CP
nonconservation can arise.
Numerical results show the electron electric
dipole moment can naturally lie within reach of experiment.
We also
present results for the corresponding contributions to
$d_{n}$.
General Formalism
We first comment on the model-independence of our analysis,
which considers only those graphs that involve CP violation from
charged Higgs exchange between
fermions. There are other possibilities,
such as quartic charged Higgs vertices involving three or four distinct
charged Higgs bosons with complex coupling, but these are strongly
dependent on model details and less amenable to
model-independent parameterization.
Secondly, one may have several charged Higgs bosons,
but barring significant degeneracy,
cancellation among contributions from the various
charged Higgs should be mild. For simplicity we only consider
contributions from the lightest charged Higgs.
We shall ignore model dependent neutral
Higgs sector CP violation contributions and
CP violation involving both neutral and charged Higgs boson,
as our concern here is with exclusively
charged Higgs sector CP violation effects.
We shall also omit terms
where the neutral Higgs bosons participate only incidentally, serving as
a leg of an internal loop but not contributing a CP phase.
We expect
that these (perhaps non-negligible) contributions
should not strongly cancel with the part studied here; thus our
analysis should furnish a reasonable lower limit to $d_{e}$ for general
theories with CP violating charged Higgs exchange between fermions.
We parametrize the charged Higgs sector CP violation as follows:
$${\cal L}={g\over\sqrt{2}}\left({m_{t}c_{t}\over M_{W}}\bar{t}_{R}b_{L}H^{+}+{m%
_{e}c_{e}\over M_{W}}\bar{\nu}_{L}e_{R}H^{+}+\bar{t}_{L}\gamma^{\alpha}b_{L}W^%
{+}_{\alpha}+\bar{\nu}_{L}\gamma^{\alpha}e_{L}W^{+}_{\alpha}\right)$$
$$-m_{e}\bar{e_{L}}e_{R}-m_{t}\bar{t_{L}}t_{R}+{\rm H.c.}$$
(1)
Note that in our model-independent analysis, one needs
not specify the origin of the CP violation (e.g., explicit or
spontaneous CP violation).
We only illustrate the most important
contribution from the top-bottom generation; our study can easily be
generalized to the three generations case. The bottom quark mass $m_{b}$
is also set to zero. The
complex mixing parameters $c_{t}$ and $c_{e}$ signal deviations from the
2HDM II. If $c_{t}c_{e}^{*}$ has a non-zero imaginary part, the phase is
intrinsic to the lagrangian and cannot be rotated away by redefinition
of any or all of the fields in Eq.(1).
The two-loop charged Higgs contribution involves
Feynman diagrams such as the one shown in Fig. 1. We first present a
simple expression for the one-loop sub-diagram with fermion in the loop,
that is, the truncated three-point Green’s function $\Gamma^{\mu\nu}=\langle 0|[H^{-}(p)A^{\mu}(k){W^{+}}^{\nu}(-q)]_{+}|0\rangle$. We note
that $\Gamma^{\mu\nu}$ is the off-shell extension of the amplitude
(and Feynman diagrams) for
$H^{-}\to W^{-}\gamma$ given in Ref.[21], and that both
$\Gamma^{\mu\nu}$ and its charge conjugate contribute to $d_{e}$.
We consider only
the (gauge-invariant) set of terms
in $\Gamma^{\mu\nu}$ involving CP violation from
charged Higgs exchange between fermions.
We have verified that our results are gauge-independent,
but the calculation simplifies greatly
in the non-linear $R_{\xi}$ gauge[26]. Since the EDM is defined
in the soft photon limit, only the leading term in $k$ is kept. We also
work to lowest
order in $(m_{e}/M_{W})$, to which approximation the separation of
the calculation into $\Gamma^{\mu\nu}$ and its insertion in the full
two-loop graph is gauge-invariant. We thus obtain:
$$\Gamma^{\mu\nu}={3eg^{2}\over 16\pi^{2}}{c_{t}\over M_{W}}[(g^{\mu\nu}q\cdot k%
-q^{\mu}k^{\nu})S+iP\varepsilon^{\mu\nu\alpha\beta}p_{\alpha}q_{\beta}]\ ,$$
(2)
$$S=\int_{0}^{1}{q_{t}(1-y)^{2}+q_{b}y(1-y)\over 1-yq^{2}/m_{t}^{2}}dy\ ,\quad P%
=\int_{0}^{1}{q_{t}(1-y)+q_{b}y\over 1-yq^{2}/m_{t}^{2}}dy\ ,$$
(3)
and the quark charges are denoted $q_{t},q_{b}$.
The above vertex is further connected to the lepton propagator to
produce EDM (see Fig. 1).
$$d_{e}=\left({3eg^{2}\over 32\pi^{2}}\right)\left({g^{2}\over 32\pi^{2}M_{W}}%
\right)\left({m_{e}\over M_{W}}\right){\rm Im}(c_{t}^{*}c_{e})\left(q_{t}F_{t}%
+q_{b}F_{b}\right)\ .$$
(4)
Here the form factors $F_{\alpha}$ for $\alpha=t,b$ are given by
$$F_{\alpha}=\int_{0}^{\infty}\int_{0}^{1}{m_{t}^{2}f_{\alpha}(2-y)\;dy\,Q^{2}dQ%
^{2}\over(M_{H^{+}}^{2}+Q^{2})(m_{t}^{2}+yQ^{2})(M_{W}^{2}+Q^{2})}\ ;\quad f_{%
t}=(1-y)\ ,\quad f_{b}=y\ .$$
(5)
The integrations can be carried
through analytically,
$$T(z)={1-3z\over z^{2}}{\pi^{2}\over 6}-\left({1\over z}-{5\over 2}\right){\rm
ln%
\ }z-{1\over z}-\left(2-{1\over z}\right)\left(1-{1\over z}\right){\rm Sp}(1-z%
)\ ,$$
(6)
$$B(z)={1\over z}+{2z-1\over z^{2}}{\pi^{2}\over 6}+\left({3\over 2}-{1\over z}%
\right){\rm ln\ }z-\left(2-{1\over z}\right){1\over z}{\rm Sp}(1-z)\ ,$$
(7)
$$F_{t}={T(z_{H})-T(z_{W})\over z_{H}-z_{W}}\ ,\quad F_{b}={B(z_{H})-B(z_{W})%
\over z_{H}-z_{W}}\ ,$$
(8)
with
$z_{H}=M_{H^{+}}^{2}/m_{t}^{2}$ and $z_{W}=M_{W}^{2}/m_{t}^{2}$,
and the Spence function is defined by
${\rm Sp}(z)=-\int_{0}^{z}t^{-1}{\rm\ln}(1-t)dt$
with the normalization ${\rm Sp}(1)=\pi^{2}/6$.
Models
To illustrate how easily this
mechanism can give rise to a measurable electron EDM,
we consider some specific models.
We first note that the
charged Higgs contribution to $d_{e}$ vanishes to two loops in the
much-studied simple extension of the Standard Model —
the two Higgs-doublet Model (2HDM)[22], with the softly broken
discrete symmetry[23] imposed
to enforce natural flavor conservation (NFC).
At one loop level single charged
Higgs exchange between fermions does not
violate CP[8]. As for the two loop contribution, setting the
scalar doublets $\phi_{i}=(\phi^{+}_{i},\phi^{0}_{i}),i=1,2$ to have
purely real VEVs $\langle\phi^{0}_{i}\rangle_{0}=v_{i}$,
in the unitary gauge the charged Higgs propagators
$\langle\phi^{+}_{1}\phi^{+{\dagger}}_{2}\rangle_{0},\langle\phi^{+}_{1}\phi^{+%
{\dagger}}_{1}\rangle_{0},\langle\phi^{+}_{2}\phi^{+{\dagger}}_{2}\rangle_{0}$
are purely real[7, 8].
Ignoring
the CKM matrix, the only[8] complex coupling and thus
CP phase in the
lagrangian involving the charged Higgs appears in
${\rm Re}[h(\phi_{1}^{{\dagger}}\phi_{2}-v_{1}v_{2})^{2}]$;
thus charged Higgs CP violation in this model
necessarily also involves the pseudoscalar neutral Higgs.
There are, however, several other simple models which can easily contain
sufficient CP violation to produce an electron EDM at an
observable level, through exchange of a charged Higgs boson between fermions.
2HDM III.
In Model III with two Higgs doublets[24, 19], we can choose a basis
so that
$\langle\phi^{0}\rangle={v\over\sqrt{2}}$ and
$\langle\phi^{\prime 0}\rangle=0$.
Then $\phi$ mimics the SM Higgs, and $\phi^{\prime}$ produces new
physics beyond the SM; the physical charged Higgs $H^{+}$ is just
$\phi^{\prime+}$.
The Yukawa Lagrangian for Model III is
$$-{\cal L}_{Y}=\eta_{ij}^{U}\bar{Q}_{iL}\tilde{\phi}U_{jR}+\eta_{ij}^{D}\bar{Q}%
_{iL}\phi D_{jR}+\xi_{ij}^{U}\bar{Q}_{iL}\tilde{\phi^{\prime}}U_{jR}+\xi_{ij}^%
{D}\bar{Q}_{iL}\phi^{\prime}D_{jR}$$
(9)
$$\quad\quad+\eta_{ij}^{E}\bar{L}_{iL}\phi E_{jR}+\xi_{ij}^{E}\bar{L}_{iL}\phi^{%
\prime}E_{jR}\quad\;+{\rm H.c.}$$
Here $i,j$ are generation indices. Coupling matrices $\eta$ and
$\xi$ are, in general, non-diagonal.
$Q_{iL}$ and $L_{iL}$ are the
left-handed SU(2) doublets for quarks and leptons.
$U_{jR}$, $D_{jR}$, and $E_{R}$ are the right-handed SU(2) singlets for
up-type and down-type quarks and charged leptons respectively.
$\langle\phi\rangle$ generates all fermion mass matrices which are
diagonalized by bi-unitary transformations, e.g.
$M_{U}={\rm diag}(m_{u},m_{c},m_{t})={v\over\sqrt{2}}({\cal L}^{U})^{\dagger}%
\eta^{U}(\cal R^{U})$.
In terms of the mass eigenstates $U$, $D$, $E$, and $N$(neutrinos), the
relevant
charged Higgs interaction is given by
$${\cal L}_{H^{+}}=-H^{+}\bar{U}\biggr{[}V_{\rm KM}{\hat{\xi}}^{D}\hbox{$1\over 2%
$}(1+\gamma^{5})-{\hat{\xi}}^{U\dagger}V_{\rm KM}\hbox{$1\over 2$}(1-\gamma^{5%
})\biggr{]}D-H^{+}\bar{N}{\hat{\xi}}^{E}\hbox{$1\over 2$}(1+\gamma^{5})E+{\rm H%
.c.}\;\;,$$
(10)
with
the CKM matrix
$V_{\rm CKM}=({\cal L}_{U})^{\dagger}({\cal L}_{D})$, and
$\hat{\xi}^{P}=({\cal L}^{P})^{\dagger}\xi^{P}({\cal R}^{P})$ (for
$P=U,D,E$).
Tree-level flavor changing neutral currents (FCNC)
are implied by non-zero off-diagonal elements of the
matrices $\hat{\xi}^{U,D,E}$.
We adopt the simple ansatz[24]
$\hat{\xi}^{U,D,E}_{ij}=\lambda_{ij}{g\sqrt{m_{i}m_{j}}}/({\sqrt{2}M_{W}})\ .$
The mass hierarchy ensures that FCNC within the first two
generations are naturally suppressed by small quark masses, while
a larger freedom is allowed for FCNC involving the third
generations. Here $\lambda_{ij}$ can be
$O(1)$ and complex.
CP is already not a symmetry even if we restrict our attention to the
flavor conserving diagonal entries of $\lambda_{ii}$.
We consider only the third generation quark contribution and set $(V_{\rm CKM})_{tb}=1$.
The parameters $c_{t}$
and $c_{e}$ in Eq.(1) are given as,
$c_{e}=-\lambda_{ee}\ ,\quad c_{t}=\lambda_{tt}^{*}\ .$
While $\lambda_{tt}$ cannot be significantly larger than $O(1)$ without
producing strong coupling to the top quark, clearly $\lambda_{ee}$ can
be much larger than $O(1)$ so that ${\rm Im}(c_{t}^{*}c_{e})\gg 1$ is quite
allowed.
3HDM.
Charged Higgs sector CP violation can occur in the
three Higgs doublet model[25, 15].
The first two doublets $\phi_{1}$ and $\phi_{2}$ are responsible for the
masses of the $b$–like quarks and the $t$–like quarks respectively.
The charged leptons $e,\mu$ and $\tau$ only couple to $\phi_{1}$.
The last doublet $\phi_{3}$ does not couple to the known fermions.
This assignment naturally preserves NFC.
The mass eigenstates $H_{1}^{+}$ and $H_{2}^{+}$ together with the
unphysical charged Goldstone boson $H_{3}^{+}$ are linear combinations of
$\phi_{i}^{+}$:
$\phi_{i}^{+}=\sum_{j=1}^{3}U_{ij}H_{j}^{+}\quad(i=1,2,3)$.
As with the CKM matrix for three quark generations in the SM,
the mixing amplitude $U_{ij}$ matrix generally
contains a single non-zero complex phase,
which gives rise to CP nonconservation through the
Yukawa couplings,
$c_{t}=U_{2i}(v/v_{2}),\;c_{e}=U_{1i}(v/v_{1})$.
In the approximation that the lightest Higgs dominates, the index $i$
refers to the lightest charged Higgs. As with the 2HDM III, ${\rm Im}(c_{t}^{*}c_{e})$ can be much larger than one — which occurs here if
$v_{1}\ll v_{2}$ (the possibility $v_{2}\ll v_{1}$ is constrained by
maintaining perturbative coupling to the top quark).
Discussion
We have analyzed the heavy fermion contribution
to the electron EDM due to CP
violation in the charged Higgs sector. From the general structure of the
typical models discussed above, we have shown that the relevant CP
violating parameter Im$(c_{t}^{*}c_{e})$ can be of order one or larger. In
Fig. 2, we show the dependence of the electron EDM on $M_{H^{+}}$ for the
case Im$(c_{t}^{*}c_{e})=1$. The size of $d_{e}$ is naturally around $10^{-26}$
e$\cdot$cm, around the current limit.
As noted above, in any specific model there may be other
contributions to $d_{e}$, but in the absence of accidental
strong cancellation, we expect that the results
presented here reasonably estimate a lower limit on $d_{e}$
which applies to a very wide class of CP violating models.
In such case, Fig. 2 may
be used, for example, to rule out $m_{H^{+}}>200\,{\rm GeV}$ for
Im$(c_{t}^{*}c_{e})\geq 1$.
The muon EDM can be easily obtained by the replacements $m_{e}\to m_{\mu}$
and Im$(c_{t}^{*}c_{e})\to$ Im$(c_{t}^{*}c_{\mu})$. In this case, the analogous
case of
Im$(c_{t}^{*}c_{\mu})=1$ would lead to
an observable muon EDM,
assuming the proposed future sensitivity down to
$d_{\mu}=10^{-24}$ e$\cdot$cm.
Finally, we can carry over the calculation for $d_{e}$ to estimate
contributes to $d_{n}$, using SU(6) relations [27]:
$d_{n}={1\over 3}(4d_{d}-d_{u})$,
where we obtain the down and up quark EDMs with replacements as made
for $d_{\mu}$, but with an additional factor $\eta_{q}$ multiplying both
$d_{d}$ and $d_{u}$ coming from QCD evolution of the quark mass and the
quark EDM [28]:
$\eta_{q}=q(m_{t},m_{b})^{16/23}q(m_{b},m_{c})^{16/25}q(m_{c},\mu)^{16/27},$ with $q(m_{a},m_{b})=\alpha_{s}(m_{a}^{2})/\alpha_{s}(m_{b}^{2})$.
There are sizable uncertainties coming from the quark masses and the
extraction of $d_{n}$ from $d_{d}$ and $d_{u}$, but the resulting neutron EDM
should be $d_{n}\approx 10^{-27}\,(m_{d}/m_{e}){\rm Im}(c_{t}^{*}c_{d})\,{\rm e}\cdot{\rm
cm}$ for $m_{H^{+}}\approx 100$ GeV (ignoring the up quark
contribution). This contribution would reach the observable limit for
${\rm Im}(c_{t}^{*}c_{d})>6$. In contrast to the case of $d_{e}$ or $d_{\mu}$,
there is a sizable contribution from the charged Higgs boson through the
three-gluon operator [14, 29]. The relative magnitudes are highly
model-dependent: in 2HDM III, the three-gluon operator may vanish even
while the two-loop contributions presented here are non-zero, if $c_{t}$
is purely real but $c_{e}$ remains complex. In the 3HDM, however, the two
contributions to $d_{n}$ are either both zero or both non-zero. In any
case, barring strong cancellations, our result places a limit of
$m_{H^{+}}>100$ GeV for ${\rm Im}(c_{t}^{*}c_{d})=6$.
D. B.-C. and W.-Y. K. are supported by a grant from the Department of
Energy, and
D. C. by a grant from the National Science Council of R.O.C. We also wish to
thank I. Phillips, T-C. Yuan and C.-S. Li who were
involved in an earlier stage of this work.
Figures
Fig. 1.
A typical two-loop Feynman diagram for the electron EDM
due to charged Higgs sector CP violation. The other diagrams for the
one-loop subgraph $H^{-}\rightarrow W^{-}\gamma$ may be found
in Ref.[21].
Fig. 2.
Model-independent contributions to $d_{e}$ versus $M_{H^{+}}$ for
Im$(c_{t}^{*}c_{e})=1/2,\,1$. The horizontal line denotes the current
experimental limit.
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An integer
construction of infinitesimals: Toward a theory of Eudoxus hyperreals
Alexandre Borovik
School of Mathematics, University
of Manchester, Oxford Street, Manchester, M13 9PL, United Kingdom
[email protected]
,
Renling Jin
Department of Mathematics, College of
Charleston, South Carolina 29424, USA
[email protected]
and
Mikhail G. Katz${}^{0}$
Department of Mathematics, Bar
Ilan University, Ramat Gan 52900 Israel
[email protected]
Abstract.
A construction of the real number system based on almost homomorphisms
of the integers ${\mathbb{Z}}$ was proposed by Schanuel, Arthan, and others. We
combine such a construction with the ultrapower or limit ultrapower
construction, to construct the hyperreals out of integers. In fact,
any hyperreal field, whose universe is a set, can be obtained by such
a one-step construction directly out of integers. Even the maximal
(i.e., On-saturated) hyperreal number system described by
Kanovei and Reeken (2004) and independently by Ehrlich (2012) can be
obtained in this fashion, albeit not in NBG. In NBG, it
can be obtained via a one-step construction by means of a definable
ultrapower (modulo a suitable definable class ultrafilter).
Key words and phrases:Eudoxus; hyperreals; infinitesimals; limit ultrapower;
universal hyperreal field
2000 Mathematics Subject Classification: Primary 26E35; Secondary 03C20
00footnotetext: Supported by the Israel Science Foundation grant
1517/12
Contents
1 From Kronecker to Schanuel
2 Passing it through an ultraproduct
3 Cantor, Dedekind, and Schanuel
4 Constructing an infinitesimal-enriched continuum
5 Formalization
6 Limit ultrapowers and definable ultrapowers
7 Universal and On-saturated Hyperreal Number Systems
1. From Kronecker to Schanuel
Kronecker famously remarked that, once we have the natural numbers in
hand, “everything else is the work of man” (see Weber [49]).
Does this apply to infinitesimals, as well?
The exposition in this section follows R. Arthan [7]. A
function $f$ from ${\mathbb{Z}}$ to ${\mathbb{Z}}$ is said to be an almost
homomorphism if and only if the function $d_{f}$ from ${\mathbb{Z}}\times{\mathbb{Z}}$
to ${\mathbb{Z}}$ defined by
$$d_{f}(p,q)=f(p+q)-f(p)-f(q)$$
has bounded
(i.e., finite) range, so that for a suitable integer $C$, we
have $|d_{f}(p,q)|<C$ for all $p,q\in{\mathbb{Z}}$. The set denoted
$${{\mathbb{Z}}}\rightarrow{{\mathbb{Z}}}$$
(1.1)
of all functions from ${\mathbb{Z}}$ to ${\mathbb{Z}}$ becomes an abelian group if we add
and negate functions pointwise:
$$(f+g)(p)=f(p)+g(p),\qquad(-f)(p)=-f(p).$$
It is easily checked that if $f$ and $g$ are almost homomorphisms then
so are $f+g$ and $-f$. Let $S$ be the set of all almost homomorphisms
from ${\mathbb{Z}}$ to ${\mathbb{Z}}$. Then $S$ is a subgroup of ${{\mathbb{Z}}}\rightarrow{{\mathbb{Z}}}$.
Let us write $B$ for the set of all functions from ${\mathbb{Z}}$ to ${\mathbb{Z}}$ whose
range is bounded. Then $B$ is a subgroup of $S$. The ‘‘Eudoxus
reals’’ are defined as follows.111The term “Eudoxus reals” has gained some
currency in the literature, see e.g., Arthan [7].
Shenitzer [46, p. 45] argues that Eudoxus anticipated 19th
century constructions of the real numbers. The attribution of such
ideas to Eudoxus, based on an interpretation involving Eudoxus,
Euclid, and Book 5 of The Elements, may be historically
questionable.
Definition 1.1.
The abelian group ${\mathbb{E}}$ of Eudoxus reals is the quotient group
$S/B$.
Elements of ${\mathbb{E}}$ are equivalence classes, $[f]$ say, where $f$ is an
almost homomorphism from ${\mathbb{Z}}$ to ${\mathbb{Z}}$, i.e., $f$ is a function from
${\mathbb{Z}}$ to ${\mathbb{Z}}$ such that $d_{f}(p,q)=f(p,q)-f(p)-f(q)$ defines a
function from ${\mathbb{Z}}\times{\mathbb{Z}}$ to ${\mathbb{Z}}$ whose range is bounded. We have
$[f]=[g]$ if and only if the difference $f-g$ has bounded range,
i.e., if and only if $|{f(p)-g(p)}|<C$ for some $C$ and all $p$ in
${\mathbb{Z}}$.
The addition and additive inverse in ${\mathbb{E}}$ are induced by the pointwise
addition and inverse of representative almost homomorphisms:
$$[f]+[g]=[f+g],\qquad-[f]=[-f]$$
where $f+g$ and $-f$ are defined by
$$(f+g)(p)=f(p)+g(p)$$
and
$$(-f)(p)=-f(p)$$
for all $p$ in ${\mathbb{Z}}$.
The group ${\mathbb{E}}$ of Eudoxus reals becomes an ordered abelian
group if we take the set $P\subset{\mathbb{E}}$ of positive elements to be
$$P=\left\{[f]\in{\mathbb{E}}:\sup_{m\in{\mathbb{N}}\subset{\mathbb{Z}}}f(m)=+%
\infty\right\}.$$
The multiplication on ${\mathbb{E}}$ is induced by composition of almost
homomorphisms. The multiplication turns ${\mathbb{E}}$ into a commutative ring
with unit. Moreover, this ring is a field. Even more
surprisingly, ${\mathbb{E}}$ is an ordered field with respect to the ordering
defined by $P$.
Theorem 1.2 (See Arthan [7]).
${\mathbb{E}}$ is a complete ordered field and is therefore isomorphic to the
field of real numbers ${\mathbb{R}}$.
The isomorphism ${\mathbb{R}}\to{\mathbb{E}}$ assigns to every real number $\alpha\in{\mathbb{R}}$, the class $[f_{\alpha}]$ of the function
$$\displaystyle f_{\alpha}:{\mathbb{Z}}$$
$$\displaystyle\longrightarrow$$
$$\displaystyle{\mathbb{Z}}$$
$$\displaystyle n$$
$$\displaystyle\mapsto$$
$$\displaystyle\lfloor\alpha n\rfloor,$$
where $\lfloor\,\cdot\,\rfloor$ is the integer part function.
In the remainder of the paper, we combine the above one-step
construction of the reals with the ultrapower or limit ultrapower
construction, to obtain hyperreal number systems directly out of the
integers. We show that any hyperreal field, whose universe is a set,
can be so obtained by such a one-step construction. Following this,
working in NBG (von Neumann-Bernays-Gödel set theory with the
Axiom of Global Choice), we further observe that by using a suitable
definable ultrapower, even the maximal (i.e., the
On-saturated)222Recall that a model $M$ is On-saturated if $M$ is
$\kappa$-saturated for any cardinal $\kappa$ in On. Here
On (or ON) is the class of all ordinals
(cf. Kunen [40, p. 17]). A hyperreal number system $\langle{\mathbb{R}},{}^{*}\!{\mathbb{R}},S\in\mathfrak{F}\rangle$ is On-saturated if it
satisfies the following condition: If $X$ is a set of equations and
inequalities involving real functions, hyperreal constants and
variables, then $X$ has a hyperreal solution whenever every finite
subset of $X$ has a hyperreal solution (see Ehrlich [17, section 9,
p. 34]).
hyperreal number system recently described by Ehrlich [17] can
be obtained in a one-step fashion directly from the integers.333Another version of such an On-saturated number system
was introduced by Kanovei and Reeken (2004, [25, Theorem
4.1.10(i)]) in the framework of axiomatic nonstandard analysis.
As Ehrlich [17, Theorem 20] showed, the ordered field
underlying an On-saturated hyperreal field is isomorphic to
J. H. Conway’s ordered field No, an ordered field Ehrlich describes as
the absolute arithmetic continuum (modulo NBG).
2. Passing it through an ultraproduct
Let now $\mathcal{Z}=\mathbb{Z}^{\mathbb{N}}$ be the ring of integer
sequences with operations of componentwise addition and
multiplication. We define a rescaling to be a sequence $\rho=\langle\rho_{n}:n\in\mathbb{N}\rangle$ of almost
homomorphisms $\rho_{n}:\mathbb{Z}\longrightarrow\mathbb{Z}$.
Rescalings are thought of as acting on $\mathcal{Z}$, hence the name.
A rescaling $\rho$ is called bounded if each of its
components, $\rho_{n}$, is bounded.
Rescalings factorized modulo bounded rescalings form a commutative
ring $\mathcal{E}$ with respect to addition and composition.
Quotients of $\mathcal{E}$ by its maximal ideals are hyperreal fields.
Thus, hyperreal fields are factor fields of the ring of rescalings of
integer sequences. This description is a tautological translation of
the classical construction, due to E. Hewitt [23], but it is
interesting for the sheer economy of the language used. We will give
further details in the sections below.
3. Cantor, Dedekind, and Schanuel
The strategy of Cantor’s construction of the real numbers444The construction of the real numbers as equivalence classes
of Cauchy sequences of rationals, usually attributed to Cantor, is
actually due to H. Méray (1869, [42]) who published three years
earlier than E. Heine and Cantor.
can be represented schematically by the diagram
$${\mathbb{R}}:=\left({\mathbb{N}}\to({\mathbb{Z}}\times{\mathbb{Z}})_{\alpha}%
\right)_{\beta}$$
(3.1)
where the subscript $\alpha$ evokes the passage from a pair of
integers to a rational number; the arrow $\to$ alludes to forming
sequences; and subscript $\beta$ reminds us to select Cauchy sequences
modulo equivalence. Meanwhile, Dedekind proceeds according to the
scheme
$${\mathbb{R}}:=\left(\mathcal{P}({\mathbb{Z}}\times{\mathbb{Z}})_{\alpha}\right%
)_{\gamma}$$
(3.2)
where $\alpha$ is as above, $\mathcal{P}$ alludes to the set-theoretic
power operation, and $\gamma$ selects his cuts. For a history of the
problem, see P. Ehrlich [16].
An alternative approach was proposed by Schanuel, and developed by
N. A’Campo [1], R. Arthan [6], [7],555Arthan’s Irrational construction of ${\mathbb{R}}$ from ${\mathbb{Z}}$
[6] describes a different way of skipping the rationals,
based on the observation that the Dedekind construction can take as
its starting point any Archimedean densely ordered commutative group.
The construction delivers a completion of the group, and one can
define multiplication by analyzing its order-preserving endomorphisms.
Arthan uses the additive group of the ring ${\mathbb{Z}}[\sqrt{2}]$, which can
be viewed as ${\mathbb{Z}}\times{\mathbb{Z}}$ with an ordering defined using a certain
recurrence relation.
T. Grundhöfer [22], R. Street [48], O. Deiser
[15, pp. 112-127], and others, who follow the formally simpler
blueprint
$${\mathbb{R}}:=({\mathbb{Z}}\to{\mathbb{Z}})_{\sigma}$$
(3.3)
where $\sigma$ selects certain almost homomorphisms from ${\mathbb{Z}}$ to
itself, such as the map
$$a\mapsto\lfloor ra\rfloor$$
(3.4)
for real $r$, modulo equivalence (think of $r$ as the ‘‘large-scale
slope’’ of the map).666One could also represent a real by a string based on its
decimal expansion, but the addition in such a presentation is highly
nontrivial due to carry-over, which can be arbitrarily long. In
contrast, the addition of almost homomorphisms is term-by-term.
Multiplication on the reals is induced by composition in ${\mathbb{Z}}\to{\mathbb{Z}}$,
see formula (1.1).
Such a construction has been referred to as the Eudoxus reals.777See footnote 1 for a discussion of the term.
The construction of ${\mathbb{R}}$ from ${\mathbb{Z}}$ by means of almost homomorphisms
has been described as “skipping the rationals ${\mathbb{Q}}$”.
We will refer to the arrow in (3.3) as the space
dimension, so to distinguish it from the time dimension
occurring in the following construction of an extension of ${\mathbb{N}}$:
$$({\mathbb{N}}\to{\mathbb{N}})_{\tau_{\!co\!f}}$$
(3.5)
where $\tau_{\!co\!f}$ identifies sequences
$f,g:{\mathbb{N}}\to{\mathbb{N}}$ which differ on a finite set of indices:
$$\{n\in{\mathbb{N}}:f(n)=g(n)\}\quad\hbox{is cofinite.}$$
(3.6)
Here the constant sequences induce an inclusion
$${\mathbb{N}}\to({\mathbb{N}}\to{\mathbb{N}})_{\tau_{\!co\!f}}.$$
Such a construction is closely related to (a version of)
the $\Omega$-calculus of Schmieden and Laugwitz [45]. The
resulting semiring has zero divisors. To obtain a model which
satisfies the first-order Peano axioms, we need to quotient it
further. Note that up to this point the construction has not used any
nonconstructive foundational material such as the axiom of choice or
the weaker axiom of the existence of nonprincipal ultrafilters.
4. Constructing an infinitesimal-enriched continuum
The traditional ultrapower construction of the hyperreals proceeds
according to the blueprint
$$({\mathbb{N}}\to{\mathbb{R}})_{\mathcal{U}}$$
where ${\mathcal{U}}$ is a fixed ultrafilter on ${\mathbb{N}}$. Replacing ${\mathbb{R}}$
by any of the possible constructions of ${\mathbb{R}}$ from ${\mathbb{Z}}$, one in
principle obtains what can be viewed as a direct construction of the
hyperreals out of the integers ${\mathbb{Z}}$. Formally, the most economical
construction of this sort passes via the Eudoxus reals.
An infinitesimal-enriched continuum can be visualized by means of an
infinite-magnification microscope as in Figure 1.
To construct such an infinitesimal-enriched field, we have to deal
with the problem that the semiring
$({\mathbb{N}}\to{\mathbb{N}})_{\tau_{\!co\!f}}$ constructed in the previous
section contains zero divisors.
To eliminate the zero divisors, we need to quotient the ring further.
This is done by extending the equivalence relation by means of a
maximal ideal defined in terms of an ultrafilter. Thus, we extend the
relation defined by (3.6) to the relation declaring $f$ and $g$
equivalent if
$$\{n\in{\mathbb{N}}:f(n)=g(n)\}\in\mathcal{U},$$
(4.1)
where $\mathcal{U}$ is a fixed ultrafilter on ${\mathbb{N}}$, and add
negatives. The resulting modification of (3.5), called an ultrapower, will be denoted
$$\mbox{I\!I\!I\!\!N}:=({\mathbb{N}}\to{\mathbb{N}})_{\tau}$$
(4.2)
and is related to Skolem’s construction in 1934 of a nonstandard model
of arithmetic [47]. We refer to the arrow in (4.2) as time to allude to the fact that a sequence that increases without
bound for large time will generate an infinite “natural” number
in IIIN. A “continuous” version of the ultrapower construction was
exploited by Hewitt in constructing his hyperreal fields in 1948
(see [23]).
The traditional ultrapower approach to constructing the hyperreals is
to start with the field of real numbers ${\mathbb{R}}$ and build the ultrapower
$$({\mathbb{N}}\to{\mathbb{R}})_{\tau}$$
(4.3)
where the subscript $\tau$ is equivalent to that of (4.2), see
e.g. Goldblatt [21]. For instance, relative to Cantor’s
procedure (3.1), this construction can be represented by the
scheme
$$({\mathbb{N}}\to({\mathbb{N}}\to({\mathbb{Z}}\times{\mathbb{Z}})_{\alpha})_{%
\beta})_{\tau}\;;$$
however, this construction employs needless intermediate procedures as
described above. Our approach is to follow instead the “skip the
rationals” blueprint
$$\text{\rm I\!I\!R}:=({\mathbb{N}}\to{\mathbb{Z}}^{\mathbb{N}})_{\sigma\tau}$$
(4.4)
where the image of each $a\in{\mathbb{N}}$ is the sequence $u^{a}\in{\mathbb{Z}}^{\mathbb{N}}$ with
general term $u_{n}^{a}$, so that $u^{a}=\langle u^{a}_{n}:n\in{\mathbb{N}}\rangle$.
Thus a general element of IIR is generated (represented) by the
sequence
$$\left\langle a\mapsto\left(n\mapsto u^{a}_{n}\right):a\in{\mathbb{N}}\right\rangle.$$
(4.5)
Here one requires that for each fixed element $n_{0}\in{\mathbb{N}}$ of the
exponent copy of ${\mathbb{N}}$, the map
$$a\mapsto u_{n_{0}}^{a}$$
is an almost homomorphism (space direction), while $\tau$
in (4.4) alludes to the ultrapower quotient in the
time-direction $n$. For instance, we can use almost homomorphisms of
type (3.4) with $r=\frac{1}{n}$. Then the sequence
$$\left\langle a\mapsto\left(n\mapsto\left\lfloor\tfrac{a}{n}\right\rfloor\right%
):a\in{\mathbb{N}}\right\rangle$$
(4.6)
generates an infinitesimal in IIR since the almost homomorphisms
are “getting flatter and flatter” for large time $n$.
Theorem 4.1.
Relative to the construction (4.4), we have a natural
inclusion ${\mathbb{R}}\subset\hbox{\rm\text{\rm I\!I\!R}}$. Furthermore, IIR
is isomorphic to the model ${}^{*}{\mathbb{R}}$ of the hyperreals obtained by
quotienting ${\mathbb{R}}^{\mathbb{N}}$ by the chosen ultrafilter, as in (4.3).
Proof.
Given a real number $r\in{\mathbb{R}}$, we choose the constant sequence given
by $u_{n}^{a}=\lfloor ra\rfloor$ (the sequence is constant in time $n$).
Sending $r$ to the element of IIR defined by the sequence
$$\left\langle a\mapsto\left(n\mapsto\left\lfloor ra\right\rfloor\right):a\in{%
\mathbb{N}}\right\rangle$$
yields the required inclusion ${\mathbb{R}}\hookrightarrow\text{\rm I\!I\!R}$. The
isomorphism $\text{\rm I\!I\!R}\to\,^{*}{\mathbb{R}}$ is obtained by letting
$$U_{n}=\lim\frac{u_{n}^{a}}{a}$$
for each $n\in{\mathbb{N}}$, and sending the element of IIR represented
by (4.5) to the hyperreal represented by the sequence $\langle U_{n}:n\in{\mathbb{N}}\rangle$.
∎
Denoting by $\Delta x$ the infinitesimal generated by the integer
object (4.6), we can then define the derivative of $y=f(x)$
at $x$ following Robinson as the real number $f^{\prime}(x)$ infinitely close
(or, in Fermat’s terminology, adequal)999See A. Weil [50, p. 1146].
to the infinitesimal ratio $\frac{\Delta y}{\Delta x}\in\text{\rm I\!I\!R}$.
Applications of infinitesimal-enriched continua range from aid in
teaching calculus ([18], [27], [28], [37],
[43]) to the Bolzmann equation (see L. Arkeryd [4, 5]); modeling of timed systems in computer science (see H. Rust
[44]); mathematical economics (see R. Anderson [3]);
mathematical physics (see Albeverio et al. [2]); etc.
A comprehensive re-appraisal of the historical antecedents of modern
infinitesimals has been undertaken in recent work by Błaszczyk et
al. [11], Borovik et al. [12], Bråting [13],
Kanovei [24], Katz & Katz [31, 29, 32, 30],
Katz & Leichtnam [33], Katz and Sherry [35, 36], and
others. A construction of infinitesimals by “splitting” Cantor’s
construction of the reals is presented in Giordano et al. [20].
5. Formalization
In this and the next sections we formalize and generalize the
arguments in the previous sections. We show that by a one-step
construction from ${\mathbb{Z}}$-valued functions we can obtain any given
(set) hyperreal field. We can even obtain a universal hyperreal field
which contains an isomorphic copy of every hyperreal field,
by a one-step construction from ${\mathbb{Z}}$-valued functions.
We assume that the reader is familiar with some basic concepts of
model theory. Consult [14] or [38] for concepts and
notations undefined here.
Let $\mathfrak{A}$ and $\mathfrak{B}$ be two models in a language
$\mathcal{L}$ with base sets $A$ and $B$, respectively. The model
$\mathfrak{B}$ is called an $\mathcal{L}$-elementary extension of the
model $\mathfrak{A}$, or $\mathfrak{A}$ is an $\mathcal{L}$-elementary
submodel of $\mathfrak{B}$, if there is an embedding $e:A\rightarrow B$, called an $\mathcal{L}$-elementary embedding, such that for any
first-order $\mathcal{L}$-sentence $\varphi(a_{1},a_{2},\ldots,a_{n})$ with
parameters $a_{1},a_{2},\ldots,a_{n}\in A$, $\varphi(a_{1},a_{2},\ldots,a_{n})$ is
true in $\mathfrak{A}$ if and only if
$\varphi(e(a_{1}),e(a_{2}),\ldots,e(a_{n}))$ is true in $\mathfrak{B}$.
Let $\mathfrak{A}$ and $\mathfrak{B}$ be two models in language $\mathcal{L}$
with base sets $A$ and $B$, respectively. Let
$$\mathcal{L}^{\prime}=\mathcal{L}\cup\{P_{R}:\exists m\in{\mathbb{N}},R%
\subseteq A^{m}\},$$
i.e., $\mathcal{L}^{\prime}$ is formed by adding to $\mathcal{L}$ an
$m$-dimensional relational symbol $P_{R}$ for each $m$-dimensional
relation $R$ on $A$ for any positive integer $m$. Let $\mathfrak{A}^{\prime}$ be
the natural $\mathcal{L}^{\prime}$-model with base set $A$, i.e., the
interpretation of $P_{R}$ in $\mathfrak{A}^{\prime}$ for each $R\subseteq A^{m}$ is
$R$. The model $\mathfrak{B}$ is called a complete elementary
extension of $\mathfrak{A}$ if $\mathfrak{B}$ can be expanded to an
$\mathcal{L}^{\prime}$-model $\mathfrak{B}^{\prime}$ with base set $B$ such that
$\mathfrak{B}^{\prime}$ is an $\mathcal{L}^{\prime}$-elementary extension of $\mathfrak{A}^{\prime}$.
It is a well-known fact that if $\mathfrak{B}$ is an ultrapower of
$\mathfrak{A}$ or a limit ultrapower of $\mathfrak{A}$,
then $\mathfrak{B}$ is a complete elementary extension of $\mathfrak{A}$.
In this section we always view the set ${\mathbb{R}}$ as the set of all Eudoxus
reals.
An ordered field is called a hyperreal field if it is a proper
complete elementary extension of ${\mathbb{R}}$. Let
$$\mathcal{L}^{\prime}=\{+,\cdot,\leqslant,0,1,P_{R}\}_{R\in\mathcal{R}}$$
where $\mathcal{R}$ is the collection of all finite-dimensional
relations on ${\mathbb{R}}$. We do not distinguish between ${\mathbb{R}}$ and the
$\mathcal{R}^{\prime}$-model
$\mathfrak{R}=({\mathbb{R}};+,\cdot,\leqslant,0,1,R)_{R\in\mathcal{R}}$. By
saying that ${}^{*}{\mathbb{R}}$ is a hyperreal field we will sometimes mean that
${}^{*}{\mathbb{R}}$ is the base set of the hyperreal field, but at other times we
mean that ${}^{*}{\mathbb{R}}$ is the hyperreal field viewed as an
$\mathcal{L}^{\prime}$-model. We will spell out the distinction when it
becomes necessary.
Recall that $S$ is the set of all bounded functions from ${\mathbb{Z}}$ to ${\mathbb{Z}}$.
For a pair of almost homomorphisms $f,g:{\mathbb{Z}}\rightarrow{\mathbb{Z}}$, we will
write $f\sim_{\sigma}g$ if and only if $f-g\in S$. Let $I$ be an
infinite set. If $F(x,y)$ is a two-variable function from ${\mathbb{Z}}\times I$ to ${\mathbb{Z}}$ and $i$ is a fixed element in $I$, we write $F(x,i)$ for
the one-variable function $F_{i}(x)=F(x,i)$ from ${\mathbb{Z}}$ to ${\mathbb{Z}}$.
Definition 5.1.
Let $I$ be any infinite set. We set
$$\displaystyle{\mathcal{A}}({\mathbb{Z}}\times I,{\mathbb{Z}})=$$
$$\displaystyle=\{F\in{\mathbb{Z}}^{{\mathbb{Z}}\times I}:\forall i\in I,\,F(x,i%
)\,\mbox{ is an almost homomorphism.}\}$$
Let $\mathcal{U}$ be a fixed non-principal ultrafilter on $I$. For a
pair of functions $f,g:I\rightarrow J$ for some set $J$, we set
$$f\sim_{\tau}g\,\mbox{ if and only if }\,\{i\in I:f(i)=g(i)\}\in\mathcal{U}.$$
Let $[f]_{\tau}=\{g\in I^{J}:g\sim_{\tau}f\}$.
Definition 5.2.
For any $F,G\in{\mathcal{A}}({\mathbb{Z}}\times I,{\mathbb{Z}})$ we will write
$$F\sim_{\sigma\tau}G\,\mbox{ if and only if }\,\{i\in I:F(x,i)\sim_{\sigma}G(x,%
i)\}\in\mathcal{U}.$$
It is easy to check that $\sim_{\sigma\tau}$ is an equivalence
relation on ${\mathcal{A}}({\mathbb{Z}}\times I,{\mathbb{Z}})$. For each $F\in{\mathcal{A}}({\mathbb{Z}}\times I,{\mathbb{Z}})$ let
$$[F]_{\sigma\tau}=\{G\in{\mathcal{A}}({\mathbb{Z}}\times I,{\mathbb{Z}}):G\sim_%
{\sigma\tau}F\}.$$
For each $F(x,y)\in{\mathcal{A}}({\mathbb{Z}}\times I,{\mathbb{Z}})$ we can consider
$[F(\cdot,y)]_{\sigma}$ as a function of $y$ from $I$ to ${\mathbb{R}}$. Thus
the map
$$\Phi:{\mathcal{A}}({\mathbb{Z}}\times I,{\mathbb{Z}})/\!\sim_{\sigma\tau}%
\rightarrow{\mathbb{R}}^{I}/\mathcal{U}$$
such that $\Phi([F]_{\sigma\tau})=[[F]_{\sigma}]_{\tau}$ is an
isomorphism from ${\mathcal{A}}({\mathbb{Z}}\times I,{\mathbb{Z}})/\!\sim_{\sigma\tau}$ to
${\mathbb{R}}^{I}/\mathcal{U}$. Hence ${\mathcal{A}}({\mathbb{Z}}\times I,{\mathbb{Z}})/\!\sim_{\sigma\tau}$ can
be viewed as an ultrapower of ${\mathbb{R}}$. Therefore, the quotient
$$\text{\rm I\!I\!R}_{I}={\mathcal{A}}({\mathbb{Z}}\times I,{\mathbb{Z}})/\!\sim%
_{\sigma\tau}$$
is a hyperreal field constructed in one step from the set of
${\mathbb{Z}}$-valued functions ${\mathcal{A}}({\mathbb{Z}}\times I,{\mathbb{Z}})$.
If the set $I$ is ${\mathbb{N}}$, then ${\mathcal{A}}({\mathbb{Z}}\times{\mathbb{N}},{\mathbb{Z}})/\!\sim_{\sigma\tau}$
is exactly the hyperreal field IIR mentioned in the previous
sections. Since $I$ can be any infinite set, we can construct a
hyperreal field of arbitrarily large cardinality in one step from a
set of ${\mathbb{Z}}$-valued functions ${\mathcal{A}}({\mathbb{Z}}\times I,{\mathbb{Z}})$.
6. Limit ultrapowers and definable ultrapowers
If we consider a limit ultrapower instead of an ultrapower, we can
obtain any (set) hyperreal field by a one-step construction from a set
of ${\mathbb{Z}}$-valued functions ${\mathcal{A}}({\mathbb{Z}}\times I,{\mathbb{Z}})|\mathcal{G}$. The reader
could consult (Keisler [38]) for the notations, definitions, and
basic facts about limit ultrapowers. The main fact we need here is
the following theorem (see Keisler [38, Theorem 3.7]).
Theorem 6.1.
If $\mathfrak{A}$ and $\mathfrak{B}$ are two models of the same
language, then $\mathfrak{B}$ is a complete elementary extension of
$\mathfrak{A}$ if and only if $\mathfrak{B}$ is a limit ultrapower of
$\mathfrak{A}$.
Given any (set) hyperreal field ${}^{*}{\mathbb{R}}$, let $\mathcal{U}$ be the
ultrafilter on an infinite set $I$ and let $\mathcal{G}$ be the filter
on $I\times I$ such that ${}^{*}{\mathbb{R}}$ is isomorphic to the limit ultrapower
$({\mathbb{R}}^{I}/\mathcal{U})|\mathcal{G}$. We can describe the limit ultrapower
$({\mathbb{R}}^{I}/\mathcal{U})|\mathcal{G}$ in one step from the set of
${\mathbb{Z}}$-valued functions ${\mathcal{A}}({\mathbb{Z}}\times I,{\mathbb{Z}})|\mathcal{G}$. For each
$F\in{\mathcal{A}}({\mathbb{Z}}\times I,{\mathbb{Z}})$, let
$$eq(F)=\{(i,j)\in I\times I:[F(x,i)]_{\sigma}=[F(x,j)]_{\sigma}\}.$$
Let
$${\mathcal{A}}({\mathbb{Z}}\times I,{\mathbb{Z}})|\mathcal{G}=\{F\in{\mathcal{A%
}}({\mathbb{Z}}\times I,{\mathbb{Z}}):eq(F)\in\mathcal{G}\}.$$
Notice that ${\mathcal{A}}({\mathbb{Z}}\times I,{\mathbb{Z}})|\mathcal{G}$ is a subset of ${\mathcal{A}}({\mathbb{Z}}\times I,{\mathbb{Z}})$. Hence
$({\mathcal{A}}({\mathbb{Z}}\times I,{\mathbb{Z}})|\mathcal{G})/\!\sim_{\sigma\tau}$ can be viewed as
a subset of ${\mathcal{A}}({\mathbb{Z}}\times I,{\mathbb{Z}})/\!\sim_{\sigma\tau}$. Again for each
$$F\in{\mathcal{A}}({\mathbb{Z}}\times I,{\mathbb{Z}})/\!\sim_{\sigma\tau}$$
let
$$\Phi([F]_{\sigma\tau})=[[F]_{\sigma}]_{\tau}.$$
Then $\Phi$ is an isomorphism from
$$({\mathcal{A}}({\mathbb{Z}}\times I,{\mathbb{Z}})|\mathcal{G})/\!\sim_{\sigma\tau}$$
to $({\mathbb{R}}^{I}/\mathcal{U})|\mathcal{G}$. Therefore, $({\mathcal{A}}({\mathbb{Z}}\times I,{\mathbb{Z}})|\mathcal{G})/\!\sim_{\sigma\tau}$ as an elementary subfield of
${\mathcal{A}}({\mathbb{Z}}\times I,{\mathbb{Z}})/\!\sim_{\sigma\tau}$ is isomorphic to the hyperreal
field ${}^{*}{\mathbb{R}}$.
Theorem 6.2.
An isomorphic copy of any (set) hyperreal field ${}^{*}{\mathbb{R}}$ can be obtained
by a one-step construction from a set of ${\mathbb{Z}}$-valued functions
${\mathcal{A}}({\mathbb{Z}}\times I,{\mathbb{Z}})|\mathcal{G}$ for some filter $\mathcal{G}$ on
$I\times I$.
7. Universal and On-saturated Hyperreal Number Systems
We call a hyperreal field IIR universal if any hyperreal
field, which is a set or a proper class of NBG, can be
elementarily embedded in IIR. Obviously a universal hyperreal
field is necessarily a proper class. We now want to construct a
definable hyperreal field with the property that any definable
hyperreal field that can be obtained in NBG by a one-step
construction from a collection of ${\mathbb{Z}}$-valued functions can be
elementarily embedded in it. In a subsequent remark we point out that
in NBG we can actually construct a definable hyperreal field so
that every hyperreal field (definable or non-definable) can be
elementarily embedded in it. Moreover, the universal hyperreal field
so constructed is isomorphic to the On-saturated hyperreal
field described in [17].
Notice that NBG implies that there is a well order $\leqslant_{V}$
on $V$ where $V$ is the class of all sets. A class $X\subseteq V$ is
called $\Delta_{0}$-definable if there is a first-order formula
$\varphi(x)$ with set parameters in the language $\{\in,\leqslant\}$
such that for any set $a\in V$, $a\in X$ if and only if $\varphi(a)$
is true in $V$. Trivially, every set is $\Delta_{0}$-definable. We
work within a model of NBG with set universe $V$ plus all
$\Delta_{0}$-definable proper subclasses of $V$. By saying that a class
$A$ is definable we mean that $A$ is $\Delta_{0}$-definable.
Let $\Sigma$ be the class of all finite sets of ordinals, i.e.,
$\Sigma=On^{<\omega}$. Notice that $\Sigma$ is a definable proper class.
Let $\mathcal{P}$ be the
collection of all definable subclasses of $\Sigma$. Notice
that we can code $\mathcal{P}$ by a definable class.101010 This is true because each definable subclass of $\Sigma$
can be effectively coded by the Gödel number of a first-order
formula in the language of $\{\in,\leqslant_{V}\}$ and a set in $V$. By
the well order of $V$ we can determine a unique code for each
definable class in $\mathcal{P}$.
Using the global choice, we can form a non-principal definable
ultrafilter $\mathcal{F}\subseteq\mathcal{P}$ such that for each
$\alpha\in On$, the definable class
$$\hat{\alpha}=\{s\in\Sigma:\alpha\in s\}$$
is in $\mathcal{F}$. Again $\mathcal{F}$ can be coded by a definable
class. Let ${\mathcal{A}}_{0}({\mathbb{Z}}\times\Sigma,{\mathbb{Z}})$ be the collection of all
definable class functions $F$ from ${\mathbb{Z}}\times\Sigma$ to ${\mathbb{Z}}$ such that
for each $s\in\Sigma$, $F(x,s)$ is an almost homomorphism from ${\mathbb{Z}}$ to
${\mathbb{Z}}$. For any two functions $F$ and $G$ in ${\mathcal{A}}_{0}({\mathbb{Z}}\times\Sigma,{\mathbb{Z}})$,
we write $F\sim_{\sigma\tau}G$ if and only if the definable class
$$\{s\in\Sigma:F(x,s)-G(x,s)\in S\}$$
is in $\mathcal{F}$. Let $[F]_{\sigma\tau}$ be the collection of all
definable classes $G$ in ${\mathcal{A}}_{0}({\mathbb{Z}}\times\Sigma,{\mathbb{Z}})$ such that
$F\sim_{\sigma\tau}G$. Then $\sim_{\sigma\tau}$ is an equivalence
relation on ${\mathcal{A}}_{0}({\mathbb{Z}}\times\Sigma,{\mathbb{Z}})$. Let
$$\text{\rm I\!I\!R}_{0}={\mathcal{A}}_{0}({\mathbb{Z}}\times\Sigma,{\mathbb{Z}}%
)/\!\sim_{\sigma\tau}.$$
By the arguments employed before, we can show that $\text{\rm I\!I\!R}_{0}$ is
isomorphic to the definable ultrapower of ${\mathbb{R}}$ modulo
$\mathcal{F}$. Hence $\text{\rm I\!I\!R}_{0}$ is a complete elementary extension of
${\mathbb{R}}$. Therefore, $\text{\rm I\!I\!R}_{0}$ is a hyperreal field. By slightly modifying
the proof of [14, Theorem 4.3.12, p. 255] we can prove the
following theorem.
Theorem 7.1.
$\text{\rm I\!I\!R}_{0}$ is a class hyperreal field, and any definable hyperreal field
${}^{*}{\mathbb{R}}$ admits an elementary imbedding into $\text{\rm I\!I\!R}_{0}$.
Proof.
We only need to prove the second part of the theorem. For notational
convenience we view $\text{\rm I\!I\!R}_{0}$ as ${\mathbb{R}}^{\Sigma}/\mathcal{F}$ instead of
${\mathcal{A}}_{0}({\mathbb{Z}}\times\Sigma,{\mathbb{Z}})/\sim_{\sigma\tau}$ in this proof.
Given a definable hyperreal field ${}^{*}{\mathbb{R}}$, recall that $\mathcal{L}$ is
the language of ordered fields, and
$$\mathcal{L}^{\prime}=\mathcal{L}\cup\{P_{R}:R\,\mbox{ is a finite-dimensional
relation on }\,{\mathbb{R}}\}.$$
Let $\Lambda_{{}^{*}{\mathbb{R}}}$ be all quantifier-free $\mathcal{L}^{\prime}$-sentences
$\varphi(r_{1},r_{2},\ldots,r_{m})$ with parameters $r_{i}\in\,^{*}{\mathbb{R}}$ such that
$\varphi(r_{1},r_{2},\ldots,r_{m})$ is true in ${}^{*}{\mathbb{R}}$. Since ${}^{*}{\mathbb{R}}$ is a
definable class, $\Lambda_{{}^{*}{\mathbb{R}}}$ is a definable class (under a proper
coding). Let $\kappa$ be the size of $\Lambda_{{}^{*}{\mathbb{R}}}$, i.e., $\kappa$
is the cardinality of ${}^{*}{\mathbb{R}}$ if ${}^{*}{\mathbb{R}}$ is a set and $\kappa=On$ if
${}^{*}{\mathbb{R}}$ is a definable proper class. Let $j$ be a definable bijection
from $\kappa$ to $\Lambda_{{}^{*}{\mathbb{R}}}$.
For each $r\in\,^{*}{\mathbb{R}}$ we need to find a definable function
$F_{r}:\Sigma\rightarrow{\mathbb{R}}$ such that the map $r\mapsto[F_{r}]_{\tau}$ is an
$\mathcal{L}^{\prime}$-elementary embedding. We define these $F_{r}$ simultaneously.
Let $s\in\Sigma$. If $s\cap\kappa=\emptyset$ let $F_{r}(s)=0$. Suppose
$s\cap\kappa\not=\emptyset$ and let $s^{\prime}=s\cap\kappa$. Notice that if
$\kappa=On$, then $s=s^{\prime}$. Let
$\varphi_{s}(r_{1},r_{2},\ldots,r_{m})=\bigwedge_{\alpha\in s^{\prime}}j(\alpha)$.
Since
$$\exists x_{1},x_{2},\ldots,x_{m}\varphi_{s}(x_{1},x_{2},\ldots,x_{m})$$
is
true in ${}^{*}{\mathbb{R}}$, it is also true in ${\mathbb{R}}$. Let
$(a_{1},a_{2},\ldots,a_{m})\in{\mathbb{R}}^{m}$ be the $\leqslant_{V}$-least $m$-tuple
such that $\varphi_{s}(a_{1},a_{2},\ldots,a_{m})$ is true in ${\mathbb{R}}$. If
$r\not\in\{r_{1},r_{2},\ldots,r_{m}\}$, let $F_{r}(s)=0$. If $r=r_{i}$ for
$i=1,2,\ldots,m$, then let $F_{r}(s)=a_{i}$. Since $\varphi_{s}$ is
quantifier-free, the functions $F_{r}$ are definable classes in NBG.
We now verify that $\Phi:\,^{*}{\mathbb{R}}\rightarrow\text{\rm I\!I\!R}_{0}$ such that
$\Phi(r)=[F_{r}]_{\tau}$ is an $\mathcal{L}^{\prime}$-elementary embedding.
Let $\phi(r_{1},r_{2},\ldots,r_{m})$ be an arbitrary $\mathcal{L}^{\prime}$-sentence
with parameters
$r_{1},r_{2},\ldots,r_{m}\in\,^{*}{\mathbb{R}}$.
Suppose that $\phi(r_{1},r_{2},\ldots,r_{m})$ is true in ${}^{*}{\mathbb{R}}$. Since
$\phi(x_{1},x_{2},\ldots,x_{m})$ defines an $m$-ary relation $R_{\phi}$ on
${\mathbb{R}}$, we have that the $\mathcal{L}^{\prime}$-sentence
$$\eta=:\forall x_{1},x_{2},\ldots,x_{m}(\phi(x_{1},x_{2},\ldots,x_{m})%
\leftrightarrow R_{\phi}(x_{1},x_{2},\ldots,x_{m}))$$
is true in ${\mathbb{R}}$. Hence $\eta$ is true
in ${}^{*}{\mathbb{R}}$ and in $\text{\rm I\!I\!R}_{0}$. One of the consequences of this is that
$R_{\phi}(r_{1},r_{2},\ldots,r_{m})$ is true in ${}^{*}{\mathbb{R}}$, hence it is in
$\Lambda_{{}^{*}{\mathbb{R}}}$. Let $\alpha\in\kappa$ be such that
$j(\alpha)=R_{\phi}(r_{1},r_{2},\ldots,r_{m})$. If $\alpha\in s$, then
$$\varphi_{s}=R_{\phi}(r_{1},r_{2},\ldots,r_{m})\wedge\bigwedge_{\beta\in s^{%
\prime},\beta\not=\alpha}j(\beta).$$
Hence
$R_{\phi}(F_{r_{1}}(s),F_{r_{2}}(s),\ldots,F_{r_{m}}(s))$ is true in ${\mathbb{R}}$ by
the definition of the $F_{r}$’s. Since $\eta$ is true in ${\mathbb{R}}$, we have
that $\phi(F_{r_{1}}(s),F_{r_{2}}(s),\ldots,F_{r_{m}}(s))$ is true in
${\mathbb{R}}$. Thus
$$\{s\in\Sigma:\phi(F_{r_{1}}(s),F_{r_{2}}(s),\ldots,F_{r_{m}}(s))\,\mbox{
is true in }\,{\mathbb{R}}\}\supseteq\hat{\alpha}.$$
Since $\hat{\alpha}$ is a
member of $\mathcal{F}$, we have that
$\phi([F_{r_{1}}]_{\tau},[F_{r_{2}}]_{\tau},\ldots,[F_{r_{m}}]_{\tau})$ is
true in ${\mathbb{R}}^{\Sigma}/\mathcal{F}$.
Suppose that $\phi(r_{1},r_{2},\ldots,r_{m})$ is false in ${}^{*}{\mathbb{R}}$. Then
$\neg\phi(r_{1},r_{2},\ldots,r_{m})$ is true in ${}^{*}{\mathbb{R}}$. Hence by the same
argument we have that
$$\neg\phi([F_{r_{1}}]_{\tau},[F_{r_{2}}]_{\tau},\ldots,[F_{r_{m}}]_{\tau})\,%
\mbox{
is true in }\,{\mathbb{R}}^{\Sigma}/\mathcal{F}$$
which implies that
$$\phi([F_{r_{1}}]_{\tau},[F_{r_{2}}]_{\tau},\ldots,[F_{r_{m}}]_{\tau})\,\mbox{
is false in }\,{\mathbb{R}}^{\Sigma}/\mathcal{F}.$$
Hence $\Phi(r)=[F_{r}]_{\tau}$
is an $\mathcal{L}^{\prime}$-elementary embedding from ${}^{*}{\mathbb{R}}$ to $\text{\rm I\!I\!R}_{0}$.
∎
Remark 7.2.
We have shown that every definable hyperreal field can be elementarily
embedded into $\text{\rm I\!I\!R}_{0}$. If we want to show that $\text{\rm I\!I\!R}_{0}$ is
universal, we need to elementarily embed every (definable or
non-definable) hyperreal field ${}^{*}{\mathbb{R}}$ into the definable hyperreal
field $\text{\rm I\!I\!R}_{0}$. Notice that there are models of NBG with
non-definable classes. The proof of Theorem 7.1 may not
work when ${}^{*}{\mathbb{R}}$ is a non-definable class because the bijection $j$
may not be definable and $F_{r}$ may not be definable. If $F_{r}$ is not
definable, $[F_{r}]_{\tau}$ may not be an element in $\text{\rm I\!I\!R}_{0}$.
The idea of making every hyperreal field embeddable into $\text{\rm I\!I\!R}_{0}$ is
that we can make $\text{\rm I\!I\!R}_{0}$ On-saturated by selecting a definable
ultrafilter $\mathcal{F}$ more carefully. Notice that NBG
implies that every proper class has the same size On. Hence
${}^{*}{\mathbb{R}}$ can be expressed as the union of On-many sets. If we can
make sure that $\text{\rm I\!I\!R}_{0}$ is On-saturated, i.e.,
$\alpha$-saturated for any set cardinality $\alpha$, then ${}^{*}{\mathbb{R}}$ can
be elementarily embedded into $\text{\rm I\!I\!R}_{0}$ although such an embedding may
be non-definable.
The ultrafilter $\mathcal{F}$ used in the construction of $\text{\rm I\!I\!R}_{0}$ in
the proof of Theorem 7.1 is a definable version of a regular ultrafilter. In order to make sure that $\text{\rm I\!I\!R}_{0}$ is
On-saturated, we need to require that $\mathcal{F}$ be a
special kind of definable regular ultrafilter called a definable good ultrafilter. The definition of a (set) good ultrafilter can be
found in [14, p. 386]. The construction of an $\alpha^{+}$-good
ultrafilter can be found in either [14, Theorem 6.1.4] or Kunen
[39]. By the same idea of constructing $\mathcal{F}$ above we
can follow the steps in [39] or [14] to construct a
definable class good ultrafilter $\mathcal{F}$ on On. Now the
ultrapower $\text{\rm I\!I\!R}_{0}$ of ${\mathbb{R}}$ modulo the definable class good
ultrafilter $\mathcal{F}$ is On-saturated. The proof of this
fact is similar to that in [14]. However, since the definition
of a definable class good ultrafilter and the proof of the saturation
property of the ultrapower modulo a definable class good ultrafilter
are long and technical, and the ideas are similar to what we have
already presented above, we will not include them in this paper.
Another way of constructing a definable On-saturated hyperreal
field $\text{\rm I\!I\!R}_{0}$ is by taking the union of an On-long definable
elementary chain of set hyperreal fields $\left\{{}^{*}{\mathbb{R}}_{\alpha}:\alpha\in On\right\}$ with the property that ${}^{*}{\mathbb{R}}_{\alpha}$ is
$|\alpha|$-saturated. However, this construction cannot be easily
translated into a “one-step” construction. Moreover, if we allow
higher-order classes, we can express $\text{\rm I\!I\!R}_{0}$ as a one-step limit
ultrapower following the same idea as in the proof of [14, Theorem
6.4.10]. However, this is not possible in NBG since all
classes allowed in a model of NBG are subclasses of $V$.
On the other hand, as we indicated above, the process of constructing
$\text{\rm I\!I\!R}_{0}$ as a definable ultrapower can be carried out in NBG,
and done so in a “one-step” fashion.
Acknowledgments
We are grateful to R. Arthan, P. Ehrlich, and V. Kanovei for helpful
comments.
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Almost partitioning 2-coloured complete 3-uniform hypergraphs into two monochromatic tight or loose cycles
Sebastián Bustamante${}^{,}$
All three authors acknowledge support by Millenium Nucleus Information and Coordination in Networks ICM/FIC RC130003.The first author was supported by CONICYT Doctoral Fellowship grant 21141116.
Department of Mathematical Engineering, University of Chile
Hiệp Hàn${}^{*,}$${}^{,}$
The second author was supported by Fondecyt Iniciación grant 11150913.
Instituto de Matemáticas, Pontificia Universidad Católica de Valparaíso
Maya Stein${}^{*,}$${}^{,}$
The third author received support by Fondecyt Regular grant 1140766.
Department of Mathematical Engineering, University of Chile
Abstract
We show that for every $\eta>0$ there exists an integer $n_{0}$ such that every $2$-colouring of the $3$-uniform complete hypergraph on $n\geq n_{0}$ vertices contains two disjoint monochromatic tight cycles of distinct colours that together cover
all but at most ${\eta n}$ vertices. The same result holds if we replace tight cycles with loose cycles.
1 Introduction
A certain type of Ramsey problems concerned with covering all vertices of the host graph instead of finding a small subgraph has gained popularity in recent years, both for graphs and hypergraphs. In particular, the problem of partitioning an edge-coloured complete (hyper-)graph into monochromatic cycles has received much attention. The recent surveys [5, 6] are a good starting point for the vast literature on the subject.
Central to the area is an old conjecture of Lehel for graphs (see [2]). It states that for every $n$, every two-colouring of the edges of the complete graph $K_{n}$ admits a
covering of the vertices of $K_{n}$ with at most two monochromatic vertex disjoint cycles of different colours. (For technical reasons, a single vertex or an edge count as a cycle.) This was confirmed for large $n$ in [13, 1], and for all $n$ by Bessy and Thomassé [3].
There is plenty of activity in determing the number of monochromatic cycles needed if $K_{n}$ is coloured with more than two colours. It is known that this number is independent of $n$, but the best known lower and upper bounds, $r+1$ and $100r\log r$, respectively, leave a considerable gap [15, 7].
For hypergraphs, much less is known. The problems transforms in the obvious way to hypergraphs, considering $r$-edge-colourings of the $k$-uniform $\mathcal{K}_{n}^{(k)}$ on $n$ vertices, the only question is which type of hypergraph cycles we would like to work with. Referring to [5] for other results, we concentrate here on loose and tight cycles (see the next section for their definitions).
For loose cycles the problem was studied in [8, 17] and the best bound, due to Sárközy [17], shows that at most $50rk\log(rk)$ disjoint loose monochromatic cycles are needed for partitioning $\mathcal{K}_{n}^{(k)}$, a number that this is independent of $n$.
In general even for the case $k=3$ the problem is wide open and it is for example unknown whether one can cover almost all vertices with two disjoint monochromatic loose cycles.
Concerning the more restrictive notion of tight cycles, the situation is even worse and to our best knowledge, nothing is known. Gyárfás [6] conjectures that there is $c=c(r,k)$ such that every $r$-coloured $\mathcal{K}_{n}^{(k)}$ has a partition into at most $c$ monochromatic tight cycles, but this is open even for the ‘easiest’ case of $3$-uniform hypergraphs and two colours.
Our main result establishes an approximate version for this case.
Theorem 1.1.
For every $\eta>0$ there exists $n_{0}$ such that if $n\geq n_{0}$ then every two-coloring of the edges of the complete 3-uniform hypergraph $\mathcal{K}_{n}^{(3)}$ admits two vertex-disjoint monochromatic tight cycles, of distinct colours,
which cover all but at most $\eta n$ vertices.
Moreover, we can choose the parity of the length of each of the cycles.
We might be interested in choosing the parity of the cycles for the following reason. If $\ell$ is even, then
any 3-uniform tight cycle on $\ell$ edges contains a loose cycle.
Hence, we can deduce that an analogue of Theorem 1.1 holds for loose cycles.
Corollary 1.2.
For every $\eta>0$ there exists $n_{0}$ such that if $n\geq n_{0}$ then every two-coloring of the edges of the complete 3-uniform hypergraph $\mathcal{K}_{n}^{(3)}$ admits two vertex-disjoint monochromatic loose cycles, of distinct colours,
which cover all but at most $\eta n$ vertices.
We believe that the error term $\eta n$ in Theorem 1.1 can be improved and that
every two-colouring of the edges of $\mathcal{K}_{n}^{(3)}$ admits two disjoint monochromatic tight cycles which cover all but at most a constant number $c$ of vertices (for some $c$ independent of $n$).
The proof of Theorem 1.1 is inspired by the work of Haxell et al. [10, 11] and relies on an application
of the hypergraph regularity lemma [4]. This reduces the problem at hand to that of finding, in any two-colouring of the edges of an almost complete 3-uniform hypergraph,
two disjoint monochromatic connected matchings which cover almost all vertices.
Here, as usual, a matching $\mathcal{M}$ in hypergraph $\mathcal{H}$ is a set of pairwise disjoint edges and $\mathcal{M}\subset\mathcal{H}$ is called connected
if between every pair $e,f\in\mathcal{M}$ there is a pseudo-path in $\mathcal{H}$ connecting $e$ and $f$, that is, there
is a sequence $(e_{1},\dots,e_{p})$ of not necessarily distinct edges of $\mathcal{H}$ such that $e=e_{1},f=e_{p}$ and $|e_{i}\cap e_{i+1}|=2$ for each $i\in[p-1]$.
(Note that these pseudo-paths may use vertices outside $V(\mathcal{M})$.) Now, we call a connected matching $\mathcal{M}$ in a $2$-coloured hypergraph a monochromatic connected matching if all edges in $\mathcal{M}$ and all edges on the connecting paths have the same colour.
So, our main contribution reduces to the following result, which might be of independent interest.
Theorem 1.3.
For every $\gamma>0$ there is $t_{0}$ such that the following holds for every $3$-uniform hypergraph $\mathcal{H}$ with $t>t_{0}$ vertices and
$(1-\gamma)\binom{t}{3}$ edges. Any two-colouring of the edges of $\mathcal{H}$ admits two disjoint monochromatic connected matchings covering at least $(1-240\gamma^{\frac{1}{6}})t$ vertices of $\mathcal{H}$.
We prove Theorem 1.3 in Section 2.
In Section 3, we introduce the regularity lemma for hypergraphs and state an embedding result from [11]. The proof of
Theorem 1.1 will be given in Section 4.
2 Monochromatic connected matchings
Before giving the proof of Theorem 1.3 we introduce some notation and auxiliary results.
Let $\mathcal{H}$ denote a $k$-uniform hypergraph, that is, a pair $\mathcal{H}=(V,E)$ with finite vertex set $V=V(\mathcal{H})$ and edge set $E=E(\mathcal{H})\subset\binom{V}{k}$, where $\binom{V}{k}$ denotes the set
of all $k$-element sets of $V$. Often $\mathcal{H}$ will be identified with its edges, that is, $\mathcal{H}\subset\binom{V}{k}$ and for an edge $\{x_{1},\dots,x_{k}\}\in\mathcal{H}$ we often omit brackets and write $x_{1}\dots x_{k}$ only.
A $k$-uniform hypergraph $\mathcal{C}$ is called an $\ell$-cycle
if there is a cyclic ordering of the vertices of $\mathcal{C}$ such that every edge consists of $k$ consecutive vertices, every vertex is contained in an edge and two consecutive edges
(where the ordering of the edges is inherited by the ordering of the vertices) intersect in exactly $\ell$ vertices. For $\ell=1$ we call the cycle loose whereas the cycle is called tight if $\ell=k-1$ (and we do not consider other values of $\ell$).
A tight path is a cycle from which one vertex and all incident edges are deleted. The length of a path, a pseudo-path or a cycle is the number of edges it contains.
As above, two edges in $\mathcal{H}$ are connected if there is a pseudo-path connecting them.
Connectedness is an equivalence relation on the edge set of $\mathcal{H}$
and the equivalence classes are called connected components.
All hypergraphs $\mathcal{H}$ considered from now on are $3$-uniform.
We will need the following result concerning the existence of perfect matchings in 3-uniform hypergraphs with high minimum vertex degree.
Theorem 2.1 ([9]).
For all $\eta>0$ there is a $n_{0}=n_{0}(\eta)$ such that for all $n>n_{0}$, $n\in 3\mathbb{Z}$, the following holds.
Suppose $\mathcal{H}$ is $3$-uniform hypergraph on $n$ vertices such that every vertex is contained in
at least $\left(\frac{5}{9}+\eta\right)\binom{n}{2}$ edges. Then $\mathcal{H}$ contains a perfect matching.
Denote by $\partial\mathcal{H}$ the shadow of $\mathcal{H}$, that is, the set of all pairs $xy$ for which there exists $z$ such that $xyz\in\mathcal{H}$. For a vertex $x$ in a hypergraph $\mathcal{H}$, let $N_{\mathcal{H}}(x)=\{y:xy\in\partial\mathcal{H}\}$. For two vertices $x,y$, let $N_{\mathcal{H}}(x,y)=\{z:xyz\in\mathcal{H}\}$. Note that if $y\in N_{\mathcal{H}}(x)$ (equivalently, $x\in N_{\mathcal{H}}(y)$) then $N_{\mathcal{H}}(x,y)\neq\emptyset$. We call all such pairs $xy$ of vertices active.
Lemma 2.2 ([10], Lemma 4.1).
Let $\gamma>0$ and let $\mathcal{H}$ be a $3$-uniform hypergraph on $t_{\mathcal{H}}$ vertices and at least $(1-\gamma)\binom{t}{3}$ edges.
Then $\mathcal{H}$ contains a subhypergraph $\mathcal{K}$ on $t_{\mathcal{K}}\geq(1-10\gamma^{1/6})t_{\mathcal{H}}$ vertices such that every vertex $x$ of $\mathcal{K}$ is in an active pair of $\mathcal{K}$ and for all active pairs $xy$ we have $|N_{\mathcal{K}}(x,y)|\geq(1-10\gamma^{1/6})t_{\mathcal{K}}$.
We are now ready to prove Theorem 1.3.
Proof of Theorem 1.3.
For given $\gamma>0$ let $\delta=10\gamma^{1/6}$ and apply Theorem 2.1 with $\eta=1/9$ to obtain $n_{0}$.
We choose $t_{0}=\max\left\{\frac{2}{\delta},\frac{n_{0}}{27\delta}\right\}$.
Suppose we are given a two-coloured $3$-uniform hypergraph $\mathcal{H}=\mathcal{H}_{\mathrm{red}}\cup\mathcal{H}_{\mathrm{blue}}$ on $t_{\mathcal{H}}>t_{0}$ vertices and
$(1-\gamma)\binom{t_{\mathcal{H}}}{3}$ edges. Apply Lemma 2.2 to $\mathcal{H}$ with parameter $\gamma$ to obtain
$\mathcal{K}$, $t:=t_{\mathcal{K}}$ with the properties stated in the lemma. We wish to find two monochromatic connected matchings covering all but at most $24\delta t\leq 24\delta t_{\mathcal{H}}$ vertices of $\mathcal{K}$.
Since every vertex is in an active pair in $\mathcal{K}$, we have
$$\displaystyle|N_{\mathcal{K}}(x)|\geq(1-\delta)t\quad\text{for all }x\in V(%
\mathcal{K}).$$
(1)
Let $\mathcal{K}=\mathcal{K}_{\mathrm{red}}\cup\mathcal{K}_{\mathrm{blue}}$ be the colouring of $\mathcal{K}$ inherited from $\mathcal{H}$.
Then a monochromatic component $\mathcal{C}$ of $\mathcal{K}$ is a connected component $\mathcal{K}_{\mathrm{red}}$ or $\mathcal{K}_{\mathrm{blue}}$.
Observation 2.3 ([11], Observation 8.2).
For every vertex $x\in V(\mathcal{K})$ there exists a monochromatic component $\mathcal{C}_{x}$ such that $|N_{\mathcal{C}_{x}}(x)|\geq(1-\delta)t.$
For each $x\in V(\mathcal{K})$ choose arbitrarily one component $\mathcal{C}_{x}$ as in Observation 2.3. Let $R=\{x\in V(\mathcal{K}):\mathcal{C}_{x}\text{ is red}\}$ and $B=\{x\in V(\mathcal{K}):\mathcal{C}_{x}\text{ is blue}\}$, and note that these two sets partition $V(\mathcal{K})$.
Observation 2.4 ([11], Observation 8.4).
If $|R|\geq 6\delta t$ (or $|B|\geq 6\delta t$, respectively), then there is a red component $\mathcal{R}$ (a blue component $\mathcal{B}$) such that $\mathcal{C}_{x}=\mathcal{R}$ ($\mathcal{C}_{x}=\mathcal{B}$) for all but at most $2\delta t$ vertices $x\in R$ ($x\in B$).
Set ${V}_{\mathrm{red}}:=\{x\in R:C_{x}=\mathcal{R}\}$ if $|{V}_{\mathrm{red}}|\geq 6\delta t$, and set ${V}_{\mathrm{blue}}:=\{x\in B:C_{x}=\mathcal{B}\}$ if $|B|\geq 6\delta t$. Otherwise let ${V}_{\mathrm{red}}$, or ${V}_{\mathrm{blue}}$, respectively, be the empty set. Our aim is to find two differently coloured disjoint connected matchings in $\mathcal{K}$ that together cover all but $12\delta t\leq 24\delta t-12\delta t$ vertices of ${V}_{\mathrm{red}}\cup{V}_{\mathrm{blue}}$.
We start by choosing a connected matching of maximal size in $\mathcal{R}\cup\mathcal{B}$. This matching decomposes into two disjoint monochromatic connected matchings, $\mathcal{M}_{\mathrm{red}}\subset\mathcal{R}$ and $\mathcal{M}_{\mathrm{blue}}\subset\mathcal{B}$, which together cover as many vertices as possible. Let ${V}_{\mathrm{red}}^{\prime}={V}_{\mathrm{red}}\setminus V(\mathcal{M}_{\mathrm%
{red}}\cup\mathcal{M}_{\mathrm{blue}})$ and ${V}_{\mathrm{blue}}^{\prime}={V}_{\mathrm{blue}}\setminus V(\mathcal{M}_{%
\mathrm{red}}\cup\mathcal{M}_{\mathrm{blue}})$. We may assume that ${V}_{\mathrm{red}}^{\prime}$ or ${V}_{\mathrm{blue}}^{\prime}$ has at least $12\delta t$ vertices, as otherwise we are done. By symmetry we may assume that
$$|{V}_{\mathrm{red}}^{\prime}|\geq 12\delta t.$$
(2)
Observe that there is no edge $xy$ with $x\in{V}_{\mathrm{red}}^{\prime}$ and $y\in{V}_{\mathrm{blue}}^{\prime}$ such that $xy\in\partial\mathcal{R}\cap\partial\mathcal{B}$. Indeed, any such edge $xy$ constitutes an active pair (by Lemma 2.2) and as $|{V}_{\mathrm{red}}^{\prime}|>\delta t+2$, there must be a vertex $z\in{V}_{\mathrm{red}}^{\prime}$ such that $xyz$ is an edge of $\mathcal{K}$. This contradicts the maximality of the matching $\mathcal{M}_{\mathrm{red}}\cup\mathcal{M}_{\mathrm{blue}}$.
Next, we claim that
$$|{V}_{\mathrm{blue}}^{\prime}|\leq 2\delta t.$$
(3)
Assume otherwise. Then, Observation 2.3 and the choice of the set ${V}_{\mathrm{red}}$ implies that the number of edges between ${V}_{\mathrm{red}}^{\prime}$ and ${V}_{\mathrm{blue}}^{\prime}$ that belong to $\partial\mathcal{R}$ is at least
$$|{V}_{\mathrm{red}}^{\prime}|\cdot(|{V}_{\mathrm{blue}}^{\prime}|-\delta t)%
\geq\frac{1}{2}|{V}_{\mathrm{red}}^{\prime}|\cdot|{V}_{\mathrm{blue}}^{\prime}|.$$
Similarly, there are at least $|{V}_{\mathrm{blue}}^{\prime}|\cdot(|{V}_{\mathrm{red}}^{\prime}|-\delta t)>%
\frac{1}{2}|{V}_{\mathrm{red}}^{\prime}|\cdot|{V}_{\mathrm{blue}}^{\prime}|$ edges between ${V}_{\mathrm{red}}^{\prime}$ and ${V}_{\mathrm{blue}}^{\prime}$ that belong to $\partial\mathcal{B}$. As there is no edge $xy$ with $x\in{V}_{\mathrm{red}}^{\prime}$ and $y\in{V}_{\mathrm{blue}}^{\prime}$ such that $xy\in\partial\mathcal{R}\cap\partial\mathcal{B}$, we have more than $|{V}_{\mathrm{red}}^{\prime}|\cdot|{V}_{\mathrm{blue}}^{\prime}|$ edges from ${V}_{\mathrm{red}}^{\prime}$ to ${V}_{\mathrm{blue}}^{\prime}$. This yields a contradiction and (3) follows.
Because of the maximality of $\mathcal{M}_{\mathrm{red}}\cup\mathcal{M}_{\mathrm{blue}}$, each edge having all its vertices in ${V}_{\mathrm{red}}^{\prime}$ is blue. Fix one such edge $xyz$. Obtain ${V}_{\mathrm{red}}^{\prime\prime}$ from ${V}_{\mathrm{red}}^{\prime}$ by deleting the at most $\delta t$ vertices $w$ with $wx\notin\partial\mathcal{R}$. Consider any edge $x^{\prime}y^{\prime}z^{\prime}$ with $x^{\prime},y^{\prime},z^{\prime}\in{V}_{\mathrm{red}}^{\prime\prime}$. As the pairs $xy$, $xx^{\prime}$, $x^{\prime}y^{\prime}$ are all active, and $|{V}_{\mathrm{red}}^{\prime\prime}|>3\delta t$, there is a vertex $v\in{V}_{\mathrm{red}}^{\prime\prime}$ that forms an edge with each of the three pairs, thus giving a pseudo-path in $\mathcal{K}[{V}_{\mathrm{red}}^{\prime\prime}]$ from $xyz$ to $x^{\prime}y^{\prime}z^{\prime}$. Denote by $\mathcal{B}^{\prime\prime}$ the blue component of $\mathcal{K}[{V}_{\mathrm{red}}^{\prime\prime}]$ that contains $xyz$, and let $\mathcal{B}^{\prime}$ be obtained from $\mathcal{B}^{\prime\prime}$ by deleting at most $2$ vertices and all incident edges, so that $|V[\mathcal{B}^{\prime}]|$ is a multiple of $3$. Then, by (2), we have
$$|V[\mathcal{B}^{\prime}]|\geq|{V}_{\mathrm{red}}^{\prime}|-\delta t-2\geq 10%
\delta t.$$
(4)
Let $x\in V[\mathcal{B}^{\prime}]$ be given. At least $|V[\mathcal{B}^{\prime}]|-\delta t$ vertices $y\in V[\mathcal{B}^{\prime}]$ are such that $xy\in\partial\mathcal{R}$, and, for each such $y$ there are at least $|V[\mathcal{B}^{\prime}]|-\delta t$ vertices $z\in V[\mathcal{B}^{\prime}]$ such that $xyz\in\mathcal{B}^{\prime}$. So, the total number of hyperedges of $\mathcal{B}^{\prime}$ that contain $x$ is at least
$$\frac{1}{2}(|V[\mathcal{B}^{\prime}]|-\delta t)^{2}\ \geq\ \frac{1}{2}\left(%
\frac{9}{10}|V[\mathcal{B}^{\prime}]|\right)^{2}\ \geq\ \frac{2}{3}{|V[%
\mathcal{B}^{\prime}]|\choose 2}.$$
Thus, Theorem 2.1 with $\eta=\frac{1}{9}$ yields a perfect matching $\mathcal{M}_{\mathrm{blue}}^{\prime}$ of $\mathcal{B}^{\prime}$.
At this point, we have three disjoint monochromatic connected matchings, one in red ($\mathcal{M}_{\mathrm{red}}\subset\mathcal{R}$) and two in blue ($\mathcal{M}_{\mathrm{blue}}\subset\mathcal{B}$ and $\mathcal{M}_{\mathrm{blue}}^{\prime}\subset\mathcal{B}^{\prime}$). Together, these matchings cover all but at most $3\delta t+2$ vertices of ${V}_{\mathrm{red}}\cup{V}_{\mathrm{blue}}$ (by (3) and by (4)). In particular, we can assume that
$\mathcal{M}_{\mathrm{blue}}$ contains at least two hyperedges,
as otherwise we can just forget about $\mathcal{M}_{\mathrm{blue}}$ and are done.
Our aim is now to dissolve the blue matching $\mathcal{M}_{\mathrm{blue}}$, and cover its vertices by new red edges, leaving at most $6\delta t$ vertices uncovered. In order to do so, let us first understand where the edges of $\mathcal{M}_{\mathrm{blue}}$ lie.
For convenience, let us call an edge in $\mathcal{K}$ good if two different pairs of its vertices $\{a,b\}$ and $\{c,d\}$ are such that $ab\in\partial\mathcal{R}$ and $cd\in\partial\mathcal{B}$. Notice that every good red edge is contained in $\mathcal{R}$ and every good blue edge is contained in $\mathcal{B}$.
First, we claim that for every edge $uvw\in\mathcal{M}_{\mathrm{blue}}$,
$$|\{u,v,w\}\cap{V}_{\mathrm{blue}}|\leq 1.$$
(5)
Indeed, otherwise there is an edge $uvw\in\mathcal{M}_{\mathrm{blue}}$ with $u,v\in{V}_{\mathrm{blue}}$. By the definition of $\mathcal{B}$, and by (2), there is an active edge $ua\in\partial\mathcal{B}$ with $a\in{V}_{\mathrm{red}}^{\prime}$. As $ua$ is an active pair, as $a$ has very large degree in $\partial\mathcal{R}$, and by (2), there is an edge $uab\in\mathcal{K}$ with $b\in{V}_{\mathrm{red}}^{\prime}$ such that $ab\in\partial\mathcal{R}$. Hence $uab$ is a good edge. Similarly, there
is a good edge $vcd$, with $c,d\in{V}_{\mathrm{red}}^{\prime}\setminus\{a,b\}$. Remove the edge $uvw$ from $\mathcal{M}_{\mathrm{blue}}$ and add edges $uab$ and $vcd$ to either $\mathcal{M}_{\mathrm{red}}$ or $\mathcal{M}_{\mathrm{blue}}$, according to their colour. The resulting matching covers more vertices than the matching $M_{A}\cup\mathcal{M}_{\mathrm{blue}}$, a contradiction. This proves (5).
Next, we claim that there is no edge $uvw\in\mathcal{M}_{\mathrm{blue}}$ with
$$|\{u,v,w\}\cap{V}_{\mathrm{blue}}|=1.$$
(6)
Assume otherwise. Then there is an edge $uvw\in\mathcal{M}_{\mathrm{blue}}$ with $u\in{V}_{\mathrm{blue}}$ and $v,w\in{V}_{\mathrm{red}}$. As in the proof of (5), we can cover $u$ with a good edge $uab$ such that $a,b\in{V}_{\mathrm{red}}^{\prime}$. Moreover, since $vw$ is an active pair, and $v$ has very large degree in $\partial\mathcal{R}$, there is an edge $vwc$ with $c\in{V}_{\mathrm{red}}^{\prime}\setminus\{a,b\}$ and $cv\in\partial R$.
Since $vw\in\partial\mathcal{B}$, the edge $vwc$ is good. So we can remove $uvw$ from $\mathcal{M}_{\mathrm{blue}}$ and add edges $uab$ and $vwc$ to $\mathcal{M}_{\mathrm{red}}\cup\mathcal{M}_{\mathrm{blue}}$, thus covering three additional vertices. This gives the desired contradiction to the choice of $\mathcal{M}_{\mathrm{red}}\cup\mathcal{M}_{\mathrm{blue}}$, and proves (6).
Putting (5) and (6) together, we know that
for every edge $uvw\in\mathcal{M}_{\mathrm{blue}}$ we have $u,v,w\in{V}_{\mathrm{red}}$. Consider any two edges $u_{1}v_{1}w_{1},u_{2}v_{2}w_{2}\in\mathcal{M}_{\mathrm{blue}}$. As before, there are vertices $a,b\in{V}_{\mathrm{red}}^{\prime}$ such that edges $v_{1}w_{1}a,v_{2}w_{2}b$ are good. Now, if there is a red edge $u_{1}u_{2}c$ with $c\in{V}_{\mathrm{red}}^{\prime}$ and $u_{1}c\in\partial\mathcal{R}$ then we can remove edges $u_{1}v_{1}w_{1},u_{2}v_{2}w_{2}$ and add the red edge $u_{1}u_{2}c$ to $\mathcal{M}_{\mathrm{red}}$ and edges $v_{1}w_{1}a,v_{2}w_{2}b$ to $\mathcal{M}_{\mathrm{red}}\cup\mathcal{M}_{\mathrm{blue}}$, according to their colour, contradicting the choice of $\mathcal{M}_{\mathrm{red}}\cup\mathcal{M}_{\mathrm{blue}}$. Therefore, for any choice of $u_{1}v_{1}w_{1},u_{2}v_{2}w_{2}\in\mathcal{M}_{\mathrm{blue}}$, we have that
all edges $$u_{1}u_{2}c$$ with $$c\in{V}_{\mathrm{red}}^{\prime}$$ and $$u_{1}c\in\partial\mathcal{R}$$ are blue.
(7)
Moreover, if there is a blue edge $u_{1}u_{2}x$ with $x\in\{v_{1},w_{1},v_{2},w_{2}\}$ then $u_{1}u_{2}$ is an active pair. In that case, we can calculate as before that an edge $u_{1}u_{2}c$ with $c\in{V}_{\mathrm{red}}^{\prime}$ and $u_{1}c\in\partial\mathcal{R}$ exists, and by (7), this edge is blue. The existence of the blue edge $u_{1}u_{2}x$ implies that we can link $u_{1}u_{2}c$ to $\mathcal{M}_{\mathrm{blue}}$ with a blue tight path. Thus, removing $u_{1}v_{1}w_{1}$ and $u_{2}v_{2}w_{2}$ from $\mathcal{M}_{\mathrm{blue}}$ and adding $v_{1}w_{1}a,v_{2}w_{2}b,u_{1}u_{2}c$ to $\mathcal{M}_{\mathrm{red}}\cup\mathcal{M}_{\mathrm{blue}}$ (where $a,b$ are as above), we obtain a contradiction to the choice of $\mathcal{M}_{\mathrm{red}}\cup\mathcal{M}_{\mathrm{blue}}$. So, for any choice of $u_{1}v_{1}w_{1},u_{2}v_{2}w_{2}\in\mathcal{M}_{\mathrm{blue}}$, we have that
all edges $$u_{1}u_{2}x$$ with $$x\in\{v_{1},w_{1},v_{2},w_{2}\}$$ are red.
(8)
We can now dissolve the edges of $\mathcal{M}_{\mathrm{blue}}$. For this, separate each hyperedge $uvw$ in $\mathcal{M}_{\mathrm{blue}}$ into an edge $uv$ and a single vertex $w$. Let $X$ be the set of all edges $uv$, and let $Y$ be the set of all vertices $w$ obtained in this way. Note that every $uv\in X$ is an active pair in $\mathcal{K}$, and therefore forms a hyperedge $uvw^{\prime}$ with all but at most $\delta t$ of the vertices $w^{\prime}\in Y$. Moreover, all but at most $\delta t$ of these hyperedges $uvw^{\prime}$ are such that $uw^{\prime}\in\partial\mathcal{R}$, because of the large degree $u$ has in $\partial\mathcal{R}$.
Consider the bipartite graph on $X\cup Y$ which has an edge between $uv\in X$ and $w^{\prime}$ whenever the hyperedge $uvw^{\prime}$ exists in $\mathcal{K}$ and $uw^{\prime}\in\partial\mathcal{R}$. Then
for each $X^{\prime}\subseteq X$ we have that $|N(X^{\prime})|\geq|Y|-2\delta t\geq|X^{\prime}|-2\delta t$. So by the defect form of Hall’s Theorem, there is a matching covering all but at most $2\delta t$ vertices of $|X^{\prime}|$. In $\mathcal{K}$, this corresponds to a matching $\mathcal{M}_{\mathrm{red}}^{\prime}$ covering all but at most $6\delta t$ vertices of $V(\mathcal{M}_{\mathrm{blue}})$. By (8), all hyperedges of $\mathcal{M}_{\mathrm{red}}^{\prime}$ are red. Furthermore, since we ensured that every hyperedge in $\mathcal{M}_{\mathrm{red}}$ contains a pair $uw^{\prime}$ that forms an ede of $\partial\mathcal{R}$, we know that $\mathcal{M}_{\mathrm{red}}$ and $\mathcal{M}_{\mathrm{red}}^{\prime}$ belong to the same red component of $\mathcal{K}$. In other words, $\mathcal{M}_{\mathrm{red}}\cup\mathcal{M}_{\mathrm{red}}^{\prime}$ and $\mathcal{M}_{\mathrm{blue}}^{\prime}$ are the two monochromatic connected matchings we had to find.
∎
3 Hypergraph regularity
In this section we introduce the regularity lemma for 3-uniform hypergraphs and state an embedding result from [11].
Graph regularity.
Let $G$ be a graph $G$ and let $X,Y\subseteq V(G)$ be disjoint.
The density of $(X,Y)$ is $d_{G}(X,Y)=\frac{e_{G}(X,Y)}{|X||Y|}$ where
$e_{G}(X,Y)$ denotes the number of edges of $G$ between $X$ and $Y$.
The bipartite graph $G$ on the partition classes $X$ and $Y$ is called $(d,\varepsilon)$-regular, if $|d_{G}(X^{\prime},Y^{\prime})-d|<\varepsilon$ holds for all
$X^{\prime}\subseteq X$ and $Y^{\prime}\subseteq Y$ of size $|X^{\prime}|>\varepsilon|X|$ and $|Y^{\prime}|>\varepsilon|Y|$.
If $d=d_{G}(X,Y)$ we say that $G$ is $\varepsilon$-regular.
Hypergraph regularity.
Let $\mathcal{H}$ be a $3$-uniform hypergraph. Let $P=P^{12}\cup P^{13}\cup P^{23}$ with $V(P)\subset V(\mathcal{H})$ be a tripartite graph which we also refer to as triad.
By $\mathcal{T}(P)$ denote the
$3$-uniform hypergraph on $V(P)$ whose edges are the triangles of $P$.
The density of $\mathcal{H}$ with respect to $P$ is
$$d_{\mathcal{H}}(P)=\frac{|\mathcal{H}\cap\mathcal{T}(P)|}{|\mathcal{T}(P)|}.$$
Similarly, for a tuple $\vec{\mathcal{Q}}=(Q_{1},\dots,Q_{r})$ of subgraphs of $P$, we define the density of $\mathcal{H}$ with respect to $\vec{\mathcal{Q}}$ as
$$d_{\mathcal{H}}(\vec{\mathcal{Q}})=\frac{|\mathcal{H}\cap\bigcup_{i\in[r]}%
\mathcal{T}(Q_{i})|}{|\bigcup_{i\in[r]}\mathcal{T}(Q_{i})|}.$$
Let $\alpha,\delta>0$ and let $r>0$ be an integer. We say that $\mathcal{H}$ is $(\alpha,\delta,r)$-regular with respect to $P$ if, for every $r$-tuple $\vec{\mathcal{Q}}=(Q_{1},\dots,Q_{r})$ of subgraphs of $P$ satisfying $|\bigcup_{i\in[r]}\mathcal{T}(Q_{i})|>\delta|\mathcal{T}(P)|$, we have $|d_{\mathcal{H}}(\vec{\mathcal{Q}})-\alpha|<\delta$. If $\alpha=d_{\mathcal{H}}(P)$ we say that $\mathcal{H}$ is $(\delta,r)$-regular with respect to $P$, and in the same situation, we say $P$ is $(\delta,r)$-regular (with respect to $\mathcal{H}$).
Moreover, if the bipartite graphs $P^{12},P^{13},P^{23}$ of an $(\alpha,\delta,r)$-regular $P=P^{12}\cup P^{13}\cup P^{23}$ are
$(1/\ell,\varepsilon)$-regular then
we say that the pair $(\mathcal{H},P)$ is an $(\alpha,\delta,\ell,r,\varepsilon)$-regular complex.
Finally, a partition
of $V$ into $V_{0}\cup V_{1}\cup\dots\cup V_{t}$ is called an equipartition if $|V_{0}|<t$ and $|V_{1}|=V_{2}|=\dots=|V_{t}|$.
We state the regularity lemma for $3$-uniform hypergraphs [4] as presented in [14].
Theorem 3.1 (Regularity Lemma for $3$-uniform Hypergraphs).
For all $\delta,t_{0}>0$, all integer-valued functions $r=r(t,\ell)$, and all decreasing sequences $\varepsilon(\ell)>0$ there exist constants $T_{0},L_{0}$ and $N_{0}$
such that every $3$-uniform hypergraph $\mathcal{H}$ with at least $N_{0}$ vertices admits a vertex equipartition
$$V(\mathcal{H})=V_{0}\cup V_{1}\cup\dots\cup V_{t}\qquad\text{ with }t_{0}\leq t%
<T_{0},$$
and, for each pair $i,j,$ $1\leq i<j\leq t,$ an edge partition of the complete bipartite graph
$$K(V_{i},V_{j})=\bigcup_{k\in[\ell]}P^{ij}_{k}\qquad\text{ with }1\leq\ell<L_{0}$$
such that
1.
all graphs $P_{k}^{ij}$ are $(1/\ell,\varepsilon(\ell))$-regular.
2.
$\mathcal{H}$ is $(\delta,r)$-regular with respect to all but at most $\delta\ell^{3}t^{3}$ tripartite graphs $P_{a}^{hi}\cup P_{b}^{hj}\cup P_{c}^{ij}$.
Note that the same partitions satisfy the conclusions of Theorem 3.1 for the complement of $\mathcal{H}$ as well.
Further, as noted in [11] by choosing a random index $k_{ij}\in[\ell]$ for each pair $(V_{i},V_{j})$ Markov’s inequality yields that
with positive probability there are less than $2\delta t^{3}$ chosen triads
which fail to be $(\delta,r)$-regular. Hence one obtains the following.
Observation 3.2.
In the partition produced by Theorem 3.1 there is a family $\mathcal{P}$ of bipartite graphs $P^{ij}=P^{ij}_{k_{ij}}$ with vertex classes $V_{i},V_{j}$, where $1\leq i<j\leq t$, such that $\mathcal{H}$ is $(\delta,r)$-regular with respect to all but at most $2\delta t^{3}$ tripartite graphs $P^{hi}\cup P^{hj}\cup P^{ij}$.
We end this section with a result from [16] and [11] which allows embedding tight paths in regular complexes.
In the following, an $S$-avoiding tight path is one which does not contain any vertex from $S$.
(Note that although Lemma 4.6 from [11] is stated slightly differently, its proof actually yields the version below.)
Lemma 3.3 ([11], Lemma 4.6).
For each $\alpha\in(0,1)$ there exist $\delta_{1}>0$ and sequences $r(\ell)$, $\varepsilon(\ell)$,
and $n_{1}(\ell)$, for $\ell\in\mathbb{N}$, with the following property.
For each $\ell\in\mathbb{N}$, and each $\delta\leq\delta_{1}$,
if $(\mathcal{H},P)$ is a $(d_{\mathcal{H}}(P),\delta,\ell,r(\ell),\varepsilon(\ell))$-complex with $d_{\mathcal{H}}(P)\geq\alpha$ and
all of the three vertex classes of $P$ have the same size $n>n_{1}(\ell)$, then there is a subgraph
$P_{0}$ on at most $27\sqrt{\delta}n^{2}/\ell$ edges of $P$ such that, for all ordered pairs of disjoint edges
$(e,f)\in(P\setminus P_{0})\times(P\setminus P_{0})$ there is $m=m(e,f)\in[3]$ such that the following holds. For every $S\subseteq V(\mathcal{H})\setminus(e\cup f)$ with
$|S|<n/(\log n)^{2}$, and for each $\ell$ with $3\leq\ell\leq(1-2\delta^{\frac{1}{4}})n$, there is an $S$-avoiding tight path from $e$ to $f$ of length $3\ell+m$ in $\mathcal{H}$.
4 Proof of Theorem 1.1
We follow a procedure suggested by Łuczak in [12] for graphs and used for tight cycles in $3$-uniform hypergraphs in [11].
Proof of Theorem 1.1.
For given $\eta>0$ we apply Lemma 1.3 with $\gamma=(\eta/480)^{6}$ to obtain $t_{0}$.
With foresight apply Lemma 3.3 with $\alpha=1/2$ to obtain $\delta_{1}$, and sequences $r(\ell)$, $\varepsilon(\ell)$,
and $n_{1}(\ell)$. Finally, apply Theorem 3.1 with $t_{0}$, $r(t,\ell)=r(\ell)$, $\varepsilon(\ell)$, $n_{1}(\ell)$ and $\delta=\min\{\delta_{1}/2,\gamma/48,(\eta/16)^{4}\}$
to obtain constants $T_{0}$, $L_{0}$ and $N_{0}$.
Given a two-colouring $\mathcal{K}_{n}=\mathcal{H}_{\mathrm{red}}\cup\mathcal{H}_{\mathrm{blue}}$ of the 3-uniform complete hypergraph $\mathcal{K}_{n}$ on $n>N_{0}$ vertices.
Apply Theorem 3.1 with the chosen constants to $\mathcal{H}_{\mathrm{red}}$ to
obtain partitions
$$V(\mathcal{K}_{n})=V_{0}\cup V_{1}\cup\dots\cup V_{t}\quad\text{and}\quad K(V_%
{i},V_{j})=\bigcup_{k\in[\ell]}P^{ij}_{k},1\leq i<j\leq t$$
with $t_{0}\leq t<T_{0}$, and $\ell<L_{0}$ which satisfy the properties detailed in Theorem 3.1.
The partitions satisfy the same properties for $\mathcal{H}_{\mathrm{blue}}$ as it is the complement hypergraph of $\mathcal{H}_{\mathrm{red}}$.
Observation 3.2 then yields a family of $(1/\ell,\varepsilon)$-regular bipartite graphs $P^{ij}=P^{ij}_{k_{ij}}$, one for each pair
$(V_{i},V_{j})$, $1\leq i<j\leq t$, such that $\mathcal{H}_{\mathrm{red}}$ (and thus also $\mathcal{H}_{\mathrm{blue}}$) is $(\delta,r)$-regular with respect to all but at most $2\delta t^{3}$ tripartite graphs $P^{ijk}=P^{ij}\cup P^{ik}\cup P^{jk}$.
We use this family to construct the reduced hypergraph $\mathcal{R}$ which has the vertex set $[t]$ and the edge set consisting of all triples $ijk$ such that $P^{ijk}$ is $(\delta,r)$-regular.
Further, colour the edge $ijk$ red if $d_{\mathcal{H}_{\mathrm{red}}}(P^{ijk})\geq 1/2$ and blue otherwise. Then
$\mathcal{R}$ has at least $\binom{t}{3}-2\delta t^{3}>(1-\gamma)\binom{t}{3}$ edges and as $d_{\mathcal{H}_{\mathrm{red}}}(P^{ijk})+d_{\mathcal{H}_{\mathrm{blue}}}(P^{ijk%
})=1$
we obtain a two-colouring of $\mathcal{R}=\mathcal{R}_{\mathrm{red}}\cup\mathcal{R}_{\mathrm{blue}}$.
Since $t\geq t_{0}$ Lemma 1.3 yields two disjoint monochromatic connected matchings
$\mathcal{M}_{\mathrm{red}}$ and $\mathcal{M}_{\mathrm{blue}}$
which cover all but at most $240\gamma^{\frac{1}{6}}t\leq\eta t/2$ vertices of $\mathcal{R}$ and in what follows we will
turn these connected matchings into disjoint monochromatic tight cycles in $\mathcal{K}_{n}$.
We start by choosing a red pseudo-path $\mathcal{Q}_{\mathrm{red}}=(e_{1},\dots,e_{p})\subset\mathcal{R}_{\mathrm{red}}$ which contains the matching $\mathcal{M}_{\mathrm{red}}$. This is possible since $\mathcal{M}_{\mathrm{red}}$ is a connected matching, and so, consecutive matching edges $g_{s},g_{s+1}\in\mathcal{M}_{\mathrm{red}}$ are connected by a red pseudo-path of length at most $\binom{t}{3}$.
The concatenation of these paths then forms a $\mathcal{Q}_{\mathrm{red}}$ as desired. In the same manner, choose a
blue pseudo-path $\mathcal{Q}_{\mathrm{blue}}=(e^{\prime}_{1},\dots,e^{\prime}_{q})\subset%
\mathcal{R}_{\mathrm{blue}}$ containing the matching $\mathcal{M}_{\mathrm{blue}}$. Note that although $\mathcal{M}_{\mathrm{red}}$ and $\mathcal{M}_{\mathrm{blue}}$ are disjoint, the two paths $\mathcal{Q}_{\mathrm{red}}$ and $\mathcal{Q}_{\mathrm{blue}}$
may have vertices in common.
For each edge $\{i,j,k\}=e_{s}\in\mathcal{Q}_{\mathrm{red}}$ let $P^{s}$ denote the triad $P^{ij}\cup P^{ik}\cup P^{jk}$ on the partition classes $V_{i}\cup V_{j}\cup V_{k}$ and recall that $P^{s}$ is $(\delta,r)$-regular (with respect to $\mathcal{H}_{\mathrm{red}}$).
Lemma 3.3 guarantees that one can find long tight paths in the complex $(\mathcal{H}_{\mathrm{red}},P^{s})$ for each matching edge $e_{s}\in\mathcal{M}_{\mathrm{red}}\subset\mathcal{Q}_{\mathrm{red}}$
which exhaust almost all vertices of $V_{i}\cup V_{j}\cup V_{k}$.
Using the connectedness of $\mathcal{Q}_{\mathrm{red}}$ and Lemma 3.3, we then want to
connect these long tight paths by short tight paths, hence obtain a tight cycle $\mathcal{C}_{\mathrm{red}}$ which covers almost all vertices of $\mathcal{K}_{n}$ spanned by $\mathcal{M}_{\mathrm{red}}$. We wish to do the same with $\mathcal{Q}_{\mathrm{blue}}$ to obtain
a tight cycle $\mathcal{C}_{\mathrm{blue}}$ which covers almost all vertices of $\mathcal{K}_{n}$ spanned by $\mathcal{M}_{\mathrm{blue}}$. The two cycles $\mathcal{C}_{\mathrm{red}}$ and $\mathcal{C}_{\mathrm{blue}}$ then exhaust most of the vertices of $\mathcal{K}_{n}$.
To keep the two cycles disjoint, however, the strategy will be slightly less straightforward. First, we will find two disjoint short tight cycles
$\mathcal{C}_{\mathrm{red}}^{\prime}$ and $\mathcal{C}_{\mathrm{blue}}^{\prime}$ in $\mathcal{K}_{n}$ visiting all triads $P^{s}$ corresponding to edges $e_{s}\in\mathcal{M}_{\mathrm{red}}$ and $P^{\prime s}$ corresponding to $e^{\prime}_{s}\in\mathcal{M}_{\mathrm{blue}}$, respectively.
Then, for each edge $e_{s}\in\mathcal{M}_{\mathrm{red}}$, and each edge $e^{\prime}_{s}\in\mathcal{M}_{\mathrm{blue}}$, we will replace parts of the cycles $\mathcal{C}_{\mathrm{red}}^{\prime}$, and of $\mathcal{C}_{\mathrm{blue}}^{\prime}$, i.e., paths corresponding to $e_{s}$ and $e^{\prime}_{s}$ by long tight paths as mentioned above.
We now give the details of this idea.
For each $s=1,\dots,p$, apply Lemma 3.3 to the complex $(\mathcal{H}_{\mathrm{red}},P^{s})$ to obtain the subgraph $P^{s}_{0}\subset P^{s}$ of “prohibited” edges
and let
$$B_{s}=(P^{s}\setminus P^{s}_{0})\cap(P^{s+1}\setminus P^{s+1}_{0})$$
which is a bipartite graph on the partition classes $V_{i}\cup V_{j}$ where $\{i,j\}=e_{s}\cap e_{s+1}$. We choose mutually distinct edges $f_{s},g_{s}\in B_{s}$, $s\in[p-1]$ which is possible due to the restriction on
$|P_{0}^{s}|$ provided $n$ is sufficiently large. Using Lemma 3.3 we then find a short tight cycle $\mathcal{C}_{\mathrm{red}}^{\prime}$ by
concatenating disjoint paths each of length at most 12 between $f_{1}$ and $f_{2}$, $f_{2}$ and $f_{3}$ $\dots$ between $f_{p-1}$ and $g_{p-1}$ and backwards between $g_{p-1}$ and $g_{p-2}$ $\dots$ and
finally between $g_{1}$ and $f_{1}$.
Note that the lemma allows the paths to be $S$-avoiding for any vertex set $S$ of size $|S|<n^{\prime}/(\log n^{\prime})^{2}$ where $n^{\prime}$ is the size of the partition classes.
Therefore, to guarantee the disjointness of the paths, we simply choose $S$ to be the vertices of the paths constructed so far which
has size at most $24p$, i.e., independent of $n^{\prime}>n/2t$. In the same way choose a short tight cycle $\mathcal{C}_{\mathrm{blue}}^{\prime}$ disjoint
from $\mathcal{C}_{\mathrm{red}}^{\prime}$ by including $V(\mathcal{C}_{\mathrm{red}}^{\prime})$ to $S$ in the applications of Lemma 3.3.
Let $S^{\prime}=V(\mathcal{C}_{\mathrm{red}}^{\prime})\cup V(\mathcal{C}_{\mathrm{%
blue}}^{\prime})$ which satisfies $|S^{\prime}|<n^{\prime}/(\log n^{\prime})^{2}$. It is easy to see that for each $e_{s}\in\mathcal{M}_{\mathrm{red}}$ there are two non-prohibited edges in $P^{s}$, connected by a subpath of $\mathcal{C}_{\mathrm{red}}^{\prime}$
which is entirely contained in $(\mathcal{H}_{\mathrm{red}},P^{s})$. Hence, by Lemma 3.3 we can replace this short path by an $S$-avoiding path in $(\mathcal{H}_{\mathrm{red}},P^{s})$ which covers all but at most $4\delta^{1/4}n^{\prime}$
vertices and having any desired parity. Doing this for all $e_{s}\in\mathcal{M}_{\mathrm{red}}$ and all $e^{\prime}_{s}\in\mathcal{M}_{\mathrm{blue}}$ and noting that $n^{\prime}\leq n/t$ we obtain two monochromatic disjoint tight cycles which cover all but at most
$$\displaystyle\big{(}|\mathcal{M}_{\mathrm{red}}|+|\mathcal{M}_{\mathrm{blue}}|%
\big{)}4\delta^{1/4}n^{\prime}+$$
$$\displaystyle\big{(}|V(\mathcal{R})|-|\mathcal{M}_{\mathrm{red}}|+|\mathcal{M}%
_{\mathrm{blue}}|\big{)}n^{\prime}+|V_{0}|$$
$$\displaystyle\leq\frac{1}{4}\eta n+\frac{1}{2}\eta n+t\leq\eta n$$
vertices of $\mathcal{K}_{n}$, and have any desired parity. This finishes the proof of the theorem.
∎
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$T$-dependence of the axion mass when the $U_{A}(1)$
and chiral symmetry breaking are tied ††thanks: Presented at Excited QCD 2019, Schladming, Austria,
Jan. 30 – Feb. 3, 2019.
Dubravko Klabučar
Davor Horvatić
Dalibor Kekez
Physics Department, Faculty of Science – PMF, University of Zagreb, Croatia
Rudjer Bošković Institute, Zagreb, Croatia
Abstract
Modulo the scale of spontaneous breaking of Peccei-Quinn symmetry, the
axion mass $m_{\rm a}(T)$ is given by the QCD topological susceptibility
$\chi(T)$ at all temperatures $T$. From an approach tying the $U_{A}(1)$ and
chiral symmetry breaking and getting good $T$-dependence of $\eta$ and
$\eta^{\prime}$ mesons, we get $\chi(T)$ for an effective Dyson-Schwinger
model of nonperturbative QCD. Comparison with lattice results for $\chi(T)$,
and thus also for $m_{\rm a}(T)$, shows good agreement for temperatures ranging
from zero up to the double of the chiral restoration temperature $T_{c}$.
\PACS
14.80.Mz , 14.80.Va , 12.38.Aw
1 Introduction
The fundamental theory of strong interactions, QCD, has the so-called Strong CP
problem. Namely, there is no experimental evidence of any CP-symmetry violation
in strong interactions, although there is in principle no reason why the QCD
Lagrangian should not include the so-called $\Theta$-term
${\cal L}^{\Theta}$, where gluon fields $F^{b}_{\mu\nu}(x)$ comprise the
CP-violating combination $Q(x)$,
$${\cal L}^{\Theta}(x)\,=\,\Theta\,\frac{g^{2}}{\,64\,\pi^{2}\,}\,\epsilon^{\mu%
\nu\rho\sigma}\,F^{b}_{\mu\nu}(x)\,F^{b}_{\rho\sigma}(x)\,\equiv\,\Theta\,Q(x)\,.$$
(1)
Admittedly, ${\cal L}^{\Theta}$ can be rewritten
as a total divergence, but, unlike in QED, this does not enable
discarding it in spite of the gluon fields vanishing sufficiently fast as
$|x|\to\infty$. This is because of nontrivial topological structures in QCD, such as instantons,
which are important for, e.g., solving of the $U_{A}(1)$ problem
and yielding the anomalously large mass of the $\eta^{\prime}$ meson.
Thus, there is no reason why the coefficient $\Theta$ of this term should be
of a very different magnitude from the coefficients of the other, CP-symmetric
terms comprising the usual CP-symmetric QCD Lagrangian. Nevertheless, the
experimental bound on the coefficient of the term is extremely low,
$|\Theta|<10^{-10}$ [1], and consistent with zero.
This is the mystery of the missing strong CP violation:
why is $\Theta$ so small?
Various proposed theoretical solutions stood the test of time with varying success.
A long-surviving solution, which is actually the preferred solution nowadays,
is a new particle beyond the Standard Model - the axion. Important is also that
axions turned out to be very interesting also for cosmology, as promising
candidates for dark matter. (See, e.g., [2, 3].)
2 Axion mass from the non-Abelian axial anomaly
Peccei and Quinn introduced [4, 5] a new axial global
symmetry $U(1)_{PQ}$ which is broken spontaneously at some scale $f_{\rm a}$.
This presumably huge [6] but otherwise presently unknown
scale is the key free parameter of axion theories, which determines the absolute
value of the axion mass $m_{\rm a}$. However, this constant cancels from
combinations such as $m_{\rm a}(T)/m_{\rm a}(0)$. Hence, useful insights and applications are possible in spite of $f_{\rm a}$ being not known.
We have often, including applications at $T>0$
[7, 8, 9, 10, 11],
employed a chirally well-behaved relativistic bound-state approach to modeling
nonperturbative QCD through Dyson-Schwinger equations (DSE) for Green’s functions
of the theory. (For reviews, see [12, 13, 14]
for example.)
Such calculations can yield model predictions on the QCD topological susceptibility,
including its temperature dependence $\chi(T)$, which are correctly related to the
QCD dynamical chiral symmetry breaking (DChSB) and restoration. It turns out that
$\chi(T)$ is precisely that factor in the axion mass $m_{\rm a}(T)$, which carries
the nontrivial $T$-dependence.
2.1 Axions as quasi-Goldstone bosons
The pseudoscalar axion field $\mbox{\large a}(x)$ arises as the
(would-be massless) Goldstone boson of the spontaneous breaking of the
Peccei-Quinn symmetry [15, 16].
The axion contributes to the total Lagrangian its kinetic term and its
interaction with the standard-model fermions. But what is important for
the resolution of the strong CP problem, is that the axion also couples
to the topological charge density operator operator $Q(x)$ in
Eq. (1). Then, the $\Theta$-term in the QCD Lagrangian
gets modified to
$${\cal L}^{\Theta\,+}_{\rm axion}\,=\,{\cal L}^{\Theta}\,+\frac{\mbox{\large a}%
(x)}{f_{\rm a}}\,Q(x)\,=\,\left(\,\Theta\,+\frac{\mbox{\large a}}{f_{\rm a}}%
\right)\,\frac{g^{2}}{64\pi^{2}}\,\epsilon^{\mu\nu\rho\sigma}F^{b}_{\mu\nu}\,F%
^{b}_{\rho\sigma}\,.$$
(2)
Through this coupling of the axion to gluons, the $U(1)_{PQ}$ symmetry is also broken
explicitly by the $U_{A}(1)$ non-Abelian, gluon axial anomaly, so that the axion
has a non-vanishing mass, $m_{\rm a}\neq 0$ [15, 16].
Gluons generate an effective axion potential, and its minimization leads to
the axion expectation value $\langle\mbox{\large a}\rangle$ such that the
modified coefficient, multiplying the topological charge density $Q(x)$,
should vanish:
$$\Theta\,+\frac{\langle\mbox{\large a}\rangle}{f_{\rm a}}\,=\,0\,.$$
(3)
The strong CP problem is thereby solved, irrespective of the initial
value of $\Theta$. Relaxation from any $\Theta$-value in
the early Universe towards the minimum at Eq. (3),
is known as misalignment production, and the the resulting axion oscillation energy
is a cold dark matter candidate (e.g., see [2, 3]).
2.2 Axion mass from anomalous $U_{A}(1)$ breaking driven by DChSB
A direct measure of the $U_{A}(1)$ symmetry breaking is the topological susceptibility
$\chi$, given by the convolution of the time-ordered product $\cal T$ of the
topological charge densities $Q(x)$ defined by Eq. (1) [or
Eq. (2)]:
$$\chi\,=\,\int d^{4}x\;\langle 0|\,{\cal T}\,Q(x)\,Q(0)\,|0\rangle\;.$$
(4)
The expansion of the effective axion
potential reveals in its quadratic term that the axion mass squared
(times $f_{\rm a}^{2}$) is equal111To a high level of accuracy, since
corrections to Eq. (5) are of the order $M_{\pi}^{2}/f_{\rm a}^{2}$
[20], where the pion mass $M_{\pi}$ is negligible.
to the QCD topological susceptibility.
This holds for all temperatures $T$:
$$m_{\rm a}^{2}(T)\,f_{\rm a}^{2}\,=\,\chi(T)\,.$$
(5)
On the other hand, in our study [11] of the $T$-dependence
of the $\eta$ and $\eta^{\prime}$ masses and the influence of the anomalous $U_{A}(1)$
breaking and restoration, we used the light-quark-sector result
[17, 18, 19]
$$\chi(T)\,=\,\frac{-\,1}{\,\,\,\frac{1}{\,m_{u}\,\langle{\bar{u}}u\,(T)\rangle}%
+\frac{1}{\,m_{d}\,\langle{\bar{d}}d\,(T)\rangle}+\frac{1}{\,m_{s}\,\langle{%
\bar{s}}s\,(T)\rangle}\,\,\,}\,+\,{\cal C}_{m}\,$$
(6)
where ${\cal C}_{m}$ is a very small correction term of higher orders in the
small current quark masses $m_{q}$ ($q=u,d,s$), and in the present context
we do not consider it further. Thus, the overwhelming part, namely the
leading term of $\chi$, is given by the quark condensates
$\langle{\bar{q}}q\rangle$ ($q=u,d,s$), which arise as order parameters
of DChSB. Their temperature dependence determines that of $\chi(T)$,
which in turn determines the $T$-dependence of the anomalous part of
the pseudoscalar meson masses in the $\eta$-$\eta^{\prime}$ complex.
This is the mechanism of Ref. [11], how DChSB and
chiral restoration drive, respectively, the anomalous breaking and
restoration of the $U_{A}(1)$ symmetry of QCD.
Now, Eqs. (5) and (6) show that
this mechanism determines also the $T$-dependence of the axion mass.
To describe $\eta^{\prime}$ and $\eta$ mesons, it is essential to include $U_{A}(1)$
symmetry breaking at least at the level of the masses. This could be done
simply [23, 24, 25], by adding the
anomalous contribution to isoscalar meson masses as a perturbation, thanks
to the fact that the $U_{A}(1)$ anomaly is suppressed in the limit of large
number of QCD colors $N_{c}$ [26, 27]. Concretely,
Ref. [25] adopted Shore’s equations [28],
where the $U_{A}(1)$-anomalous contribution to the light pseudoscalar masses
is expressed through the condensates of light quarks with non-vanishing
current masses. They are thus used also in $\chi(T)$ (6), since
this approach has recently been extended [11] to $T>0$.
This gave us our results for $\chi(T)$ depicted in Fig. 1.
Indeed, the now established smooth, crossover behavior around
the pseudocritical temperature $T_{c}$ for the chiral transition,
is obtained for the DChSB condensates of realistically massive
light quarks – i.e., the quarks with realistic
explicit chiral symmetry breaking [11].
In contrast, using in Eq. (6) the massless quark condensate
$\langle{\bar{q}}q\rangle_{0}$ (which drops sharply to zero at $T_{c}$)
instead of the “massive” ones,
would dictate a sharp transition of the second order at $T_{c}$
[10, 11] also for $\chi(T)$.
Obviously, this would imply that axions are massless for $T>T_{c}$.
In Fig. 1, we present (the leading term of) our model-calculated
[11] $\chi(T)^{1/4}$, depicted as the solid curve. Due
to Eq. (5), this is our model prediction for
$\sqrt{m_{\rm a}(T)\,f_{\rm a}}$. For temperatures
up to $T\approx 2.3\,T_{c}$, we compare it to the lattice results of Petreczky
et al., [21] and of Borsany et al.,
[22], rescaled to the relative temperature $T/T_{c}$.
3 Summary
The axion mass and its temperature dependence $m_{\rm a}(T)$
can be calculated in an effective model of nonperturbative QCD
(up to the constant scale parameter $f_{\rm a}$) as the square root
of the topological susceptibility $\chi(T)$. We obtained
it from the condensates of $u$-, $d$- and $s$-quarks and antiquarks
calculated in the SDE approach using a simplified nonperturbative
model interaction [11].
Our prediction on $m_{\rm a}(T)$ is thus supported by the fact
that our topological susceptibility also yields the $T$-dependence
of the $U_{A}(1)$ anomaly-influenced masses of $\eta^{\prime}$ and $\eta$
mesons which is consistent with experimental evidence [11].
Our result on $\chi(T)$ and the related axion mass is qualitatively
similar to the one obtained in the framework of the NJL model [29].
Our topological susceptibility is also qualitatively similar to the
pertinent lattice results [21, 22],
except that our dynamical model could so far access only much smaller
range of temperatures, $T<2.4\,T_{c}$.
On the other hand, the lattice supports the smooth crossover transition
of $\chi(T)$, which is, in our approach, the natural consequence of
employing the massive-quark condensates exhibiting crossover around the
chiral restoration temperature $T_{c}$. Hence, the (partial) $U_{A}(1)$
restoration observed in Ref. [11] must also be a
crossover, which in the present work, as well as in its longer
counterpart [30] containing a detailed analysis
of the model parameter dependence, translates into the corresponding
smooth $T$-dependence of the axion mass.
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KEK Preprint 2018-80
CHIBA-EP-234
Confinement of quarks in higher representations in view of dual superconductivity
Computing Research Center, High Energy Accelerator Research Organization (KEK)
SOKENDAI (The Graduate University for Advanced Studies), Tsukuba 305-0801, Japan
E-mail:
Ryutaro Matsudo
Department of Physics, Faculty of Science and Engineering,
Chiba University, Chiba 263-8522, Japan
E-mail:
[email protected]
Seikou Kato
Oyama National College of Technology, Oyama, Tochigi 323-0806, Japan
E-mail:
[email protected]
Kei-Ichi Kondo
Department of Physics, Graduate School of Science and Engineering, Chiba University,
Department of Physics, Graduate School of Science, Chiba University, Chiba 263-8522, Japan
E-mail:
[email protected]
Abstract:
Dual superconductor picture is one of the most promising scenarios for quark confinement.
We have proposed a new formulation of Yang-Mills theory on the lattice so that the so-called restricted field obtained from the gauge-covariant decomposition plays the dominant role in quark confinement.
This framework improves the Abelian projection in the gauge-independent manner.
For quarks in the fundamental representation, we have demonstrated some numerical evidences for the dual superconductivity.
However, it is known that the expected behavior of the Wilson loop in higher representations cannot be reproduced if the restricted part of the Wilson loop is extracted by adopting the Abelian projection or the field decomposition naively in the same way as in the fundamental representation.
In this talk, therefore, we focus on confinement of quarks in higher representations.
By virtue of the non-Abelian Stokes theorem for the Wilson loop operator, we propose suitable operators constructed from the restricted field only in the fundamental representation to reproduce the correct behavior of the original Wilson loop in higher representations.
Moreover, we perform lattice simulations to measure the static potential for quarks in higher representations using the proposed operators.
We find that the proposed operators well reproduce the expected behavior of the original Wilson loop average, which overcomes the problem that occurs in naively applying Abelian-projection to the Wilson loop operator for higher representations.
1 Introduction
The dual superconductor picture is one of the most promising scenarios for
quark confinement [1]. According to this picture, magnetic
monopoles causing the dual superconductivity are regarded as the dominant
degrees of freedom responsible for confinement. However, it is not so easy to
establish this hypothesis. Indeed, even the definition of magnetic monopoles
in the pure Yang-Mills theory is not obvious. If magnetic charges are naively
defined from electric ones by exchanging the role of the magnetic field and
electric one according to the electric-magnetic duality, one needs to
introduce singularities to obtain non-vanishing magnetic charges, as
represented by the Dirac monopole. For such configuration, however, the energy
becomes divergent.
There are two prescriptions avoiding this issue in defining magnetic
monopoles, i.e., the Abelian projection method and the
decomposition method, In the Abelian projection method [2],
the “diagonal component” of the Yang-Mills
gauge field is identified with the Abelian gauge field, and a magnetic
monopole is defined as the Dirac monopole. The energy density of this monopole
can be finite everywhere because the contribution from the singularity of a
Dirac monopole can be canceled by that of the off-diagonal components of the
gauge field. However, one needs to fix the gauge, because otherwise the
“diagonal component” is meaningless. In the
decomposition method, on the other hand, we need no more gauge fixing. The
gauge-covarint decomposition was first proposed by Cho, Duan and Ge [3]
for the $SU(2)$ Yang-Mills theory, and was extended to the $SU(N)$ Yang-Mills
theory (see [4] for review). The key ingredient in this
decomposition is the Lie-algebra valued field with a unit length which we call
the color field. Then, the monopoles can be defined by using the
gauge-invariant part proportional to the color field in the field strength
just like the Abelian field strength in the Abelian projection. Therefore, the
gauge invariance is explicitly maintained in the decomposition method.
It should be examined on the lattice whether or not these monopoles can
reproduce the expected infrared behavior of the original Wilson loop average.
In the preceding lattice studies for $SU(2)$ and $SU(3)$ Yang-Mills theory
within the Abelian projection method using the maximal Abelian (MA) gauge, it
was confirmed that (i) the string tension calculated from the diagonal part of
the original Yang-Mills field reproduces the full string tension calculated
from the original Yang-Mills fields [7, 9], and that (ii) the
string tension calculated from the magnetic monopole extracted from the
diagonal part mostly reproduce the full string tension [8, 9].
However, it is known that the resulting monopole contribution does not
reproduce the original Wilson loop average if the Abelian projection is
naively applied to the Wilson loop in higher representations [10].
This is because, in higher representations, the diagonal part of the Wilson
loop does not behave in the same way as the original Wilson loop. Poulis
heuristically found the correct way to extend the Abelian projection approach
for the adjoint representation in the $SU(2)$ Yang-Mills theory
[11].
In this talk, we propose a systematic prescription to extract the
“dominant” part of the Wilson loop average,
which can be applied to the Wilson loop operator in an arbitrary
representation of the $SU(N)$ Yang-Mills theory. To test this proposal, we
perform the numerical simulation on the lattice, and measure the Wilson loop
average in the higher representation for both the original Yang-Mills field
and the restricted field which is extracted as the dominant mode for quark
confinement by using the decomposition method, e.g., the Wilson loop averages
for the adjoint representation in the $SU(2)$ Yang-Mills theory, and for the
adjoint and sextet representations in the $SU(3)$Yang-Mills theory. We find
that the results support our claim.
2 The gauge field decomposition method and the non-Abelian Stokes
theorem
In this section, we give a brief review of the gauge field decomposition (
e.g., see [4]) and the non-Abelian Stokes theorem for the Wilson
loop in an arbitrary representation [5].
2.1 The gauge filed decomposition
We decompose the gauge link variable $U_{x,\mu}$ into the product of the two
variables $V_{x,\mu}$ and $X_{x,\mu}$ in such a way that the new variable
$V_{x,\mu}$, is transformed by the full $SU(N)$ gauge transformation
$\Omega_{x}$ as the gauge link variable $U_{x,\mu}$, while $X_{x,\mu}$
transforms as the site variable:
$$\displaystyle X_{x,\mu},U_{x,\mu}=X_{x,\mu}V_{x,\mu}\in G=SU(N),$$
(1a)
$$\displaystyle U_{x,\mu}\longrightarrow U_{x,\nu}^{\prime}=\Omega_{x}U_{x,\mu}%
\Omega_{x+\mu}^{{\dagger}},\text{ \ \ }V_{x,\mu}\longrightarrow V_{x,\nu}^{%
\prime}=\Omega_{x}V_{x,\mu}\Omega_{x+\mu}^{{\dagger}},\text{ \ }X_{x,\mu}%
\longrightarrow X_{x,\nu}^{\prime}=\Omega_{x}X_{x,\mu}\Omega_{x}^{{\dagger}}.$$
(1b)
From the physical point of view, $$V_{x,\mu}$$, which we call the restriced
field, could be the dominant mode for quark confinement, while $$X_{x,\mu}$$ is
the remainder part. The possible options of the decomposition are
discriminated by the stability subgroup of the gauge group. Here, we only
consider the maximal option.
The maximal option is obtained for the stability subgroup of the maximal torus
subgroup of $G$: $\tilde{H}=U(1)^{N-1}\subset SU(N).$ The resulting
decomposition is the gauge-invariant extension of the Abelian projection in
the maximal Abelian (MA) gauge. We introduce color fields as,
$$\bm{n}{}^{(k)}(x)=\Theta(x)H_{k}\Theta^{{\dagger}}(x)\in Lie[G/\tilde{H}]\text%
{
\ \ \ \ \ \ }(k=1,\ldots,N-1)\text{,}$$
(2)
which are expressed using a common $SU(N)$-valued field $\Theta(x)$ with the
Cartan generators $H_{k}$. The decomposition is obtained by solving the
defining equations:
$$\displaystyle D_{\mu}^{\epsilon}[V]\mathbf{n}_{x}^{(k)}$$
$$\displaystyle:=\frac{1}{\epsilon}\left[V_{x,\mu}\mathbf{n}_{x+\mu}^{(k)}-%
\mathbf{n}_{x}^{(k)}V_{x,\mu}\right]=0\text{ },$$
(3a)
$$\displaystyle\text{\ }g_{x}$$
$$\displaystyle:=e^{i2\pi q/N}\exp\left(-i\sum_{j=1}^{N-1}a_{x}^{(j)}\mathbf{n}_%
{x}^{(j)}\right)\text{,}$$
(3b)
where, the variable $$g_{x}$$ is the $$U(1)^{N}$$ part which is undetermined from
Eq.(3a) alone, $$a_{x}^{(j)}$$ are coeficients, and $$q$$ is an
integer. Note that the above defining equations correspond to the continuum
version ($$g_{x}=\mathbf{1}$$): $$D_{\mu}[\mathscr{V}]\mathbf{n}^{(k)}(x)=0$$ and $$\mathrm{tr}(\mathscr{X}_{\mu}(x)\mathbf{n}^{(k)}(x))$$ $$=0$$, respectively. These defining equations
can be solved exactly, and the solution is given by
$$\displaystyle X_{x,\mu}$$
$$\displaystyle=\widehat{K}_{x,\mu}^{{\dagger}}\det(\widehat{K}_{x,\mu})^{1/N}g_%
{x}^{-1},\text{ \ \ \ \ \ \ \ \ \ }V_{x,\mu}=X_{x,\mu}^{{\dagger}}U_{x,\mu},$$
$$\displaystyle\widehat{K}_{x,\mu}$$
$$\displaystyle:=\left(K_{x,\mu}K_{x,\mu}^{{\dagger}}\right)^{-1/2}K_{x,\mu},%
\text{ \ \ \ \ \ \ \ \ \ \ \ }K_{x,\mu}:=\bm{1}+2N\sum{}_{k=1}^{N-1}\bm{n}_{x}%
^{(k)}U_{x,\mu}\mathbf{n}_{x+\mu}^{(k)}U_{x,\mu}^{{\dagger}}.$$
(4)
In the naive continuum limit, we can reproduce the decomposition in the
continuum theory.
The color fields $\bm{n}{}^{(k)}$ are obtained by minimizing the functional $R_{\mathrm{MA}}$ for a
given configuration of the link variables $U_{x,\mu}$ with respect to the
gauge transformation
$$R_{\mathrm{MA}}[U,\{\bm{n}^{(k)}\}]=\sum_{x,\mu}\sum_{k=1}^{N-1}\operatorname{%
tr}[(D_{\mu}[U]\mathbf{n}_{x}^{(k)})^{{\dagger}}(D_{\mu}[U]\mathbf{n}_{x}^{(k)%
})].$$
(5)
If we choose this condition as the reduction condition, the definition of
monopoles is equivalent to that for the Abelian projection in the MA gauge. In
the present study for the $SU(3)$ gauge theory, we apply two additional
reduction conditions which are defined by minimizing following functionals
$$\displaystyle R_{n3}[U,\{\bm{n}^{(k)}\}]$$
$$\displaystyle=\sum_{x,\mu}\operatorname{tr}[(D_{\mu}[U]\bm{n}_{x}^{3})^{{%
\dagger}}(D_{\mu}[U]\bm{n}_{x}^{3})],\quad(\bm{n}_{x}^{3}:=\Theta_{x}T^{3}%
\Theta_{x}^{{\dagger}}),$$
(6)
$$\displaystyle R_{n8}[U,\{\bm{n}^{(k)}\}]$$
$$\displaystyle=\sum_{x,\mu}\operatorname{tr}[(D_{\mu}[U]\bm{n}_{x}^{8})^{{%
\dagger}}D_{\mu}[U]\bm{n}_{x}^{8})],\quad(\bm{n}_{x}^{8}:=\Theta_{x}T^{8}%
\Theta_{x}^{{\dagger}}).$$
(7)
2.2 Non-Abelian Stokes theorem
We can relate the decomposed field variables to a Wilson loop operator through
a version of the non-Abelian Stokes theorem (NAST) which was proposed by
Diakonov and Petrov [6]. In this version of the NAST, a Wilson loop
operator in a representation $R$ is rewritten into the surface integral form
by introducing a functional integral on the surface $S$ surrounded by the loop
$C$ as
$$\displaystyle W_{R}[\mathscr{A};C]$$
$$\displaystyle:=\int D\Omega\exp\left(ig\int_{S:\partial S=C}dS_{\mu\nu}\sum_{k%
=1}^{N-1}\Lambda_{k}F_{\mu\nu}^{(k)}\right)\text{
},$$
(8a)
$$\displaystyle F_{\mu\nu}^{(k)}$$
$$\displaystyle:=2\operatorname{tr}(\bm{n}^{(k)}\mathscr{F}_{\mu\nu}[\mathscr{V}%
]),$$
(8b)
where $$D\Omega$$ is the product of the Haar measure over the surface $$S$$,
$$\Lambda_{k}$$ is the $$k$$-th component of the highest-weight of the
representation $$R$$, the color fields are defined by $$\bm{n}^{(k)}=\Omega H_{k}\Omega^{{\dagger}}$$, and $$\mathscr{F}_{\mu\nu}[\mathscr{V}]$$ is the field
strength for the restricted field $$\mathscr{V}$$ in the continuous version.
Monopoles are defined in the same manner as the Dirac monopoles for the
Abelian-like gauge-invariant field strength $$F_{\mu\nu}^{(k)}$$. The resulting
monopoles are gauge invariant by construction. Thus we can relate the
restricted field to tied Wilson loop operator in the manifestly
gauge-invariant way.
In the actual lattice simulations, we do not follow this NAST directly.
Without performing the integration over the measure $D\Omega$, the argument of
the exponential is approximated by substituting the color field (as a
functional of the gauge field) which is obtained by solving the reduction
condition. Indeed, this approximation is commonly used for the Wilson loop in
the fundamental representation, e.g., the Abelian projection in the MA gauge
is equivalent to the field decomposition with the color fields determined by
minimizing the functional eq. 5.
The NAST can be applied not only to the fundamental representation but also to
any representation, which suggests the correct way to extract the dominant
part of the Wilson loop in higher representations as we explain in the next section.
3 Wilson loops in higher representations
In the preceding study, it was shown that the area law of the average of the
Wilson loop in the fundamental representation is reproduced by the monopole
contribution. However there is a possibility that the monopole contribution
accidentally coincides with the behavior of the original Wilson loop, and
therefore we should examine the other quantities. The Wilson loops in higher
representations are appropriate for this purpose because they have clear
physical meaning and show the characteristic behavior, e.g., the Casimir scaling.
It is known that if we adopt the Abelian projection naively to higher
representations, the monopole contributions do not reproduce the correct
behavior [10]. Thus, we have to find a more appropriate way to
extract the monopole contributions. The NAST (8a) suggests the
different operator as the dominant part of the Wilson loop in higher
representations. In the actual calculation, we approximate the NAST by
calculating the integrand of the NAST using the color fields satisfying the
reduction condition. In this approximation, the operator to be calculated is
not equivalent to the Wilson loop for the restricted field. From this point of
view, therefore, the distinct operator is suggested as the dominant part of
the Wilson loop. We can see the difference between the operator suggested by
the NAST and the Wilson loop for the restricted field by expressing it as the
surface integral form as
$$W_{R}[\mathscr{V};C]=\frac{1}{D_{R}}\sum_{\mathbf{\mu\in}\Delta_{R}}d_{\mathbf%
{\mu}}\exp\left(ig\sum_{k=1}^{N-1}\mu_{k}\int_{S}dS^{\alpha\beta}F_{\alpha%
\beta}^{(k)}\right),$$
(9)
where $D_{R}$ is the dimension of the representarion $R,$ $\Delta_{R}$ is set
of the weights of $R$, $\ \mu_{k}$ is the $k$-th component of the weight
$\mathbf{\mu,}$ and $d_{\mu}$ is the multiplicity of$\ \mathbf{\mu},$ which
satisfies $D_{R}=\sum_{\mathbf{\mu\in}\Delta_{R}}d_{\mathbf{\mu}}.$ In the
fundamental representation, all weights are equivalent to the highest weight
under the action of the Weyl group, but in higher representations there are
weights that are not equivalent to the highest weight. Therefore this
expression is not equivalent to the integrand of eq. 8a in higher
representations. For this reason we should modify eq. 9 so as to
include only the weights that is equivalent to the highest weights.
By using the untraced Wilson loop $V_{C}:=\prod_{\braket{x,\mu}\in C}V_{x,\mu}$ for the restricted field in the fundamental representation, we obtain the
Wilson loop in higher representations. For $SU(2)$, we propose the operator
for the spin-$j$ representation as
$$W_{[j]}^{SU(2)}[\mathscr{V};C]=\frac{1}{2}\operatorname{tr}(V_{C}^{2j}).$$
(10)
For $SU(3)$, we propose the operator for the representation with the Dynkin
index $[m_{1},m_{2}]$ as
$$W_{[m_{1},m_{2}]}^{SU(3)}[\mathscr{V};C]=\begin{cases}\frac{1}{6}\left(%
\operatorname{tr}(V_{C}^{m_{1}})\operatorname{tr}(V_{C}^{{\dagger}m_{2}})-%
\operatorname{tr}(V_{C}^{m_{1}}V_{C}^{{\dagger}m_{2}})\right)&\text{for}\quad m%
_{1},\text{ }m_{2}>0\\
\frac{1}{3}\operatorname{tr}(V_{C}^{m_{1}})&\text{for}\quad m_{2}=0\\
\frac{1}{3}\operatorname{tr}(V_{C}^{{\dagger}m_{2}})&\text{for}\quad m_{1}=0%
\end{cases}.$$
(11)
4 Numerical result
In order to support our claim that the dominant part of the Wilson loops in
higher representation is the operator suggested by the NAST, by using the
numerical simulation on the lattice, we check whether the string tension from
the Wilson loop in the representation $R$ of the restricted field $V$ can
reproduce that of the Yang-Mills field or not.
For this purpose, We set up the gauge configuration for the standard Wilson
action at $\beta=2.5$ on the $24^{4}$ lattice for $SU(2)$ case and at
$\beta=6.2$ on the $24^{4}$ lattice for $SU(3)$ case . For $SU(2)$ case, we
prepare $500$ configurations every $100$ sweeps after $3000$ thermalization by
using the heat bath method. For $SU(3)$ case, we prepare $1500$ configurations
every $50$ sweeps after $1000$ thermalization by using pseudo heat bath method
with over-relaxation algorithm ($20$ steps per sweep). In the measurement of
the Wilson loop average we apply the APE smearing technique for $SU(3)$ case
and the hyper-blocking for $SU(2)$ case to reduce noises and reduced the
exciting modes. The number of the smearing steps is determined so that the
ground state overlap is enhanced [12]. We have calculated the Wilson
loop average $W(R,T)$ for the rectangular loop with length $T$ and width $R$
to derive the potential $V(R,T)$ through the formula $V(R,T):=-\log(W(R,T+1)/W(R,T).$
In case of the $SU(2)$, we investigate the Wilson loop in the
adjoint representation $\mathbf{3}$ ($j=1$). The restricted field, the
extracted dominant mode for confinement, is obtained by using the
decomposition eq(4) for the color field which minimizes the
reduction condition Eq(5)($N=2).$ Figure 1 shows that
the string tension from the restricted and Yang-Mills filed in the adjoint
represent are in good agreement.
In case of $SU(3)$ we investigate the Wilson loop eq(11), i.e., (a) in
the fundamental representation $[0,1]=\mathbf{3}$, (b) in the adjoint
representation $[1,1]=\mathbf{8}$, and (c) in the sextet representation
$[0,2]=\mathbf{6}$. For each representation, we measure the Wilson loop
average for possible reduction conditions, eq(5) ($N=3)$,
eq(6), and eq(7). Figure 2 shows the static
potentials from the Wilson loop in the higher representations. Table
1 shows the string tensions which are extracted by fitting
the data with the linear potential for the infrared region. The string
tensions extracted from our proposed ”dominant” operators reproduce nearly
equal to or more than $80\%$ of the full string tension. These results
indicate that the proposed operators give actually the dominant part of the
Wilson loop average.
5 Conclusion
We have proposed a solution for the problem that the correct behavior of the
Wilson loop in higher representations cannot be reproduced if the restricted
part of the Wilson loop is naively extracted by adapting the Abelian
projection or the field decomposition in the same way as in the fundamental
representation. We have proposed the prescription to construct the operator
suitable for this purpose. We have performed numerical simulations to show
that this prescription works well in the adjoint rep. for $SU(2)$ color group,
and in the fundamental, adjoint , and sextet representations for $SU(3)$
color group. Further studies are needed in order to establish the magnetic
monopole dominance in the Wilson loop average for higher representations,
supplementary to the fundamental representation for which the magnetic
monopole dominance was established.
Acknowledgement
This work was supported by Grant-in-Aid for Scientific Research, JSPS KAKENHI
Grant Number (C) No.15K05042. R. M. was supported by Grant-in-Aid for JSPS
Research Fellow Grant Number 17J04780. The numerical calculations were in part
supported by the Large Scale Simulation Program No.16/17-20(2016-2017) of High
Energy Accelerator Research Organization (KEK), and were performed in part
using COMA(PACS-IX) at the CCS, University of Tsukuba.
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70 (2004) 014506. |
Top Higgs Yukawa Coupling Analysis from
$e^{+}e^{-}\rightarrow\bar{t}tH\rightarrow\bar{b}W^{-}bW^{+}\bar{b}b$
Hajrah Tabassam${}^{1}$ Victoria Martin${}^{2}$
1- Department of Physics
Quaid-i-Azam University Islamabad - Pakistan
2-School of Physics and Astronomy
University of Edinburgh - United Kigdom
Abstract
Electroweak symmetry breaking and the origin of mass of fermion and boson are fundamental questions to understand particle physics. The essential piece of this symmetry breaking, the Higgs boson, is the most probable particle to be discovered at the Large Hadron Collider (LHC).
If one, or more, Higgs particles are discovered,
precise measurements of all the properties of this Higgs will be very important including the measurement of Yukawa couplings of the fermions to the Higgs.
We present a study of $e^{+}e^{-}\rightarrow t\bar{t}H$ at a linear collider, with the aim of making a direct measurement of the the top-Higgs coupling, $g_{t\bar{t}H}$ using the semi-leptonic final state and $M_{H}$ of 120 GeV. We show that the top-Higgs coupling at the center of mass energy 500 GeV, can be measured with an accuracy of better than 28%.
1 Introduction
The ILC is a unique tool which could play an extremely important role in high-precision measurements. Data from the ILC will allow a determination of the profile of the Higgs bosons and their fundamental properties with a high level of confidence and will provide a unique opportunity to establish experimentally the mechanism that generates the particle masses.
The Higgs couplings to the fermions and the gauge bosons are related to the masses of these particles and the only free parameter of the model is the mass of the Higgs boson itself; there are, however, both experimental and theoretical constraints on the Higgs mass in the Standard Model.
Experimentally, the available direct information on the Higgs mass ($M_{H}$) is the lower limit $M_{H}\gtrsim 114.4$ GeV established at LEP [4].
The high accuracy of the electroweak data measured at LEP, SLC and Tevatron provides an indirect sensitivity to $M_{H}$: the Higgs boson contributes logarithmically, $\propto\log(M_{H}/M_{W})$, to the radiative corrections to the $W/Z$ boson propagators. A recent analysis, which uses the updated value of the top quark mass yields the value 85$\pm$25 GeV, with upper limit of $M_{H}\lesssim 165$ GeV [5].
Direct searches by CDF and D0 exclude a region at high mass between $156<M_{H}<177$ GeV [6].
ATLAS and CMS have also made direct searches for the Higgs boson. At the time of this workshop preliminary LHC results exclude the ranges 155-190 GeV [7] at ATLAS and 149-206 GeV [8] at CMS at 95% CL.
2 Higgs Production and Decay
In the Standard Model (SM), the profile of the Higgs particle is uniquely determined once its mass $M_{H}$ is fixed [1, 2]. The decay width, the branching ratios and the production cross sections are given by the strength of the Yukawa couplings to fermions and gauge bosons, the scale of which is set by the masses of these particles.
$$g_{fHf}\propto m_{f}$$
For a top quark mass of $M_{t}=175\;\mathrm{GeV}$, the Standard Model the Higgs Yukawa coupling is ${g_{t\bar{t}H}}=M_{t}/v=0.71$, where $v=246\;\mathrm{GeV}$ is the vacuum expectation value.
The aim of the work presented here is to determine how accurately ${g_{t\bar{t}H}}$ can be determined at the ILC through a measurement of $t\bar{t}H$ production.
There are two Higgs main production channels for $e^{+}e^{-}$ production at $\sqrt{s}=500\;\mathrm{GeV}$: the Higgs-strahlung process including $e^{+}e^{-}\rightarrow ZH$ and associated production with top pair $e^{+}e^{-}\rightarrow t\bar{t}H$ (figure 1) which we use for our analysis. The diagram where the Higgs boson is radiated from the $Z$ boson modifies only slightly the cross-section and thus are neglected in this study.
The signal has a cross section of $0.576\;\mathrm{fb}$ at $\sqrt{s}=500$ GeV due to phase space suppression but, at $\sqrt{s}=800$ GeV, it can reach the level of a few femtobarns.
3 Overview
References [11] and [12] have performed earlier feasibility studies of the top-Higgs Yukawa coupling via the process $e^{+}e^{-}\rightarrow t\bar{t}H$ at a linear collider and in [13], an analysis was performed at 800 GeV center-of-mass energy.
For the ILD detector, we perform the first complete analysis of the process for $M_{H}=120\;\mathrm{GeV}$ in the semileptonic channel where one $W$ boson from the top quark decay, $t\rightarrow bW$, decays as $W\rightarrow\ell\nu$ and the other as $W\rightarrow q\bar{q}$. A full reconstruction of the final state is conducted by reconstructing one hadronic and one leptonic $W$ decays. The top quarks and the Higgs boson are then reconstructed from $W^{\prime}$s and $b$-jets. Missing energy reconstruction has a large impact on the reconstruction of the semi-leptonic channel, and $b$-tagging has a crucial role in separating signal and background. The analysis is carried out for an integrated luminosity, $\mathcal{L}=1000\;\mathrm{fb}^{-1}$ and at $\sqrt{s}=500$ GeV. The main background processes are $t\bar{t}$ and $t\bar{t}Z$ production.
4 Monte Carlo Samples
The analysis was performed with the samples provided by the ILD optimisation group [9] with $M_{H}=120\;\mathrm{GeV}$ and $M_{t}=175\;\mathrm{GeV}$.
The samples were generated at SLAC using WHIZARD, and then simulated and reconstructed on the DESY Grid nodes. The GEANT4 based Mokka package was used for simulation in the ILD-00 detector model, which is the first simulated reference model of ILD. Reconstruction was performed using MarlinReco, PandoraPFA and LCFIVertex with the versions supplied in ILCInstall v01-07.
In table 1, the sample sizes used in the analysis are shown.
5 Event Reconstruction
We reconstruct the semi-leptonic final state of the signal process with one charged lepton (electron or muon) and a neutrino and six jets, accounting for
$\sim 20\%$ of the total cross section.
The final state in this channel has four $b$-jets, two light jets, $j$, one charged lepton, $\ell$, and missing energy for $\nu$:
$$e^{+}e^{-}\rightarrow t\bar{t}H\rightarrow W^{+}bW^{-}\bar{b}b\bar{b}%
\rightarrow\ell\nu\,2j\,4b$$
We select electrons and muons using selection criteria obtained from a single particle sample study.
Left figure 2 shows the reconstructed invariant mass of the charged lepton and the missing energy in the event.
We remove the electrons and muons from our sample and force the remaining particles into exactly six jets using the JETFinder algorithm. The LCFIVertex algorithm is run on the jets to obtain $b$-tagging information [10].
Jets passing the LCFIVertex reconstruction are sorted according to their $b$-tag value. The $b$-tag uses a neural net algorithm, producing a value between 0 and 1 for each jet: 1 being the most $b$-jet like, and 0 being the least $b$-jet like. To suppress backgrounds, and improve the reconstruction efficiency, we select events where at least four of the six jets have $b$-tag $>0.09$.
The four jets with the highest $b$-tag value are considered as the $b$-jets and the two with the lowest $b$-tag value are considered as the light jets. Hadronic $W$ candidates, shown in right figure 2, are created from pairs of light-jets in the event.
The remaining jets are considered as the $b$-jets and used to reconstruct $M_{bb},M_{\ell\nu b}$ and $M_{jjb}$.
Top-quark candidates are reconstructed using a tagged $b$-jet and a reconstructed $W$ candidate and the Higgs boson is reconstructed from a pair of tagged $b$-jets. All possible combinations of tagged $b$-jets are examined to reconstruct the Higgs boson and both top-quarks simultaneously.
To reduce combinatorial background, we define $\chi^{2}$ for each event:
$$\chi^{2}=\frac{(M_{\ell\nu b}-M_{t})^{2}}{\sigma_{\ell\nu b}^{2}}+\frac{(M_{%
jjb}-M_{t})^{2}}{\sigma_{jjb}^{2}}+\frac{(M_{bb}-M_{H})^{2}}{\sigma_{bb}^{2}}$$
(1)
where take the widths, $\sigma$, for each combinations from the reconstructed resolutions:
$$\sigma_{\ell\nu b}=20.7\pm 0.2\;\mathrm{GeV}\hskip 14.226378pt\sigma_{jjb}=23.%
5\pm 0.3\;\mathrm{GeV}\hskip 14.226378pt\sigma_{bb}=19.3\pm 0.2\;\mathrm{GeV}$$
The mass distributions of the lowest $\chi^{2}$ combination are given in figure 3.
6 Signal and Background Separation
The main backgrounds are from $t\bar{t}$ and $t\bar{t}Z$. We used the following variables to discriminate between signal and background by maximising ${S}/{\sqrt{S+B}}$ (where $S$ is signal and $B$ is background).
•
$E_{\mathrm{SumReco}}$: sum of the energies of all reconstructed particles in the event.
•
$P_{{\mathrm{Lepton}}}$: the reconstructed momentum of the identified lepton.
•
$P_{{\mathrm{miss}}}$: the reconstructed missing momentum.
•
$P_{{\mathrm{Jets}}}$: the momentum of each of the reconstructed jets.
•
$\chi^{2}$, as defined in equation (1).
•
$m_{\ell\nu}$: leptonic $W$ candidate mass
•
$m_{b\bar{b}}$: Higgs candidate mass.
•
Total reconstructed mass $m_{b\bar{b}}+m_{\ell\nu b}+m_{jjb}$.
Figure 4 shows some of these variables for signal and background.
The optimised selection on the above variables selects $(0.11\pm 0.01)\%$ of $t\bar{t}$, $(2.76\pm 0.12)\%$ of $t\bar{t}Z$, and $(7.57\pm 0.19)\%$ of signal events.
7 Measurement of $\mathbb{g_{t\bar{t}H}}$
In the Standard Model the Higgs Yukawa coupling is written in terms of the top quark mass ${g_{t\bar{t}H}}=M_{t}/v=0.71$.
To measure the top Higgs Yukawa coupling, we followed procedure given in [13].
The statistical and systematic uncertainties on the measurement of $g_{t\bar{t}H}$ can be expressed as:
$$\left(\frac{\Delta g_{t\bar{t}H}}{g_{t\bar{t}H}}\right)_{\mathrm{stat}}\approx%
\frac{1}{S_{\mathrm{stat}}(g^{2}_{t\bar{t}H})\sqrt{\epsilon\rho\mathcal{L}}}$$
$$\left(\frac{\Delta g_{t\bar{t}H}}{g_{t\bar{t}H}}\right)_{\mathrm{syst}}\approx%
\frac{1}{S_{\mathrm{syst}}(g^{2}_{t\bar{t}H})}\frac{1-\rho}{\rho}\frac{\Delta%
\sigma_{\mathrm{eff}}^{\mathrm{BG}}}{\sigma_{\mathrm{eff}}^{\mathrm{BG}}}\\
$$
Where $\epsilon=(7.57\pm 0.19)\%$ is the efficiency to select the data and $\rho=(12.5\pm 0.3)\%$ is the purity of the final selected sample. We use an integrated luminosity of $\mathcal{L}=1000\;\mathrm{fb}^{-1}$, equivalent three to four year’s running of the ILC.
The largest unknown comes from the knowledge of the background cross sections,
$\Delta\sigma_{\mathrm{eff}}^{\mathrm{BG}}/{\sigma_{\mathrm{eff}}^{\mathrm{BG}}}$, dominated by current knowledge of $t\bar{t}$ production. Following reference [13], we choose a relative uncertainty of 5%;
an ongoing study [16] will allow a better estimate of this uncertainty.
Ignoring the small contribution from Higgs radiation off the $Z$ (as shown in figure 1)
we can write:
$$\frac{d\sigma_{t\bar{t}H}}{d(g^{2}_{t\bar{t}H})}\approx\frac{\sigma_{t\bar{t}H%
}}{g^{2}_{t\bar{t}H}}=0.087\;\mathrm{fb}$$
with $\sigma_{t\bar{t}H}=0.044\;\mathrm{fb}$ being the effective cross section $7.57\%$ of the total cross section and ${g_{t\bar{t}H}}=0.71$.
Then the sensitivity factors $S_{\mathrm{stat}}$ and $S_{\mathrm{syst}}$ simply depend on cross section and on the squared coupling:
$$S_{\mathrm{stat}}(g^{2}_{t\bar{t}H})=\frac{1}{\sqrt{\sigma_{t\bar{t}H}}}\left|%
\frac{d\sigma_{t\bar{t}H}}{d(g^{2}_{t\bar{t}H})}\right|=\frac{\sqrt{\sigma_{t%
\bar{t}H}}}{g^{2}_{t\bar{t}H}}=1.50\;\mbox{fb}^{1/2}\hskip 28.452756pt$$
$$S_{\mathrm{syst}}(g^{2}_{t\bar{t}H})=\frac{1}{\sigma_{t\bar{t}H}}\left|\frac{d%
\sigma_{t\bar{t}H}}{d(g^{2}_{t\bar{t}H})}\right|=\frac{1}{g^{2}_{t\bar{t}H}}=1%
.98$$
The expected statistical and systematic uncertainties are therefore:
$$\left(\frac{\Delta g_{t\bar{t}H}}{g_{t\bar{t}H}}\right)_{\mathrm{stat}}=21.6\%%
\hskip 28.452756pt\left(\frac{\Delta g_{t\bar{t}H}}{g_{t\bar{t}H}}\right)_{%
\mathrm{sys}}=17.6\%$$
Combining the statistical and systematic uncertainties in quadrature we obtain an estimate on the overall uncertainty on ${g_{t\bar{t}H}}$ of 27.9%.
8 Results and Discussion
Figure 5 shows the distribution for the scaled signal and background samples. The main background after selection is due to the top-pair production.
The expected precision on measuring the top-Higgs Yukawa coupling is better than 28% for $M_{H}=120$ GeV,
if the knowledge of $t\bar{t}$ background normalisation is known at the 5% level.
Much of the uncertainty on the results of this analysis is due to the small cross section of the $t\bar{t}H$ signal process. Even though the signal cross section is very small at the $\sqrt{s}$ = 500 GeV, yet 28% accuracy is achieved.
Certainly there is room for improvement of the study presented.
In jet finding, we force particles into a jet which sometime don’t belong to that jet. It could affect the energy of the reconstructed jet. A scaling of the flavour tag input variables to the reconstructed jet energy can minimise these effects.
One of the areas of consideration is the improvement in the hadronic $W$ reconstruction.
A cut on the $b$-tag value of the jets is used to identify light jets which are then combined to reconstruct the $W$ boson. This cut was decided to accommodate the selection cut on $b$-tag of third and fourth jet. A re-examining of the cut on $b$-tag of fifth and sixth jet could recover the signal events with a $b$-tag $>0.09$ and hence, improve the efficiency and purity of the sample.
A cut based strategy is applied to discriminate the signal and background events. It is shown in previous studies that the use of neural network and likelihood methods can perform the signal background separation with better efficiency [12]. Hence, it is anticipated that a 2-3% improvement in the precision can be achieved by using neural network analysis.
For the current study, six fermion backgrounds are not included but reference [13] shows that the loss of precision on $g_{t\bar{t}H}$ measurement is negligible due to this background. An amendment in the analysis involves the $t\bar{t}$ background normalisation which is 5% in our analysis. If the exact background normalisation ${\Delta\sigma_{\mathrm{eff}}^{\mathrm{BG}}}/{\sigma_{\mathrm{eff}}^{\mathrm{BG%
}}}$ is used, our results might change. Once the results from the group working on measuring the cross section of top pair production [16] are ready, they can be included in the analysis.
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http://ilcsoft.desy.de/portal/data_samples
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Philippe Doublet (LAL)
http://ilcagenda.linearcollider.org/conferenceDisplay.py?confId=4983 |
Uncertainty principle in a cavity at finite temperature
A. P. C. Malbouisson${}^{a}$
(a) Centro
Brasileiro de Pesquisas Físicas/MCTI, 22290-180, Rio de
Janeiro, RJ, Brazil
[email protected] (A.P.C. Malbouisson)
Abstract
We employ a dressed state approach to perform a study on the behavior of the uncertainty principle for a system in a heated cavity. We find, in a small cavity for a given temperature, an oscillatory behavior of the momentum–coordinate product, $(\Delta\,p)\,(\Delta\,q)$, which attains periodically finite absolute minimum (maximum) values, no matter
large is the elapsed time. This behavior is in a sharp contrast with
what happens in free space, in which case, the product $(\Delta\,p)\,(\Delta\,q)$ tends asymptotically, for each temperature, to a constant value, independent of time.
PACS number(s): 03.65.Ta
Introduction
An account on the subject of an interacting particle-environment system,
can be found in Refs. zurek ; paz ; haake ; caldeira ; livro , in which the environment
is considered as an infinite set of noninteractng oscilllators.
Here we consider a similar model, treated with a different approach.
Let us therefore begin with some words about the method we employ: From a general point of view,
apart from computer calculations in lattice field theory, the must currently used
method to treat the physics of interacting particles is perturbation theory, in which
the starting point are bare fields (particles) interacting by means of gauge fields.
Actually, as a matter of principle, the idea of
bare particles associated to bare matter fields and of a gauge particle mediating the
interaction among them, is in fact an artifact of perturbation
theory and, strictly speaking, is physically meaningless. A charged physical particle is always coupled to
the gauge field, it is always ”dressed” by a cloud of quanta of the gauge
field (photons, in the case of Electrodynamics). Exactly the same type of argument
applies mutatis mutandis to a particle-environment system,
in which case we may speak of a ”dressing”
of the particle by the thermal bath, the
particle being ”dressed” by a cloud of quanta of the environment.
This should be true in general for any system in which a material particle is coupled to a
field, no matter the specific nature of the field (environment) nor of the interaction involved.
We give a treatment to this kind of system using some dressed (or renormalized)
coordinates. In terms of these new coordinates dressed states are defined, which
allow to divide the coupled system into two parts, the dressed
particle and the dressed environment, which makes unnecessary to work
directly with the concepts of bare particle, bare environment and interaction
between them. A detailed exposition of our formalism and of its meaning, for both zero- and
finite temperature can be found in Refs. linhares ; termalizacao ; termica ; emaranho ; erico .
About the physical situation we deal with, on general grounds, very precise investigations have been done
on the fundamentals of quantum physics, in particular on the validity of the Heisenberg uncertainty relation. In clerk ,
it is reported that a great deal of
effort is being made to minimize external noising factors,
such as thermal fluctuations and electricity oscillations in experiments,
in order to verify the relation, in the spirit of zero-temperature quantum physics.
However, changes in the uncertainty principle induced by temperature is an idea already explored in the literature, in particular for open systems. In Abe , the authors study with a thermo-field-dynamics formalism, the relation between the sum of information-theoretic entropies in quantum mechanics with measurements of the position and momentum of a particle surrounded by a thermal environment. It is found that this quantity cannot be made arbitrarily small but has a universal lower bound dependent on the temperature. They also show that the Heisenberg uncertainty relation at finite temperature can be derived in this context. In Hu ; Hu1 it is obtained the uncertainty relation for a quantum open system consisting of a
Brownian particle interacting with an
ohmic bath of quantum oscillators at finite temperature.
These authors claim that this allows to get some insight into the physical mechanisms involved in the environment-induced decoherence process.
Also, modifications of the uncertainty principle have been proposed in, for instance,
a cosmological context. As remarked in amelino , in quantum gravity it seems to be
needed a generalized position–momentum uncertainty principle. The authors of
ref. amelino investigate a possible connection between the generalized uncertainty principle and
changes in the area–entropy black hole formula and the black hole evaporation process.
In this report, we study the behavior of the particle-environment system contained
in a cavity of arbitrary size, under the influence of a heated environment. The environment is
composed of an infinity of oscillators, and assumed to be at a given temperature, realized by taking
an appropriate thermal distribution for its modes.
This generalizes previous works for zero temperature, as for instance in Refs. adolfo3 ; adolfo4 ; adolfo5 ,
for both inhibition of spontaneous decay in cavities and the Brownian motion.
We study the time dependent mean value for the dressed
oscillator position operator taken in a dressed coherent state. We find that in both cases,
of the environment at zero temperature or of the heated environment,
these mean values are independent of the temperature and are given by the same expression.
On the other side, the mean squared error for both the particle position and momentum
are dependent on the temperature. From them we get the time and
temperature-dependent uncertainty Heisenberg
relation. We then investigate how heating
affects the uncertainty principle in a cavity of arbitrary size. This is particularly
interesting in a small cavity, where the result is not a trivially expected one.
The model
Our approach to the problem
makes use of the notion of dressed thermal states termica ,
in the context of a model already employed in the literature, of atoms, or
more generally material particles, in the harmonic approximation, coupled to
an environment modeled by an infinite set of point-like harmonic oscillators
(the field modes). The dressed thermal state approach is an extension of the
dressed (zero-temperature) formalism already used earlier adolfo1 ; adolfo3 ; adolfo4 ; adolfo5 .
We consider a bare particle (atom, molecule,…) approximated by a harmonic
oscillator described by the bare coordinate and momentum $q_{0},p_{0}$
respectively, having bare frequency $\omega_{0}$, linearly
coupled to a set of $N$ other harmonic oscillators (the environment)
described by bare coordinate and momenta $q_{k},p_{k}$ respectively,
with frequencies $\omega_{k}$, $k=1,2,\ldots,N$. The limit
$N\rightarrow\infty$ will be undesrstood. The whole system is
supposed to reside inside a perfectly reflecting spherical cavity of radius $R$ in
thermal equilibrium with the environment, at a temperature
$T=\beta^{-1}$. The system is described by the Hamiltonian
$$H=\frac{1}{2}\left[p_{0}^{2}+\omega_{0}^{2}q_{0}^{2}+\sum_{k=1}^{N}\left(p_{k}%
^{2}+\omega_{k}^{2}q_{k}^{2}\right)\right]-q_{0}\sum_{k=1}^{N}c_{k}q_{k},$$
(1)
The Hamiltonian (1) is transformed to principal
axis by means of a point transformation,
$q_{\mu}=\sum_{r=0}^{N}t_{\mu}^{r}Q_{r}\,,\,\,\,p_{\mu}=\sum_{r=0}^{N}t_{\mu}^{%
r}P_{r};\;\mu=(0,\{k\}),\;k=1,2,...,N;\;\;r=0,...N$,
performed by an orthonormal matrix $T=(t_{\mu}^{r})$. The subscript
$r$ refers to the normal
modes. In terms of normal momenta and coordinates, the transformed
Hamiltonian reads
$H=\frac{1}{2}\sum_{r=0}^{N}(P_{r}^{2}+\Omega_{r}^{2}Q_{r}^{2})$,
where the $\Omega_{r}$’s are the normal frequencies corresponding
to the collective stable oscillation modes of the coupled
system. Using the coordinate transformation in the
equations of motion and explicitly making use of the
normalization of the matrix $(t_{\mu}^{r})$, $\sum_{\mu=0}^{N}(t_{\mu}^{r})^{2}=1$,
we get the matrix elements $(t_{\mu}^{r})$ adolfo1 .
We take $c_{k}=\eta(\omega_{k})^{u}$, where $\eta$ is a constant
independent of $k$. In this case the environment is classified
according to $u>1$, $u=1$, or $u<1$, respectively as
supraohmic, ohmic or subohmic paz ; haake ;
we take, as in adolfo1 , $\eta=2\sqrt{g\Delta\omega/\pi}$,
where $\Delta\omega$ is the interval between two
neighboring field frequencies and $g$ a fixed constant
characterizing the strenght of the coupling particle-environment.
Restricting ourselves to an ohmic environment, we get an equation for the
$N+1$ eigenfrequencies $\Omega_{r}$,
corresponding to the
$N+1$ normal collective modes adolfo1 . In this
case the eigenfrequencies
equation contains a divergence for $N\rightarrow\infty$ and a
renormalization procedure is needed. This leads to
the renormalized frequency adolfo1 (this renormalization procedure was pioneered in Thirring ),
$$\bar{\omega}^{2}=\omega_{0}^{2}-\delta\omega^{2}=\lim_{N\rightarrow\infty}(%
\omega_{0}^{2}-N\eta^{2}).$$
(2)
where we have defined the counterterm
$\delta\omega^{2}=N\eta^{2}$.
We introduce dressed or renormalized coordinates $q_{0}^{\prime}$ and
$\{q_{k}^{\prime}\}$ for, respectively, the dressed atom
and the dressed field, defined by,
$$\sqrt{\bar{\omega}_{\mu}}q_{\mu}^{\prime}=\sum_{r}t_{\mu}^{r}\sqrt{\Omega_{r}}%
Q_{r},$$
(3)
where $\bar{\omega}_{\mu}=\{\bar{\omega},\;\omega_{k}\}$. In terms of these, we
define thermal dressed states, precisely described in termalizacao ; termica .
It is worthwhile to note that our renormalized coordinates are
objects different from both the bare coordinates, $q$, and the
normal coordinates $Q$. Also, our dressed states, although being
collective objects, should not be confused with the eigenstates
of the system footnote .
In terms of our renormalized coordinates and dressed states, we can find a natural
division of the system into the dressed (physically observed) particle and
the dressed environment. The dressed particle will contain automatically
all the effects of the environment on it.
A cavity of arbitrary size at finite temperature
To study the behavior of the system in a cavity of arbitrary size, we write the initial
physical state in terms of dressed coordinates. We assume that initially the system is described by the density operator,
$\hat{\rho}(0)=\hat{\rho}_{0}´\otimes\hat{\rho}_{\beta}^{\prime}\;,$
where $\hat{\rho}_{0}´$ is the density operator associated with the oscilator $q_{0}^{\prime}$, that
can be in general in a pure or mixed state. Also,
$\hat{\rho}_{\beta}^{\prime}$ is the dressed density operator associated with the dressed field
modes. We assume thermal equilibrium for these dressed modes, thus
$$\hat{\rho}_{\beta}^{\prime}=\frac{\bigotimes_{k}e^{-\beta\hat{H}_{k}^{\prime}}%
}{{\rm Tr}\bigotimes_{k}e^{-\beta\hat{H}_{k}^{\prime}}}\,\;{\rm{with}}\;\hat{H%
}_{k}^{\prime}=\frac{1}{2}\hat{p}_{k}^{\prime 2}+\frac{1}{2}\omega_{k}^{2}\hat%
{q}_{k}^{\prime 2}\;.$$
(4)
The time evolution of the density operator is given by the Liouville-von Newman equation, whose solution,
in the case of an entropic evolution, is given by
$\hat{\rho}(t)=e^{-\frac{i}{\hbar}\hat{H}t}\hat{\rho}(0)e^{\frac{i}{\hbar}\hat{%
H}t}\;.$
Then, the time evolution of the average thermal expectation value of an operator is given by,
$$\langle\hat{A}\rangle(t)={\rm Tr}\left(\hat{A}e^{-\frac{i}{\hbar}\hat{H}t}\hat%
{\rho}(0)e^{\frac{i}{\hbar}\hat{H}t}\right)\,={\rm Tr}\left(\hat{A}(t)\hat{%
\rho}(0)\right)\,$$
(5)
where the ciclic property of the trace has been used; above,
$\hat{A}(t)=e^{\frac{i}{\hbar}\hat{H}t}\hat{A}e^{-\frac{i}{\hbar}\hat{H}t}$ is the time-dependent operator
$\hat{A}$ in the
Heisenberg representation.
Dressed coherent states in a heated environment
Let us consider a Brownian
particle embedded in a heated environment as described above. In our language,
we speak of a dressed Brownian
particle and we use the dressed state formalism.
We assume, as usual, that initially the particle and the environment are
decoupled and that the coupling is turned on suddenly at some given time, that we choose at
$t=0$.
In our formalism, we define
$|\lambda\left.\right>$ as a dressed coherent state given by,
$$|\left.\lambda,n_{1}^{\prime},n_{2}^{\prime},..;t=0\right>={\rm e}^{-|\lambda|%
^{2}/2}\sum_{n_{0}^{\prime}=0}^{\infty}\frac{\lambda^{n_{0}^{\prime}}}{\sqrt{n%
_{0}^{\prime}!}}|\left.n_{0}^{\prime}n_{1}^{\prime},..\right>\;,$$
(6)
where $n_{0}^{\prime}$ stands for the occupation number of the dressed particle
and where $n_{1}^{\prime},n_{2}^{\prime},..$ are the occupation numbers of the field modes. For
zero temperature we have $n_{1}^{\prime}=n_{2}^{\prime},..=0$.
For finite temperature,
we will do computations taking $\hat{\rho}_{0}=|\lambda\rangle\langle\lambda|$. This means,
that at the initial time, the dressed particle oscillator is in a pure coherent state.
Keeping this in mind, we consider the quantity,
$\langle\hat{q}_{0}^{\prime}\rangle(t)$, that we denote by $q_{0}^{\prime}(t)$,
$q_{0}^{\prime}(t)={\rm Tr}\left(\hat{q}^{\prime}_{0}(t)\hat{\rho}(0)\right)$.
In order to evaluate the above expression we first compute $\hat{q}^{\prime}_{0}(t)$. Using the relation
between the dressed coordinates and the normal coordinates, Eq.(3),
the expression for $\hat{H}$ in terms of the normal coordinates,
and the Baker-Campbell-Hausdorff formulas, we
get, after some steps of calculation,
$$q_{0}^{\prime}(t)=\sqrt{\frac{\hbar}{2\bar{w}_{0}}}\left(\lambda f_{00}(t)+%
\lambda^{\ast}f_{00}^{\ast}(t)\right),$$
(7)
where $f_{00}(t)$ is one of the quantities $f_{\mu\nu}(t)=\sum_{s}t_{\mu}^{s}t_{\nu}^{s}e^{-i\Omega_{s}t}$ adolfo3 .
Note that the above expression is independent of the temperature and coincides with the one
obtained previously for the the zero temperature case adolfo3 . This is because
$\hat{\rho}_{\beta}$ has even parity in the dressed momentum and position operators.
Due to the same reason, we find an entirely similar
formula for $p_{0}^{\prime}(t)$.
The situation is different for the quantity $q_{0}^{\prime 2}(t)=\langle\hat{q}_{0}^{\prime 2}\rangle(t)$.
After performing similar computations as above,
we get,
$$\displaystyle q_{0}^{\prime 2}(t,\beta)=\frac{\hbar}{2\bar{\omega}}\left[\left%
(\lambda f_{00}(t)+\lambda^{\ast}f_{00}^{\ast}(t)\right)^{2}\right.$$
$$\displaystyle\left.+2\sum_{k}|f_{0k}(t)|^{2}n_{k}^{\prime}(\beta)+1\right]\;,$$
(8)
Then from Eqs. (7) and (8) we obtain for the mean square error,
$$\displaystyle(\Delta q^{\prime}_{0})^{2}(t,\beta)$$
$$\displaystyle=$$
$$\displaystyle\langle\hat{q}^{\prime 2}_{0}\rangle(t,\beta)-(\langle\hat{q}_{0}%
^{\prime}\rangle(t))^{2}$$
(9)
$$\displaystyle=$$
$$\displaystyle\frac{\hbar}{2\bar{\omega}}+\frac{\hbar}{\bar{\omega}}\sum_{k}|f_%
{0k}(t)|^{2}n_{k}^{\prime}(\beta)\;,$$
where $n_{k}^{\prime}(\beta)$ is given by the Bose-Einstein distribution,
$n_{k}^{\prime}(\beta)=1/(e^{\hbar\beta\omega_{k}}-1)$ termica .
Analogously we obtain
the momentum mean squared error,
$$\displaystyle(\Delta p_{0})^{2}(t,\beta)$$
$$\displaystyle=$$
$$\displaystyle p_{0}^{\prime 2}(t,\beta)-(p_{0}^{\prime}(t))^{2}$$
(10)
$$\displaystyle=$$
$$\displaystyle\frac{\hbar\bar{\omega}}{2}+\hbar\bar{\omega}\sum_{k}|f_{0k}(t)|^%
{2}n_{k}^{\prime}(\beta).$$
From Eqs. (9) and (10) we obtain the time and temperature-dependent Heisenberg relation,
$$\Delta q_{0}^{\prime}(t,\beta)\Delta p_{0}^{\prime}(t,\beta)=\frac{\hbar}{2}+%
\hbar\sum_{k}|f_{0k}(t)|^{2}n_{k}^{\prime}(\beta)\;.$$
(11)
Time behavior in a cavity with a heated environment
Let us consider the time evolution of the uncertainty relation $\Delta q_{0}^{\prime}(t,\beta)\Delta p_{0}^{\prime}(t,\beta)$
given in Eq. (11), in a finite (small) cavity, characterized by the dimensionless
parameter $\delta=gR/\pi c$ and take a
coupling regime defined by a relation between $g$ and the
emission frequency $\bar{\omega}$, $g=\alpha\,\bar{\omega}$ . For instance, if we consider $\delta=0.1$,
$\alpha=0.2$ and $\bar{\omega}\approx 10^{14}/{\rm{s}}$ (in the visible red),
this corresponds to a cavity radius $R\sim 10^{-6}{\rm{m}}$. We
measure the uncertainty relation in units of $\hbar$ and call it for simplicity
$\Delta q_{0}^{\prime}(t,\beta)\Delta p_{0}^{\prime}(t,\beta)=\Delta(t,\beta)$.
Then calculations can be performed in a similar
way as in Ref. termica and we obtain (there is no confusion between the variable $t$ describing time
and the matrix elements $t_{\mu}^{r}$),
$$\displaystyle\Delta(t,\beta)=\frac{1}{2}+\sum_{k=1}^{\infty}\frac{1}{e^{(\hbar%
\beta g/\delta)k}-1}\left[(t_{0}^{0})^{2}(t_{k}^{0})^{2}\right.$$
$$\displaystyle\left.+2\sum_{l=1}^{\infty}t_{0}^{0}t_{0}^{l}t_{k}^{0}t_{k}^{l}%
\cos(\Omega_{0}-\Omega_{l})t\right.$$
$$\displaystyle\left.+\sum_{l,n=1}^{\infty}t_{0}^{l}t_{0}^{n}t_{k}^{l}t_{k}^{n}%
\cos(\Omega_{l}-\Omega_{n})t\right]\,.$$
The matrix
elements $t_{\mu}^{r}$ in the above formulas are
evaluated in termica ,
$$\displaystyle t_{k}^{0}\approx\frac{k\,g^{2}\sqrt{2\delta}}{k^{2}g^{2}-\Omega_%
{0}^{2}\delta^{2}}\,;\;\;t_{k}^{l}\approx\frac{2k\delta}{k^{2}-(l+\epsilon_{l}%
)^{2}}\frac{1}{l}$$
$$\displaystyle(t_{0}^{k})^{2}\approx\frac{2gR}{\pi ck^{2}}=\frac{2\delta}{k^{2}%
}\,,\;\;(t_{0}^{0})^{2}\approx 1-\frac{\pi gR}{3c}=1-\frac{\pi^{2}\delta}{3},$$
where $\epsilon_{l}$ is a small quantity such that $0<\epsilon_{l}<1$. Actually, for a
small cavity ($\delta\ll 1$) $\epsilon_{l}\approx\delta/k$.
Comments
Eq. (LABEL:n0tbeta)
describes the time evolution of the uncertainty relation
$\Delta q_{0}^{\prime}(t,\beta)\Delta p_{0}^{\prime}(t,\beta)\equiv\Delta(t,\beta)$ in a small cavity.
A plot of this time evolution is given in Fig. 1
for some representative values of the temperature.
The thermal uncertainty function, $\Delta(t,\beta)$, is an oscillating
function which attains periodically an absolute minimum (maximum) value,
${\rm{Min}}[\Delta(t,\beta)]$ (${\rm{Max}}[\Delta(t,\beta)]$). Since
the periodical character of $\Delta(t,\beta)$ does not involve $\beta$, the
location of these extrema on the time axis
do not depend on the temperature. Indeed, we can see from Fig.1, that
the values of these minima and maxima depend on the temperature,
$\beta^{-1}$, but appears to be, for each temperature, the same for all values of time where they occur,
in other words the values of the absolute extrema appears to be independent of time.
In the detail of Fig.1, are plotted two neighboring absolute minimum and
maximum (corresponding to $t\approx 2.3$ and $t\approx 2.5$), as functions of temperature.
We find from these figures that raising the temperature increases the
amplitude of oscillation and the mean value of the uncertainty relation and that its
lower and upper bounds, also grows with temperature.
We infer from Fig.1 that for a small cavity, in all cases an oscillatory behavior is
present for $\Delta(t,\beta)$, with the amplitude of the oscillation depending on the temperature, $T$.
For larger values of $T$ the
amplitude of the oscillation, and both, its absolute minimum and maximum values,
are larger than for lower temperatures.
This behavior of the uncertainty principle is to be contrasted with
the case of an arbitrarily large cavity (free space). In this last case, the product $(\Delta\,p)\,(\Delta\,q)$
goes, asymptotically, as $t\rightarrow\infty$ for each temperature, to a constant value $\Delta(\beta)$.
This asymptotic value depends on the temperature and grows with it, but is independent of time.
Distinctly, for a small cavity, even for $t\rightarrow\infty$, the product $(\Delta\,p)\,(\Delta\,q)$ presents
oscillations, which have larger and larger amplitudes for higher temperatures.
The result above falls in a general context of the different behaviors of quantum systems confined in
cavities, as compared to free space, in both, zero or finite temperature, the system
being investigated using the formalism presented in this report.
In adolfo3 some of
us got the expected result
that the probability, $P(t)$, that a simple cold atom in free space, excited at $t=0$ remain excited
after an elapsed time $t$, decays monotonically, going to zero as $t\rightarrow\infty$, while
in a small cavity,
$P(t)$ has an oscillatory behavior, never reaching zero. For $\bar{\omega}\sim 10^{14}$ (in the visible red),
$R\approx 10^{-6}{\rm{m}}$, in a weak (of the order of electromagnetic) coupling regime, ${\rm{Min}}\,P(t)\approx 0.98$,
which is in agreement with experimental observations Haroche3 . At finite temperature it is obtained in termica ,
for a small cavity, that the occupation number of a simple atom in a heated environment has an oscillatory
behavior with time and that its mean value increases with increasing temperature. In emaranho ; erico the
behavior of an entangled bipartite system at zero and finite temperature is investigated; taking two measures of
entanglement, it results an oscillatory behavior for a small cavity (entanglement is preserved at all times),
while it disappears as $t\rightarrow\infty$ for free space.
We hope that the result presented in this report could have some usefulness in nanophysics or in quantum
information theory. At this moment we are not able to comment about these aspects; they will be subject of future studies.
Acknowledgements: The author thanks FAPERJ and CNPq (brazilian agencies) for partial financial support.
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Phys. Rev. Lett., 58, 666 (1987) |
Uncovering latent structure in valued graphs: A variational approach
[
[
[
[
[
[
AgroParisTech and INRA, AgroParisTech and INRA
and
INRA and University Bordeaux I
M. Mariadassou
S. Robin
UMR 518 AgroParisTech/INRA MIA
16, rue C. Bernard, F-75005 Paris
France
E-mail: e2
C. Vacher
UMR 1202 Univ. Bordeaux I/INRA BioGeCo
69, route d’Arcachon, F-33612
Cestas
France
(#1; 10 2008; 4 2010)
Abstract
As more and more network-structured data sets are available, the
statistical analysis of valued graphs has become common place.
Looking for a latent structure is one of the many strategies used
to better understand the behavior of a network. Several methods
already exist for the binary case.
We present a model-based strategy to uncover groups of nodes in
valued graphs. This framework can be used for a wide span of
parametric random graphs models and allows to include covariates.
Variational tools allow us to achieve approximate maximum
likelihood estimation of the parameters of these models. We
provide a simulation study showing that our estimation method
performs well over a broad range of situations. We apply this
method to analyze host–parasite interaction networks in forest
ecosystems.
,
host–parasite interactions,
latent structure,
mixture model,
random graph,
valued graph,
variational method,
doi: 10.1214/10-AOAS361
††volume: #1††issue: 2
4
2010
{aug}
A]Mahendra Mariadassoulabel=e1][email protected],
A]Stéphane Robinlabel=e2][email protected]
and C]Corinne Vacherlabel=e3][email protected]
Ecological networks
.
1 Introduction
Data sets presenting a network structure are increasingly studied in many
different domains such as sociology, energy, communication, ecology or
biology [Albert and Barabási (2002)]. Statistical tools are therefore needed to analyze
the structure of these networks, in order to understand their
properties or
behavior. A strong attention has been paid to the study of various
topological characteristics such as degree distribution, clustering
coefficient and diameter [see, e.g., Barabási and Albert (1999), Newman, Watts and Strogatz (2002)]. These
characteristics are useful to describe networks but not sufficient to
understand its whole structure.
A natural and intuitive way to capture an underlying structure is to look
for groups of edges having similar connection profiles [Getoor and Diehl (2004),
Newman, Watts and Strogatz (2002)], which is refereed to as community detection
[Girvan and Newman (2002), Newman (2004)]. This usually turns into an unsupervised
classification (or clustering) problem which requires efficient estimation
algorithms since the data set at hand is ever increasing.
Several attempts at community detection have been proposed in the
literature: greedy algorithms for community detection [Girvan and Newman (2002) and
Newman (2004)] and clustering based on spectral analysis of the adjacency
matrix of a graph [von Luxburg, Belkin and
Bousquet (2008)]. Greedy algorithms and spectral
clustering both assume that communities are determined by a strong
within connectivity opposed to a low between connectivity. This might
be true for so-called communities but need not be true for other
groups of nodes. For example, a group of nodes loosely connected to
each other but highly connected to a specific group of hubs have the
same connection profile and form a homogeneous group but do not form
a community. In addition, they do not offer an explicit generative
model nor a criterion to select the correct number of communities.
Model-based methods are appealing by contrast: explicit modeling of the
heterogeneity between nodes gives different groups an intuitive and
easy to
understand interpretation. Several probabilistic models exists for random
graphs [see Pattison and Robins (2007) for a complete review], ranging from the seminal
Erdös–Rényi (ER) model [Erdös and Rényi (1959)] to the sophisticated Stochastic
Block Model (SBM) [Nowicki and Snijders (2001)]. The ER model assumes independent and
identically distributed edges which entails that all nodes are structurally
equivalent and, thus, there is only one community, although a big one. The
$p_{1}$ model from Holland and Leinhardt (1981) extended the ER model by assuming independent
dyads instead of edges, allowing the breakthrough from
undirected to directed graphs. But again, all nodes are structurally
equivalent in the $p_{1}$ model. Fienberg and Wasserman (1981) and Fienberg, Meyer and Wasserman (1985) lifted these
constraints by assuming the nodes are distributed among $Q$ classes with
different connectivity profiles. In this model, groups are easily
interpreted as nodes belonging to the same class. Unfortunately,
Fienberg, Meyer and Wasserman (1985) assumes class assignments are perfectly well known, which
rarely happens. The state of the art in terms of graph modeling is the SBM,
inspired by Lorrain and White (1971) and introduced by Nowicki and Snijders (2001), which takes
advantage of mixture models and unknown latent variables to allow an easy
modeling of groups without requiring them to be known in advance.
In the SBM framework, community detection boils down to three crucial
steps: assignment of nodes to groups, estimation of the model parameter and
selection of the correct number of groups. Several authors offered their
method to solve these issues using Bayesian methods. Nowicki and Snijders (2001) work with
the original SBM model. Hofman and Wiggins (2008) work in a highly constrained version
of SBM in which heterogeneity is strictly limited to intra- and
inter-community connection and thus characterized by only two parameters,
against $Q^{2}$ in the unconstrained SBM. Airoldi et al. (2008) extend the SBM
framework by allowing nodes to exhibit multiple communities. By contrast,
Daudin, Picard and Robin (2008) use a frequentist approach to estimate the parameters of the
SBM. The frequentist approach is less computation intensive than its
Bayesian counterpart, whereas the Bayesian approach is supposed to better
account for the uncertainty. With the notable exception of Nowicki and Snijders (2001),
who use MCMC to estimate the model parameter, both lines of work make heavy
use of variational techniques: either Variational EM [Jaakkola (2000)] or
Variational Bayes [Attias (2000); Beal and Ghahramani (2003); Xing, Jordan and Russell (2003); Winn, Bishop and Jaakkola (2005)]. MCMC computational cost
is prohibitive, effectively leading to severe size limitations (around 200
nodes). Furthermore, because of the complex likelihood landscape in the
SBM, good mixing of the Markov Chain is hard to achieve and
monitor. Variational approximations, by contrast, replace the
likelihood by
a simple surrogate, chosen so that the error is minimal in some
sense. Frequentist and Bayesian approach then differ only in the use of
this surrogate likelihood: Bayesians combine it to a prior distribution of
the parameter (chosen from some suitable distribution), whereas frequentists
use it directly. In all these methods, the number of groups is fixed during
the estimation procedure and must be selected using some criterion. By
contrast, Kemp, Griffiths and Tenenbaum (2004) propose an original approach where the number of
groups changes and is selected during the estimation process. Both
Bayesian and frequentist estimations approaches give the same kind of
results: an optimal number of groups and a probabilistic assignment of
nodes to groups, depending on their connection profile. However, the
Bayesian estimation strategy leads to severe constraints on the choice of
prior and hyperprior distributions. The Daudin, Picard and Robin (2008) maximum likelihood
approach does not require any prior specification and is more efficient
than MCMC estimation [Picard et al. (2007)].
Previous models are all models for binary networks, for which the only
information is the presence or absence of an edge. Binary information
certainly describes the topology of a network but is a rather poor
description. It accounts neither for the intensity of the
interaction between two nodes nor for the specific features of an
edge. The intensity of an edge may typically indicate the amount of
energy transported from one node to another, the number of passengers
or the number of common features between two nodes, whereas the
specific feature of an edge may be the phylogenetic distance between
its two ending nodes. Many networks, such as power, communication,
social, ecological or biological networks, are naturally valued and
are somehow arbitrarily transformed to a binary graph. This
transformation sometimes conceals important results [Tykiakanis, Tscharntke and
Lewis (2007)].
Extending binary models and the associated estimation procedures to
valued graphs with specific features allows more complexity, and more
relevant information with it, to be processed while estimating the
structure of the network.
We are motivated by the search of a structure in valued graphs
describing the similarity between species within an assemblage
according to their biotic interactions. In ecology, an assemblage is
defined as a taxonomically related group of species that occurs in the
same geographic area [Ricklefs and Miller (2000)]. The species composing an
assemblage usually interact with many species belonging to other
assemblages and the nature of these interactions is often very diverse
(predator–prey interactions, host–parasite interactions, mutualistic
interactions, competitive interactions). One of the questions facing
ecologists is to understand what determines with whom a species
interact. Conventional wisdom is that within an assemblage, two
closely related species should share more interactions than two
evolutionary distant species because the range of interactions of a
species is constrained by its physiological, morphological and
behavioral attributes. In several cases, this conventional wisdom
is revealed to be true. Phylogenetically related plant species have been
shown to bear similar pathogens and herbivores [Brandle and Brandl (2006); Gilbert and Webb (2007)]
and the diet’s range of predators has been shown to be
phylogenetically constrained [Cattin et al. (2004)]. This tendency for
phylogenetically related species to resemble each other is called
phylogenetic signal [Blomberg and Garland (2002)]. In other cases, no phylogenetic
signal was detected [Rezende et al. (2007); Vacher, Piou and
Desprez-Loustau (2008)]. Selection pressures exerted
by the environment might account for this absence: species have to
adapt to varying environments to survive, diverging from close
relatives in their physiology, morphology and behavior, and possibly
developing novel interactions [Bersier and Kehrli (2008); Cattin et al. (2004)]. The valued graphs
under study have species as nodes and the number of shared
interactions as edges. We use a mixture model with phylogenetic
distance between species as covariate to measure the strength of the
phylogenetic signal. This latter is defined as the decrease in the
number of selected groups due to the inclusion of the covariate. Two
different assemblages are considered. The first assemblage is composed
of 51 tree species occurring in the French forests and the second is
composed of 153 parasitic fungal species also occurring in the French
forests. The interactions considered are host–parasite interactions.
We expect to find a lower phylogenetic signal in the host range of
parasitic fungal species [Bersier and Kehrli (2008); Rossberg et al. (2006); Vacher, Piou and
Desprez-Loustau (2008)]
than in the
vulnerability of tree species to parasites [Brandle and Brandl (2006); Gilbert and Webb (2007);
Vacher, Piou and
Desprez-Loustau (2008)].
In this paper we propose an extension to the stochastic block model,
introduced in Fienberg and Wasserman (1981); Fienberg, Meyer and Wasserman (1985); Nowicki and Snijders (2001), and the methods
of Airoldi and Carley (2005)
and Daudin, Picard and Robin (2008), that deals with valued graphs and accounts for
possible covariates. We use a general mixture model describing the
connection intensities between nodes spread among a certain number of
classes (Section 2). A variational EM approach to get an
optimal, in a sense to be defined, approximation of the likelihood is
then presented in Section 3. In Section
4 we give a general estimation algorithm and derive some
explicit formulas for the most popular distributions. The quality of
the estimates is studied on synthetic data in Section 5.
Finally, the model is used to elucidate the structure of host–parasite
interactions in forest ecosystems and results are discussed in Section
6.
2 Mixture model
We now present the general extension of SBM to valued graphs and
discuss the two particular modelings used for the tree species and
fungal species interaction networks.
2.1 Model and notation
Nodes.
Consider a graph with $n$ nodes, labeled in
$\{1,\ldots,n\}$. In our model the nodes are distributed among
$Q$ groups so that each node $i$ is associated to a random vector
$\mathbf{Z}_{i}=(Z_{i1},\ldots,Z_{iQ})$, with $Z_{iq}$ being $1$ if node
$i$ belongs to group $q$ and $0$ otherwise. The $\{\mathbf{Z}_{i}\}$ are
supposed to be independent identically distributed observations from
a multinomial distribution:
$$\{\mathbf{Z}_{i}\}_{i}\mbox{ i.i.d.}\sim\mathcaligr{M}(1;\bolds\alpha),$$
(1)
where $\bolds\alpha=(\alpha_{1},\ldots,\alpha_{Q})$ and $\sum_{q}\alpha_{q}=1$.
Edges.
Each edge from a node $i$ to a node $j$ is
associated to a random variable $X_{ij}$, coding for the strength of
the edge. Conditionally to the group of each node, or
equivalently knowing the $\{\mathbf{Z}_{i}\}$, the edges are supposed to be
independent. Knowing group $q$ of node $i$ and group $\ell$ of
node $j$, $X_{ij}$ is distributed as $f(\cdot,\theta_{q\ell}):=f_{q\ell}(\cdot)$, where $f_{\theta_{q\ell}}$ is a probability
distribution known up to a finite-dimensional parameter
$\theta_{q\ell}$:
$$X_{ij}|i\in q,j\in\ell\sim f(\cdot,\theta_{q\ell}):=f_{q\ell}(\cdot).$$
(2)
Up to a relabeling of the classes, the model is identifiable and
completely specified by both the mixture proportions $\bolds\alpha$ and
the connectivity matrix $\bolds\theta=(\theta_{q\ell})_{q,\ell=1,\ldots,Q}$. We denote $\bolds\gamma=(\bolds\alpha,\bolds\theta)$ the parameter of the
model.
Directed and undirected graphs
This modeling can be applied to both directed and undirected graphs.
In the directed version, the variables $X_{ij}$ and $X_{ji}$ are
supposed to be independent conditionally to the groups to which
nodes $i$ and $j$ belong. This hypothesis is not always realistic
since, for example, the traffic from $i$ to $j$ is likely to be
correlated to the traffic from $j$ to $i$. A way to account for such
a dependency is to consider a undirected graph with edges labeled
with the bivariate variables $\{(X_{ij},X_{ji})\}_{1\leq i<j\leq n}$. All the results presented in this paper are valid for
directed graphs. The results for undirected graphs can easily be
derived and are only briefly mentioned.
2.2 Modeling the number of shared hosts/parasites
In our tree interaction network, each edge is valued with the number
of common fungal species two tree species can host. Our purpose is to
understand the structure of this network and it is natural to model
the counts $X_{ij}$ as Poisson distributed. The mixture models aims
at explaining the heterogeneity of the $X_{ij}$. However, we would
also like to account for some factors that are known to be
influential. In our network, we expect two phylogenetically related
tree species $i$ and $j$ to share a high number $X_{ij}$ of parasitic
species. As such, their average number of shared parasitic species
$\mathbb{E}[X_{ij}]$ is expected to decrease with their phylogenetic
distance $y_{ij}$. We consider three alternatives, and compare two of
them.
Poisson mixture (PM):
In this mixture, we do not account for the
covariates and $X_{ij}$ only depends on the classes of $i$ and $j$:
$$X_{ij}|i\in q,j\in\ell\sim\mathcaligr{P}(\lambda_{q\ell}).$$
$\lambda_{q\ell}$ is then the mean number of common fungal species
(or mean interaction) between a tree species from group $q$ and one
from group $\ell$ and $\theta_{q\ell}=\lambda_{q\ell}$.
Poisson regression mixture with inhomogeneous effects (PRMI):
In
this mixture, we account for the covariates via a regression model that
is specific to the classes of $i$ and $j$:
$$X_{ij}|i\in q,j\in\ell\sim\mathcaligr{P}(\lambda_{q\ell}e^{\beta_{q\ell}^{%
\intercal}\mathbf{y}_{ij}}),$$
where $\mathbf{y}_{ij}$ is a vector of covariates and $\theta_{q\ell}=(\lambda_{q\ell},\beta_{q\ell})$.
Poisson regression mixture with homogeneous effects (PRMH):
In
this mixture, the effect of the covariates does not depend on the classes
of $i$ and $j$:
$$X_{ij}|i\in q,j\in\ell\sim\mathcaligr{P}(\lambda_{q\ell}e^{\beta^{\intercal}%
\mathbf{y}_{ij}}),$$
$\theta_{q\ell}=(\lambda_{q\ell},\beta)$.
We point out that models PRMI and PRMH have different purposes. In PRMI,
the link between the covariates and the edges is locally refined
within each class $(q,\ell),$ whereas in PRMH, the
covariates compete
globally with the group structure found by PM. In PRMH, the mixture looks
for remaining structure among the residuals of the regression model. If the
structure was completely explained by the covariates, the possibly many
components found using PM would reduce to a single component when using
PRMH. To a lesser extent, we expect the number of components to be smaller
with PRMH than with PM if the phylogenetic distance explains part of the
structure. As we look for structure beyond the one already explained by the
covariates, we consider only models PM and PRMH.
The same models are used for the fungal species interaction network.
In our examples, data consist in counts, but other types of data can be
handled with similar mixture and/or regression models (see Appendix
.6 for details).
3 Likelihood and variational EM
We now address the estimation of the parameter $\bolds\gamma=(\bolds\alpha,\bolds\theta)$. We show that the standard maximum likelihood
approach cannot be applied to our model and propose an alternative
strategy relying on variational tools, namely, variational EM.
3.1 Likelihoods
Let $\mathbf{X}$ denote the set of all edges, $\mathbf{X}=\{X_{ij}\}_{i,j=1,\ldots,n}$,
and $\mathbf{Z}$ the set of all indicator variables for nodes, $\mathbf{Z}=\{\mathbf{Z}_{i}\}_{i=1,\ldots,n}$. In the mixture model literature [McLahan and Peel (2000)]
$(\mathbf{X},\mathbf{Z})$ is referred to as the complete data set, while $\mathbf{X}$ is
referred to
as the incomplete data set. The conditional independence of the edges
knowing $\mathbf{Z}$ entails the decomposition $\log{\mathbb{P}(\mathbf{Z},\mathbf{X})}=\log{\mathbb{P}(\mathbf{Z})}+\log{%
\mathbb{P}(\mathbf{X}|\mathbf{Z})}$. It then follows from
(1) and (2) that the log-likelihood of the
complete data set is
$$\log{\mathbb{P}(\mathbf{Z},\mathbf{X})}=\sum_{i}\sum_{q}Z_{iq}\log{\alpha_{q}}%
+\sum_{i\neq j}\sum_{q,\ell}Z_{iq}Z_{j\ell}\log{f_{q\ell}(X_{ij})}.$$
(3)
The likelihood of the incomplete data set can be obtained by summing
$\mathbb{P}(\mathbf{Z},\mathbf{X})$ over all possible $\mathbf{Z}$’s: $\mathbb{P}(\mathbf{X})=\sum_{\mathbf{Z}}\mathbb{P}(\mathbf{Z},\mathbf{X})$. This summation involves $Q^{n}$ terms and quickly becomes
intractable. The popular E–M algorithm [Dempster, Laird and Rubin (1977)], widely used in
mixture problems, allows to maximize $\log{\mathbb{P}(\mathbf{X})}$ without explicitly
calculating it. The E-step relies on the calculation of the conditional
distribution of $\mathbf{Z}$ given $\mathbf{X}$: $\mathbb{P}(\mathbf{Z}|\mathbf{X})$.
Unfortunately, in
the case of network data, the strong dependency between edges makes this
calculation untractable.
Undirected graphs
The closed formula (3) still holds undirected graphs,
replacing the
sum over $i\neq j$ by a sum over $i<j$. This is also true for
equations (3.2) and (11) given below.
3.2 Variational EM
We propose to use an approximate maximum likelihood strategy based on
a variational approach [see Jordan et al. (1999) or the tutorial by
Jaakkola (2000)]. This strategy is also used in Govaert and Nadif (2005) for a
biclustering problem. We consider a lower bound of the log-likelihood
of the incomplete data set
$$\mathcaligr{J}(R_{\mathbf{X}},\bolds\gamma)=\log{\mathbb{P}(\mathbf{X};\bolds%
\gamma)}-\mathit{KL}(R_{\mathbf{X}}(\cdot),\mathbb{P}(\cdot|\mathbf{X};\bolds%
\gamma)),$$
(4)
where $\mathit{KL}$ denotes the Kullback–Leibler divergence and $R_{\mathbf{X}}$ stands for
some distribution on $\mathbf{Z}$. Classical properties of the Kullback–Leibler
divergence ensure that $\mathcaligr{J}$ has a unique maximum
$\log{\mathbb{P}(\mathbf{X};\bolds\gamma)}$, which is reached for $R_{\mathbf{X}}(\mathbf{Z})=\mathbb{P}(\mathbf{Z}|\mathbf{X})$. In other words, if $\mathbb{P}(\mathbf{Z}|\mathbf{X};\bolds\gamma)$ was
tractable, the maximization of $\mathcaligr{J}(R_{\mathbf{X}},\bolds\gamma)$ with respect to
$\bolds\gamma$ would be equivalent to the maximization of
$\log{\mathbb{P}(\mathbf{X};\bolds\gamma)}$. In our case, $\mathbb{P}(\mathbf{Z}|\mathbf{X};\bolds\gamma)$ is
untractable and we maximize $\mathcaligr{J}(R_{\mathbf{X}},\bolds\gamma)$ with respect to both
$R_{\mathbf{X}}$ and $\bolds\gamma$. Jaakkola (2000) shows that $\mathcaligr{J}(R_{\mathbf{X}},\bolds\gamma)$ can be
rewritten as
$$\mathcaligr{J}(R_{\mathbf{X}},\bolds\gamma)=\mathcaligr{H}(R_{\mathbf{X}})+%
\sum_{\mathbf{Z}}R_{\mathbf{X}}(\mathbf{Z})\log{\mathbb{P}(\mathbf{X},\mathbf{%
Z};\bolds\gamma)},$$
(5)
where $\mathcaligr{H}(\cdot)$ denotes the entropy of a distribution. The last term
of (5) can be deduced from (3):
$$\displaystyle\sum_{\mathbf{Z}}R_{\mathbf{X}}(\mathbf{Z})\log{\mathbb{P}(%
\mathbf{X},\mathbf{Z};\bolds\gamma)}$$
(6)
$$\displaystyle\quad=\sum_{i}\sum_{q}\mathbb{E}_{R_{\mathbf{X}}}(Z_{iq})\log{%
\alpha_{q}}+\sum_{i\neq j}\sum_{q,\ell}\mathbb{E}_{R_{\mathbf{X}}}(Z_{iq}Z_{j%
\ell})\log{f_{q\ell}(X_{ij})},$$
where $\mathbb{E}_{R_{\mathbf{X}}}$ denotes the expectation with respect to distribution
$R_{\mathbf{X}}$. Equation (3.2) requires only the knowledge of $\mathbb{E}_{R_{\mathbf{X}}}(Z_{iq})$
and $\mathbb{E}_{R_{\mathbf{X}}}(Z_{iq}Z_{j\ell})$ for all $i,j,q,\ell$. By contrast,
$\mathcaligr{H}(R_{\mathbf{X}})$ requires all order moments of $R_{\mathbf{X}}$ and is untractable in
general. Maximization of $\mathcaligr{J}(R_{\mathbf{X}},\bolds\gamma)$ in $R_{\mathbf{X}}$ can not be achieved
without some restrictions on $R_{\mathbf{X}}$. We therefore limit the search to the
class of completely factorized distributions:
$$R_{\mathbf{X}}(\mathbf{Z})=\prod_{i}h(\mathbf{Z}_{i},\bolds\tau_{i}),$$
(7)
where $h$ denotes the multinomial distribution and $\bolds\tau_{i}$ stands
for a
vector of probabilities, $\bolds\tau_{i}=(\tau_{i1},\ldots,\tau_{iQ})$ (with
$\sum_{q}\tau_{iq}=1$). In particular, $\mathbb{E}_{R_{\mathbf{X}}}(Z_{iq})=\tau_{iq}$
and $\mathbb{E}_{R_{\mathbf{X}}}(Z_{iq}Z_{j\ell})=\tau_{iq}\tau_{j\ell}$. In addition,
the entropy is additive over the coordinates for factorized distributions,
so that $\mathcaligr{H}(R_{\mathbf{X}})=\sum_{i}\mathcaligr{H}(h(\cdot,\bolds\tau_{i}))%
=-\sum_{i}\sum_{q}\tau_{iq}\log{\tau_{iq}}$. Wrapping everything together,
$$\displaystyle\mathcaligr{J}(R_{\mathbf{X}},\bolds\gamma)$$
$$\displaystyle=$$
$$\displaystyle-\sum_{i}\sum_{q}\tau_{iq}\log{\tau_{iq}}+\sum_{i}\sum_{q}\tau_{%
iq}\log{\alpha_{q}}$$
$$\displaystyle{}+\sum_{i\neq j}\sum_{q,\ell}\tau_{iq}\tau_{j\ell}\log{f_{q\ell}%
(X_{ij})}.$$
It is immediate from (3.2) that $\mathcaligr{J}(R_{\mathbf{X}},\bolds\gamma)$ is
tractable for distributions $R_{\mathbf{X}}$ of the form (7). The
$\bolds\tau_{i}$’s must be thought of as variational parameters to be
optimized so
that $R_{\mathbf{X}}(\mathbf{Z})$ fits $\mathbb{P}(\mathbf{Z}|\mathbf{X};\bolds\gamma)$ as well as
possible; they
depend on the observed data $\mathbf{X}$. Since $R_{\mathbf{X}}$ is restricted to be
of the
form (7), $\mathcaligr{J}(R_{\mathbf{X}},\bolds\gamma)$ is a lower bound of
$\log\mathbb{P}(\mathbf{X})$.
Discussion about tighter bounds
A fully factorized $R_{\mathbf{X}}$ is only one class of distributions we can
consider. Broader distribution classes should yield tighter bound of
$\mathcaligr{J}(R_{\mathbf{X}},\bolds\gamma)$. Unfortunately, for more general
distributions, the
entropy $\mathcaligr{H}(R_{\mathbf{X}})$ may not have a simple expression anymore
rendering the
exact calculation of $\mathcaligr{J}(R_{\mathbf{X}},\bolds\gamma)$ untractable: better
accuracy is
achieved at the expense of tractability. A solution to this issue is Bethe
free energy [Yedidia, Freeman and Weiss (2005)]. We did not consider it because it relies
on an
approximation of $\mathcaligr{H}(R_{\mathbf{X}})$ which disrupts the well-behaved
properties of
$\mathcaligr{J}(R_{\mathbf{X}},\bolds\gamma)$.
Another approach comes from Leisink and Kappen (2001) and Mariadassou (2006). Starting
from an
exponential inequality, they emphasize the strong connection between fully
factorized $R_{\mathbf{X}}$ and first order linear approximation of the exponential
function. Using a higher approximation of the exponential and some
distribution $S_{\mathbf{X}}$ in addition to $R_{\mathbf{X}}$, it is possible to
derive an
even tighter bound of $\mathcaligr{J}(R_{\mathbf{X}},\bolds\gamma)$. However, the estimation
algorithm is then of complexity $\mathcaligr{O}(n^{6}Q^{6})$ instead of
$\mathcaligr{O}(n^{2}Q^{2})$ for a gain which has the same order of magnitude as
the computer numerical precision.
4 Parameter estimation
We present here the two-steps algorithm used for the parameter estimation.
4.1 Estimation algorithm
As explained in Section 3.2, the maximum likelihood
estimator of $\bolds\gamma$ is
$$\widehat{\bolds\gamma}_{\mathit{ML}}=\operatorname{\arg\max}\limits_{\bolds%
\gamma}\log{\mathbb{P}(\mathbf{X};\bolds\gamma)}=\operatorname{\arg\max}%
\limits_{\bolds\gamma}\max_{R_{\mathbf{X}}}\mathcaligr{J}(R_{\mathbf{X}},%
\bolds\gamma).$$
In the variational framework, we restrict the last optimization
problem to factorized distributions. The estimate we propose is
hence
$$\widehat{\bolds\gamma}=\operatorname{\arg\max}\limits_{\bolds\gamma}\max_{R_{%
\mathbf{X}}\mathrm{\ factorized}}\mathcaligr{J}(R_{\mathbf{X}},\bolds\gamma).$$
The simultaneous optimization with respect to both $R_{\mathbf{X}}$ and
$\bolds\gamma$ is still too difficult, so we adopt the following
iterative strategy. Denoting by $R_{\mathbf{X}}^{(n)}$ and ${\bolds\gamma}^{(n)}$
the estimates after $n$ steps, we compute
$$\cases{R_{\mathbf{X}}^{(n+1)}=\operatorname{\arg\max}\limits_{R_{\mathbf{X}}%
\mathrm{\ factorized}}\mathcaligr{J}\bigl{(}R_{\mathbf{X}},{\bolds\gamma}^{(n)%
}\bigr{)},\cr{\bolds\gamma}^{(n+1)}=\operatorname{\arg\max}\limits_{\bolds%
\gamma}\mathcaligr{J}\bigl{(}R_{\mathbf{X}}^{(n+1)},\bolds\gamma\bigr{)}.}$$
(9)
The next two sections are dedicated to each of these steps.
Initialization step
The optimization procedure (9) only ensures the
convergence toward a local optimum, so the choice of the starting
point for $\bolds\gamma$ or $R_{\mathbf{X}}$ is crucial to avoid local optima. This
choice is difficult, but, to our experience, hierarchical clustering
seems to be a good strategy to get an initial value for $R_{\mathbf{X}}$.
4.2 Optimal approximate conditional distribution $R_{\mathbf{X}}$
We consider here the optimization of $\mathcaligr{J}$ with respect to $R_{\mathbf{X}}$.
For a
given value of $\bolds\gamma$, wedenote $\widehat{\bolds\tau}$ the variational
parameter defining the distribution $\widehat{R}_{\mathbf{X}}=\break{\arg\max}_{R_{\mathbf{X}}\mathrm{\ factorized}%
}\mathcaligr{J}(R_{\mathbf{X}},\bolds\gamma)$. This amounts to maximimizing $\mathcaligr{J}(R_{\mathbf{X}},\bolds\gamma)$,given in (3.2), under the condition that, for all $i$, the
$\tau_{iq}$’s must sum to 1. The derivative of $\mathcaligr{J}(R_{\mathbf{X}},\bolds\gamma)$ with
respect to $\tau_{iq}$ is
$$-\log{\tau_{iq}}-1+\log{\alpha_{q}}+\sum_{j\neq i}\sum_{\ell}\tau_{j\ell}[\log%
{f_{q\ell}(X_{ij})}+\log{f_{\ell q}(X_{ji})}]+L_{i},$$
where $L_{i}$ denotes the $i$th Lagrange multiplier. It results from the
previous equation that the optimal variational parameter $\widehat{\bolds\tau}$
satisfies the fixed point relation
$$\widehat{\tau}_{iq}\propto\alpha_{q}\prod_{j\neq i}\prod_{\ell}[f_{q\ell}(X_{%
ij})f_{\ell q}(X_{ji})]^{\widehat{\tau}_{j\ell}}.$$
(10)
The fixed point relation (10) can be related to a mean field
approximation [see Jaakkola (2000)]. We get $\widehat{\bolds\tau}$ simply by
iterating this relation until convergence.
Undirected graphs
For a undirected graph, $\widehat{\bolds\tau}$ satisfies
$$\widehat{\tau}_{iq}\propto\alpha_{q}\prod_{j\neq i}\prod_{\ell}[f_{q\ell}(X_{%
ij})]^{\widehat{\tau}_{j\ell}}.$$
4.3 Parameter estimates
We now have to maximize $\mathcaligr{J}$ with respect to $\bolds\gamma=(\bolds\alpha,\bolds\theta)$ for a given distribution $R_{\mathbf{X}}$. Again, this amounts to
maximizing $\mathcaligr{J}(R_{\mathbf{X}},\bolds\gamma)$, given in (3.2), under
the condition that $\sum_{q}\alpha_{q}=1$. Straightforward
calculations show
that the optimal $\bolds\alpha$ and $\bolds\theta$ are given by
$$\widehat{\alpha}_{q}=\frac{1}{n}\sum_{i}\tau_{iq},\qquad\widehat{\theta}_{q%
\ell}=\operatorname{\arg\max}\limits_{\theta}\sum_{i\neq j}\tau_{iq}\tau_{j%
\ell}\log{f(X_{ij};\theta)}.$$
(11)
Poisson models
Poisson models are of particular
interest for
our interaction networks. The optimal $\lambda_{q\ell}$ for model PM
presented in Section 2.2 is straightforward:
$$\widehat{\lambda}_{q\ell}=\sum_{i\neq j}\tau_{iq}\tau_{j\ell}X_{ij}\Big{/}\sum%
_{i\neq j}\tau_{iq}\tau_{j\ell}.$$
For models PRMH and PRMI presented in the same section, there is no closed
formula for $\lambda_{q\ell}$, $\beta_{q\ell}$ or $\beta$.
However, since
the Poisson regression model belongs to the exponential family, $\mathcaligr{J}$ is
only a weighted version of the log-likelihoods of the corresponding
generalized linear model. As such, standard optimization procedures can be
used.
Exponential family
The optimal $\bolds\theta$ is not
explicit in
the general case, but has a simpler form if the distribution $f$
belongs to
the exponential family. Namely, if $f$ belongs to an exponential family
with natural parameter $\theta$,
$$f(x;\theta)=\exp{[\bolds\Psi(x)^{\prime}\theta-A(\theta)]}.$$
According to (11), we look for $\widehat{\theta}={\arg\max}_{\theta}\sum_{i\neq j}\tau_{iq}\tau_{j\ell}\bolds%
\Psi(X_{ij})^{\prime}\theta-A(\theta)$. Maximizing this quantity in $\theta$ yields
$$\sum_{i\neq j}\tau_{iq}\tau_{j\ell}\bolds\Psi(X_{ij})-\nabla A(\theta)=\mathbf%
{0}.$$
If $\nabla A$ is invertible, the optimal $\theta$ is
$$\widehat{\theta}=(\nabla A)^{-1}\biggl{[}\sum_{i\neq j}\tau_{iq}\tau_{j\ell}%
\bolds\Psi(X_{ij})\biggr{]}.$$
(12)
4.4 Choice of the number of groups
In practice, the number of groups is unknown and should be estimated.
Many criterion have been proposed to select the dimensionality $Q$ of
the latent space, ranging from AIC to ICL. AIC, BIC and their variants
[Burnham and Anderson (1998)] are based on computing the likelihood of the observed
data $\mathbb{P}(\mathbf{X}|m_{Q})$ and penalizing it with some function of $Q$. But
the use of variational EM is precisely to avoid computation of
$\mathbb{P}(\mathbf{X}|m_{Q})$, which is untractable. Given a prior distribution
$\mathbb{P}(m_{Q})$ over models, and a prior distribution $\mathbb{P}(\bolds\gamma|m_{Q})$
for each model, variational Bayes [Beal and Ghahramani (2003)] works by selecting
the model with maximum posterior $\mathbb{P}(m_{Q}|\mathbf{X})$. Estimation of
$\mathbb{P}(\mathbf{X}|m_{Q})$ is then performed using variational EM and no
penalization is required, as complex models are already penalized by
diffuse prior $\mathbb{P}(\bolds\gamma|m_{Q})$. Extension of Deviance Information
Criterion (DIC) to finite mixture distributions via variational
approximations [McGrory and
Titterington (2007)] is even more straightforward:
choosing $Q^{*}$ larger than the expected number of components and
running the algorithm, extraneous classes become void as the
algorithm converges and the selected number of groups is just the
number of nonempty classes. In the context of unknown assignments,
Biernacki, Celeux and Govaert (2000) proposed the Integrated Classification Likelihood (ICL),
which is an approximation to the complete data likelihood
$\mathbb{P}(\mathbf{X},\mathbf{Z}|m_{Q})$. Variational Bayes, BIC and ICL can all be seen
as approximations to Bayes factors. Whereas Variational Bayes
integrates out the uncertainty about the parameter and the assignment
of nodes to groups, ICL replaces them by a point estimate, computed
thanks to variational EM. Traditional model selection essentially
involves a trade-off between goodness of fit and model complexity,
whereas ICL values both goodness of fit and classification sharpness.
Nowicki and Snijders (2001) do not propose any criterion to select the number of
groups. Hofman and Wiggins (2008) use McGrory’s method but in a very specific case of
the Stochastic Block Model. They also give no clue as to how to decide that
the algorithm has converged enough. Airoldi et al. (2008) use either a
modification to BIC (for small size networks) or cross-validation (for
large size networks) to select the number of groups. Daudin, Picard and Robin (2008) use a
modification to ICL criterion. Following along the same line as
Daudin, Picard and Robin (2008), we use a modification of ICL adapted to valued graphs to
select the number of classes.
ICL criterion:
For a model $m_{Q}$ with $Q$ classes where $\bolds\theta$ involves $P_{Q}$
independent parameters, the ICL criterion is
$$\displaystyle\mathit{ICL}(m_{Q})$$
$$\displaystyle=$$
$$\displaystyle\max_{\bolds\gamma}\log\mathbb{P}(\mathbf{X},\widetilde{\mathbf{Z%
}}|\bolds\gamma,m_{Q})$$
$$\displaystyle{}-\frac{1}{2}\{P_{Q}\log[n(n-1)]-(Q-1)\log(n)\},$$
where the missing data $\mathbf{Z}$ are replaced by their prediction
$\widetilde{\mathbf{Z}}$.
Note that the penalty term $-\frac{1}{2}\{P_{Q}\log[n(n-1)]-(Q-1)\log(n)\}$ is similar to the one of BIC, where the
$\log$ term refers to number of data. In the case of graphs, the
number of data is $n$ (i.e., the number of nodes) for the vector of
proportions $\bolds\alpha$ ($Q-1$ independent parameters), whereas it is
$n(n-1)$ (i.e., the number of edges) for parameter $\bolds\theta$ ($P_{Q}$
independent parameters). For the models PM, PRMI and PRMH (detailed in
Section 2.2), $P_{Q}$ is respectively ${Q(Q+1)}/{2}$,
$Q(Q+1)$ and $1+{Q(Q+1)}/{2}$.
5 Simulation study
5.1 Quality of the estimates
Simulation parameters
We considered undirected networks of size $n=100$ and $500$ with
$Q=3$ classes. To study balanced and unbalanced proportions, we set
$\alpha_{q}\propto a^{q}$, with $a=1,0.5,0.2$. $a=1$ gives
uniform proportions, while $a=0.2$ gives very unbalanced
proportions: $\bolds\alpha=(80.6\%,16.1\%,3.3\%)$. We finally
considered symmetric connection intensities $\lambda_{pq}$, setting
$\lambda_{pp}=\lambda^{\prime}$ for all $p$ and $\lambda_{pq}=\lambda^{\prime}\gamma$ for $p\neq q$. Parameter $\gamma$ controls the difference
between within class and between class connection intensities
($\gamma=0.1$, $0.5$, $0.9$, $1.5$), while $\lambda^{\prime}$ is set so
that the mean connection intensity $\lambda$ ($\lambda=2,5$)
depends neither on $\gamma$ nor $a$. $\gamma$ close to one makes the
distinction between the classes difficult. $\gamma$ larger than one
makes the within class connectivities less intense than the between
ones. We expect the fitting to be rather easy for the combination
$\{n=500,a=1,\lambda=5,\gamma=0.1\}$ and rather difficult for
$\{n=100,a=0.2,\lambda=2,\gamma=0.9\}$.
Simulations and computations
For each combination of the parameters, we simulated $S=100$
random graphs according to the corresponding mixture model. We
fitted the parameters using the algorithm described in
Section 4. To solve the identifiability problem of the
classes, we systematically ordered them in descending estimated
proportion order: $\widehat{\alpha}_{1}\geq\widehat{\alpha}_{2}\geq\widehat{\alpha}_{3}$. For each parameter, we calculated the estimated
Root Mean Squared Error (RMSE):
$$\displaystyle\mathit{RMSE}(\widehat{\alpha}_{p})$$
$$\displaystyle=$$
$$\displaystyle\sqrt{\frac{1}{S}\sum_{s=1}^{S}\bigl{(}\widehat{\alpha_{p}}^{(s)}%
-\alpha_{p}\bigr{)}^{2}},$$
$$\displaystyle\mathit{RMSE}(\widehat{\lambda}_{pq})$$
$$\displaystyle=$$
$$\displaystyle\sqrt{\frac{1}{S}\sum_{s=1}^{S}\bigl{(}\widehat{\lambda_{pq}}^{(s%
)}-\lambda_{pq}\bigr{)}^{2}},$$
where the superscript $(s)$ labels the estimates obtained in
simulation $s$. We also calculated the mean posterior entropy
$$H=\frac{1}{S}\sum_{s}\biggl{(}-\sum_{i}\sum_{q}\tau_{iq}^{(s)}\ln\tau_{iq}^{(s%
)}\biggr{)},$$
which gives us the degree of uncertainty of the classification.
Results
Figure 1 (resp. 2) gives
the RMSE for the proportion $\alpha_{q}$ (resp. connection intensities
$\lambda_{pq}$). As expected, the $\mathit{RMSE}$ is lower when $n$ is
larger. The parameters affecting the $\mathit{RMSE}$ are mainly $a$ and
$\gamma$, whereas $\lambda$ has nearly no effect. The departures
observed for $\alpha_{1}$ and $\alpha_{3}$ in the balanced case ($a=1.0$) are due to the systematic reordering of the proportions.
Since the graph is undirected, $\lambda_{pq}=\lambda_{qp}$, so
only nonredundant parameters are considered in
Figure 2. The overall quality of the
estimates is satisfying, especially for the diagonal terms
$\lambda_{qq}$. The within intensity parameter of the smallest class
$\lambda_{33}$ is the most difficult to estimate. The worst case
corresponds to a small graph ($n=100$) with very unbalanced classes
($a=0.2$) for parameter $\lambda_{12}$. In this case, the
algorithm is unable to distinguish the two larger classes (1 and 2),
so that the estimates extra-diagonal term $\widehat{\lambda}_{12}$
is close to the diagonal ones $\widehat{\lambda}_{11}$ and
$\widehat{\lambda}_{22}$, whereas its true value is up to ten times
smaller.
Figure 3 gives the mean entropy. Not
surprisingly, the most influential parameter is $\gamma$: when $\gamma$
is close to 1, the classes are almost indistinguishable. For small
graphs ($n=100$), the mean intensity $\lambda$ has almost no effect.
Because of the identifiability problem already mentioned, we did not
consider the classification error rate.
5.2 Model selection
We considered a undirected graph of size $n=50,\break 100,500$ and
$1000$ with $Q^{\star}=3$ classes. We considered the combination $\{a=0.5,\lambda=2,\gamma=0.5\}$ which turned out to be a medium
case (see Section 5.1) and computed ICL for
$Q$ ranging from $1$ to $10$ (from $1$ to $5$ for $n=1000$) before
selecting the $Q$ maximizing ICL. We repeated this for $S=100$
simulations.
Figure 4 gives ICL as a function of $Q$, while
Table 1 returns the frequency with which each
$Q$ is selected. As soon as $n$ is larger than $100$, ICL almost
always selects the correct number of classes; for smaller graphs
($n=50$), it tends to underestimate it. The proposed criterion is
thus highly efficient.
6 Uncovering the structure of host–parasite interactions in forest
ecosystems
Here we use mixture models to highlight the factors governing with
whom a species interact in an ecosystem. The factors which may account
for species interactions are introduced as covariates in the mixture
models. The explanatory power of each factor is measured as the
decrease in the number of groups selected. Our study focuses on
host–parasite interactions in forest ecosystems. We address the two
following questions: (1) Is similarity in the parasite assemblages of
two tree species explained by their phylogenetic relatedness rather
than by the degree of overlap of their distributional range? (2) Is
similarity in the host range of two parasitic fungal species explained
by their phylogenetic relatedness rather than their common nutritional
strategy? The explanatory power of phylogenetic relatedness is
subsequently called phylogenetic signal, as in the ecological
literature [Rezende et al. (2007); Vacher, Piou and
Desprez-Loustau (2008)].
6.1 Data
Host–parasite interaction records
We considered two undirected, valued networks having parasitic fungal
species ($n=154$) and tree species ($n=51$) as nodes, respectively. Edges
strength was defined as the number of shared host species and the
number of
shared parasitic species, respectively [Mariadassou, Robin and Vacher (2010)].
The methods used for collecting data on tree–fungus interactions are
fully described in Vacher, Piou and
Desprez-Loustau (2008). Fungal species names were checked
since then in the Index Fungorum database
(www.indexfungorum.
org): 17 names
were updated, yielding to 3
new species synonymies. The fusion of synonym species accounts for the
lower number of fungal species in the present study than in the
original publication.
Phylogenetic relatedness between species
In order to verify the existence of a phylogenetic signal in the
parasite assemblages of tree species, we estimated genetic distances
between all pairs of tree species. The maximally resolved seed plant
tree of the software Phylomatic2 [Webb and Donoghue (2005)] was used to produce a
phylogenetic tree for the 51 tree species included in our study. Then,
pairwise genetic distances (in million years) were extracted by using
the cophenetic.phylo function of the R ape package [Paradis, Claude and Strimmer (2004)].
Because the phylogenetic tree was loosely resolved for gymnosperms, we
also used taxonomic distances to estimate phylogenetic relatedness
between tree species.
Since all tree species included in the study belong to the phylum
Streptophyta, we used the finer taxonomic ranks of class, order,
family and genus to calculate pairwise taxonomic distances. Based on
the NCBI Taxonomy Browser
(www.ncbi.nlm.nih.gov/Taxonomy/),
we found that the
species are evenly distributed into two taxonomic classes
(Magnoliophyta and Conipherophyta) and further subdivided in 8 orders,
13 families and 26 genera. Following Poulin (2005), we considered that
the taxonomic distance is equal to $0$ if species are the same, $1$ if
they belong to the same genus, $2$ to the same family, $3$ to the same
order, $4$ to the same taxonomic class and $5$ if their only common
point lies in belonging to the phylum Streptophyta.
In order to investigate the existence of a phylogenetic signal in the
host range of parasitic fungal species, we estimated taxonomic
distances between all pairs of fungal species. Pairwise genetic
distances could not be calculated because genetic data were not
available for all the species. Since the 153 fungal species at hand
span a wider portion of the tree of life than the tree species, we had
to use the higher order rank of kingdom. The taxonomic distance for
fungal species thus ranges from $0$ to $6$ (kingdom level) when
compared to $0$ to $5$ for trees. The taxonomy was retrieved from
Index Fungorum (www.indexfungorum.org). All fungal species
included in the study belong to the Fungi kingdom, are divided in two
phyla (Ascomycota and Basidiomycota) and further subdivided in 9
taxonomic classes, 21 orders, 48 families and 107 genera. When pairs
included a species whose taxonomic is uncertain for a given taxonomic
rank, this rank was skipped and upper ranks were used to estimate
distance.
Other explanatory factors
Other factors than phylogenetic relatedness may account for pairwise
similarities in parasite assemblages between tree species. In
particular, two tree species having overlapping distributional range
are exposed to similar pools of parasitic species and may therefore
share more parasitic species than two tree species with
nonoverlapping distributions [Brandle and Brandl (2006)]. We tested this
hypothesis by calculating the geographical distance between all pairs
of tree species. The geographical distance is the Jaccard distance
[Jaccard (1901)] computed on the profiles of presence/absence in 309
geographical units covering the entire French territory.
In the case of fungal species, other factors may also account for
similarity in host range. Here we investigated whether fungal species
having similar nutritional strategies also have similar host ranges. Fungal
species were classified into ten nutritional strategies based on their
parasitic lifestyle (biotroph or necrotroph) and on the plant organs and
tissues attacked. Five strategies (strict foliar necrotroph parasites,
canker agents, stem decay fungi, obligate biotroph parasites and root decay
fungi) accounted for 87% of the fungal species. We considered that
nutritional distance between two species equals one if the strategies are
the same and 0 otherwise.
6.2 Identification of groups of species sharing similar
interactions
Model
For both networks, we used the mixture model to define groups of tree
species and fungal species having similar interactions We assumed
that, in each network, the edge intensities were Poisson distributed.
For both networks, we considered the PM and PRMH models (see Section
2.2) using pairwise distance between species
(genetic, taxonomic, geographic or nutritional) as a covariate.
PM model: No covariate
In the absence of covariates, the ICL criterion selected 7 groups of
tree species. Two groups of tree species (T2 and T5) were exclusively
composed of species belonging to the Magnoliophyta, whereas three other
groups (T1, T3 and T4) were exclusively composed of species belonging
to the Conipherophyta. The two last groups (T6 and T7) were mixed
(Table 2). According to the mean number of interactions
per species and the parameters estimates of the model
(Table 2), they were composed of tree species having few
parasitic species and sharing few of them with other tree species.
It is noteworthy that group T2 was composed of four species belonging
to the same order (Fagales) and also to the same family (Fagaceae).
Groups T1, T3 and T4 were also composed of species belonging to the
same family (Pinaceae) since the only three coniferous species
belonging to another family were classified in groups T6 and T7. These
results confirm that two plant species with a similar evolutionary
history are likely to share the same set of parasitic species
[Brandle and Brandl (2006), Gilbert and Webb (2007), Vacher, Piou and
Desprez-Loustau (2008)].
PRMH model: Accounting for phylogenetic relatedness
When accounting for taxonomy, ICL selected only $4$ groups of tree
species. The estimated regression coefficient was $\widehat{\beta}=-0.317$,
which means that, for the mean taxonomic distance $\overline{y}=3.82$,
the mean connexion intensity is reduced of $70\%$
($e^{\widehat{\beta}\overline{y}}=0.298$). The cross classification table
(Table 3) shows that the taxonomic distance reduces the
number of class by merging groups T1 and T2 with most of the trees of T4
and T5. T’3 essentially consists of T6, T’1 of T7 and T’2 is made of trees
from T3 completed with leftovers from other classes. Interestingly and
unlike the groups obtained with no covariates, no group has species
belonging exclusively to one or the other of the taxonomic classes
(Magnoliophyta or Conipherophyta): the association between group of trees
and taxonomy was cropped out by the covariate
(Table 4). The same results hold when using the genetic
distance as a covariate instead of the taxonomic distance (results not
shown).
Therefore, the inclusion of taxonomic (or genetic) distance as a
covariate shows that the phylogenetic relatedness between tree species
accounts for a large part of the structure of tree–parasitic fungus
interactions in forest ecosystems, but not for all the structure.
Indeed, even after controlling for the evolutionary history through
the taxonomic (or genetic) distance, ICL still finds $4$ groups of
trees, whereas we would expect only one group if the phylogeny was the
sole source of structure. Below we investigate whether the
distributional overlap between tree species is another source of
structure.
PRMH model: Accounting for distributional overlap
In contrast with the taxonomic and genetic distance, the geographical
distance between species does not reduce the number of groups (not
shown). This result suggests that the current distributional overlap
between tree species does not account for the similarity in their
parasite assemblages. This result is opposite to the conventional
wisdom in the field of community ecology, which favors ecological
processes, taking place over short time scale, over evolutionary
processes, taking place over longer time scales, as the main source of
biotic interaction diversity. Our findings point out that the relative
importance of these processes might be the other way round.
6.3 Factors accounting for the host ranges of parasitic fungal
species
PM model: No covariate
The ICL criterion selected 9 groups of parasitic fungal species. The
estimates intensities $\widehat{\lambda}_{q\ell}$ range from almost zero
($1.4\times 10^{-3}$) to $12.1$, while the group proportions
$\widehat{\alpha}_{q}$ range from $1.3\%$ to $40.2\%$ (Table 7).
PRMH model: Accounting for phylogenetic relatedness
Accounting for taxonomic distance does not reduce the number of groups
(not shown), indicating a lack of phylogenetic signal in the host
range of fungal species. These results parallel those obtained with
another clustering approach [Newman (2004)] for the same tree–fungus
network [Vacher, Piou and
Desprez-Loustau (2008)]. They are congruent with the results obtained
for other bipartite networks since asymmetries in the phylogenetic
signal have been found in numerous plant–animal mutualistic networks
[Rezende et al. (2007)] and in a host–parasite network between leaf-miner moths
and parasitoid insects [Ives and Godfray (2006)]. In the latter case, the authors
also observed a lack of signal through the parasite phylogeny. In the
case of the tree–fungus network, we proposed that the very early
divergence of the major fungal phyla may account for the asymmetric
influence of past evolutionary history [Vacher, Piou and
Desprez-Loustau (2008)]: the lack of
signal through the fungal phylogeny may be the result of parasitic
fungal species splitting into two groups when the Conipherophyta and
the Magnoliophyta diverged (both groups containing Ascomycota and
Basidiomycota species) and the subsequent coevolution of each set of
fungal species with its plant phylum. Stronger selection pressures on
parasitic species than on host species might also account for the
asymmetry of the signal [Bersier and Kehrli (2008); Rossberg et al. (2006)].
PRMH model: Accounting for nutritional strategies
Fungal
Correlation analysis showed an association between the 9 groups
selected with the PM model and the nutritional type.
In particular, two groups of fungal species (F2 and F3, see Appendix
.8) contained a high proportion of root decay fungi
(100% and 75%, respectively).
However, taking the nutritional strategy as a covariate does not
reduce the number of groups, indicating the lack of ‘nutritional
signal’ in the host range of parasitic fungal species.
6.4 Goodness of fit
Since no covariate decreases the number of mixture components in the
fungus interaction network, we assessed goodness of fit only for the
tree interaction network. The goodness is assessed in two ways: in
terms of likelihood with the ICL criterion and in terms of predictive
power for the strength of an interaction. The ICL criterion is
$-2876.6$ for the base model with no class. It jumps to $-1565.6$
($\Delta\mathit{ICL}=1212.8$) when allowing a mixture structure (with $7$
classes). It jumps again to $-1449.6$ ($\Delta\mathit{ICL}=116$) when adding
the taxonomic distance as a covariate in the model (with $4$ classes).
Interestingly, adding a covariate to the $4$ class mixture model
provides a gain in goodness of fit twice as big as the gain of adding
three additional classes ($\Delta\mathit{ICL}=214.2$ against $98.2$). But
adding a covariate only requires one additional parameter ($\beta$),
against $21$ for the three additional classes.
We also assessed goodness of fit in terms of predictive power. For the PRMH
model with 4 classes and taxonomic distance as a covariate, we can predict
both the weighted degree $K_{i}=\sum_{j\neq i}X_{ij}$ of node $i$ as
$\widehat{K}_{i}=\sum_{j}\sum_{q,\ell}\tau_{iq}\tau_{jl}\lambda_{ql}e^{\beta^{%
\intercal}\mathbf{y}_{ij}}$ and the value $X_{ij}$
of a
single edge fungal as $\widehat{X}_{ij}=\sum_{q,\ell}\tau_{iq}\tau_{j\ell}\lambda_{ql}e^{\beta^{%
\intercal}\mathbf{y}_{ij}}$. The prediction of
$K_{i}$ using
$\widehat{K}_{i}$ is pretty accurate (Figure 5 left,
$R^{2}=0.94$). The prediction of $X_{ij}$ using $\widehat{X}_{ij}$ is less
accurate, but the confidence region is still pretty good
(Figure 5 right, $R^{2}=0.56$).
6.5 Conclusion
The structure of host–parasite interactions in forest ecosystems is a
complex one. Some tree species share more parasites than others and
this variability is well captured by a mixture model. However and as
shown in Table 5, the naive mixture model deceptively
captures part of the variability readily explained by other factors,
such as the phylogenetic relatedness (measured either by taxonomic or
genetic distance) and artificially increases the number of groups in
the mixture. Accounting for relevant factors decreases the number of
groups selected.
Using group reduction as a yardstick (Table 5), we
conclude that similarity in the parasite assemblages of tree species
is explained by their phylogenetic relatedness rather than their
distributional overlap, indicating the importance of evolutionary
processes for explaining the current patterns of inter-specific
interactions. Our study is however inconclusive on the relative
contribution of phylogenetic relatedness and nutritional strategy to
the similarity in the host ranges of parasitic fungal species
parasites of two parasitic fungus (Table 8 in Appendix
.8). In either case, since the PRMH model still finds
$4$ (resp. $9$) classes for the tree species (resp. fungal species)
interaction network, a significant fraction of the variability remains
unexplained by our predictors.
Appendix
.6 Other mixture models
We examine here some other classical distributions which can be used
in our framework.
Bernoulli.
In some situations such as co-authorship or
social networks, the only available information is the presence or
absence of the edge. $X_{ij}$ is then supposed to be Bernoulli
distributed:
$$X_{ij}|i\in q,j\in\ell\sim\mathcaligr{B}(\pi_{q\ell}).$$
It is equivalent to the stochastic block model of Nowicki and Snijders (2001) or
Daudin, Picard and Robin (2008).
Multinomial.
In a social network, $X_{ij}$ may specify the
nature of the relationship: colleague, family, friend, etc.
The $X_{ij}$’s can then be modeled by multinomial variables:
$$X_{ij}|i\in q,j\in\ell\sim\mathcaligr{M}(1;\mathbf{p}_{q\ell}).$$
The parameter $\theta_{q\ell}$ to estimate is the vector of
probability $\mathbf{p}_{q\ell}=(p_{q\ell}^{1},\dots,p_{q\ell}^{m})$, $m$ being the number of possible labels.
In directed random graphs, this setting allows to account for some
dependency between symmetric edges $X_{ij}$ and $X_{ji}$. We only need
to consider the equivalent undirected graphs where edge
$(i,j)$ is labeled with the couple $(X_{ij},X_{ji})$. $m=4$
different labels can the be observed: $(0,0)$ if no edge exists,
$(1,0)$ for $i\rightarrow j$, $(1,1)$ for $i\leftarrow j$ and
$(1,1)$ for $i\leftrightarrow j$.
Gaussian.
Traffic networks describe the intensity of
the traffic between nodes. The airport network is a typical example
where the edges are valued according to the number of passengers
traveling from airport $i$ to airport $j$. The intensity $X_{ij}$
of the traffic can be assumed to be Gaussian:
$$X_{ij}|i\in q,j\in\ell\sim\mathcaligr{N}(\mu_{q\ell},\sigma^{2}_{q\ell}),%
\qquad\theta_{q\ell}=(\mu_{q\ell},\sigma^{2}_{q\ell}).$$
Bivariate Gaussian.
The correlation between symmetric edges
$X_{ij}$ and $X_{ji}$ can be accounted for, considering the
undirected valued graph where edge $(i,j)$ is valued by $(X_{ij},X_{ji})$, which is assumed to be Gaussian. Denoting $\mathbf{X}_{ij}=[X_{ij}X_{ji}]^{\prime}$,
$$\mathbf{X}_{ij}|i\in q,j\in\ell\sim\mathcaligr{N}(\bolds\mu_{q\ell},\bolds%
\Sigma_{q\ell}),\qquad\theta_{q\ell}=(\bolds\mu_{q\ell},\bolds\Sigma_{q\ell}).$$
Linear regression.
When covariates are available, the
linear model, either Gaussian for real valued edges or generalized for
integer valued (e.g., Poisson or Bernoulli) allows to include them. For
example, for Gaussian valued edges, denoting $\mathbf{y}_{ij}$ the $p\times 1$
vector of covariates describing edge $(i,j)$, we set
$$X_{ij}|i\in q,j\in\ell\sim\mathcaligr{N}({\bf\beta}_{q\ell}^{\intercal}.%
\mathbf{y}_{ij},\sigma_{q\ell}^{2}).$$
Simple linear regression.
A case of specific interest for
plant ecology is the simple linear homoskedastic regression with group
specific intercept $a_{q\ell}$ but constant regression coefficient
$b$. It is
particularly useful when controlling for the effect of geography, which
is assumed to be the same for all groups of plants. We then set
$$X_{ij}|i\in q,j\in\ell\sim\mathcaligr{N}(a_{q\ell}+by_{ij},\sigma^{2}).$$
The model can again be extended to Poisson or Bernoulli valued edges
using adequate link function.
.7 Parameter estimates for other distributions
Table 6 gives the parameter estimates for the
model listed in Section .6. The estimates of the
mean parameter for Gaussian ($\mu_{q\ell}$) distributions are the same
as the estimate of the probability $\pi_{q\ell}$ in the Bernoulli
case. The results displayed in this table are all straightforward.
Note that all estimates are weighted versions of the intuitive ones.
.8 Parameter estimates for the fungus interaction network
Acknowledgments
We thank the Département Santé des Forêts (DSF) of the French
Ministère de l’Agriculture et de la Pêche for allowing us to use
their database. We thank Dominique Piou and Marie-Laure
Desprez-Loustau for checking the data and for helpful comments on the
results.
{supplement}
[id=suppA]
\stitleInteraction network between tree and fungal species
\slink[doi]10.1214/07-AOAS361SUPP
\slink[url]http://lib.stat.cmu.edu/aoas/361/supplement.csv
\sdatatype.csv
\sdescription
This file contains:
•
The adjacency matrix of interactions between tree and fungal
species.
•
The list of the tree species.
•
The list of the fungal species.
•
The matrix of genetic distances between tree species.
•
The matrix of geographical distances between tree species.
•
The matrix of taxonomic distances between fungal species.
•
The matrix of nutritional type of the fungal species.
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\MR2246363 |
A modular-invariant modified Weierstrass sigma-function as a
building block for lowest-Landau-level wavefunctions on the torus
F. D. M. Haldane
[email protected]
Department of Physics, Princeton
University, Princeton NJ 08544-0708, USA
(June 3, 2018, revised June 20, 2018)
Abstract
A “modified” variant of the Weierstrass sigma, zeta, and elliptic
functions is proposed whereby the zeta function is redefined by $\zeta(z)$ $\mapsto$
$\widetilde{\zeta}(z)$ $\equiv$ $\zeta(z)-\gamma_{2}z$, where $\gamma_{2}$ is
a lattice invariant
related to the almost-holomorphic modular invariant of the
quasi-modular-invariant weight-2 Eisenstein series. If $\omega_{i}$ is
a primitive half-period, $\widetilde{\zeta}(\omega_{i})$ = $\pi\omega_{i}^{*}/A$,
where $A$ is the area of the primitive cell of the lattice. The
quasiperiodicity of the modified sigma function is much simpler than
that of the original, and it becomes the building-block for the
modular-invariant formulation of lowest-Landau-level wavefunctions on
the torus. It is suggested that the “modified” sigma function is
more natural than the original Weierstrass form, which was formulated
before quasi-modular forms were understood. For the high-symmetry
(square and hexagonal) lattices, the modified and original sigma
functions coincide.
In 1862, Weierstrassweier , as part of his work on elliptic
functions,
introduced an odd meromorphic “zeta function”
$\zeta(z;\Lambda)$ which has simple poles with residue 1 on a lattice $\Lambda$ = $\{L\}$ =
$\{2m\omega_{1}+2n\omega_{2}\}$ in the complex plane, where $m$ and
$n$ are integers: $m,n$ $\in$ $\mathbb{Z}$.
Using the metric $d(z_{1},z_{2})$ = $|z_{1}-z_{2}|$, the lattice
has a primitive cell of area
$$A(\Lambda)=2|\omega_{1}^{*}\omega_{2}-\omega^{*}_{2}\omega_{1}|>0.$$
(1)
Here $\omega_{i}$ will mean any primitive half-period of the lattice, so
$2\omega_{i}$ $\in$ $\Lambda$, but
$2\omega_{i}/p$ $\not\in\Lambda$ for any integer $p>1$.
A pair of primitive half-periods $(\omega_{1},\omega_{2})$ is a basis of
$\Lambda$ if
$\omega_{1}+p\omega_{2}$ and $\omega_{2}+p\omega_{1}$ are primitive for
all integers $p$. Once a basis is chosen, it is useful to define
$\omega_{3}$ =$-(\omega_{1}+\omega_{2})$, which is also a primitive half-period.
The Weierstrass zeta function is defined by the convergent sum
$$\zeta(z;\Lambda)=\frac{1}{z}+{\sum_{L\neq 0}}^{\prime}\left(\frac{1}{z-L}+%
\frac{1}{L}+\frac{z}{L^{2}}\right)=\frac{1}{z}+{\sum_{L\neq 0}}^{\prime}\frac{%
z^{3}}{L^{2}(z^{2}-L^{2})},$$
(2)
where the primed sum is over all $L\in\Lambda$ except $L=0$.
Near $z$ = 0, it has a Laurent expansion
$$\zeta(z;\Lambda)=\frac{1}{z}-\sum_{k=2}^{\infty}\gamma_{2k}(\Lambda)z^{2k-1},$$
(3)
where
$\gamma_{2k}(\Lambda)$ are lattice invariants
$$\gamma_{2k}(\Lambda)={\sum_{L\neq 0}}^{\prime}\frac{1}{L^{2k}},\quad k\geq 2,$$
(4)
which are related to the Eisenstein series $G_{2k}(\omega_{2}/\omega_{1})$
= $(2\omega_{1})^{2}\gamma_{2k}(\Lambda)$.
The Weierstrass elliptic function $\wp(z;\Lambda)$ is (minus) the
derivative of the zeta function, and is an even doubly-periodic meromorphic
function of $z$ defined by
$$\wp(z;\Lambda)=-\zeta^{\prime}(z;\Lambda)\equiv-\frac{\partial}{\partial z}%
\zeta(z;\Lambda)=\frac{1}{z^{2}}+{\sum_{L\neq 0}}^{\prime}\left(\frac{1}{(z-L)%
^{2}}-\frac{1}{L^{2}}\right).$$
(5)
The elliptic function seem to have been Weierstrass’ main interest,
but the emphasis here will be on an application of the quasiperiodic
sigma function, defined in terms of the zeta function by
$$\sigma(z;\Lambda)=\lim_{\epsilon\rightarrow 0}\epsilon\exp\left(\int_{\epsilon%
}^{z}dz^{\prime}\zeta(z^{\prime};\Lambda)\right),$$
(6)
where the integral is independent of the path because the poles of the
zeta function have unit residue. The sigma function is an odd holomorphic
function with only simple zeroes which are at the lattice points
$z$ $\in$ $\Lambda$. Then
$$\zeta(z;\Lambda)=\frac{\sigma^{\prime}(z;\Lambda)}{\sigma(z;\Lambda)},\quad%
\sigma^{\prime}(0;\Lambda)=1.$$
(7)
A curious feature of the Weierstrass definitions
is the absence of a term
$O(z)$ in the Laurent expansion (3) for $\zeta(z;\Lambda)$.
Weierstrass presumably made
this choice because the definition (4) of $\gamma_{2k}(\Lambda)$ is ill-defined for $2k$
= 2, and he
was working long before Ramanujan investigated quasi-modular forms.
However, based on Ramanujan’s workram on such forms, there is a
“natural” definition of $\gamma_{2}(\Lambda)$ given by “Eisenstein summation”
$$\Gamma_{2}(\omega_{1},\Lambda)\equiv\sum_{m\neq 0}\frac{1}{(2m\omega_{1})^{2}}%
+\sum_{n\neq 0}\left(\sum_{m}\frac{1}{(2m\omega_{1}+2n\omega_{2})^{2}}\right)=%
\gamma_{2}(\Lambda)+\frac{\pi}{A(\Lambda)}\frac{\omega_{1}^{*}}{\omega_{1}},$$
(8)
where a sum over $n$ is interpreted as the limit as $N$
$\rightarrow$ $\infty$ of a sum over $n$ in the interval $[-N,N]$.
$\Gamma_{2}(\omega_{i},\Lambda)$ is related to the weight-2
quasi-modular-invariant
of the $2k$ = 2 Eisenstein series, and
$\gamma_{2}(\Lambda)$ is a lattice invariant related to the corresponding
“almost-holomorphic modular form”.
In fact knowledge of the invariant $\gamma_{2}(\Lambda)$
had already been available since 1847, fifteen years before
Weierstrass presented his work, in the
published (but apparently ignored) work of
Eisensteineis who died in 1851. This work was later briefly brought to
attention in 1891 in a lecture by Kronecker, who appears to have
belatedly recognized its
importance, but who unfortunately died just fifteen days after the lecture,
leaving extensive notes that remained unpublished, and which were believed lost until
recently rediscoveredems .
Some of Eisenstein’s results remained overlooked and forgotten
until they were reviewed in detail and publicized in a 1976 book by Weilweil .
Then defining “modified” Weierstrass functions by including the
$2k$ = $2$ term in the Laurent-series definition of
$\zeta(z;\Lambda)$ has the effect
$$\displaystyle\wp(z,\Lambda)$$
$$\displaystyle\mapsto\widetilde{\wp}(z,\Lambda)=\wp(z,\Lambda)+\gamma_{2}(%
\Lambda),$$
(9)
$$\displaystyle\zeta(z,\Lambda)$$
$$\displaystyle\mapsto\widetilde{\zeta}(z,\Lambda)=\zeta(z,\Lambda)-\gamma_{2}(%
\Lambda)z,$$
(10)
$$\displaystyle\sigma(z,\Lambda)$$
$$\displaystyle\mapsto\widetilde{\sigma}(z,\Lambda)=e^{-\frac{1}{2}\gamma_{2}(%
\Lambda)z^{2}}\sigma(z,\Lambda).$$
(11)
Note that $\gamma_{2}$ vanishes for both the hexagonal lattice ($\gamma_{4}$ = 0),
and the square lattice ($\gamma_{6}$ = 0), so in these high-symmetry
lattices, the “modified” and “original” Weierstrass functions
coincide. Specification of the pair of complex invariants $(\gamma_{4},\gamma_{6})$
(so long as they do not both vanish) uniquely parametrizes the lattice $\Lambda$.
For ease of notation, the dependence of these functions on the lattice
$\Lambda$ will now be left implicit, unless needed.
An important property of both the original and the modified Weierstrass
functions is that they depend only the lattice $\Lambda$, and do not
depend on a particular choice of basis $(\omega_{1},\omega_{2})$ (or on a
choice of orientation of the basis). The group $SL(2,\mathbb{Z})$ of
(orientation-preserving) changes of basis is known as the modular group,
and (with a slight abuse of terminology) the basis-independence of
the Weierstrass functions may be called “modular-invariance”.
The “modified” forms of the Weierstrass functions were
in fact already present in Eisenstein’s
workeis , as reviewed by Weilweil . In the
notation
used by Weil, $\widetilde{\zeta}(z)$ is “$E_{1}(z)$”, $\widetilde{\sigma}(z)$ is
“$\varphi(z)$”, and $\widetilde{\wp}(z)$ is “$E_{2}(z)$”.
A key result also found in Eisenstein’s workeis is that the
non-meromorphic function
$$\widehat{\zeta}(z,z^{*})=\widetilde{\zeta}(z)-\frac{\pi z^{*}}{A}$$
(12)
is a doubly-periodic modular-invariant
function (referred to as “$E_{2}^{*}(z)$” by Weilweil ).
In recent workrecent , there is renewed interest in this
function, which has been called
“Eisenstein’s (periodic) completion” of the
Weierstrass zeta function.
Generically, it has a lattice of simple poles
at $z$ $\in$ $\{L\}$, three lattices of
simple holomorphic zeroes
and two lattices of simple antiholomorphic zeroes.
Each lattice of holomorphic zeroes is invariant under inversion about
a pole, and they
are located at the points
$z$ $\in$ $\{L+\omega_{i}\}$, as a consequence of the relation
$$\zeta(\omega_{i})\equiv\eta_{i}=\gamma_{2}\omega_{i}+\frac{\pi\omega_{i}^{*}}{%
A};\quad\widetilde{\zeta}(\omega_{i})=\frac{\pi\omega_{i}^{*}}{A}.$$
(13)
The lattices of antiholomorphic zeroes are mapped into each other by
the inversion.
The six antiholomorphic zeroes
which are closest to a pole define the corners of a
six-sided unit cell centered on the pole, with holomorphic zeroes at
the centers of the sides. (This is similar to a “Wigner-Seitz” or
Voronoi cell, but direct inspection using the Weierstrass functions
implemented in Mathematica® shows that
the locations of the antiholomorphic zeroes are not in general given
by the Voronoi construction using the metric $d(z,z^{\prime})$ = $|z-z^{\prime}|$.)
In the limit of a simple-rectangular lattice, one pair of sides shrink
to points where two antiholomorphic zeroes combine with a holomorphic
one, leaving a rectangular unit cell with a simple antiholomorphic
zero at its corners.
Eisenstein’s periodic function (12) is a sum of
“holomorphic” (meromorphic) and antiholomorphic parts:
the “modified zeta function” is just the “holomorphic” part of
Eisenstein’s periodic completion of the zeta function.
A consequence of the relation (13) is the well-known property
$$\eta_{1}\omega_{2}-\eta_{2}\omega_{1}=\frac{\pi(\omega_{1}^{*}\omega_{2}-%
\omega_{1}^{*}\omega_{2})}{A}=\pm{\textstyle\frac{1}{2}}i\pi,$$
(14)
which is an orientation-independent version of a standard relation
between the $\eta_{i}$ and $\omega_{i}$. (Note that the formulation presented here
does not require the orientation of the basis to be chosen so
that the imaginary part of $\omega_{2}/\omega_{1}$ (or “$\tau$”) is
positive.) The explicit expression (13) involving
$\gamma_{2}(\Lambda)$
does not seem to be
as widely known in the literature on Weierstrass functions (it does
not seem to appear in the standard reference sources), but is
easily verified by
using a basis
$(\omega_{1},\omega_{2})$, with $\omega_{1}$ = $\omega_{i}$,
and evaluating the lattice sum (2) in the order used
in the definition (8) of $\Gamma_{2}(\omega_{1};\Lambda)$:
$$\eta_{1}=S(\omega_{1},\omega_{2})+\omega_{1}\Gamma_{2}(\omega_{1},\Lambda),$$
(15)
where the sum over $1/L$ is dropped because it vanishes by symmetry,
and
$$\displaystyle S(\omega_{1},\omega_{2})$$
$$\displaystyle=\sum_{n}\left(\sum_{m}\frac{1}{(2m+1)\omega_{1}+2n\omega_{2}}%
\right)=-\frac{\pi}{2\omega_{1}}\sum_{n}\tan(n\pi\omega_{2}/\omega_{1})=0.$$
(16)
The terms in the sum $S$ are odd under $n\mapsto-n$,
so it also vanishes. (This somewhat-heuristic symmetry-based argument can
be confirmed by more rigorous “$\epsilon\rightarrow 0$” treatments.)
The values of $\eta_{i}$ $\equiv$ $\zeta(\omega_{i};\Lambda)$
appear as a complication in
the expressions for the quasiperiodicity of the Weierstrass sigma function.
The equivalent quantity in the
“modified” sigma function has the much simpler form
$$\widetilde{\eta}_{i}\equiv\eta_{i}-\gamma_{2}\omega_{i}=\pi\omega_{i}^{*}/A,$$
(17)
so the modified values of the “strange numbers” $\eta_{i}$ that permeate various
sigma-function identities become the natural primitive “reciprocal half-lattice
vectors”
conjugate to the primitive “half-lattice vectors” $\omega_{i}$.
All the various Weierstrass sigma-function identities involving $\eta_{i}$ remain true
when the Weierstrass functions are replaced by their “modified”
forms and $\eta_{i}$ is replaced by $\pi\omega_{i}^{*}/A$.
In particular, for $q$ = $\exp(i\pi\tau)$ with $\tau$ = $\omega_{2}/\omega_{1}$,
$\sigma(z;\Lambda)$ has a rapidly-converging product representation
$$\sigma(z)=ze^{\eta_{1}z^{2}/2\omega_{1}}\left(\frac{\sin u}{u}\prod_{n=1}^{%
\infty}\frac{\left(q^{2n}+q^{-2n}-2\cos 2u\right)}{(q^{n}-q^{-n})^{2}}\right)%
\equiv ze^{\eta_{1}z^{2}/2\omega_{1}}\left(\frac{\vartheta_{1}(u|\tau)}{u%
\vartheta_{1}^{\prime}(0|\tau)}\right),\quad u=\frac{\pi z}{2\omega_{1}},$$
(18)
where $\vartheta_{1}(u|\tau)$ is the Jacobi theta-function with zeroes
at $\{m\pi+n\pi\tau\}$.
The modified form $\widetilde{\sigma}(z)$ is given by the same formula, with the replacement
$\eta_{1}\mapsto\widetilde{\eta}_{1}$ = $\pi\omega_{1}^{*}/A$.
Note that while the constructive definition of the original Weierstrass
sigma function (as an infinite product)
follows from the lattice sum (2), the constructive
definition of the modified sigma function is directly given by
(18) with
$\eta_{1}\mapsto\pi\omega^{*}_{1}/A$.
The expression (18) is not a constructive definition for the
original Weierstrass sigma function, as it requires a separate evaluation of
$\gamma_{2}(\Lambda)$ in order to provide $\eta_{1}$. Similarly, an
expression based on (2) does not provide a constructive
definition of the modified form, because it would then also require
a separate evaluation
of $\gamma_{2}(\Lambda)$.
The quasiperiodicity of the modified sigma function now has
the more “elegant” form
$$\widetilde{\sigma}(z+L)=\xi(L)e^{(\pi L^{*}/A)(z+\frac{1}{2}L)}\widetilde{%
\sigma}(z),$$
(19)
where $\xi(L)$ is the parity of $L$:
$\xi(L)=1$ if $\frac{1}{2}L\in\Lambda$, and $-1$ otherwise.
In the limit $|z|\rightarrow 0$, the holomorphic sigma functions (both the
modified and original forms) have the property
$$\lim_{z\rightarrow 0}\frac{\sigma(z)}{z}\rightarrow 1.$$
(20)
It is worth pointing out that the non-holomorphic function
$$\mathcal{Z}(z,z^{*})\equiv\widetilde{\sigma}(z)e^{-\frac{1}{2}\pi z^{*}z/A}$$
(21)
has the property that its absolute value $|\mathcal{Z}(z,z^{*})|$ is a function with the
lattice periodicity. This may be regarded as the nearest equivalent
to the linear function $f(z)$ = $z$ that can be made (quasi)periodic,
with
$$\lim_{z\rightarrow 0}\frac{\mathcal{Z}(L+z,L^{*}+z^{*})}{z}\rightarrow\xi(L).$$
(22)
This function is essentially an analog of the sigma function defined by using
(6) with Eisenstein’s periodic completion (12) as a
replacement for the zeta function.
For any primitive
$\omega_{i}$,
there is a symmetric variant of the sigma function defined by
$$\sigma_{i}(z)=\sigma_{i}(-z)=e^{-\eta_{i}z}\sigma(z+\omega_{i})/\sigma(\omega_%
{i}),$$
(23)
so $\sigma_{i}(\omega_{i}+L)$ = 0.
This is normalized so that $\sigma_{i}(0)$ = 1, and
there are just three
distinct functions $\sigma_{i}(z)$, $i$ = 1,2,3,
that can be labeled in a basis-dependent
way using $\omega_{1}$, $\omega_{2}$ and $\omega_{3}$.
The replacement of $\eta_{i}$ by $\widetilde{\eta}_{i}$ and $\sigma(z)$ by
$\widetilde{\sigma}(z)$ now
defines modified symmetric sigma function variants $\widetilde{\sigma}_{i}(z)$.
The symmetric variants have the same quasiperiodicity as the modified
sigma function under even translations:
$$\widetilde{\sigma}_{i}(z+2L)=e^{(\pi L^{*}/A)(2z+L)}\widetilde{\sigma}_{i}(z).$$
(24)
The product (18) provides a very efficient algorithm for
numerical computation of the modified sigma function in
finite-precision floating-point arithmetic, as terms in the product eventually
become numerically equal to unity above some $n$. An optimum
algorithm appears to involve choosing $\omega_{1}$ and $\omega_{2}$ so
that $\pm\omega_{1}$, $\pm\omega_{2}$ and $\pm\omega_{3}$, are on the
boundary of the Voronoi cell of the lattice point at the origin, with
$|\omega_{1}|$ $\leq$
$|\omega_{2}|$ $\leq$ $|\omega_{3}|$. Then use the quasiperiodicity
to reduce $z$ into this Voronoi cell, evaluate the product for
the reduced $z$, and use the quasiperiodicity to migrate the result
back to the original $z$. The accuracy of the migration process can
be enhanced if it is known that an integer multiple of $z$ is a lattice
point $L$.
In the view of this author, the various properties described above justify the view that
the modified functions are the natural ones that
Weierstrass should have defined,
and probably would have used, if knowledge of the theory of quasi-modular forms had been available
to him. Some evidence in favor of this assertion comes from an
application to the physics of electrons in a Landau level on
a two-dimensional plane that is compactified into a torus, where the
wavefunctions have a natural representation in terms of the modified
(as opposed to the original) sigma function.
This problem can be formulated as that of non-relativistic electrons with a dispersion
$\varepsilon(\bm{p})$
= $|\bm{p}|^{2}/2m$, with dynamical momentum $\bm{p}$ = $-i\hbar\bm{\nabla}-e\bm{A}(\bm{r})$, moving on a plane though which a uniform
magnetic flux density passes. This quantizes the kinetic energy to take discrete
values $(\hbar^{2}/m\ell^{2})(n+\frac{1}{2})$, $n=0,1,2,\ldots$, where $2\pi\ell^{2}$ is
the area through which one London quantum $\Phi_{0}$ = $2\pi\hbar/e$
of magnetic flux passes.
In the conventional treatment using the
so-called “symmetric gauge”, where the electromagnetic vector potential
$(A_{x},A_{y})$ is given by $A_{x}+iA_{y}$ =
$i\Phi_{0}z/(4\pi\ell^{2})$, and $z$ = $x+iy$ is a mapping from the
Euclidean plane to the complex plane, the lowest-Landau-level ($n=0$) wavefunctions have the form
$$\psi(z,z^{*})=f(z)e^{-\frac{1}{4}z^{*}z/\ell^{2}}$$
(25)
where $f(z)$ is a holomorphic function.
Quasiperiodic boundary conditions under a
Bravais lattice $\Lambda$ of translations with a
primitive region of area $A$ = $2\pi N_{\Phi}\ell^{2}$ (and integer
$N_{\Phi}$) can be imposed
(as a technical device to avoid the problems of working with an
infinite system), so that
$$\psi(z+L,z^{*}+L^{*})=\xi(L)^{N_{\Phi}}e^{K^{*}L-KL^{*}}e^{\frac{1}{4}(L^{*}z-%
Lz^{*})/\ell^{2}}\psi(z,z^{*}),$$
(26)
or
$$f(z+L)=e^{K^{*}L-KL^{*}}\left(\xi(L)e^{(\pi L^{*}/A)(z+\frac{1}{2}L)}\right)^{%
N_{\Phi}}f(z).$$
(27)
This is solved by wavefunctions with
$$f(z)\propto e^{K^{*}z}\left(\prod_{i=1}^{N_{\Phi}}\widetilde{\sigma}(z-w_{i})%
\right),\quad\sum_{i=1}^{N_{\Phi}}w_{i}=KA/\pi.$$
(28)
This wavefunction has $N_{\Phi}$
zeroes in the fundamental region.
The many-particle Slater-determinant
state with $N$ = $N_{\Phi}$ fermions that completely
fills the lowest Landau level is then given by
$$\Psi\propto e^{K^{*}Z}\widetilde{\sigma}(Z-W)\left(\prod_{i<j}\widetilde{%
\sigma}(z_{i}-z_{j})\right)\prod_{i}\left(e^{-\frac{1}{2}\pi z_{i}^{*}z_{i}/A}%
\right)^{N_{\Phi}},\quad Z=\sum_{i}z_{i},\quad W=KA/\pi.$$
(29)
Many other more-interesting model many-particle states on the torus (e.g., the
Laughlin state) can be written in an explicitly modular-invariant
way using the modified sigma function described here. These will be
described elsewhere.
The importance of these constructions using the modified sigma
function is that they are explicitly modular-invariant, unlike most
previous representations using the Jacobi elliptic function
$\vartheta_{1}(u|\tau)$, $u$ = $\pi z/2\omega_{1}$,
$\tau$ = $\omega_{2}/\omega_{1}$.
When constructing model wavefunctions
on the toroidal compactification of the plane,
an important physical requirement is that they do not depend on the
choice of basis of $\Lambda$, i.e., that they are
modular-invariant. Use of the modified sigma function ensures this property is explicit.
Whether or not the simple “modification” described here is useful in
other applications of the Weierstrass functions, it is now clear that the
modified sigma function is the variant of choice for describing lowest
Landau-level wavefunctions. However the simplicity of the formula (17),
and the appearance of “naturalness” of the “completion” by the inclusion of the
“missing” $2k$ = 2 term in the Laurent expansion (3),
suggests that all applications of the Weierstrass functions could
benefit
from the modification proposed here, which has antecedents in Eisenstein’s workeis ; weil .
Acknowledgements.
This work was supported by Department of Energy BES Grant DE-SC0002140.
References
(1)
K. Weierstrass, Mathematische Werke von Karl Weierstrass,
(Mayer & Müller, Berlin, 1895), Vol 2, p. 245.
(2)
S. Ramanujan, Trans. Cambridge Philos. Soc. 22, 159 (1916).
(3)
G. Eisenstein, J. Reine Angew. Math. 35, 137 (1847); reprinted in
G. Eisenstein, Mathematische Werke, (Chelsea, New York, 1989),
Vol. I, p. 357.
(4)
P. Charollois and R. Sczech, EMS Newsl. 9, 8 (2016).
(5)
A. Weil,
Elliptic functions according to Eisenstein and Kronecker,
2nd Edition (Springer-Verlag, Berlin, 1999). (1st Edition 1976.)
(6)
L. Rolen, Proc. Amer. Math. Soc. 144, 1453 (2016). |
The distribution of nearby stars in phase space mapped by
Hipparcos††thanks: Based on data from the Hipparcos astrometry
satellite
M. Crézé
1Centre de Données astronomique de Strasbourg, CNRS URA1280
11 rue de l’Université, F-67000 Strasbourg, France
12IUP de Vannes,
8 rue Montaigne, BP 561, 56017 Vannes Cedex, France
2
E. Chereul
1Centre de Données astronomique de Strasbourg, CNRS URA1280
11 rue de l’Université, F-67000 Strasbourg, France
1
O. Bienaymé
1Centre de Données astronomique de Strasbourg, CNRS URA1280
11 rue de l’Université, F-67000 Strasbourg, France
1
C.Pichon
3Astronomisches Institut Univ. Basel,
Venusstrasse 7,
CH-4102 Binningen, Switzerland 3
(Received / Accepted )
Abstract
Hipparcos data provide the first, volume limited and
absolute magnitude limited homogeneous tracer of stellar density and
velocity distributions in the solar neighbourhood. The density of
A-type stars more luminous than $M_{v}=2.5$ can be accurately mapped
within a sphere of 125 pc radius, while proper motions in galactic
latitude provide the vertical velocity distribution near the galactic
plane. The potential well across the galactic plane is traced
practically hypothesis-free and model-free. The local dynamical
density comes out as $\rho_{0}=0.076\pm 0.015~{}M_{\sun}~{}{\rm pc}^{-3}$ a
value well below all previous determinations leaving no room for any
disk shaped component of dark matter.
keywords: Hipparcos - Galaxy: kinematics and dynamics -
Galaxy: fundamental parameters -
Galaxy: halo -
solar neighbourhood -
Galaxy: structure -
dark matter -
\thesaurus
04 10.11.1; 10.06.2; 10.08.1; 10.19.1; 10.19.3; 12.04.1
\subtitle
I. The potential well and local dynamical mass
\offprints
M. Crézé
1 Introduction
All the data used here were collected by the Hipparcos satellite
([ESA, 1997]). Individual stellar distances within more than 125 pc
were obtained with an accuracy better than 10$\%$ for almost all
stars brighter than $m_{v}=8.$, together with accurate proper
motions. Based on these data, a unique opportunity is offered to
revisit stellar kinematics and dynamics; any subsample of sufficiently
luminous stars is completely included within well defined distance and
luminosity limits, providing a tracer of the local density-motion
equilibrium in the galaxy potential: a snapshot of the phase space. We
have selected a series of A-F dwarf samples ranging from $M_{v}=-1.0$
down to $M_{v}=4.5$. Completeness is fixed within 50 pc over the whole
magnitude range and within 125 pc at the luminous end ($M_{v}\leq 2.5$). In this series of papers we shall investigate such samples in
terms of density and velocity distribution small scale inhomogeneities
addressing the problem of cluster melting and phase mixing.
The expected first order departure to homogeneity is the potential
well across the galactic plane. This problem is well known in galactic
dynamics; it is usually referred to as “the $K_{z}$ problem”, where
$K_{z}$ means the force law perpendicular to the galactic plane. The
$K_{z}$ determination and subsequent derivation of the local mass
density $\rho_{0}$ has a long history, nearly comprehensive reviews
can be found in [Kerr & Lynden-Bell (1986)] covering the subject before 1984 and in
[Kuijken (1995)] since 1984. Early ideas were given by Kapteyn (1922),
while Oort (1932) produced the first tentative determination.
The essence of this determination is quite simple: the kinetic energy
of stellar motions in the z direction when stars cross the plane fixes
their capability to escape away from the potential well. Given a
stellar population at equilibrium in this well, its density law
$\nu(z)$ and velocity distribution at plane crossing $f(w_{0})$ are tied
to each other via the $K_{z}$ or the potential $\phi(z)$. Under quite
general conditions the relation that connects both distributions is
strictly expressed by Eq. (1).
$$\nu(\phi)=2\int_{\sqrt{2\phi}}^{\infty}{f(|w_{0}|)~{}w_{0}~{}{\rm d}w_{0}\over%
\sqrt{w_{0}^{2}-2\phi}}$$
(1)
This integral equation and its validity conditions are established and
discussed in detail by [Fuchs & Wielen (1993)] and [Flynn & Fuchs (1994)]. There is no
specification as to the form of distribution functions $f$ and $\phi$
except smoothness and separability of the z component.
Given $\nu(z)$ and $f(w_{0})$, $\phi(z)$ can be derived. Then according
to Poisson equation, the local dynamical density comes out as
$$\rho_{0}={1\over 4\pi G}({\rm d}^{2}\phi/{\rm d}z^{2})$$
(2)
Early determinations of $\rho_{0}$ ([Oort, 1960]) were as high as
$0.18\,M_{\sun}\,{\rm pc}^{-3}$ while the total mass density of stars
and interstellar matter would not exceed $0.08\,M_{\sun}\,{\rm pc}^{-3}$. The discrepancy between these two figures latter
re-emphasised by J. Bahcall and collaborators (1984ab, 1992), came
into the picture of the dark matter controversy to make it even
darker. While the local density of standard “dark halos” required
to maintain flat rotation curves of galaxies away from their center
would hardly exceed 0.01, such ten times larger densities could only
be accounted for in rather flat components which composition and
origin could not be the same. Following Bahcall’s claim, several
attempts were made to reduce the intrinsic inaccuracy of the dynamical
mass determinations in particular [Bienaymé et al (1987)], used a complete
modelisation of the galaxy evolution to tie the $K_{z}$ determination to
star counts, [Kuijken & Gilmore (1989)] used distant tracers of K type stars to
determine not the local volume density but the integrated surface
density below 1 kpc, [Flynn & Fuchs (1994)] pointed at difficulties in
relation with the underlying dynamical approach. Most concluded that
there was no conclusive evidence for a high dynamical density if all
causes of errors were taken into account.
The local mass density of observable components (the observed mass)
can hardly exceed $0.08M_{\sun}{\rm pc}^{-3}$ out of which a half
stands for stars which local luminosity function is reasonably well
known. The mass density of interstellar gas and dust is not so easily
accessed since the bulk of it, molecular hydrogen, can only be traced
via CO and the ratio CO/$\rm{H_{2}}$ is not so well established. Also
interstellar components are basically clumpy and distances poorly
known: it is not so easy to define a proper volume within which a
density makes sense. A conservative range for the observed mass
density should be $[0.06,0.10]$ including 0.04 for stars.
Nevertheless, so far the discrepancy used to be charged to the
dynamical density since the determination of this quantity involved
many difficulties, most poorly controlled :
•
Tracer homogeneity: first of all, the determination
require that a suitable tracer can be duly identified. A criterion
independent of velocity and distance should allow to detect and select
all stars matching the tracer definition. This was hardly achieved by
spectral surveys which completeness and selectivity degrade as
magnitude grows.
•
Stationarity: in order for the tracer to make dynamical
sense, it should be in equilibrium in the potential. Youngest stars
which velocities reflect the kinematics of the gas at their birth time
do not fulfill this requirement.
•
Undersampling: one searches for the bending of density
laws caused by gravitation along the z direction. This imposes not
only that tracer densities can be determined, but also that they can
be determined at scales well below the typical length of the
bending. This means that the tracer should be dense enough in the
region studied. Most determinations so far rely upon sample surveys in
the galactic polar caps, resulting in “pencil beam” samples quite
inappropriate to trace the density near the plane ([Crézé et al, 1989]). When
the bending is traced at large distances from the plane as in
[Kuijken & Gilmore (1989)], one gets information on the surface density below a
certain height. A comparison of this surface density with the local
volume density involves modeling hypotheses. Another associated effect
can be generated by a clumpy distribution of the tracer even
stationary if the tracer includes clusters or clumps the statistics of
densities and velocities may be locally biased. Other problems arise
from the velocity distribution sampling.
•
Systematic errors: out of necessity before Hipparcos,
all previous studies used photometric distances. Uncertainties in the
calibration of absolute magnitudes resulted in systematic errors in
density determinations. This is particularly true for red giants which
are not at all a physically homogeneous family and show all but normal
absolute magnitude distributions: it is striking that calibrations
based on different samples provide discrepant calibrations outside the
range permitted by formal errors. This produced both biased density
distributions and uncontrolled effects in selecting volume limited
samples.
•
Random errors: distances and velocities were also
affected by random measurement errors. Even perfectly calibrated,
standard errors of photometric distances based on intermediate band
photometry would hardly get below 20%.
Hipparcos data solve nearly all the problems quoted above: instead of
“pencil beam” tracers at high galactic latitudes which distribution
is only indirectly related to the local $K_{z}$, what we get here is a
dense probe inside the potential well.
The samples were preselected within magnitude limits inside the
completeness limit of the Hipparcos survey program
([Turon & Crifo, 1986]). It includes all stars brighter than $m_{v}$ = 8.0 (7.9
at low latitudes) with spectral types earlier than G0, stars with no
or poorly defined spectral types were included provisionally.
Within this coarse preselection, sample stars were eventually selected
on the basis of their Hipparcos derived distance, magnitude and
colour. So the only physical criterion is absolute magnitude while the
sample is distance limited on the basis of individual parallaxes.
Within 125 parsecs, Hipparcos parallaxes provide individual distances
with accuracies better than 10 percent for over ninety percent of the
sample. Distances are individual and free from calibration errors.
Samples are dense, typical inter-star distances are less than 10 pc.
This is suitable to trace density variations which typical scales are
of the order of hundred parsecs. Thanks to this high tracer density,
difficulties related to clumpiness can be monitored through
appropriate analysis of local density residuals. Stationarity
considerations can also be monitored since W velocities are available
for stars everywhere in the volume studied.
The main characteristics of the tracer samples are reviewed in section
2. In order to take full benefit of this new situation, an original
method has been developed to analyze the densities. The statistical
aspects of this method based on single star volumes (volume of the
sphere extending to the nearest neighbour) are described in section 3
yielding a model free view of the potential well. Then an estimate of
the local mass density of matter is produced (section 4).
Consequences in terms of galactic structure and dark matter
distribution are reviewed in conclusion (section 5).
2 The Hipparcos Sample
2.1 Sampling definition
The sample was pre-selected from the Hipparcos Input Catalogue
([ESA, 1992]), among the “Survey stars”. The Hipparcos Survey
delimitation is described in full detail in Turon & Crifo (1986). For
spectral types earlier than G5 the limiting magnitude is $m_{v}\leq~{}7.9+1.1sin|b|$. It is complete within those limits so
far as the apparent magnitudes were known. Inside the Hipparcos
Survey, the phase space tracer sample has been first given a coarse
limitation: spectral types later than A0 and earlier than G0,
luminosity class V through IV. Stars with uncertain or undefined
luminosity classes were kept in the sample at this stage. Within this
pre-selection, the final choice was based on Hipparcos magnitudes
($m_{v}<8.0$), colours ($-0.1~{}\leq~{}B-V~{}\leq~{}0.6$) and parallaxes
($\pi~{}\leq~{}8~{}{\rm mas}$). Then a series of subsamples were cut in
absolute magnitude and distance in such a way that each subsample is
complete for the adopted colour range within a well defined sphere.
The resulting sample characteristics are summarized in Tab.(1). The
sample designation system is straightforward: h125 means stars within
125 pc, under the apparent magnitude condition $m_{v}<8.0$, imposes
not to consider stars less luminous than $M_{v}\leq 2.5$. All stars of
sample h125 closer than 100 pc are included in sample H100 and so
on. Samples named k… come from the corresponding h… samples but
the identified cluster stars have been removed. The typical density of
each sample estimated as the inverse of the average single star volume
is given in the last column of Table 1. The corresponding
typical interstar distance of e.g. tracer h125 is 8.8 pc.
2.2 Completeness
With this method, the sample limits do not rely upon pre-Hipparcos
data except for the faintest stars next to the edge of each sphere,
some of those might have been missed due to the limited accuracy of
pre-Hipparcos magnitudes. The method may look a naive one since there
are now sophisticated algorithms to correct estimations for censored
data, (see for instance [Ratnatunga & Casertano, 1991]) but it is the only one
that does not make assumptions as to the absolute magnitude
distribution of the stars. All previous investigations did assume that
the absolute magnitudes of stars sharing a certain spectral or
photometric characteristic are normally distributed around a common
mean value.
Figures 1 and 2 give a quantitative
illustration of the risks involved: the absolute magnitude
distributions of dwarfs within four V-I ranges show an obvious
asymmetry while in the reddest part, the luminosity distribution spans
over more than two magnitudes. This means that however accurately the
apparent magnitude can be measured, the uncensoring correction would
be much more uncertain than other error factors.
With the approach adopted here, Fig. 3 shows the
$M_{v}$/distance distribution of the global sample, the neat border
follows simply the line $m_{v}=8$; the censoring effects are maximum
along this line. Our subsamples are cut along lines parallel to the
axes of this plot, so they reach the censoring line only at the bottom
right corner of each zone.
Since the accuracy of Hipparcos magnitudes is far beyond the
necessities of this study, the sampling biases can only result from
two effects: the parallax errors which, however unprecedently small
are still of the order of $10\%$ beyond 100 pc, and the stars lost at
the time of the early selection due to the inaccuracy of apparent
magnitudes available then. Concerning the second effect, it can only
result in stars being omitted near the edge of the sample. The residual
effect of apparent magnitude incompleteness can only affect the
faintest magnitudes at the edge of each sample sphere, since the
single star volume used in sections 3 and 4 as statistics for the
density estimation are local statistics, it is easy to keep such
effect under control in the study of residual, furthermore this kind
of incompleteness if important would bias the estimation towards
making the potential well even shallower than observed.
2.3 Distance errors
The other possible source of bias left is generated by distance
errors. Figure 4 shows the distribution of parallax
errors versus distance for sample h125 which, extending to 125 pc is
the most severely affected by distance errors. This is still a
moderate effect: one can see on the figure that only a handful of
parallax errors exceed 15$\%$. Parallaxe errors are Hipparcos formal
errors. All feasible external tests able to check the actual accuracy show
that these formal errors are if something a little pessimistic
([Arenou et al, 1995]) with respect to previous studies of the density
trends.
3 Mapping the tracer density and the potential well
The essential choice of this investigation is to stick as closely as
possible to observations. Whatever the velocity distribution $f(w_{0})$,
once an homogeneous tracer has been defined, that is once a selection
criterion has been defined uncensored with respect to velocities and
positions, Eq. (1) expresses the correspondence between the
tracer density law and the potential as a plain change of variables.
So if there is a potential well to be found in the data, it must
appear as a density peak. If we can map the density trend model free,
we’ll get a picture of the potential well.
3.1 Single star volume statistics
The tracer should be dense enough for the typical distances between
tracer stars to be small with respect to the typical scale of density
variations. Under this assumption, at a place were the tracer density
is $\nu$, the observed number of stars counted in any probe volume $V$
is a Poisson variate with expectation $V\cdot\nu$. Defining $v$ as
the single star volume around one specific star (that is the sphere
extending to the nearest neighbour), and introducing the quantity $x=v\cdot\nu$ we get a new variate which expectation is 1 and which
probability distribution is exponential (Eq. (3)). It is the
probability distribution of the distance between two events in a
Poisson process under unit density.
$${\rm d}P(x)=\exp(-x)~{}{\rm d}x$$
(3)
and the probability distribution of the single star volume $v$ is
$${\rm d}P(v)=\nu\cdot\exp(-\nu\cdot v)~{}{\rm d}v$$
(4)
so the expectation of the single star volume is $1/\nu$, which makes
it a suitable local statistics for the density. It is not a novel
approach to use statistics based on nearest neighbour distances to
investigate densities, but it is quite appropriate here since it turns
out to provide a parameter free Maximum Likelihood estimator for the
density by plain moving average of single star volumes. Also, single
star volumes being a local measure of the inverse density at the
position of each star, they offer the possibility to calculate
individual density residuals. Checks can be made that deviations from
average or model predicted values are randomly distributed and do not
show unmodeled systematic trends.
3.2 Maximum Likelihood estimator
In a constant density sample, the plain average of single star volumes
is a maximum likelihood statistics for the density: let $v_{i}$ be the
single star volume around star i, according to Eq. (4) the
log-likelihood of a sample of n stars $(i=1,n)$ under constant
density $\nu$ is simply given by
$$\log L=\sum_{i=1,n}(\log(\nu)-\nu\cdot v_{i})$$
(5)
and the Maximum Likelihood is reached for
$${\rm d}\log L/{\rm d}\,\nu=0\\
$$
(6)
which obvious solution is:
$$n/\nu=\sum_{i=1,n}v_{i}\\
$$
(7)
$$\nu=1/<v_{i}>$$
(8)
this makes the moving average of $v_{i}$ along a parameter a maximum
likelihood mapping of the density variations along this parameter,
provided only that variations along other parameters are negligible.
3.3 Residual statistics
Furthermore, given a model or an estimate of $\nu$ near star i the
quantities $x_{i}=\nu\cdot v_{i}$ are local “residuals” of the fit
and their distribution under the assumed model should be exactly
given, parameter-free, by Eq. (3). So the distribution of
residuals provides an immediate test of the validity of the model
including the fact that the density is smooth and do not include too
many clusters or voids. This distribution can be tested over the whole
sample or over any subset chosen to explore neglected parameters (e.g.
the completeness near the edge of the sampling volume, completeness in
apparent magnitude). In order to visualize the agreement (or
disagreement) of the observed distributions we have used two
representations: the histogram of the log residuals $y=log(x)$ which
distribution according to Eq. (3) must be given by
$${\rm d}P(y)=\exp[y-\exp(y)]{\rm d}y\\
$$
(9)
and the cumulative distribution of x:
$$H(X)=\int_{0}^{X}\exp(-x)\,{\rm d}x=1-\exp(-X)\\
$$
(10)
Let $n_{i}$ be the rank of residual $x_{i}$ in a sample of N values, then
according to Eq. (10),
$$n_{i}/N=1-\exp(-x_{i})\\
$$
(11)
$$-\log(1-n_{i}/N)=x_{i}\\
$$
(12)
In the following the residuals have been plotted as $-log(1-n_{i}/N)$
against $x_{i}$ producing what statisticians use to call exponential
probability plots. On such plot, observed residual distributions that
fail matching the expected exponential at any scale sign their
existence by departures from the diagonal.
3.4 Edge effect
Around stars close to the edge of the completeness sphere, the single
star volume is not anymore the full volume of the sphere to the
nearest neighbour. The volume must be corrected for the part of the
sphere outside the completeness volume (Fig. 5).
Defining $d_{nn}$ the distance to the nearest neighbour, $d$ the
Sun-star distance and $R$ the radius of the sample sphere, and
defining the two spheres, one centered at sun position with radius
$R$, the other at star position with radius $d_{nn}$, the corrected
volume is calculated as $V_{cor}$ according to the following formulae.
Angles $\theta_{1}$ and $\theta_{2}$ are the half apertures of the cones
intercepting the intersection of the two spheres (summit at sun
position for $\theta_{2}$, at star position for $\theta_{1}$).
$$\cos\theta_{2}=(d^{2}+R^{2}-d_{nn}^{2})/(2dR)$$
(13)
$$\sin\theta_{1}=(R/d_{nn})\sin\theta_{2}$$
(14)
$$V_{1}=(\pi/3)d_{nn}^{3}(2-3\cos\theta_{1}+\cos^{3}\theta_{1})$$
(15)
$$V_{2}=(\pi/3)R^{3}(2-3\cos\theta_{2}+\cos^{3}\theta_{2})$$
(16)
$$V_{cor}=(4/3)\pi d_{nn}^{3}-(V_{1}-V_{2})$$
(17)
3.5 Mapping the density
Single star volumes which are unbiased local estimates of the inverse
tracer density have been computed for all the tracer samples described
in Table 1. Defining x,y,z as a set of cartesian coordinates
centered at the sun position, with z positive pointing towards
$b=+90^{\circ}$, x positive towards the galactic center ($l=0^{\circ}$)
and y towards $l=90^{\circ}$, each series of values was sorted along
x, y and z successively and each resulting set smoothed by moving
average along the sort parameter. Single star volumes are averaged
over 101 neighbouring stars, producing parameter free maximum
likelihood inverse density profiles. Figures 6 through
8 show the resulting profiles along x, y and z for our
three most accurate and extended samples (namely samples h125, H100
and H80, according to definitions of section 2.1).
In these figures the x and y plots are given only as references: since
the sampling definition has spherical symmetry, any systematic effect along
the z axis which would trace a sampling artifact rather than a real trend
of the density would appear as well along the x and y axis.
At first glance only do the h125 and H100 profiles along the z axis
show significant bending. Expectedly the youngest subsamples with the
smallest velocity dispersions are also the most sensitive
tracers. This is a direct mapping of the potential well. The next
section will be dedicated to deriving quantitative consequences of
this observation. Beforehand the result should be scrutinized in the
light of two possible criticisms: biases in the inverse volume
estimates at the edges of the sample and clumpiness effects among
poorly mixed A stars.
Edge effects potentially related to completeness and/or to distance
errors can be easily controlled: would they be responsible for the
main part of the observed bending they would appear as well on the x
and y plots. There is no such thing visible even at marginal level. In
addition, any significant incompleteness at the edges would result in
a density decrease driving the single star volume statistics up at the
edges. As a result, the dynamical density derived below, once
corrected for this effect, would be even smaller.
A more serious problem comes from the clumpiness of the distribution:
young stars that dominate the most luminous samples are likely to be
partly concentrated in clusters or clumps on scales of a few parsecs
to a few ten parsecs. Such clumps must be suspected to distort the
density profile.
The fluctuations observed at small scales on figure 6 through 8 are not
significant in this respect, they are the result of random fluctuations
of single star volumes smoothed by the moving average process. The moving
average introduces a strong self correlation in the series. As a result
random fluctuations fake systematic ones on scales which are connected with
the z ranges spanned by 101 neighbouring stars, but do not reflect these
ranges in any simple way (it has something to do with the probability
of getting one, two, three … extreme peaks within adjacent strips).
Similar pictures are obtained on strictly uniform random simulations.
The only place where to check this effect is the statistics of residuals.
Real clumps which are not plain random fluctuations in a uniform density
are expected to sign their presence not by their local density which
may not be very high locally on the average, but by the frequency of
corresponding single star volume residuals (see section 3.3 for the
definition of these residuals. Such an overfrequency is
clearly visible on the bottom left wing of the histogram on figure 9a.
In terms of cumulative distribution it is even more clear on figure 9b.
A separate analysis of clustering has been performed by
means of wavelet analysis (Chereul et al. to appear as paper II of
this series) showing that no more than 8$\%$ out of stars in the h125
sample are involved in clumps. Removing only cluster members
identified through proper motions we get sample k125. The z inverse
density profile of this new sample is plotted on figure 6d, the
profile is not strongly modified by removing cluster stars. On the
residual plot (figure 9c) the residual anomaly at small scales
vanishes definitely. So the inverse density profiles turn out quite
robust to clumpiness effects. Furthermore we shall see in the following
(section 4.2 and figure 13) that local mass density estimates based on the
two samples (with or without cluster stars) are fully compatible. Also,
once a suitable density model is found, the residual statistics of
sample k125 (Figure 13b) do not show any significant feature that might signal
undetected clumpiness left.
This gives confidence that the bending is real and that figures 6cd, 7c
are the first ”image” of the galactic potential well. The remarkable
feature here is that this well is extremely shallow. We shall see in
the following that any substantial layer of dark matter contributing
significantly in the local mass density would unmistakably sign its
presence by deepening this well far beyond observed shapes. This is
true also for samples H80 and H100 which, being dominated by elder
stars are less affected by the clumpiness effect.
4 Estimating $\rho_{0}$
Solving Eqs. (1) and (2) for $\phi(z)$ can follow two
different ways: a parametric approach and a non parametric one.
Either were used on the above defined samples producing quite similar
results.
•
In the parametric approach, realistic mathematical forms
are adopted for both $\phi(z)$ and $f(w_{0})$. Free parameters for
$f(w_{0})$ can be determined by fitting the observed vertical velocity
distribution then adopting $f(w_{0})$ free parameters for the potential,
$\phi(z)$ can be derived from the observed density. The advantage of
this approach is simplicity in so far as manageable formulae can be
found to represent $f(w_{0})$. The cost is that results may be biased
by the choice of the model. An advantage may be that systematic
effects can be traced and individual residuals can be computed at the
position and velocity of each star. This approach is developed in
section 4.1.
•
At the other end one can try to produce a purely
numerical fit. Since this kind of fit is unstable to noise in the
data, some regularization should be applied. This non parametric
approach is developed in section 4.3.
4.1 Parametric approach: velocity analysis
The observed W vertical velocity distributions in the galactic plane
were derived from Hipparcos proper motions in galactic latitude and
parallaxes. For this analysis, samples were restricted to stars below
$b=\pm 10\degr$ so that $\mu_{b}/\pi$ simply reflects the W
tangential velocity.
W histograms of the four samples under consideration are shown on
Fig. 10.
All four distributions show reasonable symmetry and smoothness with
the following exceptions: h125 (Fig. 10a) shows pronounced
spikes corresponding to clusters (spikes vanish in the histogram of
k125) and a dissymmetry of large velocities (involving only 5 stars
out of 456). The plain average $W_{0}$ has been estimated for all
samples and appears quite stable suggesting that unsteady streaming
motions characteristics of very young stars do not dominate. Under
such conditions, the capability of the subsample of 536 low latitude
stars to be representative of the whole population within 125 pc is
questionable. Radial velocities have been compiled from the SIMBAD
database in order to calculate space velocities for as many stars as
possible. This resulted in a new velocity sample named h125W including
1366 stars spread in the whole sphere. The distribution of W velocity
components has been studied in the following along the same lines as
the other velocity sample.
A double gaussian centered model was fitted to each centered histogram
using a simple maximum likelihood scheme:
the model fitted is
$$\displaystyle{\rm d}P(w)=\{{\beta\over\sqrt{2\pi}~{}\sigma_{w1}}\exp[-{1\over 2%
}(w-<w>)^{2}/\sigma_{w1}^{2}]$$
$$\displaystyle+{1-\beta\over\sqrt{2\pi}~{}\sigma_{w2}}\exp[-{1\over 2}(w-<w>)^{%
2}/\sigma_{w2}^{2}]~{}\}~{}~{}{\rm d}w$$
(18)
and the maximum likelihood is searched for on $\beta$, $\sigma_{w1}$
and $\sigma_{w2}$.
There is no claim here that the kinematical mixture is isothermal. The
idea is simply that, if this mixture can be accurately represented by
a sum of gaussians, then there is a simple solution to the self
consistent dynamical equilibrium, the force law and the local volume
density can be solved for explicitly. The justification lays in the
quality of the representation. This is also justified a posteriori by
the fact that non parametric solutions do provide the same results.
Table 2 gives the maximum likelihood solutions for the four
main samples investigated here. Two tests of the quality of the fit
are given, one is the Kolmogorov-Smirnoff test: Pk is the observed
Kolmogorov maximum distance, Pk(0.8) and Pk(0.95) are the 20$\%$ and 5
$\%$ rejection thresholds for this parameter. Obviously no rejection
of the double gaussian representation can be supported at any
meaningful level.
The second test is given by probability plots (Fig. 11), velocity
ranks in each sample are plotted against the theoretical cumulative
probabilities of associated velocities: velocities distributed
according to the estimated distributions would produce a strictly
linear plot. So any deviation from linearity signs a failure of the
double gaussian to represent observations.
The likelihood surface is regular around each maximum, the dependence
in $\sigma_{w1}$ and $\sigma_{w2}$ follows a nice parabola in
log-likelihood indicating that the estimator behaves quite alike a
gaussian variate in this region. Standard errors follow. The
proportion $\beta$ is poorly determined although different values of
$\beta$ may produce quite different M.L. estimates for $\sigma_{w1}$
and $\sigma_{w2}$. But the Likelihood levels reached do not deviate
significantly from oneanother. A series of tests, not reported here,
show that adopting any of those solutions, nearly equivalent in
likelihood, do produce extremely close estimations of the dynamical
density. Expectedly, what matters here, is the quality of the
distribution representation, not the choice of specific functions.
Eventually this conclusion is comforted by the excellent agreement
between the results of the parametric and non parametric approach.
4.2 Parametric approach: density and potential analysis
Once we get models fitting W velocity distributions, which can be
represented by double gaussians, the self consistent solution of
Eqs. (1) and (2) can be expressed explicitly as
$$\displaystyle\ \nu(z)=\nu_{0}~{}[\,\beta~{}e^{-\phi(z)/\sigma_{w1}^{2}}+(1-%
\beta)~{}e^{-\phi(z)/\sigma_{w2}^{2}}\,]$$
(19)
Over the very small z range spanned by our 125 pc sphere, the
potential well can be approximated at any required accuracy by a plain
quadratic form
$$\phi(z)=\alpha\cdot(z-z_{0})^{2}$$
(20)
So, the density trend of a tracer $\nu(z)$ can be used to estimate
$\alpha$, $\nu_{0}$ and $z_{0}$, once $\beta$, $\sigma_{w1}$ and
$\sigma_{w2}$ have been derived from the velocities. The likelihood
of a set of single star volumes under the assumption that the density
trend is represented by Eqs. (19), (20), can be
expressed by Eq. (21) similar to Eq. (5).
$$\log L=\sum_{i=1,n}~{}(~{}\log\nu(z_{i})-\nu(z_{i})\cdot v_{i}~{})$$
(21)
A maximum likelihood can be searched for $\alpha$, $\nu_{0}$ and $z_{0}$.
According to the Poisson equation, the local dynamical density follows
from $\alpha$ as:
$$\rho_{0}={1\over 2\pi G}~{}~{}~{}\alpha$$
(22)
Where $1\,/(2\pi G)=37.004$ if $\alpha$ is in $({\rm km\,s}^{-1}\,{\rm pc}^{-1})^{2}$ and $\rho_{0}$ in $M_{\sun}\,{\rm pc}^{-3}$.
Table 3 summarizes the results of solutions based on the
three main samples (the separate solutions derived from samples
without identified cluster stars do not produce different results).
Three solutions are given for each sample: one is the 3-parameter
M.L. solution, the other two are M.L. solutions for $\alpha$,
$\nu_{0}$, while specific values have been forced for $z_{0}$.
All three parameter solutions produce unexpected positive values for
the z coordinate of the galactic plane. That is, the Sun would be
very slightly south of the plane, in contradiction with most accepted
values. We do not consider this result worth much consideration: for
one thing the estimated local density does not change appreciably if
$z_{0}$ is forced to 0 and even if it is pushed to a more classical $z_{0}=-10$, the estimated density decreases. For the other thing, there
is certainly some clumpiness among the youngest stars in this sample
which may blur the shape of the potential well the more so as this
well turns out shallower than expected. Since it is not possible to
evaluate accurately this effect we shall consider it as an intrinsic
limitation of this determination. Accordingly the adopted solution is
the weighted mean of M.L. solutions (weights proportional to inverse
square standard errors), but error bars are extended in order to
include all main solutions of Table 3.
$$\rho_{0}~{}=~{}0.076~{}\pm~{}0.015~{}M_{\sun}~{}{\rm pc}^{-3}$$
(23)
Based on this value and on the velocity distributions obtained in
section 4.1, predicted density models defined by Eq. (19) have
been overplotted on the z moving average profiles of all three tracer
samples (Fig. 12 a,b,c). For comparison, profiles
corresponding to arbitrary densities $\rho_{0}~{}=~{}0.10~{}M_{\sun}~{}{\rm pc}^{-3}$ and $\rho_{0}~{}=~{}0.15~{}M_{\sun}~{}{\rm pc}^{-3}$ are added. Obviously, even
under quite conservative hypotheses, none of the samples can be
reconciled with such local densities. The existence of any
substantial disk shaped dark matter would definitely push all three
curves outside the acceptable range.
The distributions of single star volume residuals under the three
density models have been scrutinized against possible unmodeled
systematic effects. Probability plots as defined in section 3.3 (Eq.
(11)) are given in Fig. 13.
The first two plots on top show the probability plot of residuals
for samples h125 and k125 . Clearly enough models based on local mass
densities higher than 0.08 generate a systematic distortion of the
cumulative distribution of residuals with respect to the expected one.
There is a small excess of low residuals even with our best fit model,
but this effect vanishes when cluster stars are removed (sample k125)
This correction does not change the conclusions on the local mass.
At this stage , the validity of the simplified assumption made on the
potential can also be tested. A square function in z has been adopted for the
potential, meaning a linear $K_{z}$. This means some homogeneity of the mass
distribution over a 100 parsecs scale. If a substantial fraction of the mass
happened to be concentrated in a very thin layer, this approximation would
not hold.
The Hipparcos samples are the first able to test this hypothesis
over such small scales. The test is provided by the statistics of single
star residuals. Would the $K_{z}$ slope change significantly within the limits
of the sample, then this statistics of subsamples at low z would not
match the expectation for the same assumed $\rho_{0}$. This test is
presented in the bottom parts of Fig. 13.
Samples h125 and H100 were split in different subsamples selected by
$z$ after restriction to a cylinder of radius $r$. Expectedly, the failure
of models based on $\rho_{0}=0.10$ and beyond is enhanced at high z
($|z|>69~{}{\rm pc}$).
But, even over the restricted range $|z|<69~{}{\rm pc}$ there is signal left in
the residuals to reject high density solutions. The low z part
of sample H100 ($|z|<55~{}{\rm pc}$) is unable to produce any discriminant signal.
These residual plots fully confirm the result that local total volume
densities beyond $0.10~{}M_{\sun}~{}{\rm pc}^{-3}$ cannot be reconciled
with the observed velocity-density distributions. Similar plots
restricted to subsamples selected by distance or magnitude or colour
do not show significant deviations that might sign the presence of
major selection effects in the sampling.
In terms of mass model the assumed linear $K_{z}$ implies that the total
mass density is constant close to the plane. Assuming that it is true
only for half of the mass, for instance the stellar distribution and
that the gas distribution follows a sharper gaussian distribution with
a scale height of 75 pc we may determine the extra forth order term in
the potential. Determining the amplitude of the potential on counts at
75 pc leads to a very small difference of 1/24 between the two
models. The resulting error on the derived $\rho_{0}$ is negligible with
respect to other uncertainties.
4.3 Non parametric approach
The non parametric approach does not rely upon any particular
mathematical model for the velocity distribution function nor the
height density profile ([Pichon & Thiébaut, 1997]). That is: ideally we should
try and solve self consistently Eqs. (1) and (2) for $\rho_{0}$, with
functions $f(|w_{0}|)$ and $\nu(\phi)$ defined on a purely numerical
basis (in the range investigated here, approximating $\phi(z)$ by a
simple square function is well justified). However a pure numerical
solution is unstable: a small departure in the measured data (e.g. due to noise) may produce drastically different solutions since these
solutions are dominated by artifacts due to the amplification of
noise. Some kind of trade off must therefore be found between the
level of smoothness imposed on the solution in order to deal with
these artifacts on the one hand, and the level of fluctuations
consistent with the amount of information in the signal on the other
hand. Finding such a balance is called the “regularization”
([Wahba & Wendelberger, 1979]) of the self consistency problem. Regularization allows
to avoid over-interpretation of the data. Here we chose to regularize
the non-parametric self consistency problem by requiring a smooth
solution. This criterion is tunable so that its importance may be
adjusted according to the noise level (i.e. the worse the data
quality is, or the sparser the sample is, the lower the informative
contents of the solution will be and the smoother the restored
distribution will be).
Under the assumption that the potential is separable, the proposed
method yields a unique mean density compatible with the measurement of
the velocity distribution perpendicular to the Galactic plane and the
corresponding measured density profile.
The discretized integral equation
From the knowledge of the level of errors in the measurements and
given the finite number of data points, it is possible to obtain
$f(w)$ by fitting the data with some model. A general approach
assumes that the solution can be described by its projection onto a
complete basis of functions with finite
support (here cubic splines) $\{e_{k}(w);k=1,\ldots,N\}$:
$$f(w)=\sum_{k=1}^{N}f_{k}\;e_{k}(w)\equiv{\mathbf{f}}\cdot{\mathbf{e}}\;.$$
(24)
where the $N$ parameters to fit are the weights $f_{k}$.
Real data correspond to discrete measurements $z_{i}$ and ${w}_{j}$ of $z$
and $w$ respectively. Following the non-parametric expansion
in Eq. (24), Eq. (1) becomes:
$$\rho_{i}\equiv\rho(z_{i})=\sum_{k=1}^{N}A_{i,k}(\alpha,z_{0})\;f_{k}\,,$$
(25)
with
$$A_{i,k}(\alpha,z_{0})=\int_{\sqrt{2\alpha z^{2}_{i}}}^{\infty}\frac{e_{k}(w)w}%
{\sqrt{w^{2}-2\alpha z^{2}_{i}}}\,{{\mathrm{d}}{w}}\,.$$
(26)
Since the relation between ${\mathbf{\tilde{\rho}}}\equiv\{\rho_{i}\}$
and ${\mathbf{f}}\equiv\{f_{k}\}$
is linear, Eq. (25) — the discretized
form of the integral equation (1) — can be written in a
matrix form:
$${\mathbf{\tilde{\rho}}}={\mathbf{A}}(\alpha,z_{0})\!\cdot\!{\mathbf{f}}$$
(27)
Here $z_{0}$ accounts for the relative offset in the position of the sun
with respect to the galactic plane as defined by the Hipparcos sample to
the measured density.
The Hipparcos sample gives access within a given sphere to a number of
stars for which the velocity, $w$, and/or the altitude, $z$, is known.
In other words the measurement yields a realization of the cumulative
height density distribution and the cumulative velocity distribution.
Indeed for each star at height $z$, within ${{\mathrm{d}}{z}}$ of the next star
(once they have been sorted in height), we can associate a volume
given by
$$\pi\,{{\mathrm{d}}{z}}\,R^{2}(1-z^{2}/R^{2})$$
(28)
where $R$ is the radius of the Hipparcos sphere; an alternative
estimated local density for this star is given by the inverse of this
volume. This density estimator yields a local density consistent
with that derived in section 3. An estimate of its cumulative
distribution is directly available from the measurements without
resorting to averaging nor binning. In order to project the modeled
distribution, $f$, into data space it is therefore best to model both
the cumulative density distribution, $R(z)$ and the cumulative
velocity distribution, $F(w)$, which satisfy respectively
$$R(z)=\int_{-\infty}^{z}\rho(z)\,{{\mathrm{d}}{z}}\,,\quad{\rm and}\quad F(w)=%
\int_{-\infty}^{w}f(w)\,{{\mathrm{d}}{w}}\,.$$
(29)
Now if $f(w)$ obeys Eq. (24), its cumulative distribution will satisfy
$$F(w)=\sum_{k=1}^{N}f_{k}\;E_{k}(w)\equiv{\mathbf{f}}\cdot{\mathbf{E}}\;,$$
(30)
where the basis of functions $\{E_{k}(w);k=1,\ldots,N\}$ is simply the
anti-derivative of $\{e_{k}(w)\}$ with respect to $w$.
Similarly the cumulative density distribution satisfies
$$\displaystyle R_{i}\equiv R(z_{i})=\sum_{k=1}^{N}\frac{1}{\sqrt{2\alpha}}C_{k}%
\left[\sqrt{2\alpha}(z_{i}-z_{0})\right]\;f_{k}\,,$$
$$\displaystyle\quad{\rm with}\quad C_{k}(Z)=\int_{0}^{Z}\int_{z}^{\infty}\frac{%
e_{k}(w)w}{\sqrt{w^{2}-z^{2}}}\,{{\mathrm{d}}{w}}\,{{\mathrm{d}}{z}}\,.$$
(31)
Note that for a cubic spline basis, the function $C_{k}$ is algebraic (though
somewhat clumsy). Eq. (30) and Eq. (31) provide
two non parametric means of relating the measurements, $R_{i}$ and $F_{i}$
to the underlying distribution, $f(w)$, with two extra parametres, $z_{0}$ and $\alpha$,
which one should adjust so that the distributions match.
Practical implementation
First we compute the best regularized fit of the cumulative velocity
distribution (we use $\chi^{2}$ fitting even though the noise is
Poissonian because it yields a simpler quadratic problem). We
therefore minimize
the quadratic form:
$$Q_{\mathrm{\mu}}({\mathbf{f}})={{(\tilde{{\mathbf{F}}}-{\mathbf{B}}\!\cdot\!{%
\mathbf{f}})}^{\bot}}\!\cdot\!{\mathbf{W}}\!\cdot\!(\tilde{{\mathbf{F}}}-{%
\mathbf{B}}\!\cdot\!{\mathbf{f}})+\mu\,{{{\mathbf{f}}}^{\bot}}\!\cdot\!{%
\mathbf{K}}\!\cdot\!{\mathbf{f}}\,,$$
(32)
where $\tilde{{\mathbf{F}}}$ is the measured cumulative velocity
distribution, ${\mathbf{K}}$ the discrete 2nd order differentiation
operator,
and ${\mathbf{B}}=\{E_{k}(w_{i})\}$ ; the
weight matrix ${\mathbf{W}}$ is the inverse of the covariance matrix of the
error in the
data and $\mu$ is the Lagrange parameter
tuning the level of regularization.
Error estimation is achieved via Monte Carlo simulation: the
regularization parameter is initialized at some fixed value (here
$\mu=10$); from this value a first cumulative distribution is
fitted. A draught corresponding to the same number of measured
velocity is generated and the procedure iterated. An estimate of the
error bars per point follows. The best fit ${\mathbf{f}}_{\mathrm{\mu}}$ which
minimizes $Q_{\mathrm{\mu}}$ in Eq. 32 is:
$${\mathbf{f}}_{\mathrm{\mu}}=({{{\mathbf{B}}}^{\bot}}\!\cdot\!{\mathbf{W}}\!%
\cdot\!{\mathbf{B}}+\mu\,{\mathbf{K}})^{-1}\!\cdot\!{{{\mathbf{B}}}^{\bot}}\!%
\cdot\!{\mathbf{W}}\!\cdot\!\tilde{{\mathbf{F}}}\,.$$
(33)
Next we
apply Eq. (31)
to project the distribution function
into cumulative density space. Calling ${\mathbf{C}}$ the matrix given
by its components $C_{i,k}=C_{k}\left[\sqrt{2\alpha}(z_{i}-z_{0})\right]/\sqrt{2\alpha}$,
we
then adjust the potential curvature $\alpha$
(together with the possible offset $z_{0}$) while minimizing
$$\displaystyle|{\mathbf{\tilde{R}}}-{\mathbf{C}}(\alpha,z_{0})\cdot{\mathbf{f}}%
_{\mu}|^{2}=$$
$$\displaystyle|{\mathbf{\tilde{R}}}-{\mathbf{C}}(\alpha,z_{0})\cdot({{{\mathbf{%
B}}}^{\bot}}\!\cdot\!{\mathbf{W}}\!\cdot\!{\mathbf{B}}+\mu\,{\mathbf{K}})^{-1}%
\!\cdot\!{{{\mathbf{B}}}^{\bot}}\!\cdot\!{\mathbf{W}}\!\cdot\!\tilde{{\mathbf{%
F}}}\,|^{2}\,,$$
(34)
with respect to $\alpha$ and $z_{0}$.
This
yields the local density of the
disk. Note that the last two steps could be carried simultaneously,
i.e. we could seek the triplet, $(\mu,\alpha,z_{0})$
which minimizes Eq. (34), or even better,
the unknown, $\mu,\alpha,z_{0}$ and $\{f_{k}\},\;k=1\cdots N$,
which minimize the quantity:
$$\displaystyle{{(\tilde{{\mathbf{F}}}-{\mathbf{B}}\!\cdot\!{\mathbf{f}})}^{\bot%
}}\!\cdot\!{\mathbf{W}}_{1}\!\cdot\!(\tilde{{\mathbf{F}}}-{\mathbf{B}}\!\cdot%
\!{\mathbf{f}})+$$
$$\displaystyle{{(\tilde{{\mathbf{R}}}-{\mathbf{C}}\!\cdot\!{\mathbf{f}})}^{\bot%
}}\!\cdot\!{\mathbf{W}}_{2}\!\cdot\!(\tilde{{\mathbf{R}}}-{\mathbf{C}}\!\cdot%
\!{\mathbf{f}})+\mu\,{{{\mathbf{f}}}^{\bot}}\!\cdot\!{\mathbf{K}}\!\cdot\!{%
\mathbf{f}}\,,$$
(35)
though
in practice the level of regularization, $\mu$, – within some range, only
affects the wings of the reconstructed density profile and this has no
consequence on the central density value, $\rho_{0}$ (a
supplementary difficulty
involved in minimizing Eq. (35)
would be to assess the relative weights between
errors in position and velocity).
Since our purpose was primarily to validate the parametric approach
described in sections 4.1 and 4.2, we also minimized
$$|{\mathbf{\tilde{\rho}}}-{\mathbf{A}}(\alpha,z_{0})\cdot{\mathbf{f}}_{\mu}|^{2}$$
(36)
instead of Eq. (34) using the nearest neighbour density estimator
described in section 3.
This lead to non parametric estimator for
sample h125 given by
$\rho_{0}=0.07-0.08~{}M_{\sun}~{}{\rm pc}^{-3}$ with $z_{0}=13\,\,{\rm pc}$.
The analysis of sample H100 gives
$\rho_{0}=0.08-0.10~{}M_{\sun}~{}{\rm pc}^{-3}$.
In practice, the parametric and non parametric analysis
therefore lead to
statistically consistent answer.
4.4 Radial force correction
The $K_{z}$ (vertical forces) determination constrains the local mass
density: precisely $\partial^{2}\Phi_{total}\over\partial z^{2}$.
A complete determination of the Poisson
equation should include the $K_{r}$ (radial force) component. This
second contribution is small and can be estimated from the A and B
Oort’s constants:
$$\Delta\Phi_{total}={\partial^{2}\Phi_{total}\over\partial z^{2}}+2(B^{2}-A^{2}%
)=4\pi G\rho_{total}$$
(37)
From the dynamical analysis of the present Hipparcos sample, we
have obtained 0.076 for the $\partial^{2}\Phi_{total}\over\partial z^{2}$ term. The term including A and B Oort’s constants ranges
between $\pm 0.004$; it is probably positive corresponding to a
rotation curve locally decreasing. This term is small and
we assume it is zero to simplify the discussion.
5 Conclusions
The consequence of estimations given in previous sections is to fix
the total mass density in the galactic disc in the neighbourhood of
the Sun in the range 0.065-0.10 $M_{\odot}~{}{\rm pc}^{-3}$ with a most
probable value by 0.076 $M_{\odot}~{}{\rm pc}^{-3}$. In conclusion we
review below the consequences of this very low value in terms of local
dynamical mass versus observed density, in terms of global
galaxy modeling and in terms of dark matter distribution.
5.1 Dynamical mass versus known matter
Previous dynamical determinations of the local mass density used to be
based on star counts a few hundred parsecs away from the galactic
plane providing constraints for the local column density integrated
over a substantial layer. Comparisons were tentatively made with the
observed density of known galactic components, namely visible stellar
populations, stellar remnants, ISM. So, on either side of the
comparison there were assumptions on the vertical distribution of
matter involved.
For the first time, the present determination is based on strictly
local data inside a sphere of more than 100 parsecs radius and
gives access directly to the total mass density around the Sun in
the galactic plane.
On the side of the known matter, the current situation is the
following: the main contributors are main sequence stars of the
disc, stellar remnants and the interstellar medium .
1) Recent determinations of the stellar disc luminosity function
have a maximum near $M_{v}$=12 and decreases quickly beyond.
Counting nearby stars Wielen et al. (1983) estimate the total mass
density of main sequence stars in the solar neighbourhood as
0.037 plus 0.008 in the form of giants and white dwarfs. Since then the
luminosity function below $M_{v}$ = 13 has been revised downwards (Reid et
al 1995, Gould et al 1996). Thus main sequence stars are expected to
contribute by 0.027-0.033$~{}M_{\odot}~{}{\rm pc^{-3}}$.
This is the most reliable part of the known matter, yet some extra
uncertainty might result from the non detection of low mass binaries.
Preliminary results based on Hipparcos parallaxes seem to imply
that stellar distances are on the average slightly larger than expected
which would mean an even smaller density (Jahreiss and Wielen, 1997).
2) Stellar remnants dominated by white dwarfs should contribute by
an additional $0.015\pm 0.005~{}M_{\odot}~{}{\rm pc^{-3}}$. This
figure is compatible with observational determination of the WD’s
luminosity function ([Sion & Liebert 1977], [Ishida et al 1982]). It
does not conflict with state of the art scenarios of star
formation.
3) The local mass density of the ISM estimated from HI and
$\rm{H_{2}}$ abundances. $\rm{H_{2}}$ is deduced from
$\rm{CO}$/$\rm{H_{2}}$ ratio. This estimation suffers very large
uncertainties. According to [Combes (1991)], a contribution about
0.04 is reasonable, but errors by a factor two or more are not
excluded. This is certainly the weakest link of this analysis.
So, 0.085 $M_{\odot}~{}{\rm pc^{-3}}$ is probably an acceptable
figure for the known mass. But the uncertainty is larger than
0.02. It does not make sense to overcomment about the observed
mass density being larger than the dynamical estimate: for one
thing the discrepancy is far below the error bars, for the other
thing there has been little effort dedicated so far to estimate
properly the smallest density compatible with observations, the
attention being always focused on upper bounds. For the moment the
only conclusion is that the density of known matter has a lower
limit of say 0.065 $M_{\odot}~{}{\rm pc^{-3}}$. In term of mass
discrepancy controversy, the main uncertainty is now on the
observation side.
5.2 Galactic mass model
In the following we adopt a simple, two component modelisation of
the galactic disc including stars and interstellar medium
(double-exponential and exponential-sech-square laws). The
formalism is standard, it is for instance the one adopted by
Gould et al (1996) (Eq. 1.1 in their paper).
A good convergence is now obtained on the main parameters of the
disc populations although there is no agreement as to the exact
decomposition in populations (see for example, [Reid & Gilmore 1983], Robin et
al 1986ab-1996, Haywood 1997ab, Gould et al 1996, etc..).
Based on the discussions quoted above the tentative structural
parameters of our model are close to Gould’s (see also [Sackett, 1997]):
stellar disc scale length 2.5 kpc, scale height 323 pc; thick disc
scale length 3.5 kpc, scale height (exponential) 656 pc. 20$\%$ of
the local stellar density is in the thick disc. The error on the
scale length has little effect on the disc mass $M_{d}$:
$M_{d}(2.5{\rm\,kpc})/M_{d}(3.5{\rm\,kpc})=1.34$ (assuming $R_{0}$=8.5 kpc)
and a factor 1.26 on the amplitude of the corresponding velocity
curves.
Adopting a total stellar mass density $\rho_{\star}=.043M_{\odot}\,{\rm pc}^{-3}$, the surface density at $R_{0}$ is then
$\Sigma_{0}=33.4~{}M_{\odot}\,{\rm pc}^{-2}$
The contributions of the bulge and stellar halo are small beyond
3 kpc of the galactic center. They are neglected in this discussion.
The interstellar matter is accounted for by a double exponential
disc 2500/80 pc its local density is set to 0.04 $M_{\odot}~{}{\rm pc}^{-3}$.
The total column density due to gas and stellar component is
$\Sigma_{0}=40~{}M_{\odot}~{}{\rm pc}^{-2}$.
A dark halo is necessary to maintain a flat rotation curve beyond
the solar galactic radius. It is modeled with a Miyamoto spheroid
that allows flattening.
This mass model is fitted to the rotation curve setting the solar
galactic radius to $R_{0}$=8.5 kpc and the velocity curve at $R_{0}$ to
$220\,{\rm km\,s^{-1}}$. The velocity curve is assumed flat beyond
5 kpc of the center to 20 kpc.
The first consequence is that the stellar disc (stars+remnants+gas)
does not contribute for more than half the mass implied by the
rotation curve at $R_{0}$=8.5 kpc: the galactic disc is far from
maximal. Even with $R_{0}$=7.5 kpc, a maximal disc would mean a
rotation curve plateau as low as 164 km s${}^{-1}$. This is excluded
by inner HI and CO rotation curves.
Filling the galactic mass distribution with a strictly spherical
dark matter halo required by the rotation curve, the dark matter
local density comes out as
$\rho_{dark\,halo}(R_{0},z=0)=0.007~{}M_{\odot}~{}{\rm pc}^{-3}$. Such a
local density of dark matter is well compatible with the above
quoted uncertainties of the dynamical and known mass. However, any
attempt to flatten the mass distribution of the dark component
result in an increase of the local halo density. Under the most
extreme hypotheses, that is considering the interstellar matter
contribution negligible and giving the stellar component a total
density $0.043~{}M_{\odot}~{}{\rm pc}^{-3}$ with the same scale length
and adopting our best dynamical estimate at its face value (0.076)
the acceptable halo scale height cannot go below 2150 pc.
So, there is just room for a spherical halo (local density:
$0.008~{}M_{\odot}~{}{\rm pc}^{-3}$). Extreme changes of the parameters
are needed to permit a significantly flatter dynamical halo. There is
no room for an important amount of dark matter in the disc. Dark
matter models assuming that the dark matter is in form of a flat
component related to a flat fractal ISM should be modified to
equivalent models where dark matter is in form of fractal structures
distributed in the halo (Pfenniger et al 1994ab, Gerhard & Silk
1996).
5.3 Summary of conclusions
1.
The potential well across the galactic plane has been traced
practically hypothesis-free and model-free, it turns out to be shallower
than expected.
2.
The local dynamical volume density comes out as
$0.076\pm 0.015~{}M_{\odot}~{}{\rm pc}^{-3}$ a value well compatible with
all existing observations of the known matter.
3.
Building a disc-thick disc mass model compatible with this
constraint as well as previous determinations of the local surface
density, it is shown that such a disc cannot be maximal, a massive
halo of dark matter is required.
4.
The dark halo should be spherical or nearly spherical in order
for its local density to remain within the range permitted between the
known matter density and the current determination of the dynamical
density.
5.
There is no room left for any disk shaped component of dark matter.
Acknowledgements.
Les travaux de MC à l’IUP de Vannes ont été rendus possibles
grâce à une aide spécifique de la Mission Scientifique et
Technique (DSPT3) et à l’assistance permanente de Robert Nadot.
CP thanks E. Thiébaut and O. Gerhard for valuable discussions and
acknowledges funding from the Swiss NF.
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An SU(2) gauge principle for the Cosmic Microwave Background: Perspectives on the Dark Sector of the Cosmological Model
Ralf Hofmann
Institut für Theoretische Physik, Universität Heidelberg,
Philosophenweg 16, D-69120 Heidelberg, Germany;
[email protected]
We review consequences for the radiation and dark sectors of the cosmological model arising
from the postulate that the Cosmic Microwave Background (CMB) is governed by an SU(2) rather than a U(1) gauge principle. We also speculate on the possibility of
actively assisted structure formation due to the de-percolation of
lump-like configurations of condensed ultralight axions with a Peccei-Quinn scale comparable to the Planck mass. The chiral-anomaly induced
potential of the axion condensate
receives contributions from SU(2)/SU(3) Yang-Mills factors of hierarchically
separated scales which act in a screened (reduced) way in confining phases.
1 Introduction
Judged by decently accurate agreement of cosmological parameter values extracted from (i) large-scale structure observing campaigns on galaxy- and
shear-correlation functions as well as redshift-space distortions towards a redshift of unity by photometric/spectroscopic surveys (sensitive
to Baryonic Accoustic Oscillations (BAO), the evolution of Dark Energy, and non-linear structure growth), for recent, ongoing, and future projects
see, e.g. [1, 2, 3, 4, 5],
(ii) fits of various CMB angular power spectra based on frequency-band optimised intensity and polarisation data collected by satellite
missions [6, 7, 8], and (iii) fits to cosmologically local luminosity-distance redshift data for Supernovae Ia (SNeIa)
[9, 10] the spatially flat standard Lambda Cold Dark Matter ($\Lambda$CDM) cosmological model is a good, robust
starting point to address the evolution of our Universe. About twenty years ago, the success of this model has
triggered a change in paradigm in accepting a present state of accelerated expansion induced by an essentially dark Universe
made of 70 % Dark Energy and 25 % Dark Matter.
While (i) and (ii) are anchored on comparably large cosmological standard-ruler co-moving distance scales, the sound horizons $r_{s,{\rm d}}$ and $r_{s,*}$,
which emerge at baryon-velocity freeze-out and recombination, respectively, during the epoch of CMB decoupling and therefore refer to very-high-redshift
physics ($z>1000$), (iii) is based on direct distance ladders linking to host galaxies at redshifts of say, $0.01<z<2.3$, which
do not significantly depend on the assumed model for Dark Energy [11] and are robust against sample variance,
local matter-density fluctuations, and directional bias [12, 13, 14, 15].
Thanks to growing data quality and
increased data-analysis sophistication to identify SNeIa [16] and SNeII [17] spectroscopically, to establish precise and independent
geometric distance indicators [18, 19] (e.g., Milky Way parallaxes, Large Magellanic Cloud (LMC) detached eclipsing binaries,
and masers in NGC 4258), tightly calibrated
period-to-luminosity relations for LMC cepheids and cepheids in SNeIa host galaxies [19], the use of
the Tip of the Red Giant Branch (TRGB) [20, 21] or the Asymptotic Giant Branch (AGB, Mira) [22] to connect to the distance ladder
independently of cepheids, the high value of the basic cosmological parameter
$H_{0}$ (Hubble expansion rate today) has (see [21], however), over the last decade and within $\Lambda$CDM,
developed a tension of up to $\sim 4.4\,\sigma$ [19] in SNeIa distance-redshift fits using the LMC geometrically
calibrated cepheid distance ladder compared to its low value
extracted from statistically and systematically accurate medium-to-high-l CMB angular power spectra [7, 8] and the BAO method
[3]. Both, the CMB and BAO probe the evolution of small radiation and matter density fluctuations in global cosmology: Fluctuations are
triggered by an initial, very-high-redshift (primordial) spectrum of scalar/tensor curvature
perturbations which, upon horizon entry, evolve linearly [23] up to late times when
structure formation generates non-linear contributions in the galaxy spectra [2, 3] and
subjects the CMB to gravitational lensing [24].
Recently, a cosmographic way of extracting $H_{0}$ through time delays of
strongly lensed high-redshift quasars, see e.g. [25, 26], almost matches the
precision of the presently most accurate SNeIa distance-redshift fits [19] – $H_{0}=71.9^{+2.4}_{-3.0}\,$ km s$\mbox{}^{-1}$Mpc$\mbox{}^{-1}$ vs. $H_{0}=(74.22\pm 1.82)\,$ km s$\mbox{}^{-1}$Mpc$\mbox{}^{-1}$ –, supporting
a high local value of the Hubble constant and rendering the local-global tension even
more significant [27]. The high local value of $H_{0}=(72\cdots 74)\,$ km s$\mbox{}^{-1}$Mpc$\mbox{}^{-1}$ [19, 27] (compared to the global value of
$H_{0}=(67.4\pm 0.5)\,$ km s$\mbox{}^{-1}$Mpc$\mbox{}^{-1}$ [8]) is stable to sources
of systematic uncertainty such as line-of-sight effects, peculiar motion (stellar kinematics), and assumptions made in the
lens model. There is good reason to expect that an improved localisation
of sources for gravitational-wave emission without an electromagnetic counterpart and the
increase of statistics in gravitational-wave events accompanied by
photon bursts (standard sirenes) within specific host galaxies will lead to luminosity-distance-high-redshift data
producing errors in $H_{0}$ comparable of those of [19],
independently of any distance-ladder calibration [28], see also [29].
Apart from the Hubble tension,
there are smaller local-global but possibly also tensions between BAO and the CMB
in other cosmological parameters such as the
amplitude of the density fluctuation power spectrum ($\sigma_{8}$)
and the matter content ($\Omega_{m}$), see [30]. Finally, there are persistent large-angle anomalies
in the CMB, already seen by the Cosmic Background Explorer (COBE) and strengthened by the Wilkinson Microwave Anisotropy Probe (WMAP) and Planck satellite,
whose ’atoms’ (a) lack of correlation in the TT two-point function, (b) a rather significant alignment of
the low multipoles ($p$-values of below 0.1 %), and (c) a dipolar modulation, which is independent of the multipole alignment (b),
indicate a breaking of statistical isotropy at low angular resolution, see [31] for a comprehensive
and complete review.
The present work intends to review and discuss a theoretical framework addressing a possibility to
resolve the above-sketched situation. The starting point is to subject the CMB to the thermodynamics of
an extended gauge principle in replacing the conventional group U(1) by SU(2). Motivated [32]
by explaining an excess in CMB line temperature at radio frequencies, see [33] and references therein,
this postulate implies a modified temperature-redshift relation which places CMB recombination to a redshift of
$\sim 1700$ rather than $\sim 1100$ and therefore significantly reduces the density of matter at that epoch.
To match $\Lambda$CDM at low $z$, one requires a release of Dark Matter from early Dark Energy
within the dark ages at a redshift $z_{p}$: lumps and vortices, formely tighly correlated within a condensate of ultralight axion particles, de-percolate into
independent pressureless (and selfgravitating) solitons due to cosmological expansion, thereby contributing actively to
non-linear structure formation at low $z$. A fit to CMB angular power spectra of a cosmological model, which incorporates these SU(2)
features on the perfect-fluid level but neglects low-$z$ radiative effects in SU(2) [34],
over-estimates TT power on large angular scales but generates a precise fit to the data for
$l\geq 30$. With $z_{p}\sim 53$ a locally favoured value of $H_{0}\sim 74\,$km s${}^{-1}$ Mpc${}^{-1}$ and a low baryon density
$\omega_{b,0}\sim 0.017$ are obtained. Moreover, the conventionally extracted near scale invariance
of adiabatic, scalar curvature perturbations comes out to be significantly
broken by an infrared enhancement of their power spectrum. Finally, the fitted redshift $\sim 6$ of
re-ionisation is compatible with the detection of the Gunn-Peterson-trough in the spectra of high-$z$
quasars [36] and distinctly differs from the high value
of CMB fits to $\Lambda$CDM [7, 8].
As of yet, there are loose ends to the SU(2) based scenario. Namely, the physics of
de-percolation requires extra initial conditions for matter density
fluctuations at $z_{p}$. In the absence of a precise modelling of the ’microscopics’ of the associated soliton
ensembles111Lump sizes could well match those of galactic dark-matter
halos, see Sec. 4. it is only a guess that
these fluctuations instantaneously follow the density fluctuations of primordial Dark Matter as assumed
in [34]. Moreover, it is necessary to investigate to what extent profiles of the axion
field (lumps of localised energy density) actively seed non-linear structure formation and whether their role in
galactic halo formation and baryon accretion meets the observational constraints (Tully-Fisher relation, etc.).
Also, radiative effects on thermal photon propagation at low
$z$ [37], which are expected to contribute or even explain the above-mentioned
’atoms’ (a), (b), and (c) of the CMB large-angle anomalies, see [38], and which could reduce the
excess in low-$l$ TT power of [34] to realistic levels, need to be incorporated into
the model. They require analyses in terms of Maxwell’s multipole-vector
formalism [31] and/or other robust and intuitive statistics to
characterise multipole alignment and the large-angle suppression of TT.
This review-type paper is organised as follows. In Sec. 2 we briefly discuss the main
distinguishing features between the conventional
$\Lambda$CDM model and cosmology based on a CMB which obeys deconfining SU(2) Yang-Mills thermodynamics. A presentation of
our recent results on angular-power-spectra fits to Planck data is carried out Sec. 3, including a discussion of
the parameters $H_{0}$, $n_{s}$, $\sigma_{8}$, $z_{\rm re}$, and $\omega_{\text{b,0}}$. In Sec. 4 we interpret the rough
characteristics of the Dark Sector employed in [34] to
match $\Lambda$CDM at low $z$. In particular, we point out that the value of $z_{p}$ seems to be consistent
with the typical dark-matter densities in the Milky Way. Finally, in
Sec. 5 we sketch what needs to be done to arrive at a solid, observationally well
backed-up judgement of whether SU(2)${}_{\rm CMB}$ based cosmology (and its extension
to SU(2) and SU(3) factors of hierarchically larger Yang-Mills scales including their nonthermal phase transitions) may provide a future
paradigm to connect local cosmology with the very early Universe. Throughout the article, super-natural units $\hbar=k_{B}=c=1$ are used.
2 SU(2)${}_{\rm CMB}$ vs. conventional CMB photon gas in $\Lambda$CDM
The introduction of an SU(2) gauge principle for the description of the CMB is motivated
theoretically by the fact that the deconfining thermodynamics of such a Yang-Mills theory
exhibits a thermal ground state, composed of densely packed (anti)caloron [39]
centers with overlapping peripheries [40, 41], which breaks SU(2) to U(1) in terms of an adjoint
Higgs mechanism [42]. Therefore, the spectrum of excitations consists
of one massless gauge mode, which can be identified with the CMB photon, and two
massive vector modes of a temperature-dependent mass on tree level: thermal
quasi-particle excitations. The interaction between these excitations is feeble [42].
This is exemplified by the one-loop polarisation tensor of the
massless mode [43, 37, 44]. As
a function of temperature, polarisation effects peak at about twice the critical temperature $T_{c}$ for
the deconfining-preconfining transition. As a function of increasing photon momentum,
there are regimes of radiative screening/antiscreening, the latter being subject to an exponential fall-off
[37]. At the phase boundary ($T\sim T_{c}$) electric monopoles [42], which occur as isolated and unresolved
defects deeply in the deconfining phase, become massless by virtue of screening due to transient dipoles [45] and therefore condense to endow the
formely massless gauge mode with a quasiparticle Meissner mass $m_{\gamma}$. This mass rises critically (with mean-field exponent)
as $T$ falls below $T_{c}$ [42]. Both, (i) radiative screening/antiscreening of massless modes and (ii) their Meissner effect are important
handles in linking SU(2)${}_{\textnormal{\tiny{cmb}}}$ to the CMB: While (i)
induces large-ange anomalies into the TT correlation [38] and
contributes dynamically to the CMB dipole [46, 47] (ii) gives rise to a nonthermal
spectrum of evanescent modes222That the deep Rayleigh-Jeans regime is indeed subject to classical wave
propagation is assured by the fact that wavelengths then are greater than the spatial scale $s\equiv\pi T|\phi|^{-2}$, separating a(n) (anti)caloron
center from its periphery where its (anti)selfdual gauge field is that of a dipole [41]. The expression for $s$ contains
the modulus $|\phi|=\sqrt{\Lambda_{\textnormal{\tiny{cmb}}}^{3}/(2\pi T)}$ of the emergent, adjoint Higgs field $\phi$ ($\Lambda_{\textnormal{\tiny{cmb}}}\sim 10^{-4}\,$eV the Yang-Mills scale
of SU(2)${}_{\textnormal{\tiny{cmb}}}$), associated with densely packed (anti)caloron centers,
and, explicitely, temperature $T$. for frequencies $\omega<m_{\gamma}$ once $T$ falls below $T_{c}$. This theoretical anomaly of the blackbody spectrum
in the Rayleigh-Jeans regime can be considered to explain the excess in CMB radio power below 1 GHz, see [33] and references therein,
thereby fixing $T_{c}=2.725\,$K and, as a
consequence of $\lambda_{c}=13.87=\frac{2\pi T_{c}}{\Lambda_{\textnormal{\tiny{cmb}}}}$ [42], the
Yang-Mills scale of SU(2)${}_{\textnormal{\tiny{cmb}}}$ to $\Lambda_{\textnormal{\tiny{cmb}}}\sim 10^{-4}\,$eV [32].
Having discussed the low-frequency deviations of SU(2)${}_{\rm CMB}$ from the conventional Rayleigh-Jeans
spectrum, which fix the Yang-Mills scale and associate with large-angle anomalies, we would now like to
review its implications for the cosmological model. Of paramount importance for the set-up of such a model
is the observtion that SU(2)${}_{\textnormal{\tiny{cmb}}}$ implies a modified temperature ($T$)-redshift
($z$) relation for the CMB which is derived from energy conservation of the
SU(2)${}_{\textnormal{\tiny{cmb}}}$ fluid in the deconfining phase in an FLRW
universe with scale factor $a$ normalised to unity today [48]. Denoting by $T_{c}=T_{0}=2.725\,$K [32]
the present CMB baseline temperature [6] and by $\rho_{{\rm SU(2)}_{\textnormal{\tiny{cmb}}}}$ and $P_{{\rm SU(2)}_{\textnormal{\tiny{cmb}}}}$ energy density and
pressure, respectively, of SU(2)${}_{\textnormal{\tiny{cmb}}}$, one has
$$a\equiv\frac{1}{z+1}=\exp\left(-\frac{1}{3}\log\left(\frac{s_{{\rm SU(2)}_{%
\textnormal{\tiny{cmb}}}}(T)}{s_{{\rm SU(2)}_{\textnormal{\tiny{cmb}}}}(T_{0})%
}\right)\right)\,,$$
(1)
where the entropy density $s_{{\rm SU(2)}_{\textnormal{\tiny{cmb}}}}$ is defined by
$$s_{{\rm SU(2)}_{\textnormal{\tiny{cmb}}}}\equiv\frac{\rho_{{\rm SU(2)}_{%
\textnormal{\tiny{cmb}}}}+P_{{\rm SU(2)}_{\textnormal{\tiny{cmb}}}}}{T}\,.$$
(2)
For $T\gg T_{0}$, Eq. (1) simplifies to
$$T=\left(\frac{1}{4}\right)^{1/3}T_{0}(z+1)\approx 0.63\,T_{0}(z+1)\,.$$
(3)
For arbitrary $T\geq T_{0}$, a multiplicative deviation $S(z)$ from linear scaling in $z+1$ can be introduced as
$${\cal S}(z)=\left(\frac{\rho_{{\rm SU(2)}_{\textnormal{\tiny{cmb}}}}(z=0)+P_{{%
\rm SU(2)}_{\textnormal{\tiny{cmb}}}}(z=0)}{\rho_{{\rm SU(2)}_{\textnormal{%
\tiny{cmb}}}}(z)+P_{{\rm SU(2)}_{\textnormal{\tiny{cmb}}}}(z)}\frac{T^{4}(z)}{%
T^{4}_{0}}\right)^{1/3}\,.$$
(4)
Therefore,
$$T={\cal S}(z)\,T_{0}(z+1)\,.$$
(5)
Fig. 1 depicts function ${\cal S}(z)$.
Amusingly, the asymptotic $T$-$z$ relation of Eq. (3) also holds for the relation between
$\rho_{{\rm SU(2)}_{\textnormal{\tiny{cmb}}}}(z)$ and the conventional CMB energy density $\rho_{\gamma}(z)$ in $\Lambda$CDM
(the energy density of a thermal U(1) photon gas, using the $T$-$z$ relation $T=T_{0}(z+1)$). Namely,
$$\rho_{{\rm SU(2)}_{\textnormal{\tiny{cmb}}}}(z)=4\,\left(\frac{1}{4}\right)^{4%
/3}\rho_{\gamma}(z)=\left(\frac{1}{4}\right)^{1/3}\rho_{\gamma}(z)\quad(z\gg 1%
)\,.$$
(6)
Therefore, the (gravitating) energy density of the CMB in SU(2)${}_{\textnormal{\tiny{cmb}}}$ is, at the same redshift $z\gg 1$,
by a factor of $\sim 0.63$ smaller than that of the $\Lambda$CDM model even though there are eight (two plus two times three)
gauge-mode polarisations in SU(2)${}_{\textnormal{\tiny{cmb}}}$ and only two such polarisations in the U(1) photon gas.
Not yet considering linear and next-to-linear perturbations
in SU(2)${}_{\textnormal{\tiny{cmb}}}$ to shape typical CMB large-angle anomalies in terms of
late-time screening/antiscreening effects [38], Eq. (5) has
implications for the Dark Sector if one wishes to maintain the successes of the standard cosmological
model at low $z$ where local cosmography and fits to distance-redshift curves produce a
consistent framework within $\Lambda$CDM. As it was shown in [34], the assumption
of matter domination at recombination, which is not unrealistic even for SU(2)${}_{\textnormal{\tiny{cmb}}}$ [34], implies that
$$z_{\Lambda{\rm CDM},*}\sim\left(\frac{1}{4}\right)^{1/3}z_{{\rm SU(2)}_{%
\textnormal{\tiny{cmb}}},*}\,$$
(7)
and, as a consequence,
$$\Omega_{\Lambda{\rm CDM},m,0}\approx 4\,\Omega_{{\rm SU(2)}_{\textnormal{\tiny%
{cmb}}},m,0}\,,$$
(8)
where $\Omega_{m,0}$ denotes today’s density parameter for nonrelativistic
matter (Dark Matter plus baryons), and $z_{*}$ is the redshift of CMB photon decoupling in either model. Since the matter sector of the
SU(2)${}_{\textnormal{\tiny{cmb}}}$ model, as roughly represented by Eq. (8), contradicts $\Lambda$CDM at low $z$ one needs to allow for a
transition between the two somewhere in the dark ages as $z$ decreases. In [34] a simple model, where the transition is sudden at a
redshift $z_{p}$ and maintains a small dark-energy residual, was introduced: a coherent axion field – a dark-energy like condensate of ultralight axion particles,
whose masses derive from U(1)${}_{A}$ anomalies [49, 50, 51, 52, 53] invoked by the topological
charges of (anti)caloron centers in the thermal ground state of SU(2)${}_{\rm CMB}$ and, in a screened way, SU(2)/SU(3) Yang-Mills
factors of higher scales – releases solitonic lumps by de-percolation due to the Universe’s expansion. Accordingly, we have
$$\Omega_{\rm ds}(z)=\Omega_{\Lambda}+\Omega_{\rm pdm,0}(z+1)^{3}+\Omega_{\rm edm%
,0}\left\{\begin{array}[]{lr}\left(z+1\right)^{3}\,,&z<z_{p}\\
\left(z_{p}+1\right)^{3}\,,&z\geq z_{p}\end{array}\right.\,.$$
(9)
Here $\Omega_{\Lambda}$ and $\Omega_{\rm pdm,0}+\Omega_{\rm edm,0}\equiv\Omega_{\rm cdm,0}$ represent today’s
density parameters for Dark Energy and Dark Matter, respectively, $\Omega_{\rm pdm,0}$ refers to primordial Dark Matter (that is, dark
matter that existed before the initial redshift of $z_{i}\sim 10^{4}$ used
in the CMB Boltzmann code) for all $z$ and $\Omega_{\rm edm,0}$ to emergent Dark Matter for $z<z_{p}$. In [34] the
initial conditions for the evolution
of density and velocity perturbations of the emergent Dark-Matter portion at $z_{p}$ are, up to a rescaling of order unity,
assumed to follow those of the primordial Dark Matter.
3 SU(2)${}_{\textnormal{\tiny{cmb}}}$ fit of cosmological parameters to Planck data
In [34] a simulation of the new cosmological model subject to SU(2)${}_{\textnormal{\tiny{cmb}}}$ and the
Dark Sector of Eq. (9) was performed using a modified version of the Cosmic Linear Anisotropy Solving System (CLASS) Boltzmann code
[54]. Best fits to the 2015 Planck data [8] on the angular power spectra of the two-point correlation functions
temperature – temperature (TT), electric-mode polarisation – electric-mode polarisation (EE), and temperature – electric-mode polarisation (TE), subject to typical
likelihood functions used by the Planck collaboration, were performed. Because temperature perturbations
can only be coherently propagated by (low-frequency) massless modes in SU(2)${}_{\textnormal{\tiny{cmb}}}$ [41] the propagation of the (massive)
vector modes was excluded in one version of the code (physically favoured). Also, entropy conservation in e${}^{+}$e${}^{-}$ annihilation invokes a slightly different
counting of relativistic degrees of freedom in SU(2)${}_{\textnormal{\tiny{cmb}}}$ for this epoch [55]. As a consequence, we have a $z$-dependent
density parameter for (massless) neutrinos given as [34]
$$\Omega_{\nu}(z)=\frac{7}{8}N_{\rm eff}\,\left(\frac{16}{23}\right)^{\frac{4}{3%
}}\Omega_{{\rm SU(2)}_{\textnormal{\tiny{cmb}}},\gamma}(z)\,,$$
(10)
where $N_{\rm eff}$ refers to the effective number of neutrino flavours (or any other extra relativistic, free-streaming, fermionic species), and $\Omega_{{\rm SU(2)}_{\textnormal{\tiny{cmb}}},\gamma}$ is the density parameter associated with the massless
mode only in the expression for $\rho_{{\rm SU(2)}_{\textnormal{\tiny{cmb}}}}(z)$ of Eq. (6). The value $N_{\rm eff}=3.046$ of the Planck result was
used as a fixed input in
[34]. The statistical goodness of the best fit was found to be comparable to the one obtained by the Planck collaboration [8],
see Figs. 2, Figs. 3, and Figs. 4 as well as the lower part of Tab. 1. There
is an excess of power in TT for $7\leq l\leq 30$, however.
This excess could be attributable to the omission of radiative effects in the low-$z$ propagation of the massless mode.
The consideration of the modified dispersion law into the CMB Boltzmann code presently is under way.
The following table was obtained in [34]:
We would like to discuss the following parameters (The value of $z_{p}$ will be discussed in Sec. 4.): (i) $H_{0}$, (ii) $n_{s}$, (iii) $\sigma_{8}$, (iv) $z_{\rm re}$,
and (v) $\omega_{\text{b,0}}$. (i): The fitted value of $H_{0}$ is in agreement with local-cosmology observation [19, 27] but
discrepant at the 4.5 $\sigma$ level with the value $H_{0}\sim(67.4\pm 0.5)$ km Mpc${}^{-1}$ s${}^{-1}$ in $\Lambda$CDM of global-cosmology fits as extracted by the
Planck collaboration [8],
a galaxy-custering survey – the Baryonic Oscillation
Spectroscopy Survey (BOSS) [3] – , and a galaxy-custering-weak-lensing survey – the Dark Energy Survey (DES) [2] –
whose distances are derived from inverse distance ladders using the sound horizons at CMB decoupling or
baryon drag as references. Such anchors assume
the validy of $\Lambda$CDM (or a variant thereof with variable Dark Energy) at high $z$, and the data analysis employs
fiducial cosmologies that are close to those of CMB fits. As a rule of thumb, the inclusion of extra relativistic degrees
of freedom within reasonable bounds [3, 2] but also sample variance, local matter-density fluctuations, or a directional bias in SNa Ia observations
[12, 13, 14, 15] cannot explain
the above-quoted tension in $H_{0}$. Note that the DES Y1 data on three two-point functions (cosmic shear auto correlation,
galaxy angular auto correlation, and galaxy-shear cross correlation in up to five redshift bins with $z\leq 1.3$), which are most
sensitive to $\Omega_{m}$ and $\sigma_{8}$ (or $S_{8}=\sigma_{8}/\Omega_{m}$),
cannot usefully constrain $H_{0}$ by itself. However, the combination of DES 1Y data with those of Planck (no lensing)
yields an increases in $H_{0}$ on the 1-2 $\sigma$ level compared
to the Planck (no lensing) data alone, and a similar tendency is seen when BAO data [3] are enriched with those of
DES Y1, see Table II in [2]. Loosely speaking, one thus may state a mild increase of $H_{0}$ with an increasing portion of late-time (local-cosmology) information in the data.
(ii): The index of the initial spectrum of adiabatic, scalar curvature perturbations $n_{s}$ is unusually low, expressing an enhancement
of infrared modes as compared to the ultraviolet ones (violation of scale invariance). As discussed in [34],
such a tilted spectrum is not consistent with single-field
slow-roll inflation and implies that the Hubble parameter during the inflationary
epoch has changed appreciably. (iii): There is a low value of $\sigma_{8}$ (initial amplitude of matter-density power spectrum at
comoving wavelength 8 h${}^{-1}$ Mpc with
$H_{0}\equiv h\,100$ km s${}^{-1}$Mpc${}^{-1}$) compared to CMB fits [8] and BAO [3] although the DES Y1 fit to cosmic shear data alone
would allow for a value $\sigma_{8}\sim 0.71$ within the 1-$\sigma$ margin [2]. This goes also for our
high value of today’s matter density parameter $\Omega_{\text{m,0}}$ (the ratio of physical matter density to the critical density).
(iv): The low value of the redshift to (instantaneous) re-ionisation $z_{\rm re}$ (and the correspondingly low optical depth $\tau_{\text{re}}$)
compared to values obtained in CMB fits [8] is consistent with the one extracted from the
observation of the Gunn-Peterson trough in the spectra of high-redshift quasars [36]. (v): The low value of
$\omega_{\text{b,0}}$ – today’s physical baryon density – could provide a theoretical solution of the missing baryon problem [34] (missing compared
to CMB fits to $\Lambda$CDM (CMB fits) and the primordial D/H (Deuterium-to-Hydrogen) ratio in Big-Bang Nucleosynthesis (BBN), the latter yielding
$\omega_{\text{b,0}}=0.02156\pm 0.00020$ [56] which is by 2.3 $\sigma$ lower than the Planck value
of $0.0225\pm 0.00016$ in Tab. 1). If this low value of $\omega_{\text{b,0}}$ could be
consolidated observationally than either the idea that BBN is determined by isolated nuclear reaction cross sections only or the
observation of a truly primordial D/H in metal-poor environments or both will have to be questioned in future ivestigations.
There is a recent claim, however, that the missing 30-40 % of baryons were
observed in highly ionized oxygen absorbers representing the warm-hot intergalactic medium when illuminated by the X-ray spectrum of a quasar with $z>0.4$
[57]. This results is disputed in [58]. In [59] the missing baryon problem is addressed by the
measurement of electron column density within the intergalactic medium using the dispersion of localised radio bursts with $z\leq 0.522$.
In projecting electron density
the measurement appeals to a flat $\Lambda$CDM CMB-fitted model (Planck collaboration [8]).
Presently, $\omega_{\text{b,0}}$ is extracted from the ASKAP data in [59]
to be consistent with CMB fits and BBN albeit subject to a 50 % error which is expected to decrease with the advent
of more powerful radio observatories such as SKA.
4 Axionic Dark Sector and galactic Dark-Matter densities
The model for the Dark Sector in Eq. (9) is motivated by the possibility that a
coherent condensate of ultra-light particles – an axion field [53] – forms selfgravitating
lumps or vortices in the course of nonthermal (Hagedorn) phase transitions due to SU(2)/SU(3)
Yang-Mills factors, governing the radiation and matter content of the very early Universe,
going confining. A portion of the thus created, large abundance of such
solitons percolates into a dark-energy like state: the contributions $\Omega_{\Lambda}$ and
$\Omega_{\rm edm,0}(z_{p}+1)^{3}$ in Eq. (9) of which the latter may de-percolate at $z=z_{p}$ into a dark-matter like
component $\Omega_{\rm edm,0}(z+1)^{3}$ for $z<z_{p}$, the expansion of the Universe increasing the average distance between the centers of
neighbouring solitons. By virtue of the U(1)${}_{A}$ anomaly, mediated by topological charge
density [49, 50, 51, 52], residing in turn within the ground state of an SU(2)/SU(3) Yang-Mills theory [42],
the mass $m_{a}$ of an axion particle due to a single such theory of Yang-Mills scale $\Lambda$ is given as [53]
$$m_{a}=\frac{\Lambda^{2}}{M_{P}}\,,$$
(11)
where we have assumed that the so-called Peccei-Quinn scale, which associates with a dynamical
chiral-symmetry breaking, was set equal to the Planck mass $M_{P}=1.221\times 10^{28}\,$eV, see [60, 61] for motivations.
Let (natural units: $c=\hbar=1$)
$$r_{c}\equiv 1/m_{a}$$
(12)
denote the Compton wavelength,
$$r_{B}\equiv\frac{M_{P}^{2}}{Mm_{a}^{2}}$$
(13)
the gravitational Bohr radius, where $M\sim 10^{12}\,M_{\odot}$ is the total dark mass of a typical (spiral) galaxy
like the Milky Way, and
$$d_{a}\equiv\left(\frac{m_{a}}{\rho_{\rm dm}}\right)^{1/3}$$
(14)
the interparticle distance where $\rho_{\rm dm}$ indicates the typical mean energy density in Dark Matter of a spiral. Following [63], we assume
$$\rho_{\rm dm}=(0.2\cdots 0.4)\,\mbox{GeV\,cm}^{-3}\,.$$
(15)
For the concept of a gravitational Bohr radius in a selfconsistent, non-relativistic potential model
to apply, the axion-particle velocity
$v_{a}$ in the condensate should be much small than unity. Appealing to the virial theorem at a distance to the gravitational center
of $r_{B}$, one has $v_{a}\sim\left(\frac{Mm_{a}}{M_{P}^{2}}\right)^{2}$ [62]. In [62], where a selfgravitating axion condensate
was treated non-relativistically by means of a non-linear and non-local Schrödinger equation to represent a typical
galactic dark-matter halo, the following citeria on the validity of such an approach were put forward (natural units: $c=\hbar=1$):
(i) $v_{a}\ll 1$. ii) $d_{a}\ll r_{c}$ is required
for the description of axion particles in terms of a coherent Bose
condensate to be realistic. (iii) $r_{B}$ should be the typical extent of a galactic Dark-Matter halo:
$r_{B}\sim 100-300\,$kpc. With $\Lambda_{\textnormal{\tiny{cmb}}}\sim 10^{-4}\,$eV one obtains $m_{a}=8.2\times 10^{-37}\,$eV, $d_{a}=(4.1\cdots 5.2)\times 10^{-34}\,$pc,
$r_{c}=3.2\times 10^{25}\,$pc, and
$r_{B}\sim 8\times 10^{24}\,$kpc. While (i) and (ii) are extremely well satified with $v_{a}\sim 10^{-28}$ and $\frac{d_{a}}{r_{c}}\sim 10^{-59}$
point (iii) is badly violated. ($r_{B}$ is about $2\times 10^{18}$
the size of the visible Universe).
Therefore, the Yang-Mills scale responsible for the axion mass that associates
with dark-matter halos of galaxies must be dramatically larger. Indeed, setting $\Lambda=10^{-2}\Lambda_{e}$ where $\Lambda_{e}=\frac{m_{e}}{118.6}$
is the Yang-Mills scale of an SU(2) theory
that could associate with the emergence of the electron of mass $m_{e}=511\,$keV [64, 65], one obtains
$m_{a}=1.5\times 10^{-25}\,$eV, $d_{a}=(2.3\cdots 2.9)\times 10^{-30}\,$pc, $r_{c}=1.7\times 10^{14}\,$pc, and
$r_{B}\sim 232\,$kpc. In addition to (i) and (ii) with $v_{a}\sim 10^{-4}$ and $\frac{d_{a}}{r_{c}}\sim 10^{-44}$ also
point (iii) is now well satisfied. If the explicit Yang-Mills scale of an SU(2) theory, which
is directly imprinted in the spectra of the excitations in the pre - and deconfining phases,
acts only in a screened way in the confining phase as far as the axial anomaly
is concerned – reducing its value by a factor of one hundred or so –, then the above axionic Dark-Sector
scenario would link the theory responsible for the emergence of the electron with galactic dark-matter halos! In addition, the axions of
$\text{SU(2)}_{\textnormal{\tiny{cmb}}}$ would provide the Dark-Energy density $\Omega_{\Lambda}$ of such a scenario.
Finally, we wish to point out that the de-percolation mechanism of axionic solitons (lumps forming out of former Dark-Energy density) in the cosmological model based on
SU(2)${}_{\textnormal{\tiny{cmb}}}$, which may be considered to underly the transition
in the Dark Sector at $z_{p}=53$ described by Eq. (9), is consistent with the Dark-Matter density in the Milky Way.
Namely, working with $H_{0}=74$ km s${}^{-1}$Mpc${}^{-1}$, see Table 1 and [19],
the total (critical) energy density $\rho_{c,0}$ of our spatially flat Universe
is at present
$$\rho_{c,0}=\frac{3}{8\pi}\,M_{P}^{2}H_{0}^{2}=1.75\times 10^{-9}\,{\rm eV}^{4}\,.$$
(16)
The portion of cosmological Dark Matter $\rho_{{\rm cdm},0}$ then is, see Table 1,
$$\rho_{{\rm cdm},0}\sim 0.35\,\rho_{c,0}$$
(17)
which yields a cosmological energy scale $E_{{\rm cdm},0}$ in association with Dark Matter of
$$E_{{\rm cdm},0}\equiv\rho^{1/4}_{{\rm DM},0}\sim 0.00497\,{\rm eV}\,.$$
(18)
On the other hand, we may imagine the percolate of axionic field profiles, which dissolves at $z_{p}$, to be associated
with densely packed dark-matter halos typical of today’s galaxies. Namely, scaling the typical Dark-Matter
energy density of the Milky Way $\rho_{\rm dm}$ of Eq. (15) from $z_{p}$ (at $z_{p}+0$
$\rho_{\rm dm}$ yet behaves like a cosmological constant) down to $z=0$ and allowing for a factor of
$(\omega_{\text{pdm,0}}+\omega_{\text{edm,0}})/\omega_{\text{edm,0}}=2.47$, see Table 1, one extracts the energy
scale for cosmological Dark Matter $E_{{\rm G},0}$ in association with galactic Dark-Matter halos and primordial dark matter as
$$E_{{\rm G},0}\equiv\left(\frac{2.47\rho_{\rm dm}}{(z_{p}+1)^{3}}\right)^{1/4}=%
(0.00879\cdots 0.0105)\,{\rm eV}\,.$$
(19)
A comparison of Eqs. (18) and (19) reveals that $E_{{\rm cdm},0}$ is smaller but comparable to
$E_{{\rm G},0}$. This could be due to the neglect of galactic-halo compactification through the missing pull
by neighbouring profiles in the axionic percolate and because of the omission of selfgravitation and
baryonic-matter accretion/gravitation during the evolution from $z_{p}=53$ to present. (That is, the use of the values
of Eq. (15) in Eq. (19) overestimates the homogeneous energy density in the percolate.) The de-percolation of axionic solitons at $z_{p}$, whose mean, selfgravitating
energy density $\rho_{\rm dm}$ in Dark Matter is nearly independent of cosmological
expansion but subject to local gravitation, could therefore be linked to cosmological Dark Matter
today within the Dark-Sector model of Eq. (9).
5 Conclusions
The present article’s goal was to address some tensions between local and global cosmology on the basis
of the $\Lambda$CDM standard model. Cracks in this model
could be identified during the last few years thanks to independent tests resting on precise
observational data and their sophisticated analysis. To reconcile these results, a change of
$\Lambda$CDM likely is required before the onset of the formation of non-linear, large-scale structure. Here, we have reviewed a proposal
made in [34], which assumes thermal photons to be governed by
an SU(2) rather than a U(1) gauge principle, and we have discussed the SU(2)${}_{\textnormal{\tiny{cmb}}}$-implied changes in cosmological
parameters and the structure of the Dark Sector. Noticeably, the tensions in $H_{0}$, the baryonic density, and the
redshift for re-ionisation are addressed in favour of local measurements. High-$z$ inputs to CMB and BAO simulations,
such as $n_{s}$ and $\sigma_{8}$, are sizeably reduced as compared to their fitted values in $\Lambda$CDM.
The Dark Sector now invokes a de-percolation of axionic
field profiles at a redshift of $z_{p}\sim 53$. This idea is roughly consistent with
typical galactic Dark-Matter halos today, such as the one of the Milky Way, being released from the percolate. Axionic
field profiles, in turn, appear to be compatible with Dark-Matter halos
in typical galaxies if (confining) Yang-Mills dynamics subject to much higher mass scales than that of SU(2)${}_{\textnormal{\tiny{cmb}}}$
is considered to produce the axion mass.
To consolidate such a scenario
two immediate fields of investigation suggest themselves: (i) A deep understanding of possible
selfgravitating profiles needs to be gained towards their role in actively assisted large-scale structure formation as well as in
quasar emergence, strong lensing, cosmic shear, galaxy clustering, and galaxy
phenomenology (Tully-Fisher, rotation curves, etc.), distinguishing spirals from ellipticals and satellites from hosts. (ii)
More directly, the CMB large-angle anomalies require an addressation in terms of radiative effects in SU(2)${}_{\textnormal{\tiny{cmb}}}$, playing out
at low redshifts, which includes a re-investigation of the CMB dipole.
Funding: None.
Conflicts of Interest: There is no conflict of interest.
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Axion field induces exact symmetry
P. [email protected]
Saha Institute of Nuclear Physics,
1/AF Bidhannagar
Calcutta 700064
()
Abstract
While no regularization is consistent with the anomalous chiral symmetry which
occurs for massless fermions,
the artificial axion-induced symmetry for massive fermions
is shown here to be consistent with a standard regularization,
even in curved spacetime,
so that it can be said to have no anomaly in gauge or gravitational fields.
Implications for $\theta$ terms are pointed out.
1 Introduction
Chiral symmetry, which is exact for massless fermions at the classical level
and approximate for light quarks, has been very useful in particle physics.
In analogy with this chiral symmetry,
an artificial chiral-like symmetry [1] was introduced
some time back for the strong interactions with massive quarks.
It implies the occurrence of a light
pseudoscalar particle, the axion [2], which has however not been detected in
spite of elaborate searches [3].
This symmetry has been suspected to be fraught with an anomaly because of the
involvement of a chiral transformation. However, this is not obvious
and has to be examined by careful regularization of the theory.
This is important because of implications for the symmetry of the theory.
While chiral symmetry of the action, corresponding to the transformation
$$\displaystyle\psi\rightarrow e^{i\alpha\gamma_{5}}\psi,\quad\bar{\psi}%
\rightarrow\bar{\psi}e^{i\alpha\gamma_{5}},$$
(1)
is broken by a non-vanishing quark mass term $m\bar{\psi}\psi$, with mass $m$,
the artificial chiral symmetry for massive fermions works by letting a new field
${\varphi}$ absorb the chiral transformation. The mass term is replaced
by
$$\displaystyle\bar{\psi}me^{i{\varphi}\gamma_{5}}\psi,$$
(2)
which is classically invariant under the transformation
$$\displaystyle\psi\rightarrow e^{i\alpha\gamma_{5}}\psi,\quad\bar{\psi}%
\rightarrow\bar{\psi}e^{i\alpha\gamma_{5}},\quad\varphi\rightarrow{\varphi}-2\alpha,$$
(3)
which is also a symmetry of the other terms of the action.
The original interaction introduced by Peccei and Quinn [1] was of the form
$$\displaystyle\bar{\psi}[\Phi{1+\gamma^{5}\over 2}+\Phi^{\dagger}{1-\gamma^{5}%
\over 2}]\psi,$$
(4)
where $\Phi$ is a complex scalar field with a symmetry breaking potential.
The artificial chiral symmetry transformation here is
$$\displaystyle\Phi\to e^{-2i\alpha}\Phi,\quad\psi\to e^{i\alpha\gamma_{5}}\psi,%
\quad\bar{\psi}\to\bar{\psi}e^{i\alpha\gamma_{5}}.$$
(5)
$\Phi$ may be taken to be of the form $\rho e^{i{\varphi}}$.
The amplitude $\rho$ of the scalar field acquires a vacuum expectation value
because of symmetry breaking, which provides a massive boson.
The phase ${\varphi}$ is the zero mode of the potential
and provides a Goldstone boson.
This is the axion, which should acquire
a mass because of the quark masses, but does not appear to exist.
The kinetic term of $\Phi$ yields
$\frac{1}{2}\rho_{0}^{2}\partial_{\mu}\varphi\partial^{\mu}\varphi$
as the kinetic term for $\varphi$, with $\rho_{0}$ the vacuum expectation
value of $\rho$.
The mass $m$ comes from $\rho_{0}$ and the coupling constant of $\Phi$.
The anomaly, which occurs for the usual chiral symmetry
and makes the divergence of the axial current $\bar{\psi}\gamma_{\mu}\gamma_{5}\psi$
nonvanishing,
appears when the singularities in the theory are handled by regularization.
Even the measure picture
of anomalies requires a regularization for actual calculations.
We shall therefore use an explicit
regularization for studying chiral transformations of fermions
in the presence of the axion coupling. This will tell us whether
the continuous symmetry (3)
introduced classically for massive quarks is anomalous
or not.
The new symmetry is expected to be broken
spontaneously by the vacuum because of the translation of the spinless field
by a c-number. The field must have a definite value in a fixed vacuum:
let it be denoted by $\varphi_{0}$. Then
$$\displaystyle\varphi^{\prime}=\varphi-\varphi_{0}$$
(6)
is the shifted field with vanishing vacuum expectation value. This is a
pseudoscalar field, so that $\bar{\psi}\varphi^{\prime}\gamma_{5}\psi$ and
$(\varphi^{\prime}{\rm~{}tr~{}}F^{\mu\nu}\tilde{F}_{\mu\nu})$ are scalars. On the other
hand, $(\varphi_{0}{\rm~{}tr~{}}F^{\mu\nu}\tilde{F}_{\mu\nu})$ is
a pseudoscalar,
like the $\theta$ term $(\theta{\rm~{}tr~{}}F^{\mu\nu}\tilde{F}_{\mu\nu})$.
If a $(\varphi_{0}{\rm~{}tr~{}}F^{\mu\nu}\tilde{F}_{\mu\nu})$ term is generated
when the fermion is integrated out, it
can modify the $\theta$ term. The possibility of this occurrence is related to
the question of an anomaly in the symmetry (3).
If the symmetry (3) is anomalous,
no regularization of the fermions will be consistent with
it. On the other hand, if any regularization is found to be consistent with
it, the symmetry has no anomaly in this quantization and
such symmetric regularizations are the ones to be preferred.
Pauli-Villars regularization will be studied in flat and in curved spacetime.
2 Pauli-Villars regularization
We recall first the simple form of the Pauli-Villars
regularization. For a fermion with mass $m$, the Lagrangian density
$$\displaystyle\bar{\psi}[i\not{D}-m]\psi+\bar{\chi}[i\not{D}-M]\chi$$
(7)
involves regulator spinor fields $\chi,\bar{\chi}$ which however are
assigned Bose statistics. The regulator mass $M$ is ultimately taken to
infinity when the regulator fields decouple. The regulator fields couple to the
gauge fields in the standard way. The regulator mass term breaks the
chiral symmetry and yields the chiral anomaly in the $M\to\infty$ limit.
This is the original form of the Pauli-Villars regularization. A more general
form with several species of regulator fields is also available [4]:
$$\displaystyle\bar{\psi}[i\not{D}-m]\psi+\sum_{j}\sum_{k=1}^{|c_{j}|}\bar{\chi}%
_{jk}[i\not{D}-M_{j}]\chi_{jk}.$$
(8)
Here $c_{j}$ are integers whose signs are related to the statistics assigned
to $\chi_{jk}$. They have to satisfy some conditions to ensure regularization
of the divergences [4]:
$$\displaystyle 1+\sum_{j}c_{j}=0,\quad m^{2}+\sum_{j}c_{j}M_{j}^{2}=0.$$
(9)
Chiral transformations work on $\chi$ as well as $\psi$. Hence
the axion coupling has to be introduced for both $\psi,\chi$,
like the gauge coupling. One has
$$\displaystyle\bar{\psi}[i\not{D}-me^{i\varphi\gamma_{5}}]\psi+\bar{\chi}[i\not%
{D}-Me^{i\varphi\gamma_{5}}]\chi.$$
(10)
This regularization (10) is invariant under the combined transformation
$$\displaystyle\psi\rightarrow e^{i\alpha\gamma_{5}}\psi,$$
$$\displaystyle\bar{\psi}\rightarrow\bar{\psi}e^{i\alpha\gamma_{5}},$$
$$\displaystyle\chi\rightarrow e^{i\alpha\gamma_{5}}\chi,$$
$$\displaystyle\bar{\chi}\rightarrow\bar{\chi}e^{i\alpha\gamma_{5}},$$
$$\displaystyle\varphi$$
$$\displaystyle\rightarrow$$
$$\displaystyle{\varphi}-2\alpha,$$
(11)
which is the Pauli-Villars extension of (3).
This means the symmetry survives when the regularization is
removed by taking the limit $M\to\infty$
and hence is not anomalous. Unlike the regularized axial current, which has
a pseudoscalar divergence arising from the masses,
the Noether current for the extended symmetry, namely
$$\displaystyle\bar{\psi}\gamma_{\mu}\gamma_{5}\psi+\bar{\chi}\gamma_{\mu}\gamma%
_{5}\chi+2\rho_{0}^{2}\partial\mu\varphi,$$
(12)
is conserved.
The non-vanishing divergence of the axial current is cancelled by that
of the $\varphi$ piece by virtue of the axion equation of motion.
It is interesting to note that
the $\varphi$ phases in (10) can be removed by a joint chiral
transformation of both the physical fermions and the regulator spinors:
$$\displaystyle\psi\to e^{-i\varphi\gamma_{5}/2}\psi,$$
$$\displaystyle\bar{\psi}\to\bar{\psi}e^{-i\varphi\gamma_{5}/2},$$
$$\displaystyle\chi\to e^{-i\varphi\gamma_{5}/2}\chi,$$
$$\displaystyle\bar{\chi}\to\bar{\chi}e^{-i\varphi\gamma_{5}/2}.$$
(13)
The derivative operators in the action produce derivatives of $\varphi$,
which appear after the transformation.
Now the transformation has a trivial Jacobian because the contribution of the
fermion field is cancelled by that of the bosonic
regulator field as in the latter case the determinant arising from
functional integration comes in the denominator. The argument for the
generalized Pauli-Villars regularization, with (10) replaced by
$$\displaystyle\bar{\psi}[i\not{D}-me^{i\varphi\gamma_{5}}]\psi+\sum_{j}\sum_{k=%
1}^{|c_{j}|}\bar{\chi}_{jk}[i\not{D}-M_{j}e^{i\varphi\gamma_{5}}]\chi_{jk},$$
(14)
involves a Jacobian with the factor $1+\sum_{j}c_{j}=0$
in the exponent [5]. Consequently the effective action does not contain
any $\varphi{\rm~{}tr~{}}F^{\mu\nu}\tilde{F}_{\mu\nu}$ in this regularization.
The conserved Noether current here is
$$\displaystyle\bar{\psi}\gamma_{\mu}\gamma_{5}\psi+\sum_{j}\sum_{k=1}^{|c_{j}|}%
\bar{\chi}_{jk}\gamma_{\mu}\gamma_{5}\chi_{jk}+2\rho_{0}^{2}\partial\mu\varphi,$$
(15)
Note that the argument may be easily extended to curved spacetime,
where the Dirac operator
$$\displaystyle i\not{D}=i\gamma^{l}e^{\mu}_{l}(\partial_{\mu}-iA_{\mu}-\frac{i}%
{2}A^{mn}_{\mu}\sigma_{mn})$$
(16)
comes with a tetrad $e^{\mu}_{l}$ and a spin connection $A^{mn}_{\mu}$ in
addition to the gauge field, but
it continues to anticommute with $\gamma^{5}$.
The regularized action is again invariant under the Pauli-Villars extension of
(3).
This means that there is not even an $R\tilde{R}$ anomaly [6] in this symmetry.
Similar results have also been found by others [7].
3 Conclusion
We have examined the chiral symmetry of the fermion action including
$\varphi$. This classical symmetry is preserved by the explicit regularization
(10).
When an acceptable regularization preserves a symmetry, one can
conclude that the quantum theory defined by the limit of that regularization
will also satisfy it. Hence, after the
fermion is integrated out, the effective action must be invariant under
translations of $\varphi_{0}$, i.e., independent of this variable.
This is indeed true of this regularization, as seen above, and there
is no $\varphi{\rm~{}tr~{}}F^{\mu\nu}\tilde{F}_{\mu\nu}$ term. This makes the
modification of a $\theta$ term by a contribution from $\varphi_{0}$ impossible.
The field $\varphi$ occurs only in the form of derivatives.
The extension of the argument to curved spacetime means that an existing
$R\tilde{R}$ term [8] too cannot be modified.
There is another way of seeing this.
The exact classical chiral symmetry could be expected to play a rôle like
the chiral symmetry that holds for massless fermions. With massless fermions,
the $\theta$ term can be modified by a chiral transformation which does not
alter the fermion action but produces an $F\tilde{F}$ term because of the chiral
anomaly. However,
the chiral symmetry which occurs in the presence of mass and $\varphi$ is not
anomalous, as we have seen.
So $\theta$ cannot be modified in that way.
It should be pointed out here that any
effect of $\theta$ terms can be removed by setting $\theta=0$ [5] or by making
it explicitly dynamical, which forces the topological charge to vanish.
A dynamical $\theta$-like term is essentially a Peccei-Quinn
$(\varphi_{0}{\rm~{}tr~{}}F^{\mu\nu}\tilde{F}_{\mu\nu})$ term without
an axion particle. However, as only fields are dynamical in field theory,
the field $\varphi$ was used in [1] and a direct
$(\varphi{\rm~{}tr~{}}F^{\mu\nu}\tilde{F}_{\mu\nu})$ term can remove $\theta$.
One may wonder how much freedom one has in choosing regularizations.
Different regularizations have been used in the past
and it is known that all do not lead to the same result. A key point is that
symmetries of the action are generally sought to be preserved by the
regularization. Thus one is always looking for Lorentz invariant and gauge
invariant regularizations for actions having such symmetries.
Regularizations which maintain all symmetries are technically natural.
We have seen that
the Pauli-Villars regularization
conveniently respects the symmetry (3) of the fermion action.
Non-symmetric regularizations are possible, but are not
preferable in any way.
Regularizations that break an
existing classical symmetry like (3) may to that extent be
termed technically unnatural.
The artificial chiral symmetry here is not anomalous
as a regularization consistent with it has been demonstrated to exist. In these
circumstances, working with regularizations inconsistent with it would be
a needless and avoidable violation.
We end with the hope that the exact chiral symmetry in the presence of axions
will be useful also in other calculations and even in curved spacetime.
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R. Peccei, & H. Quinn,
Phys. Rev. Letters
38, 1440 (1977);
Phys. Rev. D16, 1791 (1977)
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S. Weinberg,
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F. Wilczek, Phys. Rev. Letters 40, 279 (1978)
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C. Patrignani et al., Review of Particle Physics,
Chin. Phys. C40, 100001 (2016)
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L. D. Faddeev & A. A. Slavnov, Gauge Fields, (Benjamin-Cummings, Reading, 1982)
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H. Banerjee, D. Chatterjee & P. Mitra,
Phys. Letters B573, 109 (2003);
see also P. Mitra, Symmetries and symmetry breaking in field theory,
CRC Press, Florida (2014)
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T. Kimura, Prog. Theo. Phys. 42, 1191 (1969)
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G. Dvali and L. Funcke, arXiv:1608.08969 (2016)
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S. Deser, M. Duff and C. Isham, Phys. Lett. B93, 419 (1980) |
Implications of imaginary chemical potential for model building of QCD
Kouji Kashiwa
[email protected]
Yukawa Institute for Theoretical Physics,
Kyoto University, Kyoto 606-8502, Japan
Abstract
Properties of QCD at finite imaginary chemical potential are revisited to
utilize for the model building of QCD in low energy regimes.
For example, the electric holonomy which is closely related to the
Polyakov-loop drastically affects thermodynamic quantities beside
the Roberge-Weiss transition line.
To incorporate several properties at finite imaginary chemical
potential, it is important to introduce the
holonomy effects to the coupling constant of effective models.
This extension is possible by considering the entanglement vertex.
We show justifications of the entanglement vertex based on the
derivation of the effective four-fermi interaction in the
Nambu–Jona-Lasinio model and present its general form with the
local approximation.
To discuss how to remove model ambiguities in the entanglement
vertex, we calculate the chiral condensate with different
$\mathbb{Z}_{3}$ sectors and the dual quark condensate.
pacs: 11.30.Rd, 21.65.Qr, 25.75.Nq
††preprint: YITP-16-38
I Introduction
Understanding non-perturbative properties of Quantum Chromodynamics
(QCD) is one of the important and interesting subjects in nuclear and
elementary particle physics.
The lattice QCD simulation is a promising method to understand such
non-perturbative properties, but the sign problem obscures it at finite
real chemical potential ($\mu_{\mathrm{R}}$).
Therefore, low energy effective models of QCD are widely used to
investigate the QCD phase structure at finite $\mu_{\mathrm{R}}$.
In the modeling of QCD in low energy regimes, the chiral condensate
and the Polyakov-loop play an important role to describe the chiral
and confinement-deconfinement transitions.
The famous effective model is the Polyakov-loop extended
Nambu–Jona-Lasinio (PNJL) model Fukushima (2004).
In the PNJL model, the quark and gluon contributions are taken
into account through the one-loop level effective potential in usual.
There is the uncertainty how to introduce the gauge boson contribution.
Usually, the chiral part is described by using the Nambu–Jona-Lasinio
(NJL) model or similar quark models in effective models.
However, the gluon part is less clear than the quark part.
There are several effective models for the gluon contribution;
for example,
the logarithmic potential based on the strong coupling
expansion
Fukushima (2004); Hell et al. (2010),
the Ginzburg-Landau type potential
Ratti et al. (2006); Roessner et al. (2007),
the Meisinger-Miller-Ogilvie model
Meisinger et al. (2002),
the matrix model for the confinement-deconfinement transition
Dumitru et al. (2011); Kashiwa et al. (2012),
the effective potential from the Landau-gauge gluon and ghost propagators
Fukushima and Kashiwa (2013),
the holonomy potential Shuryak and Sulejmanpasic (2013) and so on.
In addition to individual parts of the quark and gluon contributions,
it is important how to control the strength of the correlation between
the quark and gluon contributions in effective models.
Importance of such correlation can be clearly seen in the
lattice QCD data at finite temperature ($T$) and the imaginary chemical
potential ($\mu_{\mathrm{I}}$) D’Elia and Sanfilippo (2009); Bonati et al. (2011).
At finite $\mu_{\mathrm{I}}$, there is the first-order transition line which is so
called the Roberg-Weiss (RW) transition and its endpoint which is so
called the RW endpoint Roberge and Weiss (1986).
In Ref. D’Elia and Sanfilippo (2009); Bonati et al. (2011), the pion mass ($m_{\pi}$)
dependence of the RW endpoint has been predicted by the lattice QCD
simulation.
At sufficiently large $m_{\pi}$, the order
of the RW endpoint is the first-order which is induced by the
first-order $\mathbb{Z}_{3}$ transition.
The first-order RW endpoint becomes the crossover by the
explicit breaking of the $\mathbb{Z}_{3}$ symmetry at moderate $m_{\pi}$.
However, there is the mysterious behavior of the RW endpoint at small
$m_{\pi}$; the order of the transition turns into the first-order!
This behavior cannot be reproduced by using simple effective model of
QCD, but it can be possible by considering the extension of the
coupling constant.
Recently, it has been proposed in
Ref. Kashiwa and Ohnishi (2015, 2016) that the
confinement-deconfinement transition can be described by using the
analogy of the topological order Wen (1990) motivated by the
recent progress done in Ref. Sato (2008) at $T=0$ QCD.
In the conjecture Kashiwa and Ohnishi (2015, 2016), the nontrivial
free-energy degeneracy at finite $\mu_{\mathrm{I}}$ plays a crucial role.
The non-trivial degeneracy is related with the drastic change of
the holonomy beside $\theta=(2k-1)k/N_{\mathrm{c}}$ in the deconfinement phase
where $\theta$ is the dimensionless chemical potential
$\theta\equiv\mu_{\mathrm{I}}/T$, $N_{\mathrm{c}}$ means the number of color and $k$ is any
integer.
Then, we can define the quantum order-parameter which is so called the
quark number holonomy on the manifold in all possible
boundary condition space which is equivalence to the $\theta$
space because the boundary condition and the
imaginary chemical potential have direct relation, see
Ref. Roberge and Weiss (1986); Kashiwa and Misumi (2013) as an example.
From these viewpoints, holonomy effects in the effective model is
crucial not only incorporating the confinement-deconfinement
transition into the model, but also understanding the
confinement-deconfinement transition itself.
In this paper, we revisit the imaginary chemical potential region
to obtain some constraints for the model building of QCD in the low
energy regime.
This paper is organized as follows.
In the next section, we summarize the RW transition and some related
topics on it at finite $\mu_{\mathrm{I}}$.
Section III explains the extension of the effective model based
on Ref. Sakai et al. (2010).
Then, we show justifications of the extension based on the derivation of
the effective four-fermi interaction.
In Sec. IV, we calculate the chiral condensate with different
$\mathbb{Z}_{3}$ sectors to discuss how to remove ambiguities of model
parameters.
Section V is devoted to summary.
II Imaginary chemical potential
It is well known that there is a special periodicity in the QCD
partition function ($Z$) as a function of $\theta$;
$$\displaystyle Z(\theta)$$
$$\displaystyle=Z\Bigl{(}\theta+\frac{2\pi k}{N_{\mathrm{c}}}\Bigr{)}.$$
(1)
This means that several quantities have the $2\pi/N_{\mathrm{c}}$
periodicity.
This periodicity is so called the Roberge-Weiss (RW) periodicity.
In the low and high $T$ region, origins of the RW periodicity is quite
different.
At low $T$, the RW periodicity is realized by only one global minimum on
the complex Polyakov-loop ($\Phi$) plane, but does not at high $T$.
The $\mathbb{Z}_{N_{\mathrm{c}}}$ images should be needed to realize the RW periodicity at high
$T$; for example, the $\mathbb{Z}_{3}$ images are $e^{i2\pi/3}$ and
$e^{i4\pi/3}$ for $\Phi=1$.
This difference of origins seems to be related with the confinement and
the deconfinement nature.
From the different realization of the RW periodicity at low and high
$T$, there should be the first-order transition and its endpoint at
$\theta=(2k-1)\pi/N_{\mathrm{c}}$.
Those are called the RW transition and the RW endpoint, respectively.
When we change $\theta$ with fixing $T$ in the confined phase, the
phase of the Polyakov-loop $\phi$ which is determined from
$$\displaystyle\Phi$$
$$\displaystyle=\frac{1}{N_{\mathrm{c}}}\mathrm{tr}{\cal P}\Bigl{[}\exp\Bigl{(}i\oint_{0}^{\beta}A_{4}(\tau,{\vec{x}})~{}d\tau\Bigr{)}\Bigr{]}=|\Phi|\hskip 1.42262pte^{i\phi},$$
(2)
is continuously rotated as the soliton solution which may be described
by the Jacobi elliptic function where ${\cal P}$ means the path ordering
operator, $g$ is the gauge coupling
constant and $\beta$ means the inverse temperature ($1/T$).
On the other hand, the Polyakov-loop phase is discontinuously
rotated in the deconfined phase.
The discontinuous point appears when we across the RW transition line.
The electric holonomy ($\nu$) is then discontinuously changed beside the RW
transition line and then we obtain
$$d\nu=\lim_{\epsilon\to 0}\Bigl{[}\nu(\theta_{\mathrm{RW}}-\epsilon)-\nu(\theta_{\mathrm{RW}}+\epsilon)\Bigr{]}\left\{\begin{array}[]{l}0~{}~{}~{}~{}~{}~{}~{}T<T_{\mathrm{RW}}\\[14.22636pt]
\dfrac{2\pi}{N_{\mathrm{c}}}~{}~{}~{}~{}T>T_{\mathrm{RW}},\\
\end{array}\right.$$
where we define $d\nu$ as $\{d\nu~{}|~{}0\leq d\nu\leq 2\pi\}$.
Just on the RW endpoint, $d\nu$ depends on its order:
If the RW endpoint is first-order, $d\nu$ should be $2\pi/N_{\mathrm{c}}$.
Because of this behavior, $\theta$-even quantities such as the entropy
density and pressure should have the cusp, but $\theta$-odd quantities
such as the quark number density have the gap at $\theta=(2k-1)\pi/N_{\mathrm{c}}$
above the RW endpoint.
The RW transition and endpoint have interesting phenomena as discussed above,
but it may have more mysterious behavior which is the triple-point
realization of the RW endpoint at small $m_{\pi}$ which is predicted by
the lattice QCD simulation D’Elia and Sanfilippo (2009); Bonati et al. (2011).
At sufficiently large $m_{\pi}$, the order
of the RW endpoint is the first-order.
This first-order RW endpoint becomes the crossover by the
explicit breaking of the $\mathbb{Z}_{3}$ symmetry when we set $m_{\pi}$ as
moderate values.
However, the order of the RW transition turns into the first-order again
at small $m_{\pi}$.
This behavior is considered as the consequence of the correlation between
the chiral and the confinement-deconfinement transition nature, but it is
still under debate.
By using simple effective models such as the standard PNJL model, this
behavior cannot be reproduced; for example, see
Ref. Sasaki et al. (2011).
To resolve this point, we need extension of the effective model.
Actually, by extending the coupling constant in the PNJL model, we can
reproduce the lattice QCD simulation Sakai et al. (2010), but the exact
form of the extended coupling constant is not discussed so far.
In the next section, we revisit this point.
III Entanglement vertex
In this study, we use the PNJL model.
The simplest two-flavor and three-color PNJL model Lagrangian density
becomes
$$\displaystyle{\cal L}$$
$$\displaystyle={\bar{q}}(i{\ooalign{\hfil/\hfil\crcr$D$}}-m_{0})q+G[({\bar{q}}q)^{2}+({\bar{q}}i\gamma_{5}{\vec{\tau}}q)^{2}]-\beta V{\cal U},$$
(5)
where $D^{\mu}=\partial^{\mu}+i\delta^{\nu}_{0}A^{\nu,a}\lambda_{a}/2$ with
Gell-Man matrices $\lambda_{a}$,
$m_{0}$ expresses the current quark mass, $G$ is the coupling
constant and $V$ does the
three-dimensional volume.
The term ${\cal U}$ expresses the gluon contribution.
Model parameters in the NJL part such as $m_{0}$, $G$ and the cutoff of
the three-dimensional momentum integration ($\Lambda$) is determined by
using the $m_{\pi}$ and pion decay constant.
The actual values used in this study are
$m_{0}=5.5$ MeV, $G=5.498$ GeV${}^{-2}$, $\Lambda=613.5$ MeV.
In this study, we use the mean-field approximation; see
Ref. Sakai et al. (2010) for the details of the calculation.
The actual thermodynamic potential can be obtained from the Lagrangian
density (5) as
$$\displaystyle\Omega$$
$$\displaystyle=-4\int\frac{d^{3}p}{(2\pi)^{3}}\Bigl{[}3E(p)+T(\ln f^{-}+\ln f^{+})\Bigr{]}$$
$$\displaystyle~{}~{}~{}+2G\sigma^{2}+{\cal U},$$
(6)
where
$$\displaystyle f^{-}$$
$$\displaystyle=1+3(\Phi+{\bar{\Phi}}e^{-\beta E^{-}(p)})e^{-\beta E^{-}(p)}+e^{-3\beta E^{-}(p)},$$
$$\displaystyle f^{+}$$
$$\displaystyle=1+3({\bar{\Phi}}+\Phi e^{-\beta E^{+}(p)})e^{-\beta E^{+}(p)}+e^{-3\beta E^{+}(p)},$$
(7)
here ${\bar{\Phi}}$ is the conjugate of $\Phi$ and
$E^{\mp}(p)=E(p)\mp\mu=\sqrt{p^{2}+M^{2}}\mp\mu$ with
$M=m_{0}-2G\sigma$ and the chemical potential $\mu=(\mu_{\mathrm{R}},\mu_{\mathrm{I}})$.
The quantity $\sigma$ becomes the chiral condensate in the present
model.
The gluon contribution ${\cal U}$ is taken into account via the
logarithmic Polyakov-loop effective potential Hell et al. (2010) as
$$\displaystyle\frac{{\cal U}}{T^{4}}$$
$$\displaystyle=-\frac{1}{2}b_{2}(T){\bar{\Phi}}\Phi$$
$$\displaystyle+b_{4}(T)\ln\Bigl{[}1-6{\bar{\Phi}}\Phi+4\Bigl{(}{\bar{\Phi}}^{3}+\Phi^{3}\Bigr{)}-3({\bar{\Phi}}\Phi)^{2}\Bigr{]},$$
(8)
where
$$\displaystyle b_{2}(T)$$
$$\displaystyle=a_{0}+a_{1}\Bigl{(}\frac{T_{0}}{T}\Bigr{)}+a_{2}\Bigl{(}\frac{T_{0}}{T}\Bigr{)}^{2}+a_{3}\Bigl{(}\frac{T_{0}}{T}\Bigr{)}^{3},$$
$$\displaystyle b_{4}(T)$$
$$\displaystyle=b_{4}\Bigl{(}\frac{T_{0}}{T}\Bigr{)}^{3},$$
(9)
with $a_{0}=3.51$, $a_{1}=-2.56$, $a_{2}=15.2$, $a_{3}=-0.62$ and
$b_{4}=-1.68$.
These parameters are determined to reproduce the lattice
QCD data such as the energy and the entropy density in the pure gauge
limit.
The parameter $T_{0}$ controls the confinement-deconfinement temperature
($T_{\mathrm{D}}$) in the pure gauge limit and thus we set $T_{0}=270$ MeV because we are interested in the quenched calculation.
In Ref. Sakai et al. (2010), the extension of the PNJL model coupling
constant has been proposed.
The extension has been done by the replacement
$$\displaystyle G\to G(\Phi)=G[1-\alpha_{1}{\bar{\Phi}}\Phi-\alpha_{2}({\bar{\Phi}}^{3}+\Phi^{3})],$$
(10)
where $\alpha_{1}$ and $\alpha_{2}$ are new parameters.
This extended coupling constant is so called the entanglement vertex.
In the extension, the $\mathbb{Z}_{3}$ symmetry of the vertex is assumed.
This interaction strongly correlates the chiral and the
confinement-deconfinement transition.
The entanglement nature is deeply related with the nonlocal properties
of the interaction which is usually hidden in the local PNJL model.
From the derivation of the NJL and also the PNJL model, the nonlocal
nature is a natural consequence.
The PNJL model can be obtained starting from QCD partition function by
using the color-current expansion and some ansatz for gluon
properties; details of the nonlocal PNJL model, for example, see
Ref. Van den Bossche (1997); Kondo (2010); Kashiwa et al. (2011) and
references therein.
The nonlocal interaction should feel effects coming from the
gluon two-point function since the interaction forms
$$\displaystyle{\cal L}_{\mathrm{int}}(x)$$
$$\displaystyle=g^{2}j_{a}^{\mu}(x)\int d^{4}y~{}W^{(2)ab}_{\mu\nu}j^{\nu}_{b}(y),$$
(11)
where
$j_{\mu}^{a}={\bar{q}}\frac{\lambda^{a}}{2}\gamma_{\mu}q$ is the
color-current, $W^{(2)}$ is the gluon two-point function without quark
loops because we use the color-current expansion around $j=0$.
Since the two-point function does not have quark loops, $W^{(2)}$ should
be $\mathbb{Z}_{N_{\mathrm{c}}}$ symmetric form.
To obtain the local version of the PNJL model, we should replace the
two-point function as the delta function since we assume $W^{(2)}$ is
very short range.
The gluon follows the adjoint representation of
$SU(N_{\mathrm{c}})$, and thus the entanglement vertex should obey the adjoint
representation nature.
To introduce the adjoint representation nature to the entanglement vertex,
we refer to the structure of the gluon one-loop effective potential and
the Bose distribution function with the electric holonomy.
Those manifest the $\mathbb{Z}_{3}$ symmetric form and other adjoint
representation natures.
So, more general form of the entanglement vertex for the color singlet
sector can be written as
$$\displaystyle G(\Phi)$$
$$\displaystyle=G_{\mathrm{N}}\Bigl{[}1+\Bigl{(}\sum_{n=1}^{8}\alpha^{\prime}_{n}C_{n}e^{-n\beta E}\Bigr{)}^{\gamma}\Bigr{]},$$
(12)
where $G_{\mathrm{N}}$, $\alpha^{\prime}_{n}$ and $\gamma$ are parameters,
$E=|{\vec{p}}|$ is the typical energy scale in the momentum space
and
$$\displaystyle C_{1}$$
$$\displaystyle=C_{7}=1-9{\bar{\Phi}}\Phi,$$
$$\displaystyle C_{2}$$
$$\displaystyle=C_{6}=1-27{\bar{\Phi}}\Phi+27({\bar{\Phi}}^{3}+\Phi^{3}),$$
$$\displaystyle C_{3}$$
$$\displaystyle=C_{5}=-2+27{\bar{\Phi}}\Phi-81({\bar{\Phi}}\Phi)^{2},$$
$$\displaystyle C_{4}$$
$$\displaystyle=2\Bigl{[}-1+9{\bar{\Phi}}\Phi-27({\bar{\Phi}^{3}+\Phi^{3}})+81({\bar{\Phi}}\Phi)^{2}\Bigr{]},$$
$$\displaystyle C_{8}$$
$$\displaystyle=1.$$
(13)
For details of functions $C_{n}$, see Ref. Sasaki and Redlich (2012).
Even if we assume the effective gluon mass such as the Gribov-Stingl
form at finite $T$ as
$$\displaystyle D^{(T,L)}_{T}$$
$$\displaystyle\propto\frac{d_{t,l}(p^{2}+d_{t,l}^{-1})}{(p^{2}+r^{2}_{t,l})^{2}},$$
(14)
we then obtain same expression of $C_{n}$ Fukushima and Kashiwa (2013),
where $D^{T}_{T}$ ($D^{L}_{T}$) are the tree-dimensional transverse
(longitudinal) gluon propagator.
Thus, we can expect that above expression is general form for the
entanglement vertex.
When we assume that the $W^{(2)}$ is short range, higher-order terms of
$e^{-\beta E}$ in Eq. (12) should be suppressed
and then only $C_{1}$ is relevant.
From here, we set $\gamma=1$ to make our discussion simple and it is not
an essential point in the level of present discussions.
Thus, we can effectively rewrite the coupling constant (12) as
the simple form
$$\displaystyle G(\Phi)$$
$$\displaystyle\sim G_{\mathrm{N}}\Bigl{(}1+\alpha{\bar{\Phi}}\Phi\Bigr{)},$$
(15)
where $G_{\mathrm{N}}$ is corresponding to $G$ in the original PNJL
model in this case because ${\bar{\Phi}}$ and $\Phi$ do not affect the
physics at $T=0$.
This form is similar to Eq. (10), but it is more general.
If we set $\alpha=-\alpha_{1}$ and $\alpha_{2}=0$ with $\gamma=1$, this
vertex is perfectly matched with the vertex (10).
From present representation, terms proportional to
$({\bar{\Phi}}^{3}+\Phi^{3})$
becomes the higher-order contribution and thus it can be safely
neglected in the level discussed here.
In the effective model construction, important operation is the
parameter fixing.
Details of parameter fixing are discussed in the next section.
Recently, it has been numerically shown that the $\mathbb{Z}_{3}$ factor
can affect the Landau-gauge gluon propagator in the deconfined phase by
using the pure gauge calculation in Ref. Silva and Oliveira (2016).
In the gluon propagator, the $\mathbb{Z}_{3}$ factors can appear via
${\cal A}_{4}^{ab}={\cal A}_{4}^{a}-{\cal A}_{4}^{b}$ when we consider the
background gauge field where ${\cal A}_{4}$ is the temporal component of the
field and $a,b$ run $1,\cdots,N_{\mathrm{c}}$ because the gluon obeys the adjoint
representation.
For $a=b$ diagonal sectors, the $\mathbb{Z}_{N_{\mathrm{c}}}$ factor cannot appear
because of the exact cancellation of the $\mathbb{Z}_{3}$ factors.
On the other hand, the non-diagonal $a\neq b$ sectors may feel the
effect via the effective gluon mass.
Also, in the confined and deconfined phases, the elementary degree of
freedom is drastically different; it is glueball in the confined phase,
but it is gluon in the deconfined phase.
Both reasons may lead to the lattice QCD predicted phenomena, but we
need more careful investigation for the lattice QCD data to conclude it.
In the short range limit of $W^{(2)}$, the color structure is extremely
simplified and thus above effect cannot appears in the local PNJL model.
To incorporate the effects, we should consider the nonlocal PNJL model,
but it seems a difficult task.
Even in the modern nonlocal PNJL
model Hell et al. (2009, 2010); Radzhabov et al. (2011); Pagura et al. (2012); Kashiwa et al. (2011); Carlomagno et al. (2013),
we implicitly use
the simplification of the color structure of the four-fermi interaction.
At present, we do not know how important this effects for the gauge
invariant observable, and thus
we assume that $W^{(2)}$ is short range to neglect the effect as a first
step.
This problem will be discussed in the nonlocal PNJL model in elsewhere.
IV Numerical results
In this section, we discuss how to remove ambiguities of the
parameters in the entanglement vertex by
using lattice QCD simulations with the quenched approximation.
The entanglement vertex correlates the chiral and
the confinement-deconfinement transition and thus the chiral condensate
with different $\mathbb{Z}_{3}$ sectors seems to be a promising quantity
to remove ambiguities of model parameters.
The existence of non-trivial $\mathbb{Z}_{3}$ sectors which is
characterized by $\phi=\frac{2\pi}{3}$ and $\frac{4\pi}{3}$ is not
trivial in the system with dynamical quarks.
Therefore, we here consider the quenched approximation since all
possible $\mathbb{Z}_{3}$ sectors are well defined.
The calculation of the chiral condensate with the different
$\mathbb{Z}_{3}$ sector was done in
Ref. Chandrasekharan and Christ (1996); Chandrasekharan and Huang (1996) and thus we
follow their calculation by using present extended model.
In the calculation Chandrasekharan and Huang (1996) , they introduce the
$T$-dependent coupling constant by hand, but this effect is automatically
introduced in the present entanglement PNJL model.
In this study, we first calculate $\Phi$ in the pure
gauge limit to consider the quenched calculation.
The Polyakov-loop with non-trivial $\mathbb{Z}_{3}$ sectors,
$\Phi_{\frac{2\pi}{3}}$ and $\Phi_{\frac{4\pi}{3}}$,
are obtained by the $\mathbb{Z}_{3}$ transformation from
the Polyakov-loop with the trivial $\mathbb{Z}_{3}$ sector, $\Phi_{0}$.
Figure 1 shows $\Phi$ with $\phi=0$ and
$\phi=\frac{2\pi}{3}$ in the pure gauge limit as a function of $T$.
After this calculation, we substitute those $\Phi$
to the PNJL model and then we minimize the thermodynamic
potential as a function of $\sigma$.
This procedure is corresponding to the quenched calculation of the PNJL
model.
Figure. 2 shows the chiral condensate with different
$\mathbb{Z}_{3}$ sectors at $\mu=0$ as a function of $T$.
The top and bottom panels of Fig. 2 show $\sigma/\sigma_{0}$
with the trivial $\mathbb{Z}_{3}$ sector $\phi=0$ and that with
$\phi=2\pi/3$ where $\sigma_{0}$ is $\sigma$ at $T=\mu=0$,
respectively.
It is natural to assume that the entanglement vertex is decreased with
increasing $T$ and thus the sign of $\alpha$ is negative
because of properties of the running coupling in QCD.
When $\alpha$ is small in the bottom panel, the chiral condensate
increases above $T_{\mathrm{D}}$, but it decreases in the large-$\alpha$
case.
There is the drastic change in the $T$-dependence of $\sigma$ and thus
we can determine the parameters in the entanglement
vertex by using precise quenched lattice QCD data.
At least in Ref. Chandrasekharan and Christ (1996); Chandrasekharan and Huang (1996),
the chiral condensate with nontrivial $\mathbb{Z}_{3}$ sectors are
decreased and thus $\alpha$ should be considerably large.
To conclude the detailed value of $\alpha$, we need more precise
quenched lattice QCD data.
Other promising quantity to remove the model ambiguities is the dual
quark condensate Bilgici et al. (2008, 2010); Bilgici .
The dual quark condensate is the order-parameter
of the $\mathbb{Z}_{N_{\mathrm{c}}}$ symmetry breaking as same
as the Polyakov-loop.
This quantity has been calculated by using the lattice QCD
simulation Bilgici et al. (2008, 2010); Bilgici ,
the Schwinger-Dyson equations Fischer (2009),
the effective model of QCD Kashiwa et al. (2009),
the functional renormalization group equations Braun et al. (2011)
and so on.
The dual quark condensate is defined as
$$\displaystyle\Sigma^{(n)}_{\sigma}$$
$$\displaystyle=-\int\frac{d\varphi}{2\pi}e^{-in\varphi}\sigma_{\varphi},$$
(16)
where $\varphi$ specifies the boundary condition of the temporal
direction for quarks, $n$ means the winding number for the temporal
direction and
$\sigma$ is the $\varphi$-dependent chiral condensate.
Usually, we take $n=1$.
The boundary condition and the imaginary chemical potential have
direct relation, $\theta=\varphi-\pi$.
In the system with dynamical quarks, there are some problems in the dual
quark condensate Benič (2013); Marquez et al. (2015); Zhang and Miao (2016), but
we can expect that the quenched calculations where the dual quark
condensate is well defined can provide the important
information for the effective model construction.
At present, quenched lattice QCD data for the dual quark condensate
are limited and thus we do not quantitatively compare our results with
the lattice data.
Figure 3 shows the $T$-dependence of $\Sigma_{\sigma}^{(1)}$
at $\mu=0$ with different values of $\alpha$.
We can find the visible $\alpha$-dependence in $\Sigma_{\sigma}^{(1)}$.
At least in quenched lattice QCD data we can
access Bilgici et al. (2008),
$\Sigma_{\sigma}^{(1)}$ increases with increasing $T$ and thus extremely
strong $\alpha$ may be excluded, but we need more quenched lattice QCD
data to conclude it.
The critical value ($\alpha_{c}$) which leads to the flat behavior of
$\Sigma_{\sigma}^{(1)}$ above $T_{\mathrm{D}}$ exists in the range
$-0.7<\alpha_{c}<-0.6$.
Finally, we propose a new quantity which can describe the
confinement-deconfinement transition based on the dual quark condensate.
The quantity is defined as
$$\displaystyle\Sigma^{(n)}_{\cal O}$$
$$\displaystyle=\int\frac{d\varphi}{2\pi}e^{-in\varphi}{\cal O}(\varphi),$$
(17)
where ${\cal O}$ is the $\theta$-even quantity.
If we consider that the dual quark condensate measures how
information of $\mathbb{Z}_{3}$ images are missed in the calculation of
Eq. (16) when the gauge configuration is fixed to the trivial
$\mathbb{Z}_{3}$ sector, there is no need to use $\sigma$ as the
integrand of Eq. (16).
It is related with the study based on the nontrivial free energy
degeneracy in Ref. Kashiwa and Ohnishi (2015, 2016).
Figure 4 shows $T$-dependence of the quantity (17)
with ${\cal O}=P$ and $n=1$ where $P$ is the pressure.
We can see that this quantity also describes the
confinement-deconfinement transition.
This behavior is, of course, trivial from the viewpoint of the RW
periodicity.
It is interesting to calculate it in the system with dynamical quarks
and some theories where we cannot easily calculate the chiral condensate.
V Summary
In this paper, we have discussed the extension of the effective model
such as the Polyakov-loop extended Nambu–Jona-Lasinio model.
Firstly, implications of QCD properties in the imaginary chemical
potential region to the effective model construction has been discussed.
Then the importance of the entanglement vertex Sakai et al. (2010)
has been explained from the derivation of the effective four-fermi
interaction from QCD partition function with some approximations and
ansatz.
Secondly, we have shown that the general form of the entanglement vertex
which should be needed to reproduce the lattice QCD prediction at finite
$\mu_{\mathrm{I}}$ D’Elia and Sanfilippo (2009); Bonati et al. (2011).
The general form is refer to the structure of the gluon one-loop
effective potential and the Bose distribution function and then it
naturally leads to the $\mathbb{Z}_{3}$ symmetric form.
Finally, the chiral condensate with different $\mathbb{Z}_{3}$ sectors
and the dual quark condensate with $n=1$
have been calculated by using the PNJL model with the entanglement
vertex.
The entanglement vertex has one parameter $\alpha$ when we assume
$\gamma=1$ at least in the next leading-order of $e^{-\beta E}$.
When $\alpha$ is mall, the chiral condensate with the non-trivial
$\mathbb{Z}_{3}$ sector increases even above $T_{\mathrm{D}}$, but it
decreases in the large-$\alpha$ case.
This result suggests that we can well remove the ambiguity of the
parameters by using the $T$-dependence of the chiral condensate
with the different $\mathbb{Z}_{3}$ sectors.
Also, we can find visible $\alpha$-dependence in the dual quark
condensate and thus this quantity should be the promising quantity to
remove the ambiguity of the parameters.
From qualitative behavior of quenched lattice QCD
data Chandrasekharan and Christ (1996); Chandrasekharan and Huang (1996); Bilgici et al. (2008),
we may exclude extremely strong and weak $\alpha$ values, but we need more
quenched lattice QCD data to conclude it.
In addition to above results, we propose new quantity which can
describe the confinement-deconfinement transition based on the dual
quark condensate.
Acknowledgements.
K.K. thanks A. Ohnishi and H. Kouno for useful comments.
K.K. is supported by Grants-in-Aid for Japan Society for the Promotion
of Science (JSPS) fellows No.26-1717.
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IDCS J1426.5+3508: Weak Lensing Analysis of a Massive Galaxy Cluster at $z=1.75$
Wenli Mo11affiliation: Department of Astronomy, University of Florida, Bryant Space Science Center, Gainesville, FL 32611, USA ,
Anthony Gonzalez11affiliation: Department of Astronomy, University of Florida, Bryant Space Science Center, Gainesville, FL 32611, USA ,
M. James Jee22affiliation: Department of Astronomy and Center for Galaxy Evolution Research, Yonsei University, 50 Yonsei-ro, Seoul 03722, Korea ,
Richard Massey33affiliation: Institute for Computational Cosmology, Durham University, South Road, Durham DH1 3LE, UK ,
Jason Rhodes44affiliation: Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA 55affiliation: California Institute of Technology, Pasadena, CA 91125, USA ,
Mark Brodwin66affiliation: Department of Physics and Astronomy, University of Missouri, 5110 Rockhill Road, Kansas City, MO 64110, USA ,
Peter Eisenhardt44affiliation: Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA ,
Daniel P. Marrone77affiliation: Steward Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721, USA ,
S. A. Stanford88affiliation: Department of Physics, University of California, Davis, One Shields Avenue, Davis, CA 95616, USA ,
and
Gregory R. Zeimann99affiliation: Department of Astronomy & Astrophysics, The Pennsylvania State University, University Park, PA 16802, USA 1010affiliation: Institute for Gravitation and the Cosmos, The Pennsylvania State University, University Park, PA 16802, USA 1111affiliation: Department of Astronomy, University of Texas, Austin, TX 78712, USA
Abstract
We present a weak lensing study of the galaxy cluster IDCS J1426.5+3508 at $z=1.75$, which is the highest redshift strong lensing cluster known and the most distant cluster for which a weak lensing analysis has been undertaken. Using F160W, F814W, and F606W observations with the Hubble Space Telescope, we detect tangential shear at $2\sigma$ significance. Fitting a Navarro-Frenk-White mass profile to the shear with a theoretical median mass-concentration relation, we derive a mass $M_{200,\mathrm{crit}}=2.3^{+2.1}_{-1.4}\times 10^{14}$ M${}_{\odot}$. This mass is consistent with previous mass estimates from the Sunyaev-Zel’dovich (SZ) effect, X-ray, and strong lensing. The cluster lies on the local SZ-weak lensing mass scaling relation observed at low redshift, indicative of minimal evolution in this relation.
Subject headings: cosmology: observations – dark matter – galaxies: clusters: individual (IDCS J1426.5+3508) – gravitational lensing: weak.
1. Introduction
Massive, high-redshift galaxy clusters, though rare, provide valuable information about cosmological parameters, structure formation, and galaxy evolution. The redshift regime $z>1.5$ represents a critical era during which significant star formation occurs in cluster galaxies (e.g, Tran et al., 2010; Brodwin et al., 2013; Bayliss et al., 2014; Webb et al., 2015b; Alberts et al., 2014). Only a handful of clusters at redshift $z>1.5$ have been confirmed to date (e.g, Newman et al., 2014; Papovich et al., 2010; Zeimann et al., 2012; Muzzin et al., 2013; Tozzi et al., 2015; Webb et al., 2015a), thanks in part to large X-ray, infrared, and Sunyaev-Zel’dovich (SZ) effect surveys (e.g., Eisenhardt et al., 2008; Fassbender et al., 2011; Wylezalek et al., 2014; Bleem et al., 2015).
In this paper we focus upon the galaxy cluster IDCS J1426.5+3508. This cluster was first discovered in the IRAC Distant Cluster Survey (IDCS). Follow-up with the Hubble Space Telescope (HST) Wide Field Camera-3 (WFC3) grism and the Low Resolution Imaging Spectrometer (LRIS, Oke et al., 1995) on the W. M. Keck Observatory spectroscopically confirms a redshift of $z=1.75$ (Stanford et al., 2012). With $100$ ks of Chandra data, Brodwin et al. (2015) obtain mass estimates based upon the X-ray temperature, gas mass, and the product of core-excised X-ray temperature and gas mass of $M_{500,T_{X}}=3.3^{+5.7}_{-1.2}\times 10^{14}$ M${}_{\odot}$, $M_{500,M_{g}}=2.3^{+0.7}_{-0.5}\times 10^{14}$ M${}_{\odot}$, and $M_{500,\mathrm{Y}_{X}}=2.6^{+1.5}_{-0.5}\times 10^{14}$ M${}_{\odot}$, respectively. In comparison, SZ observations with the Combined Array for Research in Millimeter-wave Astronomy (CARMA) indicate $M_{500,\mathrm{SZ}}=2.6\pm 0.7\times 10^{14}$ M${}_{\odot}$ (Brodwin et al., 2012), implying $M_{200}=4.1\pm 1.1\times 10^{14}$ M${}_{\odot}$ with a Duffy et al. (2008) mass-concentration relation. These mass estimates establish IDCS J1426.5+3508 as the most massive galaxy cluster confirmed at redshift $z>1.5$. For comparison, another high-redshift cluster, XDCP J0044.0-2033 ($z=1.58$), has mass derived from the Vikhlinin et al. (2009) $Y_{X}-$M scaling relation $M_{500,Y_{X}}=2.2^{+0.5}_{-0.4}\times 10^{14}$ M${}_{\odot}$ (Tozzi et al., 2015).
Moreover, IDCS J1426.5+3508 is also the most distant strong lensing galaxy cluster. A giant gravitationally lensed arc associated with the cluster was discovered in HST/WFC3 and Advanced Camera for Surveys (ACS) imaging. An initial redshift estimate based on broadband photometry yielded $z\sim 3-6$ for the arc and a lower limit of $M_{200}>2.8\times 10^{14}$ M${}_{\odot}$ for the cluster via a strong-lensing analysis (Gonzalez et al., 2012). For $\Lambda$CDM, Gonzalez et al. (2012) calculate that the expected number of giant arcs at this redshift and brightness is vanishingly small, highlighting the long-standing arc statistics problem (see Meneghetti et al., 2013, for a recent review). One explanation suggested to explain the observed arc in IDCS J1426.5+3508 is that the cluster might have a substantially more concentrated density profile—and hence an enhanced lensing cross-section—than predicted for a cluster at this epoch.
Gravitational weak lensing determines the cluster mass via the distortion in shape of background galaxies due to the gravitational potential of the cluster. Unlike mass estimates derived from X-ray or SZ observations, weak lensing measures the mass distribution independent of the dynamical state or hydrostatic equilibrium of the cluster. Thus, weak lensing is also a powerful tool for calibrating X-ray or SZ mass estimates (e.g., Marrone et al., 2012; Hoekstra et al., 2012; Jee et al., 2011). Furthermore, the mass and concentration of the cluster can in principle be determined without a prescribed mass-concentration relationship.
However, weak lensing is observationally challenging, particularly for distant clusters. Because the lensing cluster is at high redshift, the number of lensed background galaxies is small, decreasing the lensing signal. Also, the extent of the distortions to be measured is on the scale of the observational point spread function (PSF; e.g., Rhodes et al., 2007; Jee et al., 2007). Deep, space-based observations with well-understood PSFs are therefore required to obtain high fidelity measurements of the weak lensing distortion.
In this work, we demonstrate the feasibility of weak lensing analyses at $z=1.75$. IDCS J1426.5+3508 is the highest redshift cluster to have joint X-ray, SZ, and weak lensing observations, and thus provides an opportunity to compare whether the scaling relations between these mass estimators, derived at low redshift, remain valid at this epoch. In Section 2 we describe the observations and data analysis, including the selection of source galaxies. We outline the measurement of galaxy shapes and the correction for PSF distortion in those measurements in Section 3. The weak lensing shear profile is presented in Section 4. We discuss the mass estimation and SZ-weak lensing scaling relation in Sections 5 and 6, respectively, and summarize our findings in Section 7.
Throughout the paper, we adopt the nine-year Wilkinson Microwave Anisotropy Probe (WMAP9) cosmological parameters of $\Omega_{\mathrm{M}}=0.287$, $\Omega_{\Lambda}=0.713$, and $H_{0}=69.32$ km s${}^{-1}$ Mpc${}^{-1}$ (Hinshaw et al., 2013) and define $h=H_{0}/100$. All magnitudes are in the AB system (Oke & Gunn, 1983). We take the F606W image for our shape measurement and the center of the cluster to be the location of the brightest cluster galaxy (BCG), as identified by Stanford et al. (2012). Unless otherwise stated, we report masses as overdensities relative to the critical density.
2. Observations
For our analysis, we use HST data from Cycle 20 taken on 2012 December 19, and 2013 April 17-19, with the ACS F606W (V-band), F814W (I-band), and WFC3 F160W (H-band) filters for a total of 21760, 8108, and 4947 seconds, respectively. We also use Cycle 17 observations from 2010 July 8 and 2010 November 7 with the F814W and F160W filters for 4513 and 2612 seconds, respectively. Since the F160W data are mosaicked, the effective exposure time for the central region is 7559 seconds while the minimum exposure time for the outer regions is 1212 seconds. The cluster is also among the targets in HST GO-program 13677 (PI Perlmutter) which will add observations in WFC3 F140W, F105W, and F814W filters for future studies of this cluster.
2.1. HST ACS Data Reduction
We performed basic processing of the ACS data with the CALACS pipeline. Our only modification to the standard procedure was to employ the charge transfer inefficiency (CTI) correction method developed by Massey et al. (2010, 2014) to correct for CTI degradation prior to running CALACS. CTI extends the shapes of galaxies and, if uncorrected, can imitate the effects of weak lensing. Astrometric shifts, geometric distortion correction, sky subtraction, cosmic ray removal, and final image stacking were handled by the standard TweakReg and Astrodrizzle packages. Images were drizzled to a $0.03\arcsec$ pixel scale with a Gaussian kernel and pixel fraction of 1.0. Rhodes et al. (2007) showed that changing the pixel scale from the ACS native scale of $0.05\arcsec$ to $0.03\arcsec$ optimizes our ability to correct for PSF effects while the results are largely insensitive to the choice of pixel fraction.
2.2. Source Galaxy Catalog
Starting with F606W as our detection image, we extracted a catalog of 4771 sources with SExtractor (Bertin & Arnouts, 1996) with a criteria of $5$ connecting pixels brighter than a sky rms of $2$. We then obtained matched photometry for these sources in F814W and F160W by running SExtractor in dual image mode. Figure 1 is the resulting color-magnitude diagram for the two bluest filters.
We identify the foreground and cluster galaxy contamination in our catalog via color selection. One benefit of employing an F606W-selected catalog is that most cluster red sequence galaxies are non-detections. Listed below are the sequence of color and size cuts applied to distinguish among foreground galaxies, cluster members, and background galaxies. We detect 3050 objects with F606W$<28.0$, the approximate $10\sigma$ depth of our observations. We first reject $295$ sources with either F606W$<24.0$ or FWHM$>0.9\arcsec$ to exclude bright and large objects that are probable foreground galaxies. Based on a Bruzual & Charlot (2003) model of a star-forming galaxy with $z_{\mathrm{form}}=6$, we next implement a color cut of F606W-F814W $>0.5$ to eliminate $786$ additional foreground galaxies in our sample. To reject cluster members on the red sequence, we rule out $20$ objects with F814W-F160W $>3.0$ (Stanford et al., 2012). We also cut $16$ objects with signal to noise $S/N<10$.
In addition, we exclude $922$ objects based on shape measurements constraints, further discussed in Section 3. The final catalog for the shear analysis consists of 1011 sources, equivalent to $89$ arcmin${}^{-2}$. As a cross-check, we construct a radial density profile for the remaining galaxies centered on the BCG. There is no central excess evident in the radial profile, indicating that residual contamination from blue cluster galaxies is minimal.
2.3. Redshift Distribution
Given the redshift of the cluster, compiling a clean catalog of background sources is not possible with the available photometric data. However, we can control the foreground contamination in our source catalog statistically. We estimate the redshift distribution of our source catalog using the Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey Ultra Deep Survey (CANDELS-UDS) multiwavelength catalog (Santini et al., 2014) which provides infrared-detected sources to an F160W magnitude limit of $27.45$ in a range of filters. Employing the selection criteria described in Section 2.2, we found 3512 sources, 46% of which were contaminants with $z\leq 1.75$. The distribution of CANDELS-UDS redshift for sources included and excluded by our selection criteria is illustrated in Figure 2.
We estimate the effective redshift of the source population by calculating the lensing efficiency, the mean of the ratio of angular diameter distances,
$$\langle\beta\rangle=\langle\mathrm{max}(0,{D_{ls}/D_{s}})\rangle,$$
(1)
where $D_{ls}$ and $D_{s}$ are the angular diameter distances between the lens and the source and the observer and the source, respectively. We set a floor of $\beta=0$ to account for foreground sources with $z\leq 1.75$. For the subset of CANDELS-UDS sources matching our selection criteria, we compute $\langle\beta\rangle=0.086$, corresponding to an effective redshift $z_{\mathrm{eff}}=2.05$, and width $\langle\beta^{2}\rangle=0.017$.
Despite CANDELS-UDS being the deepest catalog with data in our set of filters, 12% of the galaxies in our background source catalog are fainter in F160W than the CANDELS-UDS limiting magnitude. To investigate the effects of the mismatch in detection filter threshold between our data and the CANDELS-UDS catalog, we randomly select 114 objects from a sample of $1455$ CANDELS-UDS objects near the survey limiting magnitude ($27.0<\mathrm{F160W}<27.45$) and add their redshifts to the 3512 CANDELS-UDS sources matching our selection criteria. We choose these objects because they are most likely to mimic the redshift distribution of the sources with F160W$>27.45$ missing from the CANDELS-UDS catalog. We then repeat our calculation of $\beta$ using Equation 1 with the additional 12% of galaxies. Including the randomly selected redshift sample increases $\beta$ by $3.5\%$. This source of uncertainty for $\beta$ is a subdominant contributor to the total mass error budget.
3. Shape Measurement
3.1. RRG Method
We measure galaxy distortions by applying the RRG method developed by Rhodes et al. (2000), which calculates the second- and fourth-order Gaussian-weighted moments of each galaxy to determine ellipticity and correct for convolutions with the PSF. The weak lensing shear is then derived from the average ellipticity of the sources. We choose the weight function size of each object to be $w=\max(2\sqrt{ab},6)$ where $a$ and $b$ are the semi-major and -minor axes in pixels calculated from SExtractor.
For each object, the shear $\gamma$ is related to the ellipticity $e$ of that object
$$\gamma=C\frac{e}{G},$$
(2)
where $G=1.35$ is the shear susceptibility calculated from our data and $C=0.86^{-1}$ is a calibration factor determined from the analysis of simulated images containing a known shear (Leauthaud et al., 2007). The reduced shear is then
$$g=\frac{\gamma}{(1-\kappa)}$$
(3)
where $\kappa$ is the convergence. We further incorporate the width of the redshift distribution (Seitz & Schneider, 1997). The observed shear $g^{\prime}$ is then
$$g^{\prime}=(1+(\langle\beta^{2}\rangle/\langle\beta\rangle^{2}-1)\kappa)g=(1+1.55\kappa)g.$$
(4)
In addition to the criteria presented in Section 2.2, we introduce more selection criteria to our background source catalog due to shape measurement limitations. The RRG method cannot correct for the distortion on objects smaller than $\sim 1.5$ times the RRG weight function width of $0.18\arcsec$. Therefore, we reject $730$ galaxies with a SExtractor FWHM $<0.27\arcsec$ for which we were not able to apply a PSF correction. Also, we exclude $192$ objects that encountered centroiding errors during shape measurement or for which the PSF correction did not converge. Therefore, our final catalog for weak lensing analysis includes 1011 sources.
3.2. PSF Correction
The ACS PSF is known to be both temporally and spatially variable. Thermal breathing of the telescope, dependent on the 90-minute orbit of HST, and a slow deviation in the focus on the timescale of a few weeks further complicate the PSF. In the weak lensing regime, an uncorrected PSF distortion can drown out the signal, so precise characterization is critical.
We use the publicly available TinyTim software111http://tinytim.stsci.edu/ (Krist & Hook, 1997; Krist et al., 2011) to theoretically model the PSF distortion across our image. TinyTim simulates the PSF given a focus value and takes into account the instrument, chip number, chip position, filter, and spectrum of the object. Schrabback et al. (2010) notes additional variations in the PSF dependent on relative sun angle, but finds that $97\%$ of variations can be described solely by the focus parameter.
Our F606W image is a combination of 16 separate frames. We replicate the PSF in each frame individually by specifying a focus value determined by the HST focus model222http://focustool.stsci.edu/, applying the average focus during the exposure period of each individual frame.
Adopting a modified version of TinyTim333http://community.dur.ac.uk/r.j.massey/acs/PSF, we generate a distortion-corrected grid of PSFs across the entire image with pixel scale of $0.03\arcsec$. We create the PSF model for the full stacked F606W image by combining the 16 individual PSFs weighted by the exposure time. We then interpolate the second- and fourth-order moments of the PSF model to the arbitrary positions of galaxies throughout the field of view via a third-order polynomial in $x$ and $y$.
4. Weak Lensing Analysis
The weak lensing signal is expressed as the tangential component of the shear $g_{+}$. The cross component $g_{\times}$, defined as $g_{+}$ after the object is rotated by $45^{\circ}$ and represents the null statistic and noise of our measurements and should be consistent with zero in the absence of systematic errors.
We determine the shear as a function of cluster radius by radially binning our catalog objects and calculating the average $g^{\prime}_{+}$ and $g^{\prime}_{\times}$ of all objects in each bin. The cluster center is at the location of the BCG (RA=$14^{\mathrm{h}}26^{\mathrm{m}}32^{\mathrm{s}}.95$ and Dec=$+35^{\circ}08^{\prime}23~{}6\arcsec$). The shear profile is plotted in Figure 3 where the error bars shown for the shear measurements are the statistical uncertainty of the mean shear per bin. By adding in quadrature the ratio of mean shear and uncertainty per bin for the first six bins, we calculate a significance of $2.0$ for the detection. The Pearson’s correlation coefficient between the tangential and cross shear components is small ($r=-0.12$), indicating no correlation between the two components
As a test of robustness of this detection, we also employ an alternative shape measurement from the Jee et al. (2009, hereafter J09) and RRG methods. Using the same galaxy selection, we find that the shapes obtained using the J09 and RRG methods yield consistent results: the typical absolute difference between the mean measured values in each bin is within $0.8\sigma$ for the tangential shear and $1.3\sigma$ for the cross component.
The J09 approach is designed to extend to faint magnitudes. With this method, we also perform a 2-D mass reconstruction using a more relaxed set of selection criteria. Specifically, we allow a wider range in magnitudes ($22<$F606W$<29$ and $22<$F160W$<29$), a relaxed red sequence selection (F814W-F160W$<1.8$), and a decreased size selection (F606W half-light radius $r_{h}>1.2$). The centroid of the resulting 2-D mass reconstruction is close to the location of the BCG, providing further evidence that the shear is the result of the cluster weak lensing. The shear contours are overplotted on the HST image in Figure 4. Finally, we note that the tangential shear profile using this different selection remains consistent with Figure 3 but with smaller statistical uncertainties. We continue the analysis with our original shape measurement.
5. Mass Estimation
The tangential shear is related to the density profile of the cluster assuming a Navarro-Frenk-White (NFW) profile (Navarro et al., 1997) described by the virial radius $r_{200}$, defined as the radius inside which the mass density of the halo is equal to $200$ times the critical density, and the concentration $c=r_{200}/r_{s}$, where $r_{s}$ is the scale radius. For the radial dependence of the shear in an NFW model, we refer the reader to Equations $14-17$ of Wright & Brainerd (2000).
To fit the shear profile, we utilize the maximum likelihood estimation discussed in Schneider et al. (2000). The log-likelihood function is defined as
$$\ell_{\gamma}=\sum_{i=1}^{N\gamma}{\left[\frac{|e_{t,i}-g^{\prime}_{+}(\theta_{i})|^{2}}{\sigma^{2}[g^{\prime}_{+}(\theta_{i})]}\right]+2\ln\sigma[g^{\prime}_{+}(\theta_{i})]},$$
(5)
where $N_{\gamma}$ is the number of galaxies with measured ellipticity, $e_{t,i}$ is the tangential ellipticity component, $\theta_{i}$ is the position of the $i$-th galaxy, and $g^{\prime}_{+}(\theta_{i})$ is the observed tangential shear at $\theta_{i}$. The dispersion of observed ellipticities approximated as $\sigma[g^{\prime}_{+}(\theta_{i})]\approx\sigma_{e}(1-|g^{\prime}_{+}(\theta_{i})|^{2})$, the same as that defined in Equation 14 of Schneider et al. (2000), where $\sigma_{e}=0.3$ is the intrinsic ellipticity dispersion.
We lack the necessary signal to fit the mass and concentration simultaneously as free parameters. Thus, we derive the cluster mass with the concentration defined by the mass-concentration relation from Duffy et al. (2008),
$$c=5.71\Bigg{(}\frac{M_{200,\mathrm{crit}}}{2\times 10^{12}h^{-1}M_{\odot}}\Bigg{)}^{-0.084}(1+z)^{-0.47},$$
(6)
where the mass within $r_{200}$ is $M_{200,\mathrm{crit}}=200\rho_{\mathrm{crit}}(\frac{4}{3}\pi{r_{200}^{3}})$. In this case, $M_{200,\mathrm{crit}}$ is the only free parameter in the likelihood fit, and the resultant value can be applied to directly compare IDCS J1426.5+3508 with the SZ-lensing relation defined by low redshift clusters. All objects located at radii $15\arcsec<{r}<110\arcsec$ ($130<r<950$ kpc) are included in our fit (802 sources). We exclude the strong lensing region, defined by the radius of the known strongly-lensed arc (Gonzalez et al., 2012), and extend to a radius that best constrains the $1\sigma$ mass confidence interval. We derive $M_{200,\mathrm{crit}}=2.3^{+2.1}_{-1.4}\times 10^{14}$ M${}_{\odot}$ ($M_{500,\mathrm{crit}}=1.4^{+1.3}_{-0.9}\times 10^{14}~{}M_{\odot}$) which corresponds to $r_{200}=0.68^{+0.16}_{-0.18}$ Mpc ($r_{500}=0.42^{+0.10}_{-0.11}$ Mpc). Changing the center to that of the X-ray observations (Brodwin et al., 2015) yielded a mass within $1\sigma$ of the mass derived with BCG centering. The NFW profile described by our best-fit parameters is plotted in Figure 3.
IDCS J1426.5+3508 is the only galaxy cluster in this regime to have joint weak lensing, strong lensing, SZ, and X-ray observations. Brodwin et al. (2015) find good agreement between the mass estimations from X-ray, SZ, and strong lensing. Converting from $M_{500,\mathrm{crit}}$ to $M_{200,\mathrm{crit}}$ via the Duffy et al. (2008) relation, Brodwin et al. (2012) calculate an SZ mass $M_{200,\mathrm{crit}}^{\mathrm{SZ}}=4.1\pm 1.1\times 10^{14}$ M${}_{\odot}$, Brodwin et al. (2015) estimate a mass from the X-ray gas mass of $M_{200}^{\mathrm{M_{gas}}}=3.8^{+1.1}_{-0.8}\times 10^{14}$ M${}_{\odot}$, while Gonzalez et al. (2012) projected a lower limit of $M_{200}>2.8^{+1.0}_{-0.4}\times 10^{14}$ M${}_{\odot}$ from observations of the giant arc. Our weak lensing mass of $M_{200,\mathrm{crit}}=2.3^{+2.1}_{-1.4}\times 10^{14}$ M${}_{\odot}$ falls at the lower end of these mass estimates but is consistent.
6. SZ-Weak Lensing Scaling Relation
The SZ Compton parameter $Y$ scales with mass as $Y\propto{M^{5/3}/({D_{A}^{2}E(z)^{-2/3}})}$ where $D_{A}$ is the angular diameter distance and $E(z)$ is the evolution of the Hubble parameter. We combine our weak lensing-derived mass estimate with the spherically averaged dimensionless Comptonization parameter ($Y_{\mathrm{sph},500}=7.9\pm 3.2\times 10^{-12}$) from Brodwin et al. (2012) to compare IDCS J1426.5+3508 with the mass-$Y_{\mathrm{sph}}$ scaling relation derived by Marrone et al. (2012) using 18 Local Cluster Substructure Survey (LoCuSS) clusters in the range $z=[0.164,0.290]$. Figure 5 shows our results alongside the Marrone et al. (2012) sample and scaling relation. In this figure, we match the $\Lambda$CDM cosmology assumed by Marrone et al. (2012, $\Omega_{\mathrm{M}}=0.27$, $\Omega_{\Lambda}=0.73$, $H_{0}=73$ km s${}^{-1}$ Mpc${}^{-1}$). We also include the ACT-CL J0022.2-0036 at $z=0.81$ (Miyatake et al., 2013; Reese et al., 2012), a higher redshift cluster with comparable SZ and weak lensing observations, for comparison.
Our results for IDCS J1426.5+3508 are statistically consistent with the low redshift data, indicating minimal redshift evolution in the Marrone et al. (2012) scaling relation. To quantify the redshift evolution, we fit a regression of the form
$$\Big{(}\frac{1+z}{1+z_{\mathrm{M12}}}\Big{)}^{\alpha}=\Big{(}\frac{M_{500,\mathrm{WL}}}{10^{A}10^{14}M_{\odot}}\Big{)}\Big{(}\frac{10^{-5}\mathrm{Mpc}^{2}}{Y_{\mathrm{sph},500}D_{A}^{2}E(z)^{-2/3}}\Big{)}^{B}$$
(7)
where $z_{\mathrm{M12}}=0.24$ is the average redshift of the LoCuSS sample and $A=0.367$ and $B=0.44$ are regression coefficients from Marrone et al. (2012). Including the data for the $18$ LoCuSS clusters individually, ACT-CL J0022.2-0036, and IDCS J1426.5+3508, a least-squares fit yields $\alpha=-0.1\pm 1.0$, within $1\sigma$ of zero. Thus, we find no significant evidence for evolution in the Marrone et al. (2012) scaling relation.
7. Conclusions
We have presented the weak lensing analysis of IDCS J1426.5+3508. At $z=1.75$, this is the highest redshift galaxy cluster to be studied through weak lensing. We detect a tangential shear signal at a $2\sigma$-level significance. Assuming an NFW profile and a Duffy et al. (2008) mass-concentration relation, we fit for the cluster mass with a maximum-likelihood algorithm. Our mass estimate $M_{200,\mathrm{crit}}=2.3^{+2.1}_{-1.4}\times 10^{14}$ M${}_{\odot}$ is in agreement with estimates from X-ray, SZ, and strong lensing data. We also find that the SZ and weak lensing mass estimates agree well with the local scaling relation of Marrone et al. (2012), with negligible evolution in the relation.
Gonzalez et al. (2012) initially discovered the strong-lensed arc behind IDCS J1426.5+3508. If one makes standard assumptions for the galaxy cluster mass function, cosmological model, source galaxy redshift distribution, and cluster lensing cross section, then such an arc should not exist across the entire sky. The cluster lensing cross section increases with concentration, so one plausible explanation for this arc would be if the projected concentration of the cluster mass profile dramatically exceeds the typical value predicted for a cluster of this mass at this epoch. We are unable to determine a concentration as a free parameter for this cluster with the current data. However, a higher signal-to-noise weak lensing map and additional passbands to determine photometric redshifts would enable simultaneous fitting of the mass and concentration, improving Figure 3 and helping resolve the origin of the strong arc in this cluster.
Acknowledgments
The authors thank the anonymous referee and Daniel Stern for their insightful suggestions and Audrey Galametz for her help with the CANDELS-UDS data. Support for HST GO-program 11663, 12203, and 12994 was provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. We also acknowledge funding received from the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1315138 (W.M.), NRF of Korea to CGER (M.J.J.), and Royal Society University Research Fellowship (R.M.). J.R. is supported by JPL, which is run by Caltech under a contract for NASA.
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Theory of Non-Interacting Fermions and Bosons in the Canonical Ensemble
Hatem Barghathi
Department of Physics, University of Vermont, Burlington, VT 05405, USA
Department of Physics, Missouri University of Science and Technology, Rolla, MO 65409, USA
Jiangyong Yu
Department of Physics, University of Vermont, Burlington, VT 05405, USA
Adrian Del Maestro
Department of Physics, University of Vermont, Burlington, VT 05405, USA
Abstract
We present a self-contained theory for the exact calculation of particle number counting statistics of non-interacting indistinguishable particles in the canonical ensemble. This general framework introduces the concept of auxiliary partition functions, and represents a unification of previous distinct approaches with many known results appearing as direct consequences of the developed mathematical structure. In addition, we introduce a general decomposition of the correlations between occupation numbers in terms of the occupation numbers of individual energy levels, that is valid for both non-degenerate and degenerate spectra. To demonstrate the applicability of the theory in the presence of degeneracy, we compute energy level correlations up to fourth order in a bosonic ring in the presence of a magnetic field.
I Introduction
In quantum statistical physics, the analysis of a fixed number $N$ of indistinguishable particles is difficult, even in the non-interacting limit. In such a canonical ensemble, the constraint on the total number of particles gives rise to correlations between the occupation probability and the corresponding occupation numbers of the available energy levels. As a result, the canonical treatment of non-interacting fermions and bosons is in general avoided, and it is often completely absent in introductory texts addressing quantum statistical physics Kittel (1980); Landau and Lifshitz (1980); Pathria and Beale (2011). The standard approach is to relax the fixed $N$ constraint and instead describe the system using the grand canonical ensemble. While in most cases this approximation can be trusted in the large$-N$ limit, it can explicitly fail in the low-temperature regime Bedingham (2003); Mullin and Fernández (2003). The situation is even worse if the low-temperature system under study is mesoscopic, containing a relatively small number of particles. Recent advances in ultra-cold atom experiments Wenz et al. (2013); Parsons et al. (2015); Cheuk et al. (2015); Haller et al. (2015); Pegahan et al. (2019); Mukherjee et al. (2017); Hueck et al. (2018); Mukherjee et al. (2019); Onofrio (2016); Phelps et al. (2020) are making such conditions the rule rather than the exception, where the interactions between particles are often ignored, especially in the fermionic case Giorgini et al. (2008). Thus an accurate statistical representation of these systems should be canonical enforcing the existence of fixed $N$. In a different context, particle number conservation can reduce the amount of quantum information (entanglement) that can be extracted from a quantum state Horodecki et al. (2000); Bartlett and Wiseman (2003); Wiseman and Vaccaro (2003); Wiseman et al. (World Scientific, Singapore,
2004); Vaccaro et al. (2003); Schuch et al. (2004); Dunningham et al. (2005); Cramer et al. (2011); Klich and Levitov (2008) and thus the fixed $N$ constraint requires a canonical treatment. This is reflected in the calculation of the symmetry resolved entanglement which has recently been studied in a variety of physical systems Murciano et al. (2020a); Tan and Ryu (2020); Murciano et al. (2020b); Capizzi et al. (2020); Fraenkel and Goldstein (2020); Feldman and Goldstein (2019); Barghathi et al. (2019); Bonsignori et al. (2019); Barghathi et al. (2018); Kiefer-Emmanouilidis et al. (2020); Murciano et al. (2020c); Goldstein and Sela (2018).
Given its pedagogical and now practical importance, as well as the long history of the problem of non-interacting quantum particles, the last 50 years has provided a host of results, varying from general recursive relations that govern the canonical partition functions and the corresponding occupation numbers Schmidt (1989); Borrmann and Franke (1993); Borrmann et al. (1999); Pratt (2000); Mullin and Fernández (2003); Weiss and Wilkens (1997); Arnaud et al. (1999); Schönhammer (2017); Giraud et al. (2018); Tsutsui and Kita (2016); Zhou and Dai (2018), to approximate Denton et al. (1971, 1973); Schönhammer and Meden (1996); Bedingham (2003); Svidzinsky et al. (2018); Jha and Hirata (2020) and exact results for some special cases Chase et al. (1999); Schönhammer (2000); Kocharovsky and Kocharovsky (2010); Wang and Ma (2009); Akkelin and Sinyukov (2016); Magnus et al. (2017); Grabsch et al. (2018); Kruk et al. (2020). More recently, for the case of non-degenerate energy spectra, the exact decomposition of higher-order occupation number correlations in terms of the occupation numbers of individual bosonic and fermionic energy levels have been reported Schönhammer (2017); Giraud et al. (2018).
In this paper, we present a unified framework for the calculation of physical observables in the canonical ensemble for $N$ fermions or bosons that are applicable to general non-interacting Hamiltonians that may contain degenerate energy levels. We present an analysis of the mathematical structure of bosonic and fermionic canonical partition functions in the non-interacting limit that leads to a set of recursion relations for exactly calculating the energy level occupation probabilities and average occupation numbers. Using argument from linear algebra, we show how higher-order correlations between the occupation numbers can be factorized, allowing them to be obtained from the knowledge of the occupation numbers of the corresponding energy levels, a canonical generalization of Wick’s theorem Wick (1950). The key observation yielding simplification of calculations in the canonical ensemble is that occupation probabilities and occupation numbers can be expressed via auxiliary partition functions (APFs) – canonical fermionic or bosonic partition functions that correspond to a set of energy levels that are obtained from the full spectrum of the targeted system by making a subset of the energy levels degenerate (increasing its degeneracy if its already degenerate) or alternatively by excluding it from the spectrum. Results obtained via auxiliary partition functions are validated by demonstrating that a number of previously reported formulas can be naturally recovered in a straightforward fashion within this framework.
In a non-interacting system, observables such as the average energy and magnetization can be obtained solely from the knowledge of average occupation numbers, however, the calculation of the corresponding statistical fluctuations in such quantities, i.e., specific heat and magnetic susceptibility, requires knowledge of the fluctuations in occupation numbers and (in the canonical ensemble) correlations between them. Therefore, the factorization of correlations between occupation numbers in terms of the average occupation numbers of individual energy levels provides a simplified approach to calculate quantities such as specific heat and magnetic susceptibility of a given system. Here, we highlight a method to compute such correlations by considering a degenerate system of a finite periodic chain of non-interacting bosons that is influenced by an external uniform magnetic field.
The auxiliary partition function approach presented here provides a set of tools that can be used to analyze experimental data in low-density atomic gases where the number of particles is fixed. The application of the physically relevant canonical ensemble can eliminate errors introduced via the grand canonical approximation (especially in the inferred temperature) and lead to a more accurate interpretation of experimental results, including an improved diagnosis of the role of weak interaction effects.
It is hoped that the relative simplicity of the mathematical approach presented in this paper may encourage the inclusion of the interesting topic of the canonical treatment of Fermi and Bose gases in college-level textbooks.
In Sec.II we present the general formalism of the theory, where we introduce APFs and write occupation probability distributions, occupation numbers, and their correlations in terms of the APFs. In the same section, we also show how some of the previously known results could be recovered in a straightforward fashion. In Sec.III, considering fermionic and bosonic systems, we derive the decomposition of level higher-order occupation numbers correlations into individual energy levels occupation numbers for non-degenerate and degenerate energy spectra, alike. To illustrate the applicability of the theory, In Sec.IV, we consider a bosonic ring in the presence of a magnetic field. We conclude in Sec.V.
II Non-interacting Indistinguishable particles in the canonical ensemble
As the thermodynamic properties of a system of non-interacting particles are
governed by the single-particle spectrum and the underlying particle
statistics, we begin by considering the general one-particle spectrum $\epsilon_{i}$, with
$i\in\mathcal{S}=\{1,2,,\dots,M\}$. For an unbounded spectrum $M\to\infty$. In the canonical ensemble defined by fixing the total particle number $N$, the canonical partition function $Z_{N}\equiv Z_{N}(\mathcal{S})$ for $N$ indistinguishable particles is defined by:
$$Z_{N}=\sum_{\boldsymbol{n}|_{{{N}}}}X\quantity(\boldsymbol{n}|_{{{N}}}),$$
(1)
where
$$X(\boldsymbol{n}|_{N})=\prod_{i\in\mathcal{S}}\mathrm{e}^{-\beta\epsilon_{i}n_%
{i}}$$
(2)
are the Boltzmann factors at inverse temperature $\beta=1/k_{\rm B}T$ and the components $\{n_{i}\}$ of the vector $\boldsymbol{n}|_{N}=(n_{1},\dots,n_{M})|_{N}$ are the occupation numbers for the corresponding energy levels satisfying $\sum_{i\in\mathcal{S}}n_{i}=N$. The summation in Eq. (1) runs over all the possible occupation vectors $\boldsymbol{n}|_{N}$ which, in addition to conserving the total number of particles $N$, obeys occupation limits for each of the energy levels $\epsilon_{i}$: $n_{i}\leq n_{i}^{\max}$, thus $N\leq N_{\max}=\sum_{i}n_{i}^{\max}$.
Consider the spectrum defined by $\mathcal{S}$ as the union of disjoint subsets
(subspectra) $\mathcal{S}^{(1)}$ and $\mathcal{S}^{(2)}=\mathcal{S}\setminus S^{(1)}$, i.e., $\mathcal{S}=\mathcal{S}^{(1)}\cup\mathcal{S}^{(2)}$. Under
this decomposition the Boltzmann factors in $\mathcal{S}$ can be factorized as:
$X\quantity(\boldsymbol{n}|_{N})=X\quantity(\boldsymbol{n}^{(1)}|_{k})X%
\quantity(\boldsymbol{n}^{(2)}|_{N-k})$,
where $\boldsymbol{n}^{(1)}|_{k}$ and $\boldsymbol{n}^{(2)}|_{N-k}$ represent
occupation vectors of $k=\sum_{i\in\mathcal{S}^{(1)}}n_{i}^{(1)}$ and $N-k$ particles in $\mathcal{S}^{(1)}$ and $\mathcal{S}^{(2)}$, respectively. Summing $X\quantity(\boldsymbol{n}|_{N})$ over all possible $\boldsymbol{n}^{(1)}|_{k}$ and $\boldsymbol{n}^{(2)}|_{N-k}$ gives:
$$\sum_{\boldsymbol{n}^{(1)}|_{k}}\sum_{\boldsymbol{n}^{(2)}|_{N-k}}X\quantity(%
\boldsymbol{n}|_{N})=Z_{k}\quantity(\mathcal{S}^{(1)})Z_{N-k}\quantity(%
\mathcal{S}^{(2)}),$$
(3)
where we have introduced the APFs:
$$\displaystyle Z_{k}\quantity(\mathcal{S}^{(1)})$$
$$\displaystyle=\sum_{\boldsymbol{n}^{(1)}|_{k}}X\quantity(\boldsymbol{n}^{(1)}|%
_{k})$$
(4)
$$\displaystyle Z_{N-k}\quantity(\mathcal{S}^{(2)})$$
$$\displaystyle=\sum_{\boldsymbol{n}^{(2)}|_{N-k}}X\quantity(\boldsymbol{n}^{(2)%
}|_{N-k})\,.$$
(5)
For Eqs. (4) and (5) to satisfy the restrictions imposed by the per-energy level maximum occupancies $\{n_{i}^{\max}\}$ and the fixed $N$, $k$ must satisfy $\max(0,N-N^{\max}_{2})\leq k\leq\min(N,N^{\max}_{1})$, where $N^{\max}_{1}$ and $N^{\max}_{2}$ are the maximum numbers of particles allowed in $\mathcal{S}^{(1)}$ and $\mathcal{S}^{(2)}$, respectively. Additionally, any vector
$\boldsymbol{n}|_{N}$, under these constraints can be decomposed into two allowed vectors $\boldsymbol{n}^{(1)}|_{k}$ and $\boldsymbol{n}^{(2)}|_{N-k}$ and vice versa. Thus the sum in Eq. (1) can be similarly decomposed as: $\sum_{\boldsymbol{n}|_{N}}\equiv\sum_{k=k_{\min}}^{k_{\max}}\sum_{\boldsymbol{%
n}^{(1)}|_{k}}\sum_{\boldsymbol{n}^{(2)}|_{N-k}}$ where $k_{\min}=\max(0,N-N^{(2)}_{\max})$ and $k_{\max}=\min(N,N^{(1)}_{\max})$ yielding the full partition function
$$Z_{N}\equiv\sum_{k=k_{\min}}^{k_{\max}}Z_{k}\quantity(\mathcal{S}^{(1)})Z_{N-k%
}\quantity(\mathcal{S}^{(2)})\,.$$
(6)
Employing the convention that $Z_{N}(\mathcal{S})=0$ whenever $N$ is negative, or when it exceeds the maximum number of particles set by $\mathcal{S}$, the limits in the above summation can be simplified to $\sum_{k=0}^{N}$. The above notation can be made more explicit by specifying the subset of levels that are not included in the partition function
$$Z_{N}^{\setminus\mathcal{S}^{(1)}}\equiv Z_{N}\quantity(\mathcal{S}\setminus%
\mathcal{S}^{(1)})$$
(7)
and thus for $\mathcal{S}^{(1)}=\{j_{1},j_{2},\dots,j_{\ell}\}$ containing $\ell$ levels, Eq. (6) is equivalent to
$$Z_{N}\equiv\sum_{k=0}^{N}Z_{k}(\{j_{1},j_{2},\dots,j_{\ell}\})Z_{N-k}^{%
\setminus\{j_{1},j_{2},\dots,j_{\ell}\}}.$$
(8)
To obtain a physical interpretation of Eq. (8), recall that in the canonical ensemble, the likelihood of the $N$-particle system being in a microstate defined by the occupation vector $\boldsymbol{n}|_{N}$ is given by the ratio $X(\boldsymbol{n}|_{N})/Z_{N}$. Accordingly, for the subset of energy levels with indices $\{j_{1},j_{2},\dots,j_{\ell}\}$, the joint probability distribution of the corresponding occupation numbers $\mathcal{P}_{n_{j_{1}},n_{j_{2}},\dots,n_{j_{\ell}}}$ can be obtained by performing the summation $\sum_{\boldsymbol{n}^{(2)}|_{N-k}}$, where $k=\sum_{r=1}^{\ell}n_{j_{r}}$, yielding
$$\mathcal{P}_{n_{j_{1}},n_{j_{2}},\dots,n_{j_{\ell}}}=\frac{\mathrm{e}^{-\beta%
\sum_{r=1}^{\ell}\epsilon_{j_{r}}n_{j_{r}}}}{Z_{N}}Z_{N-k}^{\setminus\{j_{1},j%
_{2},\dots,j_{\ell}\}}.$$
(9)
It follows that the probability $P_{k}(\{j_{1},j_{2},\dots,j_{\ell}\})$ of finding $k$ particles in $\{j_{1},j_{2},\dots,j_{\ell}\}$ and, of course, $N-k$ particles in $\mathcal{S}\setminus\{j_{1},j_{2},\dots,j_{\ell}\}$, can be obtained by applying the summation $\sum_{\boldsymbol{n}^{(1)}|_{k}}$:
$$P_{k}(\{j_{1},j_{2},\dots,j_{\ell}\})=\frac{Z_{k}(\{j_{1},j_{2},\dots,j_{\ell}%
\})Z_{N-k}^{\setminus\{j_{1},j_{2},\dots,j_{\ell}\}}}{Z_{N}},$$
(10)
where the normalization of $P_{k}(\{j_{1},j_{2},\dots,j_{\ell}\})$ is guaranteed by
Eq.(8).
So far the analysis of the partition functions has been completely general and we have not specified what type of particles are being described. Now, let’s be more specific and consider a system that solely consists of either fermions or bosons.
II.1 An Inverted Analogy Between Fermionic and Bosonic Statistics
Having developed an intuition for the general structure of the canonical partition function under a bipartition into sub-spectra, we now observe how this can provide insights
into the relationship between fermionic ($n_{i}^{\max}=1$) and bosonic ($n_{i}^{\max}\to\infty$) statistics of $N$ non-interacting particles. To distinguish the two cases we introduce a new subscript on the partition function ($F$ for fermions and $B$ for bosons).
If the set $\mathcal{S}^{(1)}$ represent a single energy level with an index $j_{1}=j$ then using our convention we have:
$$Z_{\!{\scriptscriptstyle F}{\!,k}}(\{j\})=\begin{cases}\mathrm{e}^{-\beta%
\epsilon_{j}k}&0\leq k\leq 1\\
0&\text{otherwise}\end{cases}$$
(11)
for fermions and
$$Z_{\!{\scriptscriptstyle B}{\!,k}}(\{j\})=\begin{cases}\mathrm{e}^{-\beta%
\epsilon_{j}k}&k\geq 0\\
0&\text{otherwise}\end{cases}$$
(12)
for bosons. Substituting into Eq. (8) we immediately find:
$$\displaystyle Z_{\!{\scriptscriptstyle F}{\!,N}}$$
$$\displaystyle=Z_{\!{\scriptscriptstyle F}{\!,N}}^{\setminus\{{j}\}}+\mathrm{e}%
^{-\beta\epsilon_{j}}Z_{\!{\scriptscriptstyle F}{\!,N-1}}^{\setminus\{{j}\}}$$
(13)
$$\displaystyle Z_{\!{\scriptscriptstyle B}{\!,N}}$$
$$\displaystyle=\sum_{k=0}^{N}\mathrm{e}^{-\beta\epsilon_{j}k}Z_{\!{%
\scriptscriptstyle B}{\!,N-k}}^{\setminus\{{j}\}}\,.$$
(14)
The relations in Eq. (13) and (14) formally describe the procedure for generating the canonical partition function of the $N$ particle system after introducing an energy level $\epsilon_{j}$ to the preexisting spectrum $\mathcal{S}\setminus\{j\}$.
Examining the structure of Eq. (13) suggests a simple matrix form: $\boldsymbol{Z}=\mathsf{A}\boldsymbol{Z}^{\setminus\{j\}}$, where $\boldsymbol{Z}=(Z_{\!{\scriptscriptstyle F}{\!,0}},Z_{\!{\scriptscriptstyle F}%
{\!,1}},\dots)$, $\boldsymbol{Z}^{\setminus\{j\}}=(Z_{\!{\scriptscriptstyle F}{\!,0}}^{\setminus%
\{{j}\}},Z_{\!{\scriptscriptstyle F}{\!,1}}^{\setminus\{{j}\}},\dots)$ and the matrix $\mathsf{A}_{n,m}=\delta_{n,m}+\mathrm{e}^{-\beta\epsilon_{j}}\delta_{n,m+1}$ is bidiagonal and can be inverted such that:
$$Z_{\!{\scriptscriptstyle F}{\!,N}}^{\setminus\{{j}\}}=\sum_{k=0}^{N}(-1)^{k}%
\mathrm{e}^{-\beta\epsilon_{j}k}Z_{\!{\scriptscriptstyle F}{\!,N-k}}.$$
(15)
Comparing this expression with Eq. (14), we observe an identical structure apart from exchanging the factor $\mathrm{e}^{-\beta\epsilon_{j}}$ with $\quantity(-\mathrm{e}^{-\beta\epsilon_{j}})$. Thus we can obtain the inversion of Eq. (14) by replacing $\mathrm{e}^{-\beta\epsilon_{j}}$ with $\quantity(-\mathrm{e}^{-\beta\epsilon_{j}})$ in Eq. (13), i.e.,
$$Z_{\!{\scriptscriptstyle B}{\!,N}}^{\setminus\{{j}\}}=Z_{\!{\scriptscriptstyle
B%
}{\!,N}}-\mathrm{e}^{-\beta\epsilon_{j}}Z_{\!{\scriptscriptstyle B}{\!,N-1}}.$$
(16)
The relations Eq. (15) and Eq. (16) exemplify the elimination of an energy level as they represent the inverse of Eq. (13) and Eq. (14) respectively.
If we absorb the negative signs in Eqs. (15) by shifting the energy $\epsilon_{j}$ by $\pm i\pi/\beta$, we can write $Z_{\!{\scriptscriptstyle F}{\!,N}}^{\setminus\{{j}\}}=\sum_{k=0}^{N}\mathrm{e}%
^{-\beta\epsilon^{\prime}_{j}k}Z_{\!{\scriptscriptstyle F}{\!,N-k}}$, where $\epsilon^{\prime}_{j}=\epsilon_{j}\pm i\pi/\beta$. As the general bipartition into sub-spectra introduced in Eq. (6) holds for energy levels with mixed statistics, and doesn’t require real entries for the $\epsilon_{i}$, we can build the bosonic partition functions $Z_{\!{\scriptscriptstyle B}{\!,N}}^{\prime}(\{j_{1},j_{2},\dots,j_{\ell}\})$ using the shifted energies $\{\epsilon_{j_{1}}^{\prime},\epsilon_{j_{2}}^{\prime},\dots,\epsilon_{j_{\ell}%
}^{\prime}\}$ and then combine it with the $Z_{\!{\scriptscriptstyle F}{\!,N}}$ to generate a mixed-form:
$$Z_{\!{\scriptscriptstyle F}{\!,N}}^{\setminus\{{j_{1},j_{2},\dots,j_{\ell}}\}}%
=\sum_{k=0}^{N}Z_{\!{\scriptscriptstyle B}{\!,k}}^{\prime}(\{j_{1},j_{2},\dots%
,j_{\ell}\})Z_{\!{\scriptscriptstyle F}{\!,N-k}}\,.$$
(17)
The effect of shifting the single particle spectrum by a constant $\omega$ on the canonical partition function $Z_{N}$ is captured by a rescaling factor $\mathrm{e}^{-\beta\omega N}$ and the resulting $N$ particle partition function of the shifted spectrum is $\mathrm{e}^{-\beta\omega N}Z_{N}$. Using $Z_{\!{\scriptscriptstyle B}{\!,k}}^{\prime}(\{j_{1},j_{2},\dots,j_{\ell}\})={%
\rm{e}}^{\pm i\pi k}Z_{\!{\scriptscriptstyle B}{\!,k}}(\{j_{1},j_{2},\dots,j_{%
\ell}\})$ then yields
$$Z_{\!{\scriptscriptstyle F}{\!,N}}^{\setminus\{{j_{1},j_{2},\dots,j_{\ell}}\}}%
=\sum_{k=0}^{N}(-1)^{k}Z_{\!{\scriptscriptstyle B}{\!,k}}(\{j_{1},j_{2},\dots,%
j_{\ell}\})Z_{\!{\scriptscriptstyle F}{\!,N-k}}.$$
(18)
Starting from Eq. (16) and following the same argument, we obtain an equivalent expression for bosons
$$Z_{\!{\scriptscriptstyle B}{\!,N}}^{\setminus\{{j_{1},j_{2},\dots,j_{\ell}}\}}%
=\sum_{k=0}^{N}(-1)^{k}Z_{\!{\scriptscriptstyle F}{\!,k}}(\{j_{1},j_{2},\dots,%
j_{\ell}\})Z_{\!{\scriptscriptstyle B}{\!,N-k}}.$$
(19)
The last two equations can be seen as a generalization of Eqs. (15) and (16). However, they also recover the symmetry between fermionic and bosonic statistics. To this end, we show how to obtain the partition function of a given spectrum using the APFs of two complementary subsets of the spectrum through Eq. (8). Also, Eqs. (18) and (19) show that this relation can be inverted, i.e., we can calculate the APFs of a subset of energy level using the APFs of its complement with the opposite statistics and the partition functions of the full spectrum.
II.2 Energy Level Occupations and Correlations
For fermions, the Pauli exclusion principle restricts the number of particles occupying an energy level $\epsilon_{j}$ to $n_{j}=0$ or $1$. Equivalently, $n_{j}^{p}=n_{j}$ for any $p>0$, simplifying the calculation of energy level occupation numbers and the correlations between them, including higher moments. More specifically, the average $\expectationvalue{n_{j_{1}}^{p_{1}}n_{j_{2}}^{p_{2}}\dots n_{j_{\ell}}^{p_{%
\ell}}}_{\!{\scriptscriptstyle F}{\!,N}}=\expectationvalue{n_{j_{1}}n_{j_{2}}%
\dots n_{j_{\ell}}}_{\!{\scriptscriptstyle F}{\!,N}}$, where $p_{i}>0,\ i\in\{1,\dots,\ell\}$ and in general
$$\displaystyle\expectationvalue{n_{j_{1}}n_{j_{2}}\dots n_{j_{\ell}}}_{\!{%
\scriptscriptstyle F}{\!,N}}=\\
\displaystyle\sum_{\boldsymbol{n}(\{j_{1},j_{2},\dots,j_{\ell}\})}n_{j_{1}}n_{%
j_{2}}\dots n_{j_{\ell}}\mathcal{P}_{\!\!{\scriptscriptstyle F}{\!,n_{j_{1}},n%
_{j_{2}},\dots,n_{j_{\ell}}}}$$
(20)
using the probability:
$$\mathcal{P}_{\!\!{\scriptscriptstyle F}{\!,n_{j_{1}},n_{j_{2}},\dots,n_{j_{%
\ell}}}}=\frac{\mathrm{e}^{-\beta\sum_{r=1}^{\ell}\epsilon_{j_{r}}n_{j_{r}}}}{%
Z_{\!{\scriptscriptstyle F}{\!,N}}}Z_{\!{\scriptscriptstyle F}{\!,N-\sum_{r=1}%
^{\ell}n_{j_{r}}}}^{\setminus\{{j_{1},j_{2},\dots,j_{\ell}}\}},$$
(21)
defined in Eq. (9). The only term that survives in Eq. (20) has $n_{j_{1}}=n_{j_{2}}=\dots=n_{j_{\ell}}=1$ giving:
$$\expectationvalue{n_{j_{1}}n_{j_{2}}\dots n_{j_{\ell}}}_{\!{\scriptscriptstyle
F%
}{\!,N}}=\frac{\mathrm{e}^{-\beta\sum_{r=1}^{\ell}\epsilon_{j_{r}}}}{Z_{\!{%
\scriptscriptstyle F}{\!,N}}}Z_{\!{\scriptscriptstyle F}{\!,N-\ell}}^{%
\setminus\{{j_{1},j_{2},\dots,j_{\ell}}\}}.$$
(22)
Following the same procedure, Eq. (20) can be generalized to describe both correlations and anti-correlations between energy levels:
$$\displaystyle\expectationvalue{\prod_{r=1}^{\ell}{\quantity[n_{j_{r}}\gamma_{j%
_{r}}+(1-n_{j_{r}})(1-\gamma_{j_{r}})]}}_{\!{\scriptscriptstyle F}{\!,N}}=\\
\displaystyle\frac{1}{Z_{\!{\scriptscriptstyle F}{\!,N}}}\mathrm{e}^{-\beta%
\sum_{r=1}^{\ell}\epsilon_{j_{r}}\gamma_{j_{r}}}Z_{\!{\scriptscriptstyle F}{\!%
,N-\sum_{r=1}^{\ell}\gamma_{j_{r}}}}^{\setminus\{{j_{1},j_{2},\dots,j_{\ell}}%
\}},$$
(23)
where $\gamma_{j_{r}}=1,0$. For the latter with $\gamma_{j_{r}}=0$, the occupation numbers in Eq. (20) have been replaced with their complements, $1-n_{j_{r}}$.
Thus, we find that fermionic level occupations and correlations can be directly written in terms of APFs without resorting to the usual definition $\expectationvalue{n_{j_{1}}n_{j_{2}}\dots n_{j_{\ell}}}_{\!{\scriptscriptstyle
F%
}{\!,N}}=\frac{1}{Z_{\!{\scriptscriptstyle F}{\!,N}}}\frac{\partial^{\ell}Z_{%
\!{\scriptscriptstyle F}{\!,N}}}{\partial(-\beta\epsilon_{j_{1}})\dots\partial%
(-\beta\epsilon_{j_{\ell}})}$ which yields equivalent results.
When considering a single level $(\ell=1)$, the occupation probability of the $j^{\rm th}$ fermionic level immediately follows
$$\expectationvalue{n_{j}}_{\!{\scriptscriptstyle F}{\!,N}}=\frac{\mathrm{e}^{-%
\beta\epsilon_{j}}Z_{\!{\scriptscriptstyle F}{\!,N-1}}^{\setminus\{{j}\}}}{Z_{%
\!{\scriptscriptstyle F}{\!,N}}},$$
(24)
with the associated probability
$$\mathcal{P}_{\!\!{\scriptscriptstyle F}{\!,n_{j}}}=\frac{\mathrm{e}^{-\beta%
\epsilon_{j}n_{j}}}{Z_{\!{\scriptscriptstyle F}{\!,N}}}Z_{\!{%
\scriptscriptstyle F}{\!,N-n_{j}}}^{\setminus\{{j}\}}.$$
(25)
For bosons, the occupation numbers $\expectationvalue{n_{j}}_{\!{\scriptscriptstyle B}{\!,N}}$ can be calculated from the corresponding occupation probability distribution $\mathcal{P}_{n_{j}}$ which are obtained from Eq. (9):
$$\mathcal{P}_{\!\!{\scriptscriptstyle B}{\!,n_{j}}}=\frac{\mathrm{e}^{-\beta%
\epsilon_{j}n_{j}}}{Z_{\!{\scriptscriptstyle B}{\!,N}}}Z_{\!{%
\scriptscriptstyle B}{\!,N-n_{j}}}^{\setminus\{{j}\}},$$
(26)
and for the $\ell$-point correlations:
$$\mathcal{P}_{\!\!{\scriptscriptstyle B}{\!,n_{j_{1}},n_{j_{2}},\dots,n_{j_{%
\ell}}}}=\frac{\mathrm{e}^{-\beta\sum_{r=1}^{\ell}\epsilon_{j_{r}}n_{j_{r}}}}{%
Z_{\!{\scriptscriptstyle B}{\!,N}}}Z_{\!{\scriptscriptstyle B}{\!,N-\sum_{r=1}%
^{\ell}n_{j_{r}}}}^{\setminus\{{j_{1},j_{2},\dots,j_{\ell}}\}}.$$
(27)
However, unlike the fermionic case, such an approach requires performing the unrestricted summation $\sum_{\boldsymbol{n}(\{j_{1},j_{2},\dots,j_{\ell}\})}$.
An alternative method which avoids this difficulty can be developed by exploiting the inverted analogy between fermionic and bosonic statistics introduced in Sec.II.1. In the fermionic case, the occupation number of an energy level $\epsilon_{j}$ is proportional to the APF $Z_{\!{\scriptscriptstyle F}{\!,N-1}}^{\setminus\{{j}\}}$ (Eq. (24)) which corresponds to the actual spectrum of the system missing the energy level $\epsilon_{j}$.
This suggests a route forward for bosons via the analogous inversion of doubly including the energy level $\epsilon_{j}$ instead of removing it, i.e., we construct an APF where this level is twofold-degenerate. We denote the corresponding $N$-boson APF by $Z_{\!{\scriptscriptstyle B}{\!,N}}^{\cup\quantity{{j}}}$ and distinguish the two levels using the dressed indices $j^{\scriptscriptstyle(0)}$ and $j^{\scriptscriptstyle(1)}$ such that the resulting combined spectrum has level indices $\quantity{1,\dots j-1,j^{\scriptscriptstyle(0)},j^{\scriptscriptstyle(1)},j+1,%
\dots,M}$ where $\epsilon_{j}=\epsilon_{j^{\scriptscriptstyle(0)}}=\epsilon_{j^{%
\scriptscriptstyle(1)}}$.
Returning to the general definition of the canonical partition function in Eq. (1), we can write $Z_{\!{\scriptscriptstyle B}{\!,N}}^{\cup\quantity{{j}}}=\sum_{\boldsymbol{n}^{%
\prime}|_{N}}X(\boldsymbol{n}^{\prime}|_{N})$, where the occupation vectors $\boldsymbol{n}^{\prime}|_{N}$ have one extra component: $n_{j}$ in $\boldsymbol{n}|_{N}$ is replaced by $n_{j^{\scriptscriptstyle(0)}}$ and $n_{j^{\scriptscriptstyle(1)}}$. The modified Boltzmann factors are:
$$X(\boldsymbol{n}^{\prime}|_{N})=\mathrm{e}^{-\beta\epsilon_{j}[n_{j^{%
\scriptscriptstyle(0)}}+n_{j^{\scriptscriptstyle(1)}}]}\prod_{i\neq j^{%
\scriptscriptstyle(0)},j^{\scriptscriptstyle(1)}}\mathrm{e}^{-\beta\epsilon_{i%
}n_{i}}$$
(28)
and thus their value is dependent only on the total occupancy of the $j^{\rm th}$ level, $n_{j^{\scriptscriptstyle(0)}}+n_{j^{\scriptscriptstyle(1)}}$. As a result, $X(\boldsymbol{n}|_{N})=X(\boldsymbol{n}^{\prime}|_{N})$ for any occupation vectors $\boldsymbol{n}|_{N}$ and $\boldsymbol{n}^{\prime}|_{N}$ with all $i\neq j$ components equal as well as $[\boldsymbol{n}|_{N}]_{j}=n_{j}=n_{j^{\scriptscriptstyle(0)}}+n_{j^{%
\scriptscriptstyle(1)}}$. For fixed $\boldsymbol{n}|_{N}$, the number of vectors $\boldsymbol{n}^{\prime}|_{N}$ that satisfies the previous conditions is equal to $n_{j}+1$, or, the number of ways in which $n_{j}$ bosons can occupy two energy levels. The APF can then be written in terms of the original occupation vector $\boldsymbol{n}|_{N}$ by inserting a frequency factor to account for the extra level $j$:
$$\displaystyle Z_{\!{\scriptscriptstyle B}{\!,N}}^{\cup\quantity{{j}}}$$
$$\displaystyle=\sum_{\boldsymbol{n}|_{N}}(n_{j}+1)X(\boldsymbol{n}|_{N})$$
(29)
$$\displaystyle=Z_{\!{\scriptscriptstyle B}{\!,N}}\expectationvalue{n_{j}+1}_{\!%
{\scriptscriptstyle B}{\!,N}}$$
and thus we can write:
$$\expectationvalue{n_{j}}_{\!{\scriptscriptstyle B}{\!,N}}=Z_{\!{%
\scriptscriptstyle B}{\!,N}}^{\cup\quantity{{j}}}/Z_{\!{\scriptscriptstyle B}{%
\!,N}}-1.$$
(30)
Applying Eq. (16), gives $Z_{\!{\scriptscriptstyle F}{\!,N}}^{\cup\quantity{{j}}}=Z_{\!{%
\scriptscriptstyle B}{\!,N}}+\mathrm{e}^{-\beta\epsilon_{j}}Z_{\!{%
\scriptscriptstyle F}{\!,N-1}}^{\cup\quantity{{j}}}$ which can be substituted into Eq. (30) to arrive at
$$\expectationvalue{n_{j}}_{\!{\scriptscriptstyle B}{\!,N}}=\frac{\mathrm{e}^{-%
\beta\epsilon_{j}}Z_{\!{\scriptscriptstyle B}{\!,N-1}}^{\cup\quantity{{j}}}}{Z%
_{\!{\scriptscriptstyle B}{\!,N}}},$$
(31)
which is in the same form as Eq. (24) for fermions.
To generalize this expression to $\ell$-level correlations with $\ell>1$ we examine the numerator of Eq. (31), recalling that we have added an extra copy of energy level $j$ to the partition function for $N-1$ particles such that it now appears $m_{j}+1=2$ times in the associated spectrum:
$$\displaystyle\mathrm{e}^{-\beta\epsilon_{j}}Z_{\!{\scriptscriptstyle B}{\!,N-1%
}}^{\cup\quantity{{j}}}$$
$$\displaystyle=\mathrm{e}^{-\beta\epsilon_{j}}\sum_{\boldsymbol{n}^{\prime}|_{N%
-1}}X(\boldsymbol{n}^{\prime}|_{N-1})$$
$$\displaystyle=\!\!\sum_{\boldsymbol{n}|_{N-1}}\!\!\binom{\tilde{n}_{j}+(m_{j}+%
1)-1}{(m_{j}+1)-1}\mathrm{e}^{-\beta\epsilon_{j}}X(\boldsymbol{n}|_{N-1})$$
$$\displaystyle=\sum_{\boldsymbol{n}|_{N,n_{j}\geq 1}}n_{j}X(\boldsymbol{n}|_{N}%
)\,.$$
(32)
Here we obtain the second line using the same trick as in Eq. (29) to convert $X(\boldsymbol{n}^{\prime}|_{N-1})$ into $X(\boldsymbol{n}|_{N-1})$ by accounting for the degeneracy where $[\boldsymbol{n}|_{N-1}]_{j}=\tilde{n}_{j}\geq 0$ is the total number of particles occupying the level $j$ as it appears in the Boltzmann factor $X(\boldsymbol{n}|_{N-1})=\mathrm{e}^{-\beta\epsilon_{j}\tilde{n}_{j}}\prod_{i%
\neq j}\mathrm{e}^{-\beta\epsilon_{i}n_{i}}$. The complicated looking binomial coefficient
$\binom{\tilde{n}_{j}+(m_{j}+1)-1}{(m_{j}+1)-1}=\left(\kern-3.0pt\left(\genfrac%
{}{}{0.0pt}{}{\tilde{n}_{j}}{m_{j}+1}\right)\kern-3.0pt\right)$ is the multiset coefficient that counts the number of ways $\tilde{n}_{j}$ bosons can be distributed amongst the $m_{j}+1$ levels with energy $\epsilon_{j}$. Finally, the last line is obtained by using the
fact that $\mathrm{e}^{-\beta\epsilon_{j}}X(\boldsymbol{n}|_{N-1})=X(\boldsymbol{n}|_{N})$ where $[\boldsymbol{n}|_{N}]_{j}=n_{j}=\tilde{n}_{j}+1\geq 1$.
Eq. (32) can be immediately extended to the case where we add copies of not 1 but $\ell$ levels $\quantity{j_{1},\dots,j_{\ell}}$:
$$\mathrm{e}^{-\beta\sum_{r=1}^{\ell}\epsilon_{j_{r}}}Z_{\!{\scriptscriptstyle B%
}{\!,N-\ell}}^{\cup\quantity{{j_{1},\dots,j_{\ell}}}}=\sum_{\boldsymbol{n}|_{N%
}}\quantity(\prod_{r=1}^{\ell}{n}_{j_{r}})X(\boldsymbol{n}|_{N})$$
(33)
where the conditions $n_{j_{r}}\geq 1$ in the occupancy vector can be neglected as any $n_{j_{r}}=0$ terms do not contribute to the sum due to the multiplicative string. Finally, we can write the desired result:
$$\expectationvalue{n_{j_{1}}n_{j_{2}}\dots n_{j_{\ell}}}_{\!{\scriptscriptstyle
B%
}{\!,N}}=\frac{\mathrm{e}^{-\beta\sum_{r=1}^{\ell}\epsilon_{j_{r}}}}{Z_{\!{%
\scriptscriptstyle B}{\!,N}}}Z_{\!{\scriptscriptstyle B}{\!,N-\ell}}^{\cup%
\quantity{{j_{1},j_{2},\dots,j_{\ell}}}}.$$
(34)
An immediate extension of Eq. (34) will turn out to be useful, which introduces an APF with higher-order degeneracy. We consider $m_{j_{r}}$ extra copies of the $r^{\rm th}$ level $\epsilon_{j_{r}}$ with $r\in\{1,\dots\ell\}$ and find:
$$\displaystyle\expectationvalue{\prod_{r=1}^{\ell}{\binom{n_{j_{r}}-q_{j_{r}}+m%
_{j_{r}}}{m_{j_{r}}}}}_{\!{\scriptscriptstyle B}{\!,N}}=\\
\displaystyle\frac{1}{Z_{\!{\scriptscriptstyle B}{\!,N}}}\mathrm{e}^{-\beta%
\sum_{r=1}^{\ell}\epsilon_{j_{r}}q_{j_{r}}}Z_{\!{\scriptscriptstyle B}{\!,N-%
\sum_{r=1}^{\ell}q_{j_{r}}}}^{\cup\quantity{{j_{1}^{\scriptscriptstyle(1)},%
\dots,j_{1}^{\scriptscriptstyle(m_{j_{1}})},\dots j_{\ell}^{\scriptscriptstyle%
(1)},\dots,j_{\ell}^{\scriptscriptstyle(m_{j_{\ell}})}}}}$$
(35)
where the $q_{j_{r}}\leq m_{j_{r}}$ allow for the added freedom of choice of how many particles are associated with each of the degenerate levels and allow us to write the left-hand side in terms of the desired occupations $n_{j}=[\boldsymbol{n}|_{N}]_{j}$. Note: to simplify notation we only include a superscript on the levels in the bosonic APF and only if we are adding more than one extra copy per original level.
Eqs. (23) and (35) are the major results of this section, and demonstrate that for both fermions and bosons, $\ell$-level correlations can be written in terms of APFs for $N-\ell$ particles with $\ell$ energy levels removed (added) for fermions (bosons).
II.3 Recovering Known Results Via the APF Theory
In this section, we illustrate the utility of our auxiliary expressions in simplifying the derivation of known recursion relations that govern the canonical partition functions and occupation numbers for fermions and bosons.
Beginning with fermionic statistics, if we use the definition of $\expectationvalue{n_{j}}_{\!{\scriptscriptstyle F}{\!,N}}$ in Eq. (24) to substitute for $Z_{\!{\scriptscriptstyle F}{\!,N}}^{\setminus\{{j}\}}$ and $Z_{\!{\scriptscriptstyle F}{\!,N-1}}^{\setminus\{{j}\}}$ in Eq. (13),
we obtain the well-known recursion relation for occupation numbers Schmidt (1989); Borrmann et al. (1999)
$$\expectationvalue{n_{j}}_{\!{\scriptscriptstyle F}{\!,N+1}}=\frac{Z_{\!{%
\scriptscriptstyle F}{\!,N}}}{Z_{\!{\scriptscriptstyle F}{\!,N+1}}}\mathrm{e}^%
{-\beta\epsilon_{j}}\left(1-{\expectationvalue{n_{j}}_{\!{\scriptscriptstyle F%
}{\!,N}}}\right)\,.$$
(36)
$\expectationvalue{n_{j}}_{\!{\scriptscriptstyle F}{\!,N}}$ can be written explicitly in term of the partition functions by substituting for $Z_{\!{\scriptscriptstyle F}{\!,N-1}}^{\setminus\{{j}\}}$ in Eq. (24) using Eq. (15) as:
$$\expectationvalue{n_{j}}_{\!{\scriptscriptstyle F}{\!,N}}=\frac{1}{Z_{\!{%
\scriptscriptstyle F}{\!,N}}}\sum_{k=1}^{N}(-1)^{k-1}\mathrm{e}^{-\beta%
\epsilon_{j}k}Z_{\!{\scriptscriptstyle F}{\!,N-k}},$$
(37)
and using the canonical condition $\sum_{i}\expectationvalue{n_{i}}_{\!{\scriptscriptstyle F}{\!,N}}=N$ we find the original 1993 result of Borrmann and Franke Borrmann and Franke (1993):
$$Z_{\!{\scriptscriptstyle F}{\!,N}}=\frac{1}{N}\sum_{k=1}^{N}(-1)^{k-1}C_{k}Z_{%
\!{\scriptscriptstyle F}{\!,N-k}},$$
(38)
where $C_{k}=\sum_{j}\mathrm{e}^{-\beta\epsilon_{j}k}$.
Bosonic statistics can be treated in analogy to the fermionic case. Assigning the roles played by Eqs. (24), (13) and (15) to Eqs. (31), (16) and (14)111Here we replace $Z_{\!{\scriptscriptstyle B}{\!,x}}$ with $Z_{\!{\scriptscriptstyle B}{\!,x}}^{\cup\quantity{{j}}}$ and $Z_{\!{\scriptscriptstyle B}{\!,x}}^{\setminus\{{j}\}}$ with $Z_{\!{\scriptscriptstyle B}{\!,x}}$ in (16) and (14), respectively, we obtain the known bosonic equivalents Schmidt (1989); Borrmann et al. (1999); Borrmann and Franke (1993):
$$\displaystyle\expectationvalue{n_{j}}_{\!{\scriptscriptstyle B}{\!,N+1}}$$
$$\displaystyle=\frac{Z_{\!{\scriptscriptstyle B}{\!,N}}}{Z_{\!{%
\scriptscriptstyle B}{\!,N+1}}}\mathrm{e}^{-\beta\epsilon_{j}}\quantity(1+%
\expectationvalue{n_{j}}_{\!{\scriptscriptstyle B}{\!,N}})$$
(39)
$$\displaystyle\expectationvalue{n_{j}}_{\!{\scriptscriptstyle B}{\!,N}}$$
$$\displaystyle=\frac{1}{Z_{\!{\scriptscriptstyle B}{\!,N}}}\sum_{k=1}^{N}%
\mathrm{e}^{-\beta\epsilon_{j}k}Z_{\!{\scriptscriptstyle B}{\!,N-k}}$$
(40)
$$\displaystyle Z_{\!{\scriptscriptstyle B}{\!,N}}$$
$$\displaystyle=\frac{1}{N}\sum_{k=1}^{N}C_{k}Z_{\!{\scriptscriptstyle B}{\!,N-k%
}}\,.$$
(41)
Due to the Pauli exclusion principle for fermionic statistics, the occupation number $\expectationvalue{n_{j}}_{\!{\scriptscriptstyle F}{\!,N}}$ of an energy level $j$ gives us direct access to the occupation probability distribution $\mathcal{P}_{\!\!{\scriptscriptstyle F}{\!,n_{j}}}$ of the level. Despite the absence of any such simplification in the bosonic case, the occupation probability distribution $\mathcal{P}_{\!\!{\scriptscriptstyle B}{\!,n_{j}}}$ of a single energy level can be related to the corresponding partition functions as
$$\mathcal{P}_{\!\!{\scriptscriptstyle B}{\!,n_{j}}}=\mathrm{e}^{-\beta\epsilon_%
{j}n_{j}}\frac{Z_{\!{\scriptscriptstyle B}{\!,N-n_{j}}}}{Z_{\!{%
\scriptscriptstyle B}{\!,N}}}-\mathrm{e}^{-\beta\epsilon_{j}(n_{j}+1)}\frac{Z_%
{\!{\scriptscriptstyle B}{\!,N-n_{j}-1}}}{Z_{\!{\scriptscriptstyle B}{\!,N}}},$$
(42)
which is obtained by using Eq. (26) to substitute for $Z_{\!{\scriptscriptstyle B}{\!,N-n_{j}}}^{\setminus\{{j}\}}$ in Eq. (16), after replacing $N$ with $N-n_{j}$ Weiss and Wilkens (1997). In summary, within the unified framework of APFs, it is straightforward to obtain most of the well-known general relations in the fermionic and bosonic canonical ensemble that were previously derived using a host of different methods. This highlights the utility of this approach as a unifying framework when studying $N$ indistinguishable non-interacting particles.
II.4 General Expressions for Probabilities and Correlations
De facto, the APFs can also be used to generalize previous results, and in a form that is highly symmetric with respect to particle statistics. To accentuate this, let us introduce the notation:
$$\zeta=\begin{cases}+1&\Leftrightarrow B\Leftrightarrow\text{bosons}\\
-1&\Leftrightarrow F\Leftrightarrow\text{fermions}\end{cases}\,.$$
(43)
Then, the recursive relations for energy level correlations can be obtained using Eqs. (13) and (16) for fermions and bosons, respectively. The number of initial values of correlations needed is equal to the number of the involved points, e.g., for the two-level correlations we find:
$$\displaystyle\expectationvalue{n_{i}n_{j}}_{\!{\scriptscriptstyle\zeta}{\!,N+2}}$$
$$\displaystyle=\frac{Z_{\!{\scriptscriptstyle\zeta}{\!,N}}}{Z_{\!{%
\scriptscriptstyle\zeta}{\!,N+2}}}\mathrm{e}^{-\beta(\epsilon_{i}+\epsilon_{j}%
)}\quantity(1-\expectationvalue{n_{i}n_{j}}_{\!{\scriptscriptstyle\zeta}{\!,N}})$$
$$\displaystyle\;+\zeta\frac{Z_{\!{\scriptscriptstyle\zeta}{\!,N+1}}}{Z_{\!{%
\scriptscriptstyle\zeta}{\!,N+2}}}\quantity(\mathrm{e}^{-\beta\epsilon_{i}}+%
\mathrm{e}^{-\beta\epsilon_{j}})\expectationvalue{n_{i}n_{j}}_{\!{%
\scriptscriptstyle\zeta}{\!,N+1}}$$
(44)
Further, for the set of levels $\mathcal{S}_{\ell}=\{j_{1},\dots,j_{\ell}\}$, Eqs. (21) and (27), define the joint probability distributions of the occupation numbers in terms of the APFs $Z_{\!{\scriptscriptstyle F}{\!,k}}^{\setminus{\mathcal{S}_{\ell}}}$ and $Z_{\!{\scriptscriptstyle B}{\!,k}}^{\setminus{\mathcal{S}_{\ell}}}$, which can be re-expressed using Eq. (18) and Eq. (19) as
$$\mathcal{P}_{\!\!{\scriptscriptstyle\zeta}{\!,n_{j_{1}},\dots,n_{j_{\ell}}}}=%
\frac{\mathrm{e}^{-\beta E_{\rm{tot}}}}{Z_{\!{\scriptscriptstyle\zeta}{\!,N}}}%
\sum_{k=0}^{N-n_{\rm{tot}}}(-1)^{k}Z_{\!{\scriptscriptstyle-\zeta}{\!,k}}(%
\mathcal{S}_{\ell})Z_{\!{\scriptscriptstyle\zeta}{\!,N-n_{\rm{tot}}-k}}\\
$$
(45)
where, $E_{\rm{tot}}=\sum_{r=1}^{\ell}\epsilon_{j_{r}}n_{j_{r}}$ and $n_{\rm{tot}}=\sum_{r=1}^{\ell}n_{j_{r}}$. To avoid confusion we note again the convention in use that $Z_{\!{\scriptscriptstyle-\zeta}{\!,k}}(\mathcal{S_{\ell}})=0$ whenever $k$ exceeds the maximum number of particles set by $\mathcal{S_{\ell}}$ for fixed particle statistics.
Now, as the level occupation correlations of fermions and bosons are represented by the APFs $Z_{\!{\scriptscriptstyle F}{\!,k}}^{\setminus{\mathcal{S}_{\ell}}}$ and $Z_{\!{\scriptscriptstyle B}{\!,k}}^{\cup{\mathcal{S}_{\ell}}}$ in Eqs. (22) and (34), we can write
$$\expectationvalue{n_{j_{1}}\dots n_{j_{\ell}}}_{\!{\scriptscriptstyle\zeta}{\!%
,N}}=\frac{\mathrm{e}^{-\beta\sum_{r=1}^{\ell}\epsilon_{j_{r}}}}{Z_{\!{%
\scriptscriptstyle\zeta}{\!,N}}}\sum_{k=0}^{N-\ell}\zeta^{k}Z_{\!{%
\scriptscriptstyle B}{\!,k}}(\mathcal{S}_{\ell})Z_{\!{\scriptscriptstyle\zeta}%
{\!,N-\ell-k}}$$
(46)
where we note that the $k$-particle bosonic partition function for the levels $\mathcal{S}_{\ell}$ appears in both fermionic and bosonic correlations.
Alternatively, the APFs $Z_{\!{\scriptscriptstyle F}{\!,k}}^{\setminus{\mathcal{S}_{\ell}}}$, $Z_{\!{\scriptscriptstyle B}{\!,k}}^{\setminus{\mathcal{S}_{\ell}}}$ and $Z_{\!{\scriptscriptstyle B}{\!,k}}^{\cup{\mathcal{S}_{\ell}}}$ can be computed recursively using Eq. (38) and Eq. (41). This has the potential to simplify the calculation of the related full joint probability distribution, as it only requires calculating the corresponding APF with a number of particles in the range $k=0,\dots,N$.
II.4.1 Simplification for degenerate levels
The expressions derived in the previous section have a very simple form when the involved levels are degenerate, i.e., if we consider correlations between the set of levels $\{j^{(0)},\dots,j^{(\ell-1)}\}$, where $\epsilon_{j^{(s)}}=\epsilon_{j}$ for $s\in\{0,\dots,\ell-1\}$. The canonical partition function of $N$ bosons in $\ell$ degenerate energy levels is
$$Z_{\!{\scriptscriptstyle B}{\!,N}}(\{j^{(0)},\dots,j^{(\ell-1)}\})=\tbinom{N+%
\ell-1}{\ell-1}\mathrm{e}^{-\beta\epsilon_{j}N}\,,$$
(47)
and thus such correlations for fermions and bosons can be computed directly from Eq. (46) as
$$\expectationvalue{n_{j^{(0)}}\dots n_{j^{(\ell-1)}}}_{\!{\scriptscriptstyle%
\zeta}{\!,N}}=\frac{1}{Z_{\!{\scriptscriptstyle\zeta}{\!,N}}}\sum_{k=\ell}^{N}%
\zeta^{k-\ell}\tbinom{k-1}{\ell-1}\mathrm{e}^{-\beta\epsilon_{j}k}Z_{\!{%
\scriptscriptstyle\zeta}{\!,N-k}}$$
(48)
where we have shifted the summation. The numerical complexity of calculating Eq. (48) differs from that of Eq. (37) or Eq. (40) by the number of multiplications and additions needed to calculate the extra factor $\tbinom{k-1}{\ell-1}$, which can be viewed as a polynomial in $k$ of degree $\ell-1$. Thus calculating Eq. (48) requires additional $\approx(N-\ell)(\ell-1)$ multiplications, and a similar number of additions which still scales linearly with $N$ for a moderate value of $\ell$.
II.4.2 Expectation values of higher moments of degenerate levels
Let us now focus on the case of bosons and revisit Eq. (35) considering a single energy level $\epsilon_{j}$:
$$\expectationvalue{{\tbinom{n_{j}-q+m_{j}}{m_{j}}}}_{\!{\scriptscriptstyle B}{%
\!,N}}\!\!=\!\frac{1}{Z_{\!{\scriptscriptstyle B}{\!,N}}}\mathrm{e}^{-q\beta%
\epsilon_{j}}Z_{\!{\scriptscriptstyle B}{\!,N-q}}^{\cup\quantity{{j^{%
\scriptscriptstyle(1)},\dots,j^{\scriptscriptstyle(m_{j})}}}}.$$
(49)
Similarly, using Eq. (6), we can write the APF $Z_{\!{\scriptscriptstyle B}{\!,N-q}}^{\cup\quantity{{j^{\scriptscriptstyle(1)}%
,\dots,j^{\scriptscriptstyle(m_{j})}}}}$ in terms of the system partition function and the bosonic APF $Z_{\!{\scriptscriptstyle B}{\!,N-q}}(\{j^{\scriptscriptstyle(1)},\dots,j^{%
\scriptscriptstyle(m_{j})}\})$ of $m_{j}$ degenerate levels. As a result, we obtain
$$\expectationvalue{{\tbinom{n_{j}-q+m_{j}}{m_{j}}}}_{\!{\scriptscriptstyle B}{%
\!,N}}=\frac{1}{Z_{\!{\scriptscriptstyle B}{\!,N}}}\sum_{k=q}^{N}\tbinom{k+m_{%
j}-q-1}{m_{j}-1}\mathrm{e}^{-\beta\epsilon_{j}k}Z_{\!{\scriptscriptstyle B}{\!%
,N-k}},$$
(50)
where $q\leq m_{j}$. Note that if we set both of $m_{j}=\ell$ and $q=\ell$, then comparing with Eq. (48) we see that
$$\expectationvalue{{\binom{n_{j^{(s)}}}{\ell}}}_{\!{\scriptscriptstyle B}{\!,N}%
}=\expectationvalue{n_{j^{(0)}}\dots n_{j^{(\ell-1)}}}_{\!{\scriptscriptstyle B%
}{\!,N}},$$
(51)
for $s\in\{0,\dots,\ell-1\}$. This demonstrates that the correlations between degenerate levels can be expressed in terms of moments of the occupation number of any of the degenerate levels. This also helps to simplify the calculation of such moments, for example, we can write
$$\expectationvalue{n_{j}^{2}}_{\!{\scriptscriptstyle B}{\!,N}}=%
\expectationvalue{{\tbinom{n_{j}}{2}}}_{\!{\scriptscriptstyle B}{\!,N}}+%
\expectationvalue{{\tbinom{n_{j}+1}{2}}}_{\!{\scriptscriptstyle B}{\!,N}}$$
which can be simplified using Eq. (50) as
$$\expectationvalue{n_{j}^{2}}_{\!{\scriptscriptstyle B}{\!,N}}=\frac{1}{Z_{\!{%
\scriptscriptstyle B}{\!,N}}}\sum_{k=1}^{N}\quantity(2k-1)\mathrm{e}^{-\beta%
\epsilon_{j}k}Z_{\!{\scriptscriptstyle B}{\!,N-k}}.$$
(52)
In the same fashion, we can write
$$\expectationvalue{n_{j}^{3}}_{\!{\scriptscriptstyle B}{\!,N}}=%
\expectationvalue{{\tbinom{n_{j}}{3}}}_{\!{\scriptscriptstyle B}{\!,N}}+4%
\expectationvalue{{\tbinom{n_{j}+1}{3}}}_{\!{\scriptscriptstyle B}{\!,N}}+%
\expectationvalue{{\tbinom{n_{j}+2}{3}}}_{\!{\scriptscriptstyle B}{\!,N}}$$
and thus
$$\expectationvalue{n_{j}^{3}}_{\!{\scriptscriptstyle B}{\!,N}}=\frac{1}{Z_{\!{%
\scriptscriptstyle B}{\!,N}}}\sum_{k=1}^{N}\quantity(3k^{2}-3k+1)\mathrm{e}^{-%
\beta\epsilon_{j}k}Z_{\!{\scriptscriptstyle B}{\!,N-k}}.$$
(53)
III Decomposition of level correlations into occupation numbers
In the previous section, we illustrated that the joint probability distributions of the occupation numbers and corresponding level correlations can be represented by auxiliary partition functions. The APF is distinguished from the actual partition function of the $N$-particle system through either the inclusion or exclusion of a set of levels from/to the complete spectrum under study. The resulting complexity of performing an actual calculation thus depends on the size of the modified set as demonstrated by, e.g. Eq. (46).
It is known that the resulting complexity can be reduced by relating higher-order correlations between non-degenerate levels to the related level-occupation numbers Schönhammer (2017); Giraud et al. (2018), representing an approach similar to Wicks theorem which only holds in the grand canonical ensemble. In this section, we directly obtain many known results using the APF method and, more importantly, generalize them to deal with degenerate energy levels.
Before we introduce the general and systematic approach to this problem, let us return to Eqs. (23) and (35) and consider two specific examples of increasing difficulty.
III.1 Examples
III.1.1 Two-level correlations
Consider the expectation value of two bosonic energy levels $j_{1}$ and $j_{2}$ where
$\epsilon_{j_{1}}\neq\epsilon_{j_{2}}$: $\expectationvalue{n_{j_{1}}n_{j_{2}}}_{\!{\scriptscriptstyle B}{\!,N}}$. Employing Eq. (35) with $m_{j_{1}}=m_{j_{2}}=1$, $q_{j_{1}}=0$ and $q_{j_{2}}=1$, we find
$$\expectationvalue{\quantity(n_{j_{1}}+1)n_{j_{2}}}_{\!{\scriptscriptstyle B}{%
\!,N}}=\frac{\mathrm{e}^{-\beta\epsilon_{j_{2}}}}{Z_{\!{\scriptscriptstyle B}{%
\!,N}}}Z_{\!{\scriptscriptstyle B}{\!,N-1}}^{\cup\quantity{{j_{1},j_{2}}}},$$
(54)
and upon exchanging the values of $q_{j_{1}}$ and $q_{j_{2}}$, we have
$$\expectationvalue{n_{j_{1}}\quantity(n_{j_{2}}+1)}_{\!{\scriptscriptstyle B}{%
\!,N}}=\frac{\mathrm{e}^{-\beta\epsilon_{j_{1}}}}{Z_{\!{\scriptscriptstyle B}{%
\!,N}}}Z_{\!{\scriptscriptstyle B}{\!,N-1}}^{\cup\quantity{{j_{1},j_{2}}}}.$$
(55)
Next, eliminating the APF from the two equations yields our final result:
$$\expectationvalue{n_{j_{1}}n_{j_{2}}}_{\!{\scriptscriptstyle B}{\!,N}}=-\frac{%
\mathrm{e}^{\beta\epsilon_{j_{1}}}\expectationvalue{n_{j_{1}}}_{\!{%
\scriptscriptstyle B}{\!,N}}-\mathrm{e}^{\beta\epsilon_{j_{2}}}%
\expectationvalue{n_{j_{2}}}_{\!{\scriptscriptstyle B}{\!,N}}}{\mathrm{e}^{%
\beta\epsilon_{j_{1}}}-\mathrm{e}^{\beta\epsilon_{j_{2}}}}.$$
(56)
Similarly, if the levels are fermionic, we use Eq. (23) with $\quantity(\gamma_{j_{1}},\gamma_{j_{2}})=\quantity(0,1)$ and $\quantity(1,0)$ to find:
$$\expectationvalue{n_{j_{1}}n_{j_{2}}}_{\!{\scriptscriptstyle F}{\!,N}}=\frac{%
\mathrm{e}^{\beta\epsilon_{j_{1}}}\expectationvalue{n_{j_{1}}}_{\!{%
\scriptscriptstyle F}{\!,N}}-\mathrm{e}^{\beta\epsilon_{j_{2}}}%
\expectationvalue{n_{j_{2}}}_{\!{\scriptscriptstyle F}{\!,N}}}{\mathrm{e}^{%
\beta\epsilon_{j_{1}}}-\mathrm{e}^{\beta\epsilon_{j_{2}}}}.$$
(57)
These known results Schönhammer (2017); Giraud et al. (2018) are thus obtainable within the APF approach with a few lines of algebra by generating a set of independent equations. We now extend this idea to three energy levels.
III.1.2 Three-level correlations
The previous example for bosons is modified by adding a third level $j_{3}$ with $m_{j_{3}}=1$ and $\epsilon_{j_{3}}$ that is different than both of $\epsilon_{j_{1}}$ and $\epsilon_{j_{2}}$. Setting $\quantity(q_{j_{1}},q_{j_{2}},q_{j_{3}})=\quantity(1,0,1)$ and $\quantity(1,1,0)$ in Eq. (35) gives the two equations
$$\displaystyle\expectationvalue{n_{j_{1}}\quantity(n_{j_{2}}+1)n_{j_{3}}}_{\!{%
\scriptscriptstyle B}{\!,N}}$$
$$\displaystyle=\frac{\mathrm{e}^{-\beta\quantity(\epsilon_{j_{1}}+\epsilon_{j_{%
3}})}}{Z_{\!{\scriptscriptstyle B}{\!,N}}}Z_{\!{\scriptscriptstyle B}{\!,N-2}}%
^{\cup\quantity{{j_{1},j_{2},j_{3}}}}$$
(58)
$$\displaystyle\expectationvalue{n_{j_{1}}n_{j_{2}}\quantity(n_{j_{3}}+1)}_{\!{%
\scriptscriptstyle B}{\!,N}}$$
$$\displaystyle=\frac{\mathrm{e}^{-\beta\quantity(\epsilon_{j_{1}}+\epsilon_{j_{%
2}})}}{Z_{\!{\scriptscriptstyle B}{\!,N}}}Z_{\!{\scriptscriptstyle B}{\!,N-2}}%
^{\cup\quantity{{j_{1},j_{2},j_{3}}}}$$
(59)
respectively. Solving for $\expectationvalue{n_{j_{1}}n_{j_{2}}n_{j_{3}}}_{\!{\scriptscriptstyle B}{\!,N}}$ leads to
$$\expectationvalue{n_{j_{1}}n_{j_{2}}n_{j_{3}}}_{\!{\scriptscriptstyle B}{\!,N}%
}=-\frac{\mathrm{e}^{\beta\epsilon_{j_{2}}}\expectationvalue{n_{j_{1}}n_{j_{2}%
}}_{\!{\scriptscriptstyle B}{\!,N}}-\mathrm{e}^{\beta\epsilon_{j_{3}}}%
\expectationvalue{n_{j_{1}}n_{j_{3}}}_{\!{\scriptscriptstyle B}{\!,N}}}{%
\mathrm{e}^{\beta\epsilon_{j_{2}}}-\mathrm{e}^{\beta\epsilon_{j_{3}}}}.$$
(60)
which can be further broken down into single-level occupation numbers by application of Eq. (56).
A slightly modified approach can be used if two of the energy levels are degenerate,
$\epsilon_{j_{2}}=\epsilon_{j_{3}}\neq\epsilon_{j_{1}}$. As above, we denote degenerate levels via superscript and we relabel $j_{2}=j_{2}^{(0)}$ and $j_{3}=j_{2}^{(1)}$.
Then it is clear that $\expectationvalue{n_{j_{1}}\!\!n_{j_{2}^{(0)}}}_{\!{\scriptscriptstyle B}{\!,N%
}}=\expectationvalue{n_{j_{1}}\!\!n_{j_{2}^{(1)}}}_{\!{\scriptscriptstyle B}{%
\!,N}}$ and Eq. (60) is not immediately applicable. However, this can be resolved by replacing one of the choices of $\quantity(q_{j_{1}}\!\!\!\!,q_{j_{2}^{(0)}},q_{j_{2}^{(1)}})$, say $\quantity(1,1,0)$, with $\quantity(0,1,1)$, which gives
$$\displaystyle\expectationvalue{\!n_{j_{1}}\!\!\!n_{j_{2}^{(0)}}n_{j_{2}^{(1)}}%
\!}_{\!{\scriptscriptstyle B}{\!,N}}=\\
\displaystyle-\frac{\mathrm{e}^{\beta\epsilon_{j_{1}}}\!\expectationvalue{\!n_%
{j_{1}}\!\!n_{j_{2}^{(1)}}\!}_{\!{\scriptscriptstyle B}{\!,N}}\!-\mathrm{e}^{%
\beta\epsilon_{j_{2}}}\expectationvalue{n_{j_{2}^{(0)}}n_{j_{2}^{(1)}}\!\!}_{%
\!{\scriptscriptstyle B}{\!,N}}}{\mathrm{e}^{\beta\epsilon_{j_{1}}}-\mathrm{e}%
^{\beta\epsilon_{j_{2}}}}.$$
(61)
We now turn to the general decomposition of $\ell$-level correlations into functions of the occupation numbers of the energy levels for both the non-degenerate and degenerate spectra.
III.2 A Systematic Approach
In the following we show that Eqs. (6), (18) and (19) can be directly employed to systematically relate higher-order correlations to levels occupation numbers. We begin by considering the set of levels $\mathcal{S}_{\ell}=\{j_{1},j_{2},\dots,j_{\ell}\}$ and relate the corresponding $\ell$-point correlations to the $r$-point correlations for any nonempty subset of levels $\mathcal{S}_{r}=\{i_{1},i_{2},\dots,i_{r}\}\subset\mathcal{S}_{\ell}$.
Imposing fermionic level statistics on the spectrum $\mathcal{S}$, we can relate the correlations $\expectationvalue{n_{j_{1}}n_{j_{2}}\dots n_{j_{\ell}}}_{\!{\scriptscriptstyle
F%
}{\!,N}}$ to $\expectationvalue{n_{i_{1}}n_{i_{2}}\dots n_{i_{r}}}_{\!{\scriptscriptstyle F}%
{\!,N}}$ by relating their corresponding APFs, i.e., $Z_{\!{\scriptscriptstyle F}{\!,N-\ell}}^{\setminus{\mathcal{S}_{\ell}}}$ and $Z_{\!{\scriptscriptstyle F}{\!,N-r}}^{\setminus{\mathcal{S}_{r}}}$. This can be achieved by building $Z_{\!{\scriptscriptstyle F}{\!,N-1}}^{\setminus{\mathcal{S}_{r}}}$ via Eq. (6) to combine the APFs of $\mathcal{S}\setminus\mathcal{S}_{\ell}$ with that of $\mathcal{S}_{\ell}\setminus\mathcal{S}_{r}$ through
$$Z_{\!{\scriptscriptstyle F}{\!,N-r}}^{\setminus{\mathcal{S}_{r}}}=\sum_{k=0}^{%
\ell-r}Z_{\!{\scriptscriptstyle F}{\!,k}}^{\setminus{\mathcal{S}_{r}}}(%
\mathcal{S}_{\ell})Z_{\!{\scriptscriptstyle F}{\!,N-r-k}}^{\setminus{\mathcal{%
S}_{\ell}}}.$$
(62)
We note the upper limit in the previous summation is $\ell-r$, where in general it should be $k_{\max}=\min(\ell-r,N-r)$ (Eq. (6)). This is because the fermionic APF $Z_{\!{\scriptscriptstyle F}{\!,k}}^{\setminus{\mathcal{S}_{r}}}(\mathcal{S}_{%
\ell})$ cannot describe more than $\ell-r$ particles, as this is the number of levels that are in $\mathcal{S}_{\ell}\setminus\mathcal{S}_{r}$. Further, for $N<\ell$ the correlations $\langle n_{j_{1}}n_{j_{2}}\dots n_{j_{\ell}}\rangle_{N}=0$ for general particle statistics. As a result, we only consider $\ell$-points correlations with $N\geq\ell$. Moreover, we can obtain $Z_{\!{\scriptscriptstyle F}{\!,k}}^{\setminus{\mathcal{S}_{r}}}(\mathcal{S}_{%
\ell})$ by removing the contribution of the levels $\mathcal{S}_{r}$ from the APF of $\mathcal{S}_{\ell}$, using Eq. (18), as
$$Z_{\!{\scriptscriptstyle F}{\!,k}}^{\setminus{\mathcal{S}_{r}}}(\mathcal{S}_{%
\ell})=\sum_{m=0}^{k}(-1)^{m}Z_{\!{\scriptscriptstyle B}{\!,m}}(\mathcal{S}_{r%
})Z_{\!{\scriptscriptstyle F}{\!,k-m}}(\mathcal{S}_{\ell}).$$
(63)
If we isolate the last term in the summation in Eq. (62), i.e., $Z_{\!{\scriptscriptstyle F}{\!,\ell-r}}^{\setminus{\mathcal{S}_{r}}}(\mathcal{%
S}_{\ell})Z_{\!{\scriptscriptstyle F}{\!,N-\ell}}^{\setminus{\mathcal{S}_{\ell%
}}}$, while substituting for $Z_{\!{\scriptscriptstyle F}{\!,k}}^{\setminus{\mathcal{S}_{r}}}(\mathcal{S}_{%
\ell})$, using Eq. (63) in all other terms, we find
$$\displaystyle Z_{\!{\scriptscriptstyle F}{\!,N-r}}^{\setminus{\mathcal{S}_{r}}}=$$
$$\displaystyle\sum_{k=0}^{\ell-r-1}\sum_{m=0}^{k}(-1)^{m}Z_{\!{%
\scriptscriptstyle B}{\!,m}}(\mathcal{S}_{r})Z_{\!{\scriptscriptstyle F}{\!,k-%
m}}(\mathcal{S}_{\ell})Z_{\!{\scriptscriptstyle F}{\!,N-r-k}}^{\setminus{%
\mathcal{S}_{\ell}}}$$
$$\displaystyle+Z_{\!{\scriptscriptstyle F}{\!,\ell-r}}^{\setminus{\mathcal{S}_{%
r}}}(\mathcal{S}_{\ell})Z_{\!{\scriptscriptstyle F}{\!,N-\ell}}^{\setminus{%
\mathcal{S}_{\ell}}}.$$
(64)
After changing the order of the summations and shifting the indexes $k\to k-r+1$ and $m\to m-r$, we obtain the unwieldy expression
$$\displaystyle Z_{\!{\scriptscriptstyle F}{\!,N-r}}^{\setminus{\mathcal{S}_{r}}}$$
$$\displaystyle=(-1)^{r-1}\!\!\sum_{m=r}^{\ell-1}(-1)^{m+1}Z_{\!{%
\scriptscriptstyle B}{\!,m-r}}(\mathcal{S}_{r})$$
$$\displaystyle\times\sum_{k=m-1}^{\ell-2}\!\!\!\!Z_{\!{\scriptscriptstyle F}{\!%
,k-m+1}}(\mathcal{S}_{\ell})Z_{\!{\scriptscriptstyle F}{\!,N-k-1}}^{\setminus{%
\mathcal{S}_{\ell}}}$$
$$\displaystyle+Z_{\!{\scriptscriptstyle F}{\!,\ell-r}}^{\setminus{\mathcal{S}_{%
r}}}(\mathcal{S}_{\ell})Z_{\!{\scriptscriptstyle F}{\!,N-\ell}}^{\setminus{%
\mathcal{S}_{\ell}}}.$$
(65)
Next, we substitute for $Z_{\!{\scriptscriptstyle F}{\!,N-\ell}}^{\setminus{\mathcal{S}_{\ell}}}$ and $Z_{\!{\scriptscriptstyle F}{\!,N-r}}^{\setminus{\mathcal{S}_{r}}}$, using Eq. (22) and the fully occupied fermionic APF $Z_{\!{\scriptscriptstyle F}{\!,\ell-r}}^{\setminus{\mathcal{S}_{r}}}(\mathcal{%
S}_{\ell})=\mathrm{e}^{-\beta\sum_{j_{\nu}\in\mathcal{S}_{\ell}\setminus%
\mathcal{S}_{r}}\epsilon_{j_{\nu}}}$. Multiplying the result by $\mathrm{e}^{-\beta\sum_{i_{\nu}\in\mathcal{S}_{r}}\epsilon_{i_{\nu}}}/Z_{\!{%
\scriptscriptstyle F}{\!,N}}$ yields:
$$Y_{\!{\scriptscriptstyle F}{\!,0}}(\mathcal{S}_{\ell})+\sum_{m=r}^{\ell-1}A_{m%
}(\mathcal{S}_{r})Y_{\!{\scriptscriptstyle F}{\!,m}}(\mathcal{S}_{\ell})=b_{F}%
(\mathcal{S}_{r})$$
(66)
where
$$\displaystyle Y_{\!{\scriptscriptstyle F}{\!,0}}(\mathcal{S}_{\ell})$$
$$\displaystyle=\expectationvalue{n_{j_{1}}n_{j_{2}}\dots n_{j_{\ell}}}_{\!{%
\scriptscriptstyle F}{\!,N}}$$
(67)
$$\displaystyle Y_{\!{\scriptscriptstyle F}{\!,1\leq m\leq\ell-1}}(\mathcal{S}_{%
\ell})$$
$$\displaystyle=\frac{(-1)^{m+1}}{Z_{\!{\scriptscriptstyle F}{\!,N}}}\!\!\!\!%
\sum_{k=m-1}^{\ell-2}\!\!\!\!Z_{\!{\scriptscriptstyle F}{\!,k-m+1}}(\mathcal{S%
}_{\ell})Z_{\!{\scriptscriptstyle F}{\!,N-k-1}}^{\setminus{\mathcal{S}_{\ell}}}$$
(68)
are independent of $\mathcal{S}_{r}$. Therefore, for each choice of the subset
$\mathcal{S}_{r}$ we can write the linear nonhomogeneous equation
(66) in the $\ell$ variables $Y_{\!{\scriptscriptstyle F}{\!,m}}$ with coefficients:
$A_{0}=1$, $A_{0<m<r}=0$ and $A_{r\leq m\leq\ell-1}(\mathcal{S}_{r})=(-1)^{r-1}\mathrm{e}^{-\beta\sum_{i_{%
\nu}\in\mathcal{S}_{r}}\epsilon_{i_{\nu}}}Z_{\!{\scriptscriptstyle B}{\!,m-r}}%
(\mathcal{S}_{r})$. The homogeneity of the linear equation is violated by the term $b_{F}(\mathcal{S}_{r})=\expectationvalue{n_{i_{1}}n_{i_{2}}\dots n_{i_{r}}}_{%
\!{\scriptscriptstyle F}{\!,N}}$.
With this formulation, we observe that for any of the $2^{\ell}-2$ choices of $\mathcal{S}_{r}$, we can write a linear equation in the same $\ell$ variables $Y_{\!{\scriptscriptstyle F}{\!,m}}$, where $Y_{\!{\scriptscriptstyle F}{\!,0}}=\expectationvalue{n_{j_{1}}n_{j_{2}}\dots n%
_{j_{\ell}}}_{\!{\scriptscriptstyle F}{\!,N}}$ is the $\ell$-point correlation while the remaining $\ell-1$ variables are auxiliary, and depend symmetrically on the levels in $\mathcal{S}_{\ell}$. Also, the $r$-point correlation $\expectationvalue{n_{i_{1}}n_{i_{2}}\dots n_{i_{r}}}_{\!{\scriptscriptstyle F}%
{\!,N}}$ of the levels in $\mathcal{S}_{r}$ plays the role of the nonhomogeneous term in the linear equation and the coefficients $A_{m}$ of the equation can be determined by the bosonic APFs of $\mathcal{S}_{r}$. An analogous expression can be obtained for bosonic statistics, with the same coefficients $A_{m}$
$$Y_{\!{\scriptscriptstyle B}{\!,0}}(\mathcal{S}_{\ell})+\sum_{m=r}^{\ell-1}A_{m%
}(\mathcal{S}_{r})Y_{\!{\scriptscriptstyle B}{\!,m}}(\mathcal{S}_{\ell})=b_{B}%
(\mathcal{S}_{r}),$$
(69)
where, in this case, the variables are
$$\displaystyle Y_{\!{\scriptscriptstyle B}{\!,1\leq m\leq\ell-1}}(\mathcal{S}_{%
\ell})=\\
\displaystyle\frac{(-1)^{m+1}}{Z_{\!{\scriptscriptstyle B}{\!,N}}}\sum_{k=m-1}%
^{\ell-2}(-1)^{k}Z_{\!{\scriptscriptstyle F}{\!,k-m+1}}(\mathcal{S}_{\ell})Z_{%
\!{\scriptscriptstyle B}{\!,N-k-1}}^{\cup{\mathcal{S}_{\ell}}},$$
(70)
with $Y_{\!{\scriptscriptstyle B}{\!,0}}(\mathcal{S}_{\ell})=\quantity(-1)^{\ell-1}%
\expectationvalue{n_{j_{1}}n_{j_{2}}\dots n_{j_{\ell}}}_{\!{\scriptscriptstyle
B%
}{\!,N}}$ and $b_{B}(\mathcal{S}_{r})=(-1)^{r-1}\expectationvalue{n_{i_{1}}n_{i_{2}}\dots n_{%
i_{r}}}_{\!{\scriptscriptstyle B}{\!,N}}$ (see Appendix A for a complete derivation).
III.2.1 Non-degenerate levels
Consider the set $\mathcal{S}_{\ell}$ specifying a set of distinct energy levels and choose $\mathcal{S}_{r=1}$ such that it contains only one of the $\ell$ levels in $\mathcal{S}_{\ell}$. We can use Eq. (66) or (69) to construct a set of $\ell$ linear equations each corresponding to one level $j_{s}\in\mathcal{S}_{\ell}$ with energy $\epsilon_{j_{s}}$. For fermions the equations are
$$\expectationvalue{n_{j_{s}}}_{\!{\scriptscriptstyle F}{\!,N}}=\sum_{m=0}^{\ell%
-1}\mathrm{e}^{-m\beta\epsilon_{j_{s}}}Y_{\!{\scriptscriptstyle F}{\!,m}}(%
\mathcal{S}_{\ell}),$$
(71)
where the coefficients $A_{m}$ were obtained from the single-level bosonic APF $Z_{\!{\scriptscriptstyle B}{\!,m}}(\{j_{s}\})=\mathrm{e}^{-m\beta\epsilon_{j_{%
s}}}$ in Eq. (12). Therefore, using the set of the $\ell$ independent linear equations in $\ell$ variables defined by Eq. (71), we can solve for $Y_{\!{\scriptscriptstyle F}{\!,0}}(\mathcal{S}_{\ell})=\expectationvalue{n_{j_%
{1}}n_{j_{2}}\dots n_{j_{\ell}}}_{\!{\scriptscriptstyle F}{\!,N}}$ as
$$\displaystyle\expectationvalue{n_{j_{1}}\dots n_{j_{\ell}}}_{\!{%
\scriptscriptstyle F}{\!,N}}$$
$$\displaystyle=\frac{\begin{vmatrix}\expectationvalue{n_{j_{1}}}_{\!{%
\scriptscriptstyle F}{\!,N}}&\mathrm{e}^{-\beta\epsilon_{j_{1}}}&\dots&\mathrm%
{e}^{-\beta(\ell-1)\epsilon_{j_{1}}}\\
\vdots&\vdots&\ddots&\vdots\\
\expectationvalue{n_{j_{\ell}}}_{\!{\scriptscriptstyle F}{\!,N}}&\mathrm{e}^{-%
\beta\epsilon_{j_{\ell}}}&\dots&\mathrm{e}^{-\beta(\ell-1)\epsilon_{j_{\ell}}}%
\\
\end{vmatrix}}{\begin{vmatrix}1&\mathrm{e}^{-\beta\epsilon_{j_{1}}}&\dots&%
\mathrm{e}^{-\beta(\ell-1)\epsilon_{j_{1}}}\\
\vdots&\vdots&\ddots&\vdots\\
1&\mathrm{e}^{-\beta\epsilon_{j_{\ell}}}&\dots&\mathrm{e}^{-\beta(\ell-1)%
\epsilon_{j_{\ell}}}\end{vmatrix}}.$$
(72)
This result was recently obtained by Giraud, Grabsch and Texier Giraud et al. (2018), using the properties of the Schur functions and it can be simplified using Vandermonde determinants:
$$\displaystyle\expectationvalue{n_{j_{1}}n_{j_{2}}\dots n_{j_{\ell}}}_{\!{%
\scriptscriptstyle F}{\!,N}}$$
$$\displaystyle=\sum_{s=1}^{\ell}(-1)^{s-1}\expectationvalue{n_{j_{s}}}_{\!{%
\scriptscriptstyle F}{\!,N}}\mathrm{e}^{-\beta\sum_{j_{i}\neq j_{s}}\epsilon_{%
j_{i}}}$$
$$\displaystyle\quad\times\frac{\mathbb{V}^{\setminus\{j_{s}\}}\quantity(\mathrm%
{e}^{-\beta\epsilon_{j_{1}}},\dots,\mathrm{e}^{-\beta\epsilon_{j_{\ell}}})}{%
\mathbb{V}\quantity(\mathrm{e}^{-\beta\epsilon_{j_{1}}},\dots,\mathrm{e}^{-%
\beta\epsilon_{j_{\ell}}})},$$
(73)
where
$$\mathbb{V}\quantity(\alpha_{1},\dots,\alpha_{\ell})=\begin{vmatrix}1&\alpha_{1%
}&\dots&\alpha_{1}^{\ell-1}\\
\vdots&\vdots&\ddots&\vdots\\
1&\alpha_{\ell}&\dots&\alpha_{\ell}^{\ell-1}\end{vmatrix}=\prod_{i<j}\quantity%
(\alpha_{j}-\alpha_{i}).$$
(74)
Thus the fermionic $\ell$-level correlation can be simplified as:
$$\expectationvalue{n_{j_{1}}n_{j_{2}}\dots n_{j_{\ell}}}_{\!{\scriptscriptstyle
F%
}{\!,N}}=\sum_{r=1}^{\ell}\frac{\expectationvalue{n_{j_{r}}}_{\!{%
\scriptscriptstyle F}{\!,N}}}{\prod_{k\neq r}\quantity[1-\mathrm{e}^{\beta%
\quantity(\epsilon_{j_{k}}-\epsilon_{j_{r}})}]}.$$
(75)
This expression was also recently derived using an elegant second quantization schemeSchönhammer (2017).
An equivalent procedure can be performed for bosons, again using the set $\mathcal{S}_{r=1}$ to yield $\ell$ linearly independent equations
$$\expectationvalue{n_{j_{r}}}_{\!{\scriptscriptstyle B}{\!,N}}=\sum_{m=0}^{\ell%
-1}\mathrm{e}^{-m\beta\epsilon_{j_{r}}}Y_{\!{\scriptscriptstyle B}{\!,m}}(%
\mathcal{S}_{\ell}),$$
(76)
such that $\expectationvalue{n_{j_{1}}n_{j_{2}}\dots n_{j_{\ell}}}_{\!{\scriptscriptstyle
B%
}{\!,N}}=\quantity(-1)^{\ell-1}Y_{\!{\scriptscriptstyle B}{\!,0}}(\mathcal{S}_%
{\ell})$ can also be expressed in terms of determinants
leading toGiraud et al. (2018)
$$\expectationvalue{n_{j_{1}}n_{j_{2}}\dots n_{j_{\ell}}}_{\!{\scriptscriptstyle
B%
}{\!,N}}=\quantity(-1)^{\ell-1}\sum_{r=1}^{\ell}\frac{\expectationvalue{n_{j_{%
r}}}_{\!{\scriptscriptstyle B}{\!,N}}}{\prod_{k\neq r}\quantity[1-\mathrm{e}^{%
\beta\quantity(\epsilon_{j_{k}}-\epsilon_{j_{r}})}]}.$$
(77)
III.2.2 Degenerate levels
Up until this point we have considered $\ell$-point level correlations in two opposite regimes: (I) when all $\ell$-levels are degenerate Eq. (48) provides a direct route to the correlations through the determination of all partition functions up to $N$ particles, and (II) when all $\ell$ levels are distinct, an associated set of $\ell$ linear equations yields the correlations in terms of individual level occupation numbers. The independence of these linear equations, and thus the existence of a unique solution, is violated in the presence of degeneracy.
We now study the most general possible $\ell$-level correlation function defined by the set $\mathcal{S}_{\ell}$ which could include both degenerate and non-degenerate levels. Consider the subset of level indices $\mathcal{S}_{m_{i}}=\{i^{(0)},\dots,i^{(m_{i}-1)}\}\subset\mathcal{S}_{\ell}$ which contains $m_{i}>1$ degenerate levels (for $m_{i}=1$, we reproduce the non-degenerate analysis discussed above). As a result, the corresponding $m_{i}$ equations, out of the total set of $\ell$ linear equations defined by Eqs. (71) and (76) for fermions and bosons respectively are identical, as they are distinguished from each other only via the energies of the involved levels and their occupation numbers. Moreover, $\mathcal{S}_{\ell}$, could contain multiple subsets of degenerate energy levels, further complicating the problem. Eqs. (75) and (77) can not be applied in this case, as is apparent from their vanishing denominators whenever $\epsilon_{j_{k}}=\epsilon_{j_{r}}$.
To resolve the complication introduced by degenerate subsets, we generalize the procedure in Sec.III.1.2 to treat the case where a three-level correlation contained a subset of 2 degenerate levels. More explicitly, we relate the $\ell$-points correlations not only to the occupation numbers of the degenerate subsets, but to all of the $m_{i}$ distinct $r$-points correlations between the degenerate levels with $2\leq r\leq m_{i}$. This is useful as we have already introduced Eq. (48), which simplifies the calculation of the correlations between degenerate energy levels for fermionic and bosonic statistics. In addition, it will result in the generation of $m_{i}$ new independent equations that could be used to calculate $\ell$-points correlations.
Accordingly, for any choice of $\mathcal{S}_{r}=\{i^{(0)},\dots,i^{(r-1)}\}\subset\mathcal{S}_{m_{i}}\subset%
\mathcal{S}_{\ell}$, the corresponding coefficients in the constructed linear equations are $A_{0}=1$, $A_{0<m<r}=0$ and
$$A_{r\leq m\leq\ell-1}(\mathcal{S}_{r})=(-1)^{r-1}\binom{m-1}{r-1}\mathrm{e}^{-%
m\beta\epsilon_{i}},$$
(78)
where $\epsilon_{i}$ is the energy of all degenerate levels in $\mathcal{S}_{m_{i}}$ and
we have substituted for $Z_{\!{\scriptscriptstyle B}{\!,m-r}}(\{i^{(0)},\dots,i^{(r)}\})=\binom{m-1}{r-%
1}\mathrm{e}^{-(m-r)\beta\epsilon_{i}}$, a bosonic partition function of $r$ degenerate levels.
The non-homogeneities $b_{\zeta}(\mathcal{S}_{r})=(-\zeta)^{r-1}\expectationvalue{n_{i^{(1)}}n_{i^{(2%
)}}\dots n_{i^{(r-1)}}}_{\!{\scriptscriptstyle\zeta}{\!,N}}$ can be
calculated using Eq. (48), and thus, the original set of $m_{i}$ identical equations, can now be replaced with the following $m_{i}$ independent equations
$$Y_{\!{\scriptscriptstyle\zeta}{\!,0}}(\mathcal{S}_{\ell})+\sum_{m=r}^{\ell-1}(%
-1)^{r-1}\binom{m-1}{r-1}\mathrm{e}^{-m\beta\epsilon_{i}}Y_{\!{%
\scriptscriptstyle\zeta}{\!,m}}(\mathcal{S}_{\ell})=b_{\zeta}(\mathcal{S}_{r}),$$
(79)
for fermions $(\zeta=-1)$ and bosons $(\zeta=+1)$.
To illustrate how this works in practice, consider the $4$-point correlation of the levels $\{j_{1^{(0)}},j_{1^{(1)}},j_{1^{(2)}},j_{2}\!\!\}$, labeling distinct energies $\epsilon_{1}$ and $\epsilon_{2}$. The resulting new set of $\ell$ equations can be solved for both fermions and bosons to give:
$$\displaystyle\expectationvalue{n_{j_{1^{(0)}}}n_{j_{1^{(1)}}}n_{j_{1^{(2)}}}n_%
{j_{2}\!\!}}_{\!{\scriptscriptstyle\zeta}{\!,N}}$$
$$\displaystyle=(-\zeta)\frac{\begin{vmatrix}\expectationvalue{n_{j_{1^{(0)}}}n_%
{j_{1^{(1)}}}n_{j_{1^{(2)}}}}_{\!{\scriptscriptstyle\zeta}{\!,N}}&0&0&\mathrm{%
e}^{-3\beta\epsilon_{j_{1}}}\\
(-\zeta)\expectationvalue{n_{j_{1^{(0)}}}n_{j_{1^{(1)}}}}_{\!{%
\scriptscriptstyle\zeta}{\!,N}}&0&-\mathrm{e}^{-2\beta\epsilon_{j_{1}}}&-2%
\mathrm{e}^{-3\beta\epsilon_{j_{1}}}\\
\expectationvalue{n_{j_{1^{(0)}}}}_{\!{\scriptscriptstyle\zeta}{\!,N}}&\mathrm%
{e}^{-\beta\epsilon_{j_{1}}}&\mathrm{e}^{-2\beta\epsilon_{j_{1}}}&\mathrm{e}^{%
-3\beta\epsilon_{j_{1}}}\\
\expectationvalue{n_{j_{2}}\!\!}_{\!{\scriptscriptstyle\zeta}{\!,N}}&\mathrm{e%
}^{-\beta\epsilon_{j_{2}}}&\mathrm{e}^{-2\beta\epsilon_{j_{2}}}&\mathrm{e}^{-3%
\beta\epsilon_{j_{2}}}\\
\end{vmatrix}}{\begin{vmatrix}1&0&0&\mathrm{e}^{-3\beta\epsilon_{j_{1}}}\\
1&0&-\mathrm{e}^{-2\beta\epsilon_{j_{1}}}&-2\mathrm{e}^{-3\beta\epsilon_{j_{1}%
}}\\
1&\mathrm{e}^{-\beta\epsilon_{j_{1}}}&\mathrm{e}^{-2\beta\epsilon_{j_{1}}}&%
\mathrm{e}^{-3\beta\epsilon_{j_{1}}}\\
1&\mathrm{e}^{-\beta\epsilon_{j_{2}}}&\mathrm{e}^{-2\beta\epsilon_{j_{2}}}&%
\mathrm{e}^{-3\beta\epsilon_{j_{2}}}\\
\end{vmatrix}}.$$
(80)
IV Applications
To illustrate the applicability of our results for degenerate non-interacting systems of particles with fixed number, and highlight the practical usage of Eq. (80) we consider a one-dimensional tight-binding chain of $N$ spinfull bosons hopping over $L$ lattice sites. Code, scripts, and data used to produce all figures for the bosonic chain results can be found online rep (2020). The bosonic chain is subject to a static external magnetic field $B$ applied along the $z$-axis. Spin-$S$ bosons are described by the Hamiltonian
$$\widehat{H}=-t\sum_{\alpha,\sigma}\left(\hat{a}^{\dagger}_{\alpha+1,\sigma}%
\hat{a}_{\alpha,\sigma}+\text{h.c.}\right)-h\sum_{\alpha,\sigma,\sigma^{\prime%
}}\hat{a}^{\dagger}_{\alpha,\sigma}{S}^{z}_{\sigma,\sigma^{\prime}}\hat{a}_{%
\alpha,\sigma^{\prime}},$$
(81)
where $\hat{a}^{\dagger}_{\alpha,\sigma}$ and $\hat{a}_{\alpha,\sigma}$ are creation and annihilation operators for a boson at site $\alpha$ with $\sigma\in\{-S,\dots,0,\dots,S\}$ satisfying $[a_{\alpha,\sigma},a^{\dagger}_{\alpha^{\prime},\sigma^{\prime}}]=\delta_{%
\alpha,\alpha^{\prime}}\delta_{\sigma,\sigma^{\prime}}$ and $t$ measures the hopping amplitude. ${S}^{z}_{\sigma,\sigma^{\prime}}=\sigma\delta_{\sigma,\sigma^{\prime}}$ are the matrix elements of the diagonal $z$-projection of the spin-$S$ representation of the spin operator $\hat{\boldsymbol{S}}$. Here, $h=g{\rm{\mu_{B}}}B$, where $g$ is the corresponding spin-$S$ $g$-factor and $\mu_{B}$ is the Bohr magneton. We employ periodic boundary conditions, such that $\hat{a}_{L+1,\sigma}=\hat{a}_{1,\sigma}$, and to avoid having an unbalanced non-degenerate excited state, we fix the parity of $L$ to be odd.
The tight-binding Hamiltonian in Eq. (81) can be diagonalized
$$\widehat{H}=\sum_{j,\sigma}\epsilon_{j,\sigma}n_{j,\sigma},$$
(82)
where $\hat{n}_{j,\sigma}$ counts the number of bosons with energy
$$\epsilon_{j,\sigma}=-2t\cos\quantity(\frac{2\pi j}{L})-h\sigma,$$
(83)
and $j$ runs over the finite set $\mathcal{S}=\{-\frac{L-1}{2},\dots,0,\dots,\frac{L-1}{2}\}$ when $L$ is odd such that $\absolutevalue{\mathcal{S}}=L$. An examination of the single-particle spectrum shows that each energy level, except the ground state $\epsilon_{0,\sigma}=-2t-h\sigma$, is $2$-fold degenerate, where $\epsilon_{-j,\sigma}=\epsilon_{j,\sigma}$; a result of the right-left symmetry of the chain. Turning off the magnetic field and fixing $S>0$, gives rise to an extra degeneracy factor of $(2S+1)$ that affects all levels.
For all numerical results presented in this section, we fix $L=1001$, $N=1000$ and measure the inverse temperature $\beta=\frac{1}{k_{\rm B}T}$ in units of $1/t$, where $T$ is the absolute temperature and $k_{\rm{B}}$ is the Boltzmann constant.
IV.1 Spinless bosons $(S=0)$
We begin with the study of spinless bosons, where the model is insensitive to the applied magnetic field and we can drop the subscript $\sigma$ without loss of generality. Using the single-particle spectrum defined in Eq. (83) with $\sigma=0$ and $h=0$
in combination with Eq. (27), we calculate the joint probability distribution $\mathcal{P}_{\!\!{\scriptscriptstyle B}{\!,n_{0},n_{1}}}$ of the occupation numbers of the ground state and the first excited state, where, we choose the level $j=1$ out of the two degenerate levels $j=\pm 1$. Note that we do not bother to use the superscript notation to distinguish degenerate level indices $(1\equiv 1^{(0)},-1\equiv 1^{(1)})$ here as there are no ambiguities due to the sign of the index $j$.
The calculation proceeds by obtaining the APFs $Z_{\!{\scriptscriptstyle B}{\!,k}}^{\setminus\{{0,1}\}}$ for $0\leq k\leq N$ using the recursion relation Eq. (41), where the factors $C_{k}$ are calculated using the spectrum $\mathcal{S}\setminus\quantity{0,1}$. The resulting distribution is
$$\mathcal{P}_{\!\!{\scriptscriptstyle B}{\!,n_{0},n_{1}}}=\frac{\mathrm{e}^{-%
\beta\quantity(\epsilon_{0}n_{0}+\epsilon_{1}n_{1})}}{Z_{\!{\scriptscriptstyle
B%
}{\!,N}}}Z_{\!{\scriptscriptstyle B}{\!,N-n_{0}-n_{1}}}^{\setminus\{{0,1}\}}.$$
(84)
where $Z_{\!{\scriptscriptstyle B}{\!,N}}$ can be found by enforcing normalization.
We expect Eq. (84) to exhibit interesting features at low temperature where the particles are mostly occupying the ground state with some fluctuations amongst the low lying energy levels. To obtain an estimate of low in this context, we choose a value of the inverse temperature $\beta$ such that the ground state has a macroscopic occupation corresponding to 50% of the particles. We compare the ratio of the Boltzmann factors of having all particles in the ground state with that of having $N/2$ particles in the first excited state and the rest in the ground state. Setting the ratio of these factors $\mathrm{e}^{-\beta\quantity(\epsilon_{0}-\epsilon_{1})N/2}$ to $\sim 0.1$, suggests $\beta\sim 100/t$. The results are illustrated in the left panel of Fig. 1,
where the relative broadness of the distribution can be attributed to the degeneracy of the first exited level $j=\pm 1$.
If we now calculate the three-level joint probability distribution $\mathcal{P}_{\!\!{\scriptscriptstyle B}{\!,n_{0},n_{1},n_{-1}}}$ and consider the fixed slice with $n_{-1}=0$, as presented in the right panel of Fig. 1, we see that the distribution becomes significantly sharper, as blocking the level $j=-1$ makes the resulting non-normalized conditional distribution more sensitive to the conservation of the total number of particles.
We now turn to the calculation of the two-level connected correlation function for our bosonic system
$$\mathcal{C}(n_{i},n_{j})=\expectationvalue{n_{i}n_{j}}_{\!{\scriptscriptstyle B%
}{\!,N}}-\expectationvalue{n_{i}}_{\!{\scriptscriptstyle B}{\!,N}}%
\expectationvalue{n_{j}}_{\!{\scriptscriptstyle B}{\!,N}}.$$
(85)
The first step is to obtain the system partition function $Z_{\!{\scriptscriptstyle B}{\!,k}}$, recursively, using Eq. (41) starting from $Z_{\!{\scriptscriptstyle B}{\!,0}}$ up to $Z_{\!{\scriptscriptstyle B}{\!,N}}$. The occupation numbers $\expectationvalue{n_{j}}_{\!{\scriptscriptstyle B}{\!,N}}$ can then be easily calculated using Eq. (40). All that remains is to calculate the two-points correlations using Eq. (56) for the non-degenerate levels. For the correlations between degenerate levels ($\expectationvalue{n_{-j}n_{j}}_{\!{\scriptscriptstyle B}{\!,N}}$), we use Eq. (48), with $\zeta=+1$.
The results for $\mathcal{C}(n_{i},n_{j})$ for all levels $i$ and $j$ at inverse temperature $\beta=1/t$ are shown as a heat-map in the lower panel of Fig. 2.
Here, we chose temperature $\beta=1/t$ in order to distribute the correlations amongst higher energy levels. The upper panel of Fig. 2 shows $\mathcal{C}(n_{i},n_{j})$ as a function of $n_{i}$ for fixed $0\leq j\leq 500$ corresponding to horizontal cuts through the lower panel. The red open circles are correlations $\mathcal{C}(n_{-j},n_{j})$ for the degenerate levels
obtained from Eq. (48) demonstrating consistency with the rest of the graph. In the positive quadrant of the correlation heat-map, we use the values of $\expectationvalue{\tbinom{n_{j}}{2}}_{\!{\scriptscriptstyle B}{\!,N}}$ (marked with black circles) instead of $\expectationvalue{n_{j}^{2}}_{\!{\scriptscriptstyle B}{\!,N}}$, were the former is also consistent with surrounding data, as expected from Eq. (51), where $\expectationvalue{n_{-j}n_{j}}_{\!{\scriptscriptstyle B}{\!,N}}=%
\expectationvalue{\tbinom{n_{j}}{2}}_{\!{\scriptscriptstyle B}{\!,N}}$ and the symmetry $\mathcal{C}(n_{i},n_{j})=\mathcal{C}(n_{-i},n_{j})$ due to the degeneracy.
IV.2 Spin-$1$ bosons $(S=1)$
To illustrate the utility of auxiliary partitions functions in studying correlations in a highly degenerate spectrum, we consider the case of spin-$1$ bosons. In the absence of a magnetic field ($h=0$), each level picks up a degeneracy factor of $2S+1=3$, such that the ground state is three-fold degenerate and all of the excitation levels are six-fold degenerate.
Degeneracy effects are apparent in the two-level connected correlations $\mathcal{C}(n_{i,\sigma},n_{j,\sigma^{\prime}})$ at $\beta=1/t$, which we calculate for various values of $h$ as shown in Fig.3.
The top-left panel of the figure presents $\mathcal{C}(n_{i,\sigma},n_{j,\sigma^{\prime}})$ for $h=0$, where the choice of $\sigma$ and $\sigma^{\prime}$ matters only in the presence of a magnetic field. A comparison with the heat-map of Fig. 2, shows an overall broadening and a reduction of one order of magnitude in the maximum of $\mathcal{C}(n_{i,\sigma},n_{j,\sigma^{\prime}})$ for $S=1$, as compared to $S=0$ case.
Removing the energy-spin degeneracy by applying a strong magnetic field of $h=5t$, results in a splitting of the spectrum into three bands, each with bandwidth $4t$ and separated from each other via an energy bandgap of $t$. In this case, we first focus on correlations between the levels in the lower energy band ($\sigma=1$, bottom-left panel of Fig. 3), and see a partial recovery of the spinless bosons case (Fig. 2). Correlations that involve higher energy levels ($\sigma=0$ and $\sigma=-1$) are orders of magnitude weaker, at the considered temperatures, as shown in the right panels of Fig .3.
Finally, we turn to correlations between a set of levels that is partially degenerate, an interesting feature of spin-1 bosons. We consider the four-level (disconnected) correlations $\expectationvalue{n_{i,1}n_{j,1}n_{j,0}n_{j,-1}}_{\!{\scriptscriptstyle B}{\!,%
N}}$ between the levels $\epsilon_{i,1}$, $\epsilon_{j,1}$, $\epsilon_{j,0}$ and $\epsilon_{j,-1}$, where, in the absence of a magnetic field, the last three levels are degenerate for any $j$. We employ the bosonic version
of Eq. (80) with $\zeta=1$ with results shown in Fig. 4.
According to Eq. (35), the results that we obtain, in this case, also represent
$\expectationvalue{n_{i,\sigma}\tbinom{n_{j,\sigma^{\prime}}}{3}}_{\!{%
\scriptscriptstyle B}{\!,N}}$, for any $\sigma$ and $\sigma^{\prime}$ $\in\{1,0,-1\}$. For the fully degenerate case $i=-j$, we use Eq. (48). Once more, Eq. (35) guarantees that $\expectationvalue{n_{i,1}n_{-i,1}n_{-i,0}n_{-i,-1}}_{\!{\scriptscriptstyle B}{%
\!,N}}=\expectationvalue{\tbinom{n_{i,\sigma}}{4}}_{\!{\scriptscriptstyle B}{%
\!,N}}$. Therefore, we use $\expectationvalue{\tbinom{n_{i,\sigma}}{4}}_{\!{\scriptscriptstyle B}{\!,N}}$ instead of $\expectationvalue{n_{i,\sigma}^{4}}_{\!{\scriptscriptstyle B}{\!,N}}$, for the diagonal elements of the 4-level disconnected correlations presented in the lower panel of Fig. 4. The consistency of our calculations using different equations and methods described herein is demonstrated in the upper panel in analogy with Fig. 2.
V Discussion
In summary, we have presented a statistical theory of non-interacting identical quantum particles in the canonical ensemble, providing a unified framework that symmetrically captures both fermionic and bosonic statistics. Table 1 includes a listing of our most important results for fermions and bosons. We achieve this by: (1) Representing correlations (Eqs. (23) and (35)) and joint probability distributions ((21) and (27)) via auxiliary partition functions. (2) Deriving general relations between the canonical partition function of a given spectrum and that of the auxiliary partition function describing a spectral subset, as captured by Eqs. (8), (18) and (19).
These key equations can be manipulated to simplify the derivation of the known recursive relations for partition functions in the canonical ensemble and lead immediately to generalizations, and more importantly, provide useful formulas for calculating the correlations between degenerate energy levels and for calculating higher moments of the occupation numbers distribution. Also, Eqs. (23) and (35) can be used to reduce the complexity order of the desired correlations, or, to relate them to the occupation numbers of the involved levels and correlations between entirely degenerate levels (see Eq. (48)). Moreover, the ability to manipulate the way an auxiliary partition function is built out of other ones, allows us to construct a systematic approach towards the decomposition of many-energy level correlations in terms of individual level occupancies. This reflects the additional constraints between energy levels due to fixed $N$ even in the absence of interactions that are not present in a grand canonical description. Thus, we present an approach to working in the canonical ensemble that includes a generalization of Wick’s theorem, where we obtain previous results for non-degenerate levels Schönhammer (2017); Giraud et al. (2018) and extend them to the case of a degenerate spectrum.
Interestingly, despite the substantial difference between fermionic and bosonic statistics, the resulting formulas show evident similarity. If we compare Eqs. (36), (37) and (38). with Eqs. (39), (40) and (41), respectively, we see that the differences between the fermionic and the bosonic formulas can be captured by simple $\pm 1$ factors. In view of the current theory, such similarity is associated with the interplay between fermionic and bosonic auxiliary partition functions. The inverted symmetry between the two distinct statistics is apparent via a comparison of Eqs. (13) and (14) with Eqs. (15) and (16) reflective of the fact that adding a fermionic energy level to the partition function is similar to excluding a bosonic one and vice-versa.
The presented formulas for combining and resolving auxiliary partition functions allows for their construction via different routes which we have utilized to obtain exact expressions for the decomposition of correlations in terms of single-level occupation numbers. These different forms may also have value in overcoming the known numerical instabilities of the recursive formula for the fermionic partition function due to influence alternating signs Schönhammer (2017). In addition, the simplicity of the presented theory suggests a possible generalization to cover different energy-levels occupation-constraints beyond the fermionic and bosonic ones.
We envision the results presented herein could have applications in the computation of entanglement entropy in the presence of super-selection rules, as well as in modelling cold atom experiments. In the context of quantum information, the spectrum of the reduced density matrix corresponding to a mode bipartition of a state of conserved number $N$ of itinerant particles on a lattice can be associated with that of a fictional entanglement Hamiltonian. For non-interacting particles, the entanglement entropy can be obtained via the so-called correlation matrix method Peschel (2003); Peschel and Eisler (2009); Eisler and Peschel (2017); Peschel (2012) which requires the evaluation of the canonical partition function of the resulting non-interacting entanglement Hamiltonian. For trapped ultra-cold atoms at low densities where $N$ is fixed and interactions can be neglected, the analysis of experimental results in the physically correct canonical ensemble provides improved thermometry, especially for the case of fermions.
Finally, the ability to directly study level statistics in the canonical ensemble for bosons and fermions may have pedagogical value in the teaching of statistical mechanics, where the more physical concept of a fixed number of particles is quickly jettisoned and replaced with a grand canonical reservoir for the sake of simplifying derivations.
Acknowledgements.We thank D. Clougerty for bringing our attention to Ref. [Denton et al., 1973] and K. Schönhammer for discussions at an early stage of this work. This research was supported in part by the National Science Foundation (NSF) under awards DMR-1553991 (A.D.) and DMR-1828489 (H.B.). All computations were performed on the Vermont Advanced Computing Core supported in part by NSF award No. OAC-1827314.
Appendix A The derivation of Eq. (69)
Starting with the sets of levels $\mathcal{S}_{r}=\{i_{1},i_{2},\dots,i_{r}\}\subset\mathcal{S}_{\ell}=\{j_{1},j%
_{2},\dots,j_{\ell}\}$ and using Eq. (19) we have
$$Z_{\!{\scriptscriptstyle B}{\!,N-r}}^{\cup{\mathcal{S}_{r}}}=\sum_{k=0}^{\ell-%
r}(-1)^{k}Z_{\!{\scriptscriptstyle F}{\!,k}}^{\setminus{\mathcal{S}_{r}}}(%
\mathcal{S}_{\ell})Z_{\!{\scriptscriptstyle B}{\!,N-r-k}}^{\cup{\mathcal{S}_{%
\ell}}}.$$
(86)
Next, we substitute for $Z_{\!{\scriptscriptstyle F}{\!,k}}^{\setminus{\mathcal{S}_{r}}}(\mathcal{S}_{%
\ell})$, using Eq. (63), except for the last term $(-1)^{\ell-r}Z_{\!{\scriptscriptstyle F}{\!,\ell-r}}^{\setminus{\mathcal{S}_{r%
}}}(\mathcal{S}_{\ell})Z_{\!{\scriptscriptstyle B}{\!,N-\ell}}^{\cup{\mathcal{%
S}_{\ell}}}$ which we separate from the rest of the previous summation, thus we obtain
$$\displaystyle Z_{\!{\scriptscriptstyle B}{\!,N-r}}^{\cup{\mathcal{S}_{r}}}=$$
$$\displaystyle\sum_{k=0}^{\ell-r-1}\!\!\sum_{m=0}^{k}(-1)^{m+k}Z_{\!{%
\scriptscriptstyle B}{\!,m}}(\mathcal{S}_{r})Z_{\!{\scriptscriptstyle F}{\!,k-%
m}}(\mathcal{S}_{\ell})Z_{\!{\scriptscriptstyle B}{\!,N-r-k}}^{\cup{\mathcal{S%
}_{\ell}}}$$
$$\displaystyle+(-1)^{\ell-r}Z_{\!{\scriptscriptstyle F}{\!,\ell-r}}^{\setminus{%
\mathcal{S}_{r}}}(\mathcal{S}_{\ell})Z_{\!{\scriptscriptstyle B}{\!,N-\ell}}^{%
\cup{\mathcal{S}_{\ell}}}.$$
(87)
If we rearrange the summations and perform the indexes change $k\to k-r+1$ and $m\to m-r$, we get
$$\displaystyle Z_{\!{\scriptscriptstyle B}{\!,N-r}}^{\cup{\mathcal{S}_{r}}}\!\!=$$
$$\displaystyle\sum_{m=r}^{\ell-1}(-1)^{m+1}Z_{\!{\scriptscriptstyle B}{\!,m-r}}%
(\mathcal{S}_{r})$$
$$\displaystyle\times\sum_{k=m-1}^{\ell-2}\!\!\!\!(-1)^{k}Z_{\!{%
\scriptscriptstyle F}{\!,k-m+1}}(\mathcal{S}_{\ell})Z_{\!{\scriptscriptstyle B%
}{\!,N-k-1}}^{\cup{\mathcal{S}_{\ell}}}$$
$$\displaystyle+(-1)^{\ell-r}Z_{\!{\scriptscriptstyle F}{\!,\ell-r}}^{\setminus{%
\mathcal{S}_{r}}}(\mathcal{S}_{\ell})Z_{\!{\scriptscriptstyle B}{\!,N-\ell}}^{%
\cup{\mathcal{S}_{\ell}}}.$$
(88)
Now, using Eq. (34), we substitute for $Z_{\!{\scriptscriptstyle B}{\!,N-\ell}}^{\cup{\mathcal{S}_{\ell}}}$ and $Z_{\!{\scriptscriptstyle B}{\!,N-r}}^{\cup{\mathcal{S}_{r}}}$ as well as the APF $Z_{\!{\scriptscriptstyle F}{\!,\ell-r}}^{\setminus{\mathcal{S}_{r}}}(\mathcal{%
S}_{\ell})=\mathrm{e}^{-\beta\sum_{j_{\nu}\in\mathcal{S}_{\ell}\setminus%
\mathcal{S}_{r}}\epsilon_{j_{\nu}}}$. After multiplying the resulting equation by $\frac{(-1)^{r-1}\mathrm{e}^{-\beta\sum_{i_{\nu}\in\mathcal{S}_{r}}\epsilon_{i_%
{\nu}}}}{Z_{\!{\scriptscriptstyle B}{\!,N}}}$, we can write
$$Y_{\!{\scriptscriptstyle B}{\!,0}}+\sum_{m=r}^{\ell-1}A_{B,m}(\mathcal{S}_{r})%
Y_{\!{\scriptscriptstyle B}{\!,m}}=b_{B}(\mathcal{S}_{r}).$$
(89)
where
$$\displaystyle Y_{\!{\scriptscriptstyle B}{\!,1\leq m\leq\ell-1}}=\frac{(-1)^{m%
+1}}{Z_{\!{\scriptscriptstyle B}{\!,N}}}\!\!\!\!\sum_{k=m-1}^{\ell-2}\!\!\!\!(%
-1)^{k}Z_{\!{\scriptscriptstyle F}{\!,k-m+1}}(\mathcal{S}_{\ell})Z_{\!{%
\scriptscriptstyle B}{\!,N-k-1}}^{\cup{\mathcal{S}_{\ell}}},$$
(90)
$Y_{\!{\scriptscriptstyle B}{\!,0}}=(-1)^{\ell-1}\expectationvalue{n_{j_{1}}n_{%
j_{2}}\dots n_{j_{\ell}}}_{\!{\scriptscriptstyle B}{\!,N}}$, $A_{0}=1$, $A_{0<m<r}=0$ and $A_{r\leq m\leq\ell-1}(\mathcal{S}_{r})=(-1)^{r-1}\mathrm{e}^{-\beta\sum_{i_{%
\nu}\in\mathcal{S}_{r}}\epsilon_{i_{\nu}}}Z_{\!{\scriptscriptstyle B}{\!,m-r}}%
(\mathcal{S}_{r})$. Also, the term $b_{B}(\mathcal{S}_{r})=(-1)^{r-1}\expectationvalue{n_{i_{1}}n_{i_{2}}\dots n_{%
i_{r}}}_{\!{\scriptscriptstyle B}{\!,N}}$.
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Here we replace $Z_{\tmspace-{.1667em}{\scriptscriptstyle B}{\tmspace-{.1667em},x}}$
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(2012). |
Collisions of Main-Sequence Stars and the
Formation of Blue Stragglers in Globular Clusters
James C. Lombardi, Jr.11affiliation: Center for Radiophysics and Space Research, Cornell
University,
Ithaca, NY 14853. Email: [email protected]. 22affiliation: Department of Astronomy, Cornell University. ,
Frederic A. Rasio33affiliation: Department of Physics, MIT 6-201, Cambridge, MA 02139.
Email: [email protected]. and
Stuart L. Shapiro44affiliation: Center for Astrophysics & Relativity,
326 Siena Drive, Ithaca, NY 14850
Abstract
We report the results of new SPH calculations of parabolic collisions
between two main-sequence stars in a globular cluster. Such
collisions are directly relevant to the formation of blue stragglers.
In particular, we consider parent stars of mass $M/M_{TO}=0.2,0.5,0.75$,
and $1$, where $M_{TO}$ is the cluster turnoff mass (typically about
$0.8\,M_{\odot}$). Low-mass stars (with $M=0.2M_{TO}$ or $0.5M_{TO}$) are
assumed to be fully convective and are therefore modeled as
$n=1.5$ polytropes. Stars at the turnoff (with $M=M_{TO}$) are
assumed to be mostly radiative and are modeled as $n=3$ polytropes.
Intermediate-mass stars (with $M=0.75M_{TO}$) are modeled as composite
polytropes consisting
of a radiative core with polytropic index $n=3$ and a convective envelope with
$n=1.5$.
We focus our study on the question of hydrodynamic mixing of helium
and hydrogen, which plays a crucial role in determining the observable
characteristics of blue stragglers. In all cases we find that
there is negligible hydrodynamic mixing of helium into the outer
envelope of the merger remnant. The amount of hydrogen mixed into the core of
the merger depends
strongly on the entropy profiles of the two colliding stars. For two
stars with nearly equal masses (and hence entropy profiles) very little
hydrodynamic mixing occurs at all, especially if they are close to the
turnoff point.
This is because the hydrogen-rich material from
both stars maintains, on average, a higher specific entropy than the
helium-rich material. If the two parent stars are close to turnoff,
very little hydrogen is present at the center of the merger remnant
and the main-sequence lifetime of the blue straggler could be very short.
In contrast, during a collision between two stars of
sufficiently different masses (mass ratio $q\lesssim 0.5$),
the hydrogen-rich material originally in the smaller star maintains,
on average, a lower specific
entropy than that of the more massive star and therefore settles
preferentially near the center of the merger remnant.
Through this process, moderately massive blue stragglers
(with masses $M_{TO}\lesssim M_{BS}\lesssim 1.5M_{TO}$) can obtain a significant supply
of fresh hydrogen fuel, thereby extending their main-sequence lifetime.
Based on our results we conclude that, in contrast to what has been
done in previous studies, blue stragglers formed by direct stellar
collisions should not necessarily be assumed to have initially homogeneous
composition profiles. However, we also demonstrate that the final merged
configurations produced by our hydrodynamic calculations, although very close
to hydrostatic equilibrium,
are usually far from thermal equilibrium. Therefore, it is possible
that convective or rotationally-induced mixing could occur on a thermal
timescale, as the merger remnant recontracts to the main-sequence.
celestial mechanics, stellar dynamics – globular clusters:
general – hydrodynamics – stars: blue stragglers – stars: evolution
– stars: interiors – stars: rotation
1 Introduction and Motivation
Blue stragglers are stars that appear along an extension of the main-sequence
(hereafter MS), beyond the turnoff point in the color-magnitude diagram of
a cluster.
It is generally believed that they are more massive objects (mass
$M_{BS}>M_{TO}$)
formed by the merger of two MS stars (each of mass $<M_{TO}$). Merging
can occur through a physical collision, or following the coalescence of the
two components of a close binary system (Leonard 1989; Livio 1993; Stryker
1993;
Bailyn & Pinsonneault 1995). Clear evidence for
binary coalescence has been found in the form of contact binaries among
blue stragglers in the low-density globular clusters NGC 5466 (Mateo et
al. 1990) and M71 (Yan & Mateo 1994), as well as in open clusters
(Kalużny & Ruciński 1993; Milone & Latham 1994; Jahn, Kalużny
& Ruciński 1995). Evidence for stellar collisions
comes from recent detections by HST of large numbers of blue stragglers
concentrated in the cores of some of the densest clusters,
such as M15 (De Marchi & Paresce 1994; Guhathakurta et al. 1995)
and M30 (Yanny et al. 1994), and from the apparent
lack of binaries in such dense systems (Shara et al. 1995).
Collisions can happen directly between two single stars only in the cores
of the densest clusters, but even in somewhat lower-density clusters they
can also happen indirectly, during resonant interactions involving primordial
binaries (Sigurdsson, Davies, & Bolte 1994; Sigurdsson & Phinney 1995;
Davies & Benz 1995). Observational evidence for the existence of primordial
binaries in globular clusters is now well established (Hut et al. 1992;
Cote et al. 1994).
Benz & Hills (1987) performed the first
three-dimensional calculations of direct collisions between two MS
stars. An important conclusion of their pioneering study was that stellar
collisions could lead to thorough mixing of the fluid.
The mixing of fresh hydrogen fuel into the core
of the merger would reset the nuclear clock of the blue straggler,
allowing it to remain on the MS for up to $\sim 10^{9}\,$yr after its
formation.
In subsequent work it was generally assumed that blue stragglers
resulting from stellar collisions were nearly homogeneous, therefore starting
their life close to the zero-age MS, but with an
anomalously high helium abundance coming from the (evolved) parent stars.
In contrast, little hydrodynamic mixing would be expected to occur during the
much gentler process of binary coalescence, which could take place on a
stellar evolution time scale rather than on a dynamical time scale
(Mateo et al. 1990; Bailyn 1992; but see Rasio & Shapiro 1995, and Rasio
1995).
On the basis of these ideas, Bailyn (1992) suggested a way of distinguishing
observationally between the two possible formation processes. The helium
abundance in the envelope of a blue straggler, which reflects the degree
of mixing during its formation process, can affect its observed position in
a color-magnitude diagram. Blue stragglers made from collisions
would have a higher helium abundance in their outer layers than those made
from binary mergers, and this would generally make them appear somewhat
brighter
and bluer. The analysis was carried out by Bailyn & Pinsonneault (1995)
who performed detailed stellar evolution calculations for blue stragglers
assuming various initial profiles. To represent the collisional case, they
again assumed chemically homogeneous initial
profiles with enhanced helium abundances, calculating the total helium mass
from
the age of the cluster and the masses of the parent stars.
In this paper we re-examine the question of mixing in stellar collisions.
We improve on the previous work of Benz & Hills (1987)
by adopting more realistic stellar models,
and by performing numerical calculations with increased spatial resolution.
We use the smoothed particle hydrodynamics (SPH) method, with $N=3\times 10^{4}$
particles for most of our calculations (Benz & Hills 1987 also used SPH, but
with
only $N=1024$ particles).
The colliding stars in our calculations are modeled as polytropes or
composite polytropes (Chandrasekhar 1939;
Rappaport, Verbunt, & Joss 1983; Ruciński 1988), and we adopt a
simple ideal gas equation of state. The polytropic index $n$
relates the pressure and density profiles in the star according to $P\propto\rho^{1+1/n}$. The adiabatic index $\Gamma_{1}={5\over 3}$ for an ideal
gas and we write the equation of state $P=A\rho^{\Gamma_{1}}$. Here
$A$ is a physical parameter related to the local specific entropy $s$
according to $A\propto\exp((\Gamma_{1}-1)s/k)$, where $k$ is Boltzmann’s
constant.
When $\Gamma_{1}\neq 1+1/n$ the quantity $A$, and hence the entropy $s$, has a
non-zero gradient.
Benz & Hills (1987) used $n=1.5$, $\Gamma_{1}={5\over 3}$ polytropic
models to represent MS stars. Unfortunately, such models
apply only to very low-mass MS stars with large convective envelopes.
For Population II MS stars, the effective polytropic index (defined in
terms of the degree of central mass concentration) is close to $n=1.5$
only for a mass $M\lesssim 0.4\,M_{\odot}$ (see Lai, Rasio, & Shapiro 1994,
Table 3). The object formed by a merger of two such low-mass stars would
hardly be recognizable as a blue straggler, since it would lie below,
or not far above, the MS turnoff point (typically $M_{TO}\simeq 0.8M_{\odot}$)
in a color-magnitude diagram.
Stars near the MS turnoff point have very shallow convective envelopes and are
much better described by $n=3$, $\Gamma_{1}={5\over 3}$ polytropes
(Eddington’s “standard model”, see, e.g., Clayton 1983).
These stars have a density profile much more centrally concentrated
than that of an $n=1.5$ polytropes, and this fact has important
consequences for the hydrodynamics of collisions.
Population II MS stars with masses in the intermediate range
$0.4\,M_{\odot}\lesssim M\lesssim 0.8\,M_{\odot}$ can be modeled by composite polytropes
with a polytropic index of $n=3$ for the radiative core and
$n=1.5$ for the convective envelope.
Stars close to the MS turnoff point in a cluster are the most relevant to
consider for stellar collision calculations, for two reasons.
First, as the cluster evolves via two-body relaxation,
the more massive MS stars will tend to concentrate
in the dense cluster core, where the collision rate is highest
(see, e.g., Spitzer 1987). Second, collision rates can be increased
dramatically
by the presence of a significant fraction of primordial binaries in the
cluster,
and the more massive MS stars will preferentially tend to be exchanged into
such a binary, or collide with another star, following a dynamical interaction
between two binaries or between a binary and a single star
(Sigurdsson & Phinney 1995).
Lai, Rasio & Shapiro (1993) have calculated collisions between MS stars
modeled by $n=3$, $\Gamma_{1}=5/3$ polytropes, but they focused on
high-velocity (hyperbolic) collisions more relevant to galactic nuclei
than to globular clusters.
The velocity dispersion of globular cluster stars is typically
$\sim 10{\,{\rm km}\,{\rm s}^{-1}}$, which is much smaller than the escape velocity from the stellar
surface. For example, a star of mass $M=0.8\,M_{\odot}$
and radius $R=R_{\odot}$ has an escape velocity $(2GM/R)^{1/2}=552{\,{\rm km}\,{\rm s}^{-1}}$. For this reason, we consider only parabolic collisions in this
paper, i.e., all initial trajectories are assumed to have zero orbital energy.
Our paper is organized as follows. In §2 we describe our
implementation of the SPH method and the numerical setup of our
calculations. In §3, we present the models used for MS stars,
detailing their assumed structure and chemical composition profiles.
We also describe the initial ($t=0$) configuration of the trajectory.
Our results are presented in §4. After describing the results for two
typical collisions in detail, we then characterize the rotation states
and the final profiles of all our merger remnants. We also present a
general method, which does not depend on our particular choice of
initial chemical composition profiles, for calculating the final
profile of any passively advected quantity in the merger remnant. We
conclude our results with an analysis of the numerical accuracy of our
simulations. Finally, in §5, we discuss the astrophysical
implications of our results as well as directions for future work.
2 Numerical Method and Conventions
2.1 The SPH Code
Our numerical calculations are done using the smoothed particle
hydrodynamics (SPH) method (see Monaghan 1992 for a recent review). We
used a modified version of the code developed by Rasio (1991)
specifically for the study of stellar interactions (see Rasio &
Shapiro 1995 and references therein). Since SPH is a Lagrangian method,
in which particles are used to represent fluid elements,
it is ideally suited for the study
of hydrodynamic mixing. Indeed, with the assumption that the gas remains
fully ionized throughout the dynamical evolution,
chemical abundances are passively advected quantities.
Therefore, the chemical
composition in the final fluid configuration can be determined after
the completion of a calculation simply by noting the original and final
positions of all SPH particles and by assigning particle abundances
according to an initial profile.
Associated with each SPH particle
$i$ is its position ${\bf r}_{i}$, velocity ${\bf v}_{i}$, mass $m_{i}$ and a
purely numerical “smoothing length” $h_{i}$ specifying the local
spatial resolution. An estimate of the fluid density at ${\bf r}_{i}$ is
calculated from the masses, positions, and smoothing lengths of
neighboring particles as a local weighted average,
$$\rho_{i}=\sum_{j}m_{j}W_{ij},$$
(1)
where the symmetric weights $W_{ij}=W_{ji}$ are calculated from the
method of Hernquist and Katz (1989), as
$$W_{ij}={1\over 2}\left[W(|{\bf r}_{i}-{\bf r}_{j}|,h_{i})+W(|{\bf r}_{i}-{\bf r%
}_{j}|,h_{j})\right].$$
(2)
Here $W(r,h)$ is an interpolation kernel, for which we use the second-order
accurate form of Monaghan and Lattanzio (1985),
$$W(r,h)={1\over\pi h^{3}}\cases{1-{3\over 2}\left({r\over h}\right)^{2}+{3\over
4%
}\left({r\over h}\right)^{3},&$0\leq{r\over h}<1$,\cr{1\over 4}\left[2-\left({%
r\over h}\right)\right]^{3},&$1\leq{r\over h}<2$,\cr 0,&${r\over h}\geq 2$.\cr}$$
(3)
In addition to passively advected scalar quantities (such as the hydrogen and
helium mass fractions $X_{i}$ and $Y_{i}$)
each particle $i$ also carries the local entropy variable
$A_{i}$. The specific entropy $s_{i}$ at ${\bf r}_{i}$ is
related to $A_{i}$ by
$$s_{i}-s_{o}={k\over\Gamma_{1}-1}\ln A_{i},$$
(4)
where $k$ is Boltzmann’s constant and $s_{o}$ is a fiducial constant. Neglecting
radiation pressure, the
pressure at ${\bf r}_{i}$ is estimated as
$$p_{i}=A_{i}\,\rho_{i}^{\Gamma_{1}},$$
(5)
where $\Gamma_{1}={5\over 3}$ is the ratio of specific heats for a fully
ionized ideal gas. As a self-consistency test, we check that
throughout the dynamical evolution the vast majority of particles have
$p_{i}$ remaining much larger than the radiation pressure ${1\over 3}aT_{i}^{4}$, where $a$ is the radiation constant and $T_{i}$ is the local
temperature (approximated by assuming an ideal gas).
An SPH code must solve the equations of motion of a large number $N$ of
Lagrangian fluid particles. Particle positions are updated according
to
$$\dot{\bf r}_{i}={\bf v}_{i}.$$
(6)
The velocity of particle $i$ is updated according to
$$m_{i}\dot{\bf v}_{i}={\bf F}^{(Grav)}_{i}+{\bf F}^{(SPH)}_{i}$$
(7)
where ${\bf F}^{(Grav)}_{i}$ is the gravitational force calculated by a
particle-mesh convolution algorithm (Hockney and Eastwood 1988, Wells
et al. 1990) based on Fast Fourier Transforms (FFT) on a $128^{3}$
grid, and
$${\bf F}^{(SPH)}_{i}=-\sum_{j}m_{i}m_{j}\left[\left({p_{i}\over\rho_{i}^{2}}+{p%
_{j}\over\rho_{j}^{2}}\right)+\Pi_{ij}\right]\nabla_{i}W_{ij}.$$
(8)
Here $\Pi_{ij}$ is an artificial viscosity term, while the rest of
equation (8) represents one of many possible SPH-estimators for the
local pressure-gradient force $-m_{i}(\nabla p/\rho)_{i}$ (see, e.g., Monaghan
1985).
For the artificial viscosity, a symmetrized version of the form
proposed by Monaghan (1989) is adopted,
$$\Pi_{ij}={-\alpha\mu_{ij}c_{ij}+\beta\mu_{ij}^{2}\over\rho_{ij}},$$
(9)
where $\alpha$ and $\beta$ are constant parameters,
$c_{ij}=(c_{i}+c_{j})/2$, and
$$\mu_{ij}=\cases{{({\bf v}_{i}-{\bf v}_{j})\cdot({\bf r}_{i}-{\bf r}_{j})\over h%
_{ij}(|{\bf r}_{i}-{\bf r}_{j}|^{2}/h_{ij}^{2}+\eta^{2})},&when $({\bf v}_{i}-%
{\bf v}_{j})\cdot({\bf r}_{i}-{\bf r}_{j})<0$,\cr 0,&when $({\bf v}_{i}-{\bf v%
}_{j})\cdot({\bf r}_{i}-{\bf r}_{j})\geq 0$,\cr}$$
(10)
with $h_{ij}=(h_{i}+h_{j})/2$. We have used $\alpha=1$, $\beta=2$ and
$\eta^{2}=0.01$, which provides a good description of shocks (Monaghan
1989, Hernquist and Katz 1989).
To complete the evolution equations of the fluid, $A_{i}$ is evolved
according to a discretized version of the first law of thermodynamics:
$${dA_{i}\over dt}={\gamma-1\over 2\rho_{i}^{\gamma-1}}\,\sum_{j}m_{j}\,\Pi_{ij}%
\,\,({\bf v}_{i}-{\bf v}_{j})\cdot\nabla_{i}W_{ij}.$$
(11)
Equation (11) has the advantage that the total entropy is strictly
conserved in the absence of shocks (ie. when $\Pi_{ij}=0$), and the
disadvantage that the total energy is only approximately conserved
(Rasio 1991; Hernquist 1993). Both total energy and angular momentum
conservation are monitored throughout the integrations as a measure of
numerical accuracy, and these quantities are conserved typically at the
percent level.
The dynamical equations are integrated using a second-order explicit
leap-frog scheme. Such a low order scheme is appropriate because
pressure gradient forces are subject to numerical noise. We calculate
the timestep as $\Delta t=C_{N}\,{\rm Min}(\Delta t_{1},\Delta t_{2})$ where
$\Delta t_{1}={\rm Min}_{i}\,(h_{i}/\dot{\bf v}_{i})^{1/2}$, $\Delta t_{2}={\rm Min}_{i}(h_{i}/(c_{i}^{2}+v_{i}^{2})^{1/2})$ and the Courant number $C_{N}=0.8$.
Other details of our implementation, as well as a number of test-bed
calculations using our SPH code are presented in Lombardi, Rasio,
& Shapiro (1995b).
Twenty of the twenty-three calculations reported here employ
$N=3\times 10^{4}$ equal-mass particles, while the remaining three
calculations (cases U, V and W in Table 2 below) use $N=1.8\times 10^{4}$
equal-mass particles. Unequal-mass SPH particles, sometimes used to
allow for higher resolution in low density regions, tend to settle
spuriously to preferred regions in the gravitational potential due to
numerical discreteness effects. Therefore, calculations with
equal-mass particles are best suited for studying mixing. In all
cases, time-dependent, individual particle smoothing lengths $h_{i}$
insure that the spatial resolution remains acceptable throughout the
dynamical evolution and that each particle interacts with a constant
number of neighbors $N_{N}\simeq 64$. With these resources, the numerical
integration of the SPH equations typically takes about 2 CPU hours per
time unit (eq. [12]) on an IBM SP-2 supercomputer.
2.2 Choice of Units
Throughout this paper, numerical results are given in units where
$G=M_{TO}=R_{TO}=1$, where $G$ is Newton’s gravitational constant and
$M_{TO}$ and $R_{TO}$ are the mass and radius of a terminal-age MS
(TAMS) star at the cluster turnoff point. The units of time, velocity, and
density are then
$$\displaystyle t_{u}$$
$$\displaystyle=$$
$$\displaystyle\left({R_{TO}^{3}\over GM_{TO}}\right)^{1/2}=1782\,{\rm s}\times%
\left({M_{TO}\over 0.8M_{\odot}}\right)^{-1/2}\left({R_{TO}\over R_{\odot}}%
\right)^{3/2}$$
(12)
$$\displaystyle v_{u}$$
$$\displaystyle=$$
$$\displaystyle\left({GM_{TO}\over R_{TO}}\right)^{1/2}=391\,{\rm km}\,{\rm s}^{%
-1}\times\left({M_{TO}\over 0.8M_{\odot}}\right)^{1/2}\left({R_{TO}\over R_{%
\odot}}\right)^{-1/2}$$
(13)
$$\displaystyle\rho_{u}$$
$$\displaystyle=$$
$$\displaystyle{M_{TO}\over R_{TO}^{3}}=5.90\,{\rm g}\,{\rm cm}^{-3}\times\left(%
{M_{TO}\over 0.8M_{\odot}}\right)\left({R_{TO}\over R_{\odot}}\right)^{-3}.$$
(14)
Furthermore, the units of temperature and specific entropy are chosen to be
$$\displaystyle T_{u}$$
$$\displaystyle=$$
$$\displaystyle{GM_{TO}m_{H}\over kR_{TO}}=1.85\times 10^{7}\,{\rm K}\times\left%
({M_{TO}\over 0.8M_{\odot}}\right)\left({R_{TO}\over R_{\odot}}\right)^{-1}$$
(15)
$$\displaystyle s_{u}$$
$$\displaystyle=$$
$$\displaystyle{k\over M_{TO}}=8.68\times 10^{-50}\,{\rm erg}\,{\rm K}^{-1}\,{%
\rm g}^{-1}\times\left({M_{TO}\over 0.8M_{\odot}}\right)^{-1}$$
(16)
where $m_{H}$ is the mass of hydrogen and $k$ is Boltzmann’s constant.
2.3 Determination of the Bound Mass and Termination of the Calculation
The iterative procedure used to determine the total amount of
gravitationally bound mass $M$ of a merger remnant is the same as in
Rasio (1991). Namely, particles with negative specific
enthalpy with respect to the bound fluid’s center of mass are
considered bound. During all of the stellar collisions we considered,
only a small fraction (typically a few percent)
of the mass is ejected and becomes gravitationally unbound.
Some SPH particles, although bound, are ejected so
far away from the system center of mass that it would take many
dynamical times for them to rain back onto the central remnant and settle
into equilibrium. Rather than wait for those particles (which would
allow for more spurious diffusion in the central region,
see §4.5), we terminate the calculation once we are confident that
at least the inner $95\%$ of the mass has settled into equilibrium. We
confirm this by two stability tests. First, we check that the
specific entropy $s$ increases from the center to the surface of the
merger remnant, a sufficient (and necessary for non-rotating stars)
condition for convective stability (see the discussion surrounding
eq. [18]). For rotating merger remnants we also check another
dynamical stability criteria, namely that the specific angular momentum
increases from the poles to the equator along surfaces of constant
entropy (Tassoul 1978).
3 Initial Data
We consider parent MS stars of masses $M=0.2,0.5,0.75$ and $1M_{TO}$.
The stellar radii are taken from the results of evolution calculations
for Population II stars by D’Antona (1987).
Table 1 lists the values we adopt, as well as the radii enclosing
mass fractions $0.9$ and $0.95$.
For the $0.5M_{TO}$ star, we adopt the value $0.29R_{TO}$ for the radius
of the interface between the radiative and convective zones.
These values correspond to an interpolation of the
results of D’Antona (1987) for an age $t=15\,$Gyr.
The $M=0.2$ and $0.5M_{TO}$ stars are modeled as $n=1.5$ polytropes,
whereas the $M=M_{TO}$ stars are $n=3$ polytropes. The $0.75M_{TO}$
stars are modeled as composite polytropes consisting of a radiative
core with index $n=3$ and a convective envelope with $n=1.5$. Figure 1
shows the specific entropy profiles of these models. The convective
regions have constant specific entropy. Note that the
specific entropy in the $M=0.2$ and $0.5M_{TO}$ stars is
everywhere smaller than the minimum specific entropy in the two more
massive stars. This fact plays a central role in understanding the dynamics
of the merger involving either an $M=0.2$ or $0.5M_{TO}$ star with a
more massive star.
We have used the stellar evolution code developed by Sienkiewicz and
collaborators (cf. Sienkiewicz, Bahcall, & Paczyński 1990) to
compute the chemical composition profile in the radiative zones of the
$M=0.75$ and $1M_{TO}$ parent stellar models. We evolved MS stars of
total mass $M=0.6$ and $0.8\,M_{\odot}$, primordial helium abundance
$Y=0.25$, and metallicity $Z=0.001$ for a time $t\simeq 15\,$Gyr. This
brought the $M=0.8\,M_{\odot}$ star to the point of hydrogen exhaustion
at the center. In the convective regions of our $M=0.5,0.75$ and $1M_{TO}$ parent stars, we set a constant helium abundance $Y=0.25$. For
the $M=0.2M_{TO}$ star, we set $Y=0.24$ everywhere. Figure 2 shows
the resulting profiles, which are used to assign the helium abundance to all
the SPH particles in the calculations. The final column in Table 1
gives the total mass fraction of helium in each of the parent stars.
Although the composition profiles do not affect the hydrodynamics of
a collision in any way, they are needed to determine the chemical composition
profile of the merger remnant. In §4.4, we present a method for
applying our results to arbitrary initial composition profiles.
The stars are initially non-rotating and separated by at least 4 times
the radius of the larger star, which allows tidal effects to be
neglected in the initial configuration. The initial velocities are
calculated by approximating the stars as point masses on an orbit with
zero orbital energy and a pericenter separation $r_{p}$. The Cartesian
coordinate system is chosen such that these hypothetical point masses
of mass $M_{1}$ and $M_{2}$ would reach pericenter at positions
$x_{i}=(-1)^{i}(1-M_{i}/(M_{1}+M_{2}))r_{p}$, $y_{i}=z_{i}=0$, where $i=1,2$ and
$i=1$ refers to the more massive star. The orbital plane
is chosen to be $z=0$. With these choices, the center of mass
resides at the the origin.
4 Results
Table 2 lists the values of the most important initial parameters and
final results for all the calculations we performed.
The first column gives the label by which the calculation is
referred to in this paper. The second and third columns give the
masses $M_{1}$ and $M_{2}$ of the colliding stars, in units of
$M_{TO}\simeq 0.8\,M_{\odot}$. Column 4 gives the ratio $r_{p}/(R_{1}+R_{2})$,
where $r_{p}$ is the pericenter separation for the initial orbit and
$R_{1}+R_{2}$ is the sum of the two (unperturbed) stellar radii. This ratio
has the value $0$ for a head-on collision, and $1$ for a grazing encounter.
Note, however, that an encounter with $r_{p}/(R_{1}+R_{2})\gtrsim 1$ can still
lead to a direct collision in the outer envelopes of the two stars
because of the large tidal deformations near pericenter. We did not
attempt to perform any calculations for $r_{p}/(R_{1}+R_{2})>1$ here, for
reasons discussed in §5.
Column 5 gives the initial separation $r_{0}$ in units of $R_{TO}$.
Column 6 gives the final time $t_{f}$ at which the calculation was terminated,
in the unit of equation (12); see §2.3 for a discussion of how the values
of $t_{f}$ were obtained.
Column 7 gives the number $n_{p}$ of successive pericenter interactions that
the stars experience before merging. In general, $n_{p}$ increases with $r_{p}$,
and it is only for very nearly head-on collisions that the two stars merge
immediately after the first impact ($n_{p}=1$ in that case).
Column 8 gives the mass-loss fraction $1-M/(M_{1}+M_{2})$, where $M$
is the mass of the bound fluid in the final merged configuration.
Column 9 gives the ratio $T/|W|$
of rotational kinetic energy to gravitational binding energy of the (bound)
merger remnant in its center-of-mass frame at time $t_{f}$.
Columns 10 and 11 give the velocity
components $V_{x}$ and $V_{y}$ in the units of equation (13) for the merger
remnant’s center of mass at
time $t_{f}$ in the system’s center-of-mass frame.
Since the amount of mass ejected during a parabolic collision is
very small, the merger remnant
never acquires a large recoil velocity. The largest
value of 0.035 in our calculations occurs for case M
and corresponds to a physical speed of about
$14{\,{\rm km}\,{\rm s}^{-1}}$ (for $M_{TO}=0.8\,M_{\odot}$ and $R_{TO}=R_{\odot}$).
This may be large enough to eject the object from the cluster core,
but not to eject it from the entire cluster.
4.1 Discussion of the Results for Two Typical Cases
One of our calculations involving two TAMS stars (Case C) has already
been described by Lombardi, Rasio & Shapiro (1995a).
In this section we discuss in some detail the results of two other
representative cases (E and G).
Figure 3 illustrates the dynamical evolution for Case E:
a TAMS star ($M_{1}=M_{TO}$) collides with a slightly less massive star
($M_{2}=0.75\,M_{TO}$). The initial separation is $r_{0}=5R_{TO}$ and the
parabolic trajectory has a pericenter separation $r_{p}=0.25(R_{1}+R_{2})$.
The first collision at time $t\simeq 4$ disrupts the outer layers
of the two stars,
but leaves their inner cores essentially undisturbed. The two
components withdraw to apocenter at $t\simeq 7$, and by $t\simeq 10$
are colliding for the second, and final, time ($n_{p}=2$).
The merger remnant undergoes some large-amplitude oscillations which
damp away quickly due to shock dissipation.
The final ($t=41$) equilibrium configuration (see
Figure 4) is an axisymmetric, rapidly rotating object ($T/|W|=0.07$).
Figure 5 shows SPH-particle values of the angular velocity $\Omega$ as
a function of radius $r$ in the equatorial plane. We see clearly that
the large envelope of the merger remnant is differentially rotating. The
uniformly rotating core contains only about 15% of the mass.
The angular velocity drops to half its central value
near $r=1.1R_{TO}$, and 80% of mass is enclosed
within the isodensity surface with this equatorial radius. Only about
2% of the total mass is ejected during this collision, and the ejection
is nearly isotropic. As a result, the final recoil
velocity of the merger remnant in the orbital plane is only about $0.007$.
Figure 6 displays the thermal energy $U$, kinetic energy $T$,
gravitational potential energy $W$ and total energy $E=U+T+W$ as a
function of time $t$ for case E. The total energy is conserved to within
$2\%$. Dips in the potential energy $W$ correspond either to
a collision of the two components before final merging or to a maximum
contraction during the subsequent oscillations of the merger remnant.
The criterion we use to distinguish
collisions (which should be included in the number of interactions
$n_{p}$ before the stars merge) from oscillations is that the first local
maximum of $W$ which is lower than the previous local maximum occurs
immediately after the final merging.
The idea behind this criterion is that a collision without merger
ultimately tends to increase the system’s gravitational potential
energy, whereas a merger will decrease the potential energy. For
example, in Figure 6, the local maximum of $W$ at $t\simeq 11$ is
lower than the one at $t\simeq 6$, so that the dips in $W$ at $t\simeq 4$ and $11$ account for the number $n_{p}=2$ of interactions given in Table 2 for
case E. The remaining dips at $t\simeq 12$ and $15$ correspond to the
peak contraction of the merger remnant during oscillations. The value
$n_{p}=2$ obtained here in this way agrees with what one gets
simply by direct visual inspection of the system at various times.
In some cases, however, visual inspection can be
subjective since it is often difficult to recognize two
components connected by a bridge of high-density material
just prior to final merging.
Figure 7 illustrates the dynamical evolution for case G, which involves
a TAMS star ($M_{1}=M_{TO}$) and a low-mass MS star with
$M_{2}=0.5\,M_{TO}$ on a head-on parabolic trajectory with initial
separation $r_{0}=5$. The initial collision occurs at time $t\simeq 4$,
and the stars never separate again. The resulting isodensity surfaces of
the final equilibrium configuration are essentially spherically
symmetric (Figure 8). About 6% of the total mass becomes
gravitationally unbound following the collision, and it is ejected
preferentially in the $+x$-direction. Of this ejected material, 95%
originated in the more massive ($M=M_{TO}$) star.
Figures 9(a) and (b) show the entropy profiles for the final
configurations in cases E and G. Except over the outer few percent of
the mass, where equilibrium has not yet been reached (see §2.3), the specific entropy $s$ is an increasing function of the
interior mass fraction $m/M$. Here $m$ is
the mass inside an isodensity surface, and $M$ is the total bound mass
of the merger remnant. The scatter of the points in Figure 9 is real, since
isodensity surfaces and surfaces of constant entropy do not coincide. The
especially small scatter in Figure 9(b)
demonstrates that the entropy does tend towards spherical symmetry in
non-rotating merger remnants, despite the strong angular dependence of
the shock-heating due to the geometry of the collision.
Even though the density and entropy profiles of both the merger remnant
and parent stars are spherically symmetric in case G, this does not
imply that the chemical composition must also share this symmetry.
Indeed, the effects of anisotropic shock-heating are always evident in
the final spatial distribution of the chemical composition.
On a constant-entropy surface in the final configuration,
particles which have been
shock heated the most necessarily had the lowest entropy prior to the
collision. Since lower entropy material generally has higher helium
abundance (see Figures 1 and 2), shock-heated regions tend to have
higher helium abundances. Generally, fluid elements which reside in the
orbital plane, and especially those which lie along the collision axis in
$r_{p}=0$ cases, are
shielded the least from the shock.
Figure 10
displays the angular distribution of the helium
abundance for the merger remnants of cases E and G, near the interior
mass fractions $m/M=0.25,0.5$ and $0.75$.
The helium abundance
$Y$ peaks in Figure 10(b) and (d) at the polar angle $\theta={\pi\over 2}$ (the equatorial plane), as well as in Figure 10(c) at $\phi=0$ (the
collision axis).
In the $r_{p}\neq 0$ cases, shear in the
differentially rotating merger remnant tends to make the profiles
axisymmetric (see Fig. 10(a)). However, no dynamical motions exist
to circulate the fluid along the meridional directions, and
consequently, on an isodensity surface, the fractional helium abundance
increases from the poles to the equators for both rotating and non-rotating
merger
remnants (cf. Fig. 10(b and d)). Meridional circulation will smooth out these
deviations from compositional spherical symmetry over a timescale much longer
than that treatable by our purely dynamical
code (see related discussion in §5). As a
practical concern, we note that stellar evolution codes, which can use our
merger remnants as initial data, usually assume
spherical symmetry. For these reasons, we often average out the
angular dependence when presenting composition, and other, profiles.
Figures 11(a) and (b) show the helium mass fraction $Y$ as a function of
the interior mass fraction $m/M$ for the final merged configuration in
cases E and G, respectively. The points correspond to the final SPH
particle values, with the long-dashed curve representing their
average. The spread in the points is due to the mixing of the fluid as
well as the fact that the final profiles are not spherically
symmetric. Only a small amount of the observed mixing is due to the
spurious diffusion of SPH particles, i.e., diffusion which is purely a
numerical artifact of the SPH scheme (see §4.5). In case E
(Figure 11(a)), there is a small amount of hydrogen in the core, with the
innermost 1% of the mass being 85% helium and the inner 25% being
60% helium. For both cases E and G, it is immediately apparent that
the helium enrichment in the outer layers is minimal since the fractional
helium abundance is just barely above $Y=0.25$, the value in the outer
layers of the parent stars.
The horizontal line at the bottom of Figure 11(b) corresponds to the
particles in case G which originated in the less massive parent star, star 2,
all of which have a helium abundance $Y=0.25$. Although these
particles are spread over the entire range $0<m/M<1$ in the merger
remnant, they are found preferentially near the center. Of all the
particles which originated in star 2, 69% ultimately end up with
$m/M<0.25$, while only 6% end in the range $0.75<m/M<1$. Essentially,
the entire star 2 has sunk to the center of the merger remnant,
displacing the material in
star 1 and leaving only a small amount of shock heated gas in the
remnant’s outer envelope. Consequently, the hydrogen enrichment in the core
is quite pronounced; all of the innermost 22% of the mass originated
in star 2 and is therefore 75% hydrogen. Furthermore, the helium
abundance jumps to a maximum average abundance exceeding $Y=0.7$ near
$m/M=0.3$. The subsequent stellar evolution of an object with such an
atypical chemical abundance profile could be quite peculiar.
4.2 Rotational Properties of the Merger Remnants
The collisions with $r_{p}\neq 0$ result in rapidly, differentially
rotating merger remnants. Rotating fluid configurations with
$T/|W|\gtrsim 0.14$ are secularly
unstable, and those with $T/|W|\gtrsim 0.26$ are dynamically unstable
(Chandrasekhar 1969; Shapiro and Teukolsky 1983, Chap. 7).
The final merged configurations
are, by definition, dynamically stable, but they could in principle be
secularly unstable. Although some of our calculations produce
merger remnants close to
the secular instability limit, none of them exceed it.
However, extrapolation of our results to larger values of $r_{p}$
suggests that secular instabilities could well develop
in some merger remnants.
Table 3 lists the values of the central angular velocity $\Omega_{0}$ in the
equatorial plane as well as other quantities characterizing the rotation in
the outer layers of the merger remnants. Specifically, for the two mass
fractions $m/M=0.9$ and $m/M=0.95$, we give the values of
the polar and equatorial radii $r_{p}$ and $r_{e}$, the angular velocity
$\Omega$ in the equatorial plane, and the ratio $\Omega^{2}r_{e}/g$ of
centrifugal to gravitational acceleration in the equatorial plane.
We see that $\Omega^{2}r_{e}/g$ can be a significant fraction of unity,
indicating that some configurations are rotating near break-up.
The central angular velocity $\Omega_{0}$ is typically an order of magnitude
larger than the angular velocity $\Omega$ at $m/M=0.95$.
Figure 12 shows contours of the specific angular momentum $\Omega\varpi^{2}$, where $\varpi$ is the cylindrical radius measured from the
rotation axis, in the vertical $(x,z)$ plane for several representative
cases. The outermost bounding curves correspond to
$m/M=0.95$. Clearly, the merger remnants are not barotropic since the condition
$d\Omega/dz=0$ is not satisfied everywhere. The
implications of this result will be discussed in §5.
4.3 Interior Structure of the Merger Remnants
Figures 13(a)–(g) show the variation of the density $\rho$, relative specific
entropy $s-s_{o}$, helium fraction $Y$ and temperature $T$ as a function of
$m/M$ for all merger remnants. The density and entropy profiles are
fundamental in the sense that they do not depend on the assumed initial
helium profiles. The entropy and helium profiles have been averaged
over isodensity surfaces. The temperature profile is calculated from
the entropy and helium profiles by setting the pressure $p=\rho kT/\mu$ equal
to
$p=A\rho^{\Gamma_{1}}$, solving for $T$ and using equation (4). Here the mean
molecular weight $\mu$ is given by
$$\mu=m_{H}(2X+{3\over 4}Y+{1\over 2}Z)^{-1},$$
(17)
where $m_{H}$ is the mass of hydrogen and $X$, $Y$ and $Z$ are the
fractional abundances of hydrogen, helium and metals. For Population II
stars, $Z\sim 10^{-4}-10^{-3}$ and the precise value does not
significantly affect the calculated temperature profiles.
Note the peculiar shapes of some of the temperature and helium profiles
in Figure 13. For example, often the temperature or helium abundance
reaches its maximum value somewhere other than the center of the star.
Although these configurations are very close to hydrostatic equilibrium,
it is clear that they are not in thermal equilibrium (see related discussion
in §5). These unusual profiles suggest that we look at the condition for
convective stability more carefully. For a non-rotating star, this condition
can be written simply as
$${ds\over dr}>0,$$
(18)
where $s$ is the local specific entropy (see, eg., Landau & Lifshitz
1957, §4). When written in terms of temperature and composition
gradients, equation (18) becomes, for an ideal gas,
$${1\over T}{dT\over dr}>{1\over T}\left({dT\over dr}\right)_{ad}+{1\over\mu}{d%
\mu\over dr}$$
(19)
which is known as the Ledoux criterion (Kippenhahn & Weigert 1990, Chap. 6).
Here the subscript $ad$ denotes that the derivative is to be taken at
constant entropy. Most of our merger remnants
have composition gradients, and it is in the regions where $d\mu/dr>0$
that equation (19) requires $dT/dr>0$ for stability. For chemically homogeneous
stars,
the second term on the right-hand side of equation (19) vanishes, and
the familiar Schwarzschild criterion results.
Although equation (18) is quite general, it does require slight
modification for rotating stars (see Tassoul 1978, Chap. 7).
Figure 13 (a, d, and f) demonstrate that merger remnants formed from
equal mass parent stars have composition profiles which mimic those of
the parents, as can be seen by comparing the resulting helium profiles
to the corresponding parent profiles in Figure 2. In Figure 13(f), all
of the merger remnants have $Y=0.25$ for all $m/M$, which is simply
because the fully convective parents stars in these cases had $Y=0.25$
everywhere.
We see from Figure 13(c) that the central specific entropy of the
merger remnants increases with $r_{p}$, which can explain the
qualitatively different shapes of the corresponding helium abundance
profiles. This increase occurs because the number of interactions
$n_{p}$, and hence the level of shock heating in star 2 (the smaller
star), increases with $r_{p}$. The shock-heating in the central region
of star 1 is less sensitive to $n_{p}$, since the outer envelope absorbs
the brunt of the shock. For case G (solid line), $n_{p}=1$ and much of
star 2 is able to maintain a lower specific entropy than the minimum
value in star 1. Since low-entropy material tends to sink to the
bottom of the gravitational potential well, the merger remnant’s core
consists entirely of fluid originally from star 2 and therefore with a
helium abundance $Y=0.25$. For case H (long-dashed line), $n_{p}=2$ and,
although the central fractional helium abundance is still $0.25$, there
is enough shock heating for the fluid at small $m/M$ to be affected by
contributions from both stars. For case I (short-dashed line), $n_{p}=3$
and the additional shock heating is sufficient to prevent most of the
fluid from star 2 from reaching the center of the remnant, which
consequently is not significantly replenished with hydrogen.
Figure 13(g) displays the profiles for merger remnants resulting from
collisions between two stars of masses $M_{1}=M_{TO}$ and $M_{2}=0.2M_{TO}$.
In the head-on case, the less massive star (star 2) plummets so
quickly to the center that there is significant
shock-heating in the core of star 1, where the highest fractional
helium abundance resides. This
causes the helium-rich material to be spread throughout a larger region of
the merger remnant, and the resulting helium profile is not as sharply
peaked as in cases V and W.
4.4 Calculating Final Profiles
In order to keep our results useful for future applications,
we now present a simple and general method for constructing the
final composition profiles in our calculated merger remnants,
corresponding to any assumed initial composition profiles.
From our results, we extract some simple functions
which can be applied to transform, for example, any given initial
helium abundance profiles into a final helium abundance profile for
the merger remnant. Indeed, these transfer functions allow one to find
the final profile of any passively advected quantity,
provided only that the profiles of that quantity in the parent stars are both
known and spherically symmetric.
Table 4 and Figure 14 establish a correlation between the initial and
final mass fractions of a fluid particle. They also demonstrate that
the details of how the fluid elements are mixed during a collision can
be quite complicated. Table 4 treats six of the collisions between
equal-mass stars. The parent stars and merger remnant are partitioned
into zones according to interior mass fractions. For every zone in the
final configuration, we list the fraction of particles which originated
in each of the initial zones. Although there is definitely a preferred
final mass fraction $m/M$ for a given initial mass fraction $m_{i}/M_{i}$,
there is always a range of $m/M$ obtainable. In Figure 14, which is
for case G, this range of mass fractions is evident in the spread of
points around a preferred average. The lower band of points
surrounding the solid line correspond to particles which originated in
star 1, while the upper band surrounding the dashed line correspond to
particles which originated in star 2. The lines correspond to the
average initial mass fraction $\langle m_{i}/M_{i}\rangle$ for stars
$i=1,2$ as a function of the final mass fraction $m/M$, obtained by
binning values of $m/M$. In contrast, note that if the parent stars
were completely mixed by the collision then the points would be
distributed uniformly over the entire plot with an average initial mass
fraction $\langle m_{i}/M_{i}\rangle={1\over 2}$ for all $m/M$.
Let us define $p_{1}=p_{1}(m/M)$ to be the probability that a particle with
final mass fraction $m/M$ originated in star 1. Obviously, $1-p_{1}$ is
then the probability that the particle originated in star 2. With
this definition we can approximate the final profile of any passively
advected quantity $Q$ according to
$$Q\left({m\over M}\right)\simeq p_{1}\left({m\over M}\right)Q_{1}\left(\left.%
\left\langle{m_{1}\over M_{1}}\right\rangle\right|_{m\over M}\right)+\left(1-p%
_{1}\left({m\over M}\right)\right)Q_{2}\left(\left.\left\langle{m_{2}\over M_{%
2}}\right\rangle\right|_{m\over M}\right),$$
(20)
where $Q_{i}$ are the initial (spherically symmetric) profiles for that
quantity in the parent stars $i=1,2$. The quantities $\left.\langle m_{i}/M_{i}\rangle\right|_{m/M}$ which appear in equation (20) are the average
initial mass fractions, such as the ones in Figure 14, evaluated at the
final mass fraction $m/M$. If all particles at $m/M$ came from a
single value of $m_{i}/M_{i}$, then equation (20) would be exact. In addition,
if the initial profiles are linear over the range of $m_{i}/M_{i}$ which
contributes to the abundance at $m/M$, then the above relationship is
exact.
Figures 15(a)–(g) give the average mass fractions $\langle m_{i}/M_{i}\rangle$ as a function of $m/M$ for all of our collisions, while
Figure 16 gives the function $p_{1}=p_{1}(m/M)$ for the 12 collisions
involving parent stars of unequal mass. For collisions involving two
identical stars we necessarily have $\langle m_{1}/M_{1}\rangle=\langle m_{2}/M_{2}\rangle$ and $p_{1}={1\over 2}$ for all $m/M$. The solid, long
dashed and short-dashed lines correspond to pericenter separations
$r_{p}$ of $0,0.25$, and $0.5(R_{1}+R_{2})$, respectively; in Figure 15(f) the
dot-dashed and dotted lines refer to case S ($r_{p}=0.75(R_{1}+R_{2})$) and
case T ($r_{p}=0.95(R_{1}+R_{2})$), respectively. Note that the horizontal
line $\langle m_{i}/M_{i}\rangle={1\over 2}$ would correspond to the fluid
of star $i$ being completely mixed throughout the merger remnant, which
is not the case for any of our calculations. For collisions involving
equal-mass stars, if there were no shock heating and no mass loss then
every particle would have identical initial and final mass fractions
(i.e., $m_{i}/M_{i}=m/M$) so that the merger remnant’s helium profile would
be the same as in the parent stars. In the equal-mass cases A, B,
C, J, K and L, we do find $\langle m_{i}/M_{i}\rangle\simeq m/M$ and the
final helium profiles are indeed quite similar to the parent profile, as shown
in §4.3.
Along with equation (20), the functions of Figures 15 and 16 provide
the means for approximating the final profile of any passively advected
quantity. As a concrete example of how to use this method, we will now
calculate the fractional helium abundance at $m/M=0.28$ in the merger
remnant of case G, using the same initial profiles as shown in
Figure 2. From the solid lines corresponding to case G in
Figures 15(c) and 16, we find that$\left.\langle m_{1}/M_{1}\rangle\right|_{0.28}=0.05$, $\left.\langle m_{2}/M_{2}\rangle\right|_{0.28}=0.65$ and $p_{1}(0.28)=0.62$. Therefore,
$$Y(0.28)\simeq 0.62\times Y_{1}(0.05)+(1-0.62)\times Y_{2}(0.65)=0.70,$$
(21)
where we have used $Y_{1}(0.05)=0.97$ and $Y_{2}(0.65)=0.25$, obtained from
the solid and short-dashed lines of Figure 2, respectively. By
repeating this calculation for other values of $m/M$, we construct the
approximate helium profile shown as the dashed line in Figure 17. Also
shown for comparison is the “exact” profile (solid line) constructed
by considering the individual helium abundance carried by each particle
(the same curve which appears in Figure 11(b)). We consider the
agreement to be quite good, given the simplicity of the approximation
scheme and the fact that it does not require access to large data files
containing information on all $N=3\times 10^{4}$ particles.
4.5 Effects of Spurious Diffusion on Observed Mixing
In all SPH calculations, numerical noise can lead to spurious, or
artificial, diffusion of SPH particles. In order to estimate how much
of the observed mixing is in fact caused by particle diffusion, we have
performed a series of systematic tests to evaluate quantitatively the
effects of spurious transport in SPH calculations (Lombardi, Rasio, &
Shapiro 1995b). In particular these tests measure, as a function of
the neighbor number $N_{N}$ and local noise level $v_{rms}$ (the root
mean square particle velocity deviation from the local mean), dimensionless
spurious diffusion coefficients defined by
$$D\equiv{n^{1/3}\over c_{s}}{d\Delta r_{s}^{2}\over dt}$$
(22)
where $\Delta r_{s}$ is the distance traveled by a particle due to
spurious diffusion,
$n$ is the local number density of SPH particles, and $c_{s}$ is
the local sound speed.
Once measured, these diffusion coefficients can be applied to each
particle in our simulations by monitoring its local values of
$v_{rms}$, $n$ and $c_{s}$ and then estimating how far
that particle has spuriously diffused by numerically integrating
$$\Delta r_{s}^{2}=\int{D{n^{-1/3}\over c_{s}}dt}.$$
(23)
From the local density gradient at the particle’s final position we can
then estimate the equivalent mass fraction corresponding to this
displacement according to
$$\Delta m_{s}\simeq{\Delta r_{s}\over 3^{1/2}}|{\bf\nabla}\rho|.$$
(24)
By repeating this procedure for all the particles, we arrive at an
average spurious diffusion distance $\langle\Delta r_{s}\rangle$ and
mass fraction $\langle|\Delta m_{s}|\rangle$, as well as a
root-mean-square spurious displacement $\langle\Delta r_{s}^{2}\rangle^{1/2}$ and mass fraction $\langle\Delta m_{s}^{2}\rangle^{1/2}$.
This method of estimating spurious diffusion distances will be referred
to as Method I.
In the case of a head-on collision, another method (which we call Method II)
can be used, which exploits the axisymmetry around the collision axis
(the x-axis).
If we make the assumption that the entire dynamical evolution remains
axisymmetric (this would not be the case if, e.g., Rayleigh-Taylor
instabilities were to develop), then
a particle should always remain in
the plane containing the collision axis and the particle’s initial
position. The extent to which a particle diffuses in the direction
perpendicular to this plane provides an estimate of the spurious diffusion
distance. Assuming the diffused amount is the same in all three directions,
the total
spurious diffusion distance is then estimated simply by multiplying by
$3^{1/2}$. We finally convert the spurious diffusion distance to
an equivalent mass fraction exactly as in Method I (see eq. [24]).
The results of the two methods applied to our calculations are given
in Table 5. When two numbers are given, the second one is
calculated by Method II. Included in this table are the average
spurious diffusion distance $\langle\Delta r_{s}\rangle$, the root-mean-square
diffusion distance $\langle\Delta r_{s}^{2}\rangle^{1/2}$, the
average equivalent mass fraction $\langle|\Delta m_{s}|\rangle/M$,
and the root-mean-square mass fraction $\langle\Delta m_{s}^{2}\rangle^{1/2}/M$.
It is clear that the two methods are generally in good agreement,
and that, when expressed in terms of $m/M$, the effects of
spurious diffusion are always small.
Also listed in Table 5
are the observed average ($\langle|\Delta m_{o}|\rangle/M$)
and root-mean-square ($\langle\Delta m_{o}^{2}\rangle^{1/2}/M$) total deviation
in final mass fraction.
The last column then subtracts from the total square deviation
$\langle\Delta m_{o}^{2}\rangle$ the contribution $\langle\Delta m_{s}^{2}\rangle$
from spurious diffusion.
For example, in case G, we observe a root-mean-square change in the
interior mass fraction of ${\langle\Delta m_{o}^{2}\rangle^{1/2}/M}=0.09$ over the entire calculation. Using Method I, we estimate that
the root-mean-square change in interior mass fraction due to spurious
diffusion is ${\langle\Delta m_{s}^{2}\rangle^{1/2}/M}\simeq 0.036$,
while Method II gives an estimate of $0.057$ for this quantity. We
therefore believe that the physical root-mean-square spread (i.e., the
spread in a calculation free of spurious diffusion) would be
approximately $0.08$ or $0.07$, depending on whether Method I
or Method II is more accurate.
5 Summary and Discussion
The main results of this paper can be summarized as follows. We have
demonstrated that the typical merger remnants produced by collisions
are rapidly and differentially rotating, and are far from chemically
homogeneous, with composition profiles that can be rather peculiar in
certain cases. For example, it often happens that the maximum helium
abundance does not occur at the center of the remnant
(cf. Fig. 11(b)). The merger remnants produced by our dynamical
calculations, although very close to hydrostatic equilibrium, are
usually far from thermal equilibrium, as discussed below. In
particular, we have shown that the merger remnants are not barotropes
(ie., the condition $d\Omega/dz=0$ is generally not satisfied), and
that their temperature profiles can have positive gradients ($dT/dr>0$)
in certain regions.
At a qualitative level, many of our results can be understood very
simply in terms of the requirement of convective stability of the final
merger remnant. If entropy production in shocks could be neglected
(which may be reasonable for parabolic collisions, especially in the
head-on case), then one could predict the qualitative features of the
remnant’s composition profile simply by observing the composition and
entropy profiles of the parent stars. Convective stability requires
that the specific entropy $s$ increase from the center to the surface
($ds/dr>0$) in the final hydrostatic equilibrium configuration.
Therefore, in the absence of shock-heating, fluid elements conserve
their entropy and the final composition profile of a merger remnant
could be determined simply by combining mass shells in order of
increasing entropy, from the center to the outside. Many of our results
follow directly. For example, in the case of a collision between two
identical stars, it is obvious why the composition profile of the
merger remnant remains very similar to that of the parent stars. For
two stars of very different masses, the much lower-entropy material of
the lower-mass star tends to concentrate at the center of the final
configuration, leading to the unusual composition and temperature
profiles seen in Figure 13 (c, e, and g).
Regions where the dynamical stability criterion $ds/dr>0$ (eq. [18]) is
satisfied can nevertheless be thermally, or secularly, unstable. The
small vertical oscillations (at the local Brunt-Väisälä frequency
$\Omega_{BV}\propto[ds/dr]^{1/2}$) of a fluid element in such a region
have amplitudes that grow slowly, and mixing will occur on a timescale
comparable to the local radiative damping time (see, e.g., Kippenhahn
& Weigert, Chap. 6). The thermal instability can be of two types.
When $d\mu/dr>0$ and $dT/dr>0$ (as in Figure 13 (c, e, and g)), a
so-called thermohaline instability can develop, allowing fingers of
the high-$\mu$ material to penetrate down into the lower-$\mu$, colder
material below (see, e.g., Ulrich 1972). When such mixing occurs in the
stellar core, it tends
to increase the central helium abundance and therefore decrease the
time that the merger remnant can remain on the MS.
When $d\mu/dr<0$ but $dT/dr<(dT/dr)_{ad}$, so-called semiconvection
occurs (Spruit 1992). In terms of easily computed SPH variables, this
criterion is equivalent to
$$0<{1\over A}{dA\over dr}<-{\Gamma_{1}\over\mu}{d\mu\over dr}$$
(25)
where $A$ is related to specific entropy by equation (4). We have
tested our merger remnants formed from head-on collisions and found that
this instability is typically present. Figure 18 shows, as a function
of the final mass fraction $m/M$, the fraction $f_{sc}$ of gas which is
semiconvective for six of our merger remnants. In all cases no
semiconvective instability exists in the outer $\sim$20% of the mass,
so that we do not expect this mixing mechanism can increase the helium
abundance of the outer layers. Figure 18 does demonstrate, however,
that some merger remnants (those of cases A, D and J) have a unstable
region which extends to the center, and these remnants therefore have a
means of mixing hydrogen into their cores. For instance in case A, we
see that the inner $\sim$40% could be significantly affected. As
semiconvection slowly attempts to decrease the central helium fraction,
it must compete against hydrogen burning. In addition, the right hand
side of equation (25) changes as the fluid mixes, so that the details
of this complicated process can only be followed numerically with a
stellar evolution code.
For a rotating, chemically homogeneous star, stable thermal equilibrium
requires $d\Omega/dz=0$, where $\Omega$ is the angular velocity and $z$
is measured along the rotation axis (the Goldreich-Schubert stability
criterion; see, e.g., Tassoul 1978, Chap. 7).
From the representative set of specific angular momentum
contours presented in Figure 12, it is therefore evident that the
merger remnants of cases Q, R, S and T (which are chemically
homogeneous, since their parent stars were fully mixed) cannot be in
thermal equilibrium. In chemically inhomogeneous stars, regions with a
sufficiently large and stabilizing composition gradient ($d\mu/dr<0$)
can in principle still be thermally stable even with $d\Omega/dz\neq 0$. However, it seems unlikely that the composition profiles generated
dynamically by a collision would conspire to keep the remnants
everywhere thermally stable.
Deupree (1990) has shown that stars with rapidly rotating cores and
slowly rotating envelopes can have their MS lifetime extended beyond
that of their non-rotating counterparts. The fact that much of the
angular momentum is hidden deep in the remnant’s interior suggests a
possible explanation for why observations of blue stragglers in open
clusters such as M67 find no signs of rapid rotation (Peterson, Carney,
& Latham 1984; Mathys 1991). Recently, Leonard & Livio (1995) have
argued that the spin-down timescale of blue stragglers due to magnetic
breaking should be on the order of only $10^{5}$ years, which is much
less than the thermal timescale of approximately $10^{7}$ years, so that
initially rapidly rotating merger remnants may not be a problem for the
collisional formation scenario.
Stellar encounters with separations $r_{p}$ larger than those considered
in this paper are difficult to compute directly with SPH since the
amount of energy $\Delta E$ dissipated during the first interaction is
then so small that the integration time until the next pericenter
passage can be several orders of magnitude larger than the hydrodynamic
time. We believe that our results can be safely extrapolated all the
way to values of $r_{p}\simeq 1.2(R_{1}+R_{2})$. For instance, from our
results for collisions of equal-mass stars it seems very likely that
the helium profile of the merger remnant will always mimic that of the
parent stars. In Figure 13(f), note that the density, entropy and
temperature profiles also seem to be converging onto a fixed profile,
and that the profiles for the $r_{p}=0.5,0.75$ and $0.95(R_{1}+R_{2})$ cases
all look very similar. For $r_{p}\gtrsim 1.2(R_{1}+R_{2})$ the encounter is
better described as a tidal capture than a collision, i.e., the amount
of energy dissipated is sufficient to form a bound system, but no
direct collision occurs, even in the outer layers of the stars. The
maximum value $r_{p}=r_{cap}$ for tidal capture can be calculated
accurately from linear perturbation theory (Press & Teukolsky 1977;
McMillan, McDermott, & Taam 1987). For two identical $0.8\,M_{\odot}$ MS
stars and a relative velocity at infinity $v_{\infty}=10{\,{\rm km}\,{\rm s}^{-1}}$, McMillan
et al. (1987) find $r_{cap}/(R_{1}+R_{2})\simeq 1.4$, which leaves little
room for “clean” tidal captures. In addition, the long-term evolution
of a tidal-capture binary may well lead to merging of the two stars
even if the initial interaction is in the linear regime (for recent
discussions, see Mardling 1995a,b and Kumar & Goodman 1995).
It must be stressed that the amount of mixing determined by SPH
calculations is always an upper limit. Indeed, some of the mixing
observed in a calculation will always be a numerical artifact.
Low-resolution SPH calculations in particular tend to be very noisy and
the noise can lead to spurious diffusion of SPH particles, independent
of any real physical mixing of fluid elements. In §4.5 we introduced
two simple methods to evaluate quantitatively the effects of spurious
diffusion in our calculations. The results suggest that spurious
diffusion does not significantly corrupt our results. This is seen for
example in the near agreement of numbers in the last two columns of
Table 5. However, both methods are approximate. The diffusion
coefficients used in Method I have been measured in the absence of
artificial viscosity. The presence of artificial viscosity in our
collision calculations may slightly change the effective values of
these coefficients. Method II assumes that we can neglect any
nonaxisymmetric instabilities in a head-on collision and that spurious
diffusion is isotropic (which may not be true in the presence of strong
entropy gradients).
Nevertheless, the reasonable agreement between the two methods
(cf. Table 5) gives us confidence that these simplifying assumptions
are generally satisfied to a good approximation. The general question
of spurious transport in SPH calculations will be addressed in a
separate paper (Lombardi et al. 1995b).
Benz & Hills (1987) performed the first fully three-dimensional
calculations of collisions between two identical MS stars. In contrast
to the present work, they considered only $n=1.5$ polytropes, which are
mostly relevant for collisions of very low-mass stars ($M_{1}=M_{2}\lesssim 0.4M_{\odot}$). Their calculations for parabolic collisions indicated a
higher level of mixing than we find in similar cases (cases P, Q, R, S
and T in our Table 1). We have also computed a number of collisions
using $N=1024$ SPH particles (as in Benz & Hills 1987, rather than
$N=3\times 10^{4}$ particles as in most of our other calculations), and
found substantially higher levels of spurious diffusion. We conclude
that a significant part of the mixing observed by Benz & Hills (1987)
was a numerical artifact. However, we should note that our
higher-resolution calculations for $n=1.5$ polytropes (cases P, Q, R,
S and T) do exhibit a generally higher level of mixing than observed
for other models (e.g., cases A, B and C). This is not surprising since
we expect parent stars of constant entropy, which are only marginally
stable against convection, to be easier to mix than those with
significant positive entropy gradients (stable stratifications). In
addition, the more homogeneous density profile of $n=1.5$ polytropes
leads to a better distribution of the impact energy throughout the
entire mass of fluid. Benz & Hills (1992) have performed calculations
of collisions between $n=1.5$ polytropes with a mass ratio
$M_{2}/M_{1}=0.2$, using $3500$ SPH particles per star. These calculations
are not directly relevant to blue straggler formation, given the very
low masses of the MS stars involved ($M_{1}\lesssim 0.4\,M_{\odot}$, hence
$M_{2}<0.1\,M_{\odot}$), and, since none of our calculations model such low
masses, no direct comparison is possible.
There are a number of ways by which our results could be improved or
extended. For instance, the equation of state could be extended to
include radiation, partial ionization and electron degeneracy
corrections. We do not expect these corrections to be significant.
More accurate initial composition profiles could be also used.
Note, however, that our results can be applied to
arbitrary initial profiles by the method of §4.4.
Incidentally, profiles of ${}^{7}$Li would be particularly interesting to
consider since this element is destroyed at
temperatures $T\gtrsim 10^{6}K$ and is therefore an
observationally measurable indicator of mixing (see, e.g., Hobbs &
Mathieu 1991 and Pritchet & Glaspey 1991).
Our dynamical calculations and the determination of hydrodynamical
mixing are only the first step in modeling blue straggler formation.
The merger remnants, which are much larger than normal equilibrium MS
stars of the same mass, will recontract to the MS on a thermal
timescale ($\sim 10^{7}\,$yr). As they evolve, other mixing processes
such as meridional circulation and convection may well be important.
Calculations of this thermal relaxation phase using the results of
dynamical calculations, such as those presented in this paper, as
initial conditions, will be necessary in order to make detailed
predictions for the observable parameters of blue stragglers.
Recently, Sills, Bailyn and Demarque (1995, hereafter SBD) have begun
investigating the consequences of blue stragglers being born unmixed.
To create their unmixed initial model of a blue straggler formed by the
collision of two TAMS stars, SBD relaxes a non-rotating TAMS star whose
mass has been artificially doubled but which is otherwise unchanged.
The subsequent stellar evolution is contrasted to that of a fully mixed
(ie. chemically homogeneous) blue straggler. SBD finds that the high
central helium concentration in the unmixed models causes the time
spent on the MS ($\sim 5\times 10^{7}$ yr) to be drastically shorter than
for the fully mixed counterparts ($\sim 5\times 10^{8}$ yr), making it
difficult to account for the observed numbers blue stragglers in the
core of NGC 6397. In addition, such unmixed blue stragglers are
neither bright nor blue enough to explain the observations. A blue
straggler population consisting purely of non-rotating, unmixed merger
remnants of two TAMS parent stars is therefore not sufficient to
explain the core blue stragglers in NGC 6397.
Future work which follows the approach of SBD would clearly be
beneficial. A number of factors need to be considered in more detail.
For instance, it is unrealistic to expect that all collisional blue
stragglers are born only from TAMS parent stars. Instead, attention
must also be given to collisions between unequal mass parent stars,
which form merger remnants with profiles that are neither homogeneous
nor like that of the parents. That is, such remnants are neither fully
mixed nor unmixed. Since these blue stragglers have an enhanced
hydrogen abundance in their cores, they will presumably remain on the
MS for a longer time and have a different position on a color magnitude
diagram than their unmixed counterparts. Moreover, most blue
stragglers will be formed rapidly rotating, especially in the stellar
core, which acts to extend their postponed residence on the MS (Deupree
1990). In addition, the density and specific entropy profiles shown in
Figure 13 can be used to specify the structure of the blue straggler
and improve upon SBD’s somewhat ad hoc profiles obtained by
artificially scaling the mass of an equilibrium star.
Support for this work was provided by NSF Grant AST 91–19475,
NASA Grant NAG 5–2809, and NASA Grant HF-1037.01-92A.
Computations were performed at the Cornell Theory Center, which
receives major funding from the NSF and IBM Corporation, with
additional support from the New York State Science and Technology
Foundation and members of the Corporate Research Institute.
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Writing and Reading antiferromagnetic Mn${}_{2}$Au: Néel spin-orbit torques and large anisotropic magnetoresistance.
S. Yu. Bodnar
Institut für Physik, Johannes Gutenberg-Universität, Staudinger Weg 7, 55128 Mainz, Germany
L. Šmejkal
Institut für Physik, Johannes Gutenberg-Universität, Staudinger Weg 7, 55128 Mainz, Germany
Institute of Physics, Academy of Sciences of the Czech Republic, Cukrovarnicka 10, 162 00 Praha 6, Czech Republic
Faculty of Mathematics and Physics, Charles University, Department of Condensed Matter Physics, Ke Karlovu 5,
12116 Praha 2, Czech Republic
I. Turek
Faculty of Mathematics and Physics, Charles University, Department of Condensed Matter Physics, Ke Karlovu 5,
12116 Praha 2, Czech Republic
T. Jungwirth
Institute of Physics, Academy of Sciences of the Czech Republic, Cukrovarnicka 10, 162 00 Praha 6, Czech Republic
School of Physics and Astronomy, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom
O. Gomonay
Institut für Physik, Johannes Gutenberg-Universität, Staudinger Weg 7, 55128 Mainz, Germany
J. Sinova
Institut für Physik, Johannes Gutenberg-Universität, Staudinger Weg 7, 55128 Mainz, Germany
A.A. Sapozhnik
Institut für Physik, Johannes Gutenberg-Universität, Staudinger Weg 7, 55128 Mainz, Germany
H.-J. Elmers
Institut für Physik, Johannes Gutenberg-Universität, Staudinger Weg 7, 55128 Mainz, Germany
M. Kläui
Institut für Physik, Johannes Gutenberg-Universität, Staudinger Weg 7, 55128 Mainz, Germany
M. Jourdan
[email protected]
Institut für Physik, Johannes Gutenberg-Universität, Staudinger Weg 7, 55128 Mainz, Germany
(December 7, 2020)
††preprint:
Antiferromagnets are magnetically ordered materials which exhibit no net moment and thus are insensitive to magnetic fields. Antiferromagnetic spintronics Jun16 aims to take advantage of this insensitivity for enhanced stability, while at the same time active manipulation up to the natural THz dynamic speeds of antiferromagnets Kam11 is possible, thus combining exceptional storage density and ultra-fast switching. However, the active manipulation and read-out of the Néel vector (staggered moment) orientation is challenging. Recent predictions have opened up a path based on a new spin-orbit torque Zel14 , which couples directly to the Néel order parameter. This Néel spin-orbit torque was first experimentally demonstrated in a pioneering work using semimetallic CuMnAs Wad16 . Here we demonstrate for Mn${}_{2}$Au, a good conductor with a high ordering temperature suitable for applications, reliable and reproducible switching using current pulses and read-out by magnetoresistance measurements. The symmetry of the torques agrees with theoretical predictions and a large read-out magnetoresistance effect of more than ${\bf\simeq 6}$ ${\bf\%}$ is reproduced by ab initio transport calculations.
For the key application operations of reading and writing in antiferromagnets, different approaches have been previously put forward. Initial experiments on spin-valve structures with an antiferromagnet (AFM) as the active layer manipulated the Néel vector by an exchange-spring effect with a ferromagnet (FM) and read-out via Tunneling-Anisotropic Magnetoresistance (T-AMR) measurements Par11 . Other related experiments were based on the same effect Fin14 , or on a FM to AFM phase transition Mar14 . However, the most promising approach is to use current induced spin-orbit torques for switching the Néel vector. It exhibits superior scaling and its counterpart in ferromagnets is already established and considered among the most efficient switching mechanisms for memory applications. Gam11 ; Bra14 .
Only two compounds, CuMnAs and Mn${}_{2}$Au, are known to provide at room temperature the collinear commensurate antiferromagnetic order and specific crystal structure, which is predicted to result in the staggered spin accumulation in the sublattice structure, leading to bulk Néel spin-orbit torques allowing for current induced switching of the Néel vector Zel14 .
Semimetallic CuMnAs was grown previously by molecular beam epitaxy (MBE) with a Néel temperature of $\simeq 500$ K Wad13 and current-induced switching of these samples was recently demonstrated for the first time Wad16 ; Grz17 . However, for spintronics applications the compound Mn${}_{2}$Au provides several advantages, as it is a good metallic conductor and does not contain toxic components. Furthermore, its magnetic ordering temperature is well above 1000 K Bar13 , providing the necessary thermal stability for applications. Mn${}_{2}$Au shows a simple antiferromagnetic structure with the collinear magnetic moments in the (001)-plane Shi10 ; Bar13 ; Bar16 . Thin film samples were previously grown in (101)-orientation by MBE Wu12 and Fe/Mn${}_{2}$Au(101) bilayers showed AMR effects of up $2.5$ $\%$ in a 14 T rotating magnetic field Wu16 .
While Mn${}_{2}$Au was the first compound for which current-induced internal staggered spin-orbit torques were predicted Zel14 , corresponding experimental evidence has been missing. Here we report current induced Néel vector switching in Mn${}_{2}$Au(001) epitaxial thin films, which is is easily read-out by a large AMR.
Our Al${}_{2}$O${}_{3}$/Ta(10 nm)/Mn${}_{2}$Au(75 nm)/Ta(3 nm) samples were prepared by sputtering as described elsewhere Jou15 and patterned into a star-structure as shown in Fig. 1.
This geometry allows for electric writing of the Néel vector orientation by pulsing currents along the two perpendicular directions $I_{pulse}^{1}$ and $I_{pulse}^{2}$ and for electric read-out by measuring either the transversal resistivity $\rho_{xy}$, i. e. the Planar Hall Effect (PHE), or the longitudinal resistivity $\rho_{long}$, corresponding to the AMR of the samples. Depending on the orientation of the patterned structure, the pulse currents can be sent along different crystallographic directions.
The AMR of a single domain sample is given by
$${\rm AMR_{hkl}}=\frac{\rho_{long}(\phi=0^{\rm o})-\rho_{long}(\phi=90^{\rm o})%
}{\bar{\rho}_{long}}=\frac{\Delta\rho_{long}}{\bar{\rho}_{long}},$$
(1)
where $\rho_{long}$ is longitudinal resistivity, $\phi$ is the angle between the Néel vector and current direction and [hkl] is the Néel vector orientation in the basis of the tetragonal conventional unit cell.
The PHE usually observed in ferromagnetic materials scales with the AMR and shows a dependence on the angle $\phi$ given by Tom75 ; See11 :
$$\rho_{xy}=\Delta\rho_{long}\,\sin\phi\cos\phi$$
(2)
Thus also in antiferromagnets $\rho_{xy}$ has its maximum value and changes sign if $\phi$ switches from $+45^{\rm o}$ to $-45^{\rm o}$.
To study the switching, trains of $100$ current pulses with a pulse length of $1$ ms and a delay between the pulses of 10 ms were applied. As after a pulse train thermal relaxation behaviour on a time scale of $1$ s after was observed, the read-out was performed with a delay of $10$ s.
Fig. 2 shows the transversal resistivity $\rho_{xy}$ versus the number of applied pulse trains.
First, a pulse current density of $1.4\times 10^{7}$ A/cm${}^{2}$ was applied along the [1$\bar{1}$0] direction, resulting in a small change of the corresponding Hall voltage. Without reaching saturation after $50$ pulse trains the pulse current direction was switched to [110], resulting in a reversal of the corresponding change of the transversal resistivity. This sequence could be reproduced several times. Increased pulse current densities of $1.7\times 10^{7}$ A/cm${}^{2}$ and $1.8\times 10^{7}$ A/cm${}^{2}$ resulted in larger changes of the corresponding Hall voltages. By increasing the number of pulse trains applied along the [110] direction to $500$, a trend towards saturation of the Hall voltage was obtained.
Internal field like spin-orbit torques are expected to generate reversible switching between distinct stable states if the current is injected along biaxial easy directions Zel14 ; Roy16 . However, we observed reversible switching to stable states for pulse currents along both the crystallographic [110] and [100] axes (rotated star-pattern). Thus we conclude that the in-plane magnetic anisotropy of our Mn${}_{2}$Au thin films is weak. This is consistent with our calculations of the magnetocrystalline anisotropy energy (MAE), which is almost negligible within the ab-plane (see Methods).
An example of the resulting changes of the transversal and longitudinal resistivities generated by pulse currents along the [100] and [010] directions is displayed in Fig. 3.
The upper panel (Fig. 3a) shows the longitudinal resistivity probed after each of the first 1600 pulse trains consisting of 100 pulses each with a current density $1.0\times 10^{7}$ A/cm${}^{2}$. For the first sequences only small variations of the longitudinal resistivity were observed. However, with the application of subsequent pulse trains the magnitude of the effect increased. This training-like behaviour may be associated with the motion and pinning of AFM domain walls in the sample. After 1600 pulse trains a constant resistance change of $\Delta\rho_{long}/{\bar{\rho}}=2.5$ % induced by 100 pulse trains was reached, which is an order of magnitude higher than what is observed for CuMnAs Wad16 .
To check the origin of these changes, the transversal resistivity of the sample was measured and also showed reproducible pulse current induced changes (Fig. 3b). The increase of the transversal resistivity induced by 100 pulse trains amounted to $\rho_{xy}=0.5$ ${\rm\mu\Omega cm}$.
Based on these numbers the identification of the longitudinal and transversal resistivities with the AMR and PHE can be verified: If both effects originate from the same anisotropic electron scattering, they have to be related by equation (2). We assume a switching of the Néel-vector in parts of the sample corresponding to a change of $\phi$ in equation (2) from $+45$ ${}^{\rm o}$ to $-45$ ${}^{\rm o}$, i. e.
$$\Delta\rho_{xy}=\rho_{xy}(+45^{\rm o})-\rho_{xy}(-45^{\rm o})=\Delta\rho_{long},$$
(3)
Thus we find that $\rho_{xy}=0.5$ ${\rm\mu\Omega cm}$ corresponds again to $\Delta\rho_{long}/{\bar{\rho}}=2.5$ %. This consistency of the longitudinal and transversal resistivities provides strong evidence for an intrinsic electronic origin of the pulse current induced changes of the magnetoresistance signals.
After two more pulse current direction reversals reproducing the previous behaviour of the sample, the pulse current direction was kept along [100] for 800 additional pulse trains. This resulted in a sign reversal of the PHE. Although a small offset of the transversal voltage due to e. g. imperfections of the patterned structure is possible, the magnitude of the PHE indicates that the Néel vector now switched in the majority of the sample. After about 300 pulse trains along the [100] direction a beginning saturation of the PHE resistivity appeared, but was not completed when after 500 additional pulse trains the sample broke. A maximum transversal resistivity of $\rho_{xy}=1.25$ ${\rm\mu\Omega cm}$ was reached, which based on equation (3) corresponds to an AMR of $6.25$ $\%$. This is one of the largest found in metallic magnetic thin films, and its size bodes well for easy read-out of the antiferromagnetic state as necessary for device applications. While small variations exist between samples, we observe consistently larger AMR effects for pulse currents along the [100] than for the [110] directions.
To understand the origin of the magnetoresistance effects, we calculated the AMR of single domain Mn${}_{2}$Au, assuming a complete $90^{\rm o}$ switching of the Néel vector (see Methods). In general, AMR originates from the effects of spin-orbit coupling on the band structure and from scattering from an extrinsic disorder potential Ran08 . When incorporating the effects of realistic disorder in the calculations (see Methods), two types were considered: Off-stoichiometry and inter-site swapping between Mn and Au atoms. Experimentally, the former was analyzed by energy dispersive x-ray spectroscopy (EDX) of $500$ nm thick Mn${}_{2}$Au films resulting in a stoichiometry of $66.2\pm 0.3$ $\%$ Mn and $33.8\pm 0.3$ $\%$ of Au, which indicates a slight Au excess. Also a small degree of inter-site disorder can be expected, but its quantification is experimentally not accessible. Thus we simulated a slight excess of Au randomly distributed over the Mn sites and random Mn - Au swapping.
We calculated the AMR for two crystal directions of the Néel vector, AMR${}_{100}$ and AMR${}_{110}$. Fig. 4a, shows the results for different degrees of disorder in Mn${}_{2}$Au.
Large AMR values consistent with our experiments were obtained for small degrees of disorder reaching a maximum value of $6.3$ % for 0.5 % excess of Au. Moreover, we obtain $AMR_{100}>AMR_{110}$, which reproduces the experimentally observed trend for the two crystalline directions. The corresponding calculated residual resistivities as shown in Fig. 4b are consistent with the experimentally obtained values ($\simeq 8$ ${\rm\mu\Omega cm}$ Jou15 ), corroborating the relevance of the simulated type of disorder.
In summary, in-plane switching of the Néel vector in the antiferromagnetic metal Mn${}_{2}$Au by current pulses was realized using intrinsic spin-orbit torques. Consistent measurements of the AMR and PHE showed pulse current direction dependent reversible changes, providing direct evidence for Néel vector switching. Easy read-out of the switching is provided by a large amplitude of the AMR of more than 6%, which is more than an order of magnitude higher than previously observed for other antiferromagnetic systems and one of the highest AMR amplitudes found for metallic magnetic thin films. We can reproduce the magnitude of the effect theoretically by including realistic disorder and, in particular, find the same dependence of the amplitude on the crystallographic directions in the experiment as in the calculation. With the basic principles of writing and read-out demonstrated, combined with a theoretical understanding of the underlying spin-orbit torques, and the large magnetoresistive effects, the metallic compound Mn${}_{2}$Au is a prime candidate to enable future AFM spintronics.
Acknowledgments: This work is supported by the German Research Foundation (DFG) through the Transregional Collaborative Research Center SFB/TRR173 Spin+X, Projects A03 and A05. J.S., L.S., O.G., and T.J. acknowledge the support of the Alexander von Humboldt Foundation, the ERC Synergy Grant SC2 (No. 610115), the Ministry of Education of the Czech Republic Grant No. LM2015087, and the Grant Agency of the Czech Republic grant no. 14-37427G, L.S. acknowledges support from the Grant Agency of the Charles University, no. 280815. Access to computing and storage facilities owned by parties and projects contributing to the National Grid Infrastructure MetaCentrum provided under the program ”Projects of Large Research, Development, and Innovations Infrastructures” (CESNET LM2015042), is greatly appreciated. The work of I.T. was supported by the Czech Science Foundation (Grant No. 14-37427G).
Author contributions
S.Yu.B. and M.J. mainly wrote the paper and performed the transport measurements; L.S. performed the AMR and anisotropy calculations and wrote the corresponding part of the manuscript, I.T. developed the codes for the transport calculations, S.Yu.B. and A.A.S. prepared the samples, H.-J.E, M.K., O.G., T.J., I.T, and J.S. discussed the results and contributed to the writing of the manuscript; M.J. coordinated the project.
Methods
Measurement Procedure
The current pulse trains were generated by a Keithley 2430 Source Meter. After each pulse train a delay of $10$ s for thermal relaxation was followed by a measurement of the transversal or longitudinal voltage across the central part of the patterned structure resulting from a probe current density of $10^{4}$ A/cm${}^{2}$. Typically this procedure was repeated several times before the pulse trains were sent along the perpendicular direction of the cross structure, keeping the probing contacts unchanged.
Based on time resolved resistivity measurements during the application of pulse trains and temperature dependent resistivity measurements of Mn${}_{2}$Au thin films the local temperature of the relevant sample region was estimated: Values of up to $600$ K in the stable regime and $\simeq 800$ K at current densities, which finally destroyed the samples, were obtained. The local temperature rapidly decays after the application of a current pulse train, therefore thermoelectric voltages can be neglected.
MAE and AMR calculation
The MAE was calculated using the FLAPW (Full Potential Linearized Augmented Plane Wave) method in combination with the GGA (Generalized Gradiend Approximation). We found the MAE in line with previous reports Shi00 $\simeq 1$ ${\rm\mu}$eV per formula unit, which is at the resolution limit of our method based on the magnetic force theorem.
To calculate the AMR in Mn${}_{2}$Au ab initio we employed the fully relativistic Dirac tight binding-linear muffin-tin orbital plus coherent potential approximation (FRD-TB-LMTO+CPA) method in combination with the Kubo formula Tur02 ; Tur14 . We used the s-, p-, and d-type orbitals in the basis and the LSDA (Local Spin Density Approximation) Vosko-Wilk-Nusair exchange-correlation potential parametrization Vos80 . The ground-state magnetization and density of states was reproduced consistently with a previous reports Shi10 ; Zel14 . In the transport calculation we used up to $10^{8}$ k-points in the Brillouin zone and for the residual resistivity calculations we set the imaginary part of the complex energy to $\Gamma=10^{-5}$ Ry.
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WIS/02/16-FEB-DPPA
IPMU-16-0022
UT-16-9
S-folds and 4d $\mathcal{N}{=}\,3$ superconformal field theories
Ofer Aharony${}^{1}$ and Yuji Tachikawa${}^{2,3}$
1.
Department of Particle Physics and Astrophysics,
Weizmann Institute of Science, Rehovot 7610001, Israel
2.
Department of Physics, Faculty of Science,
University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan
3.
Kavli Institute for the Physics and Mathematics of the Universe,
University of Tokyo, Kashiwa, Chiba 277-8583, Japan
S-folds are generalizations of orientifolds in type IIB string theory, such that the geometric identifications are accompanied by non-trivial S-duality transformations.
They were recently used by García-Etxebarria and Regalado to provide the first construction of four dimensional ${\cal N}{=}3$ superconformal theories.
In this note, we classify the different variants of these ${\cal N}{=}3$-preserving S-folds, distinguished by an analog of discrete torsion,
using both a direct analysis of the different torsion classes and the compactification of the S-folds to three dimensional M-theory backgrounds.
Upon adding D3-branes, these variants lead to different
classes of ${\cal N}{=}3$ superconformal field theories. We also analyze the holographic
duals of these theories, and in particular clarify the role of discrete gauge and global
symmetries in holography.
Contents
1 Introduction and summary of results
2 Preliminary analysis
2.1 Three general properties of ${\cal N}=3$ SCFTs
2.2 D3-branes on S-folds and their discrete symmetries
3 Holographic dual
3.1 AdS duals of CFTs with discrete symmetries
3.2 Discrete torsion
3.3 Generalized Pfaffians
3.4 Realization of the $\mathbb{Z}_{k}$ gauge theory in F-theory
4 Comparison to M-theory
4.1 Discrete fluxes in M-theory on $(\mathbb{C}^{3}\times T^{2})/\mathbb{Z}_{k}$
4.2 M2-brane and D3-brane charges
5 Special cases and $\mathcal{N}{=}\,4$ enhancement
1 Introduction and summary of results
Field theories with superconformal symmetry are useful laboratories for learning about the behavior of quantum field theories in general, and strongly coupled field theories in particular. This is because the superconformal symmetry allows many computations to be performed in these theories, using methods such as localization, integrability, and the superconformal bootstrap.
The field theories (above two dimensions) about which the most is known are ${\cal N}=4$ superconformal field theories (SCFTs) in four dimensions, that have been called the ‘harmonic oscillator of quantum field theories’. These theories have an exactly marginal deformation, and it is believed that they are all gauge theories with some gauge group $G$, such that the exactly marginal deformation is the gauge coupling constant. In these theories many observables have already been computed as functions (trivial or non-trivial) of the coupling constant, and there is a hope that they can be completely solved.
Four dimensional theories with ${\cal N}=2$ superconformal symmetry have also been extensively studied. Some of these theories have exactly marginal deformations and corresponding weak coupling limits (at least for some sector of the theory), while others do not. We do not yet have a full classification of ${\cal N}=2$ SCFTs (for rank-1 theories, see [1, 2, 3]),
though a large class of theories, called class S, has been constructed following [4].
Many observables can be computed also in ${\cal N}=2$ SCFTs, at least if they have a weak coupling limit, they are connected by renormalization group flows to theories that have such limits, or they are one of the theories of class S.
Four dimensional ${\cal N}=3$ theories should naively provide an intermediate class of theories, that is more general than ${\cal N}=4$, but such that more computations can be made than in general ${\cal N}=2$ theories. ${\cal N}=3$ SCFTs (that are not also ${\cal N}=4$ SCFTs) have no exactly marginal deformations [5, 6] and thus no weak coupling limits that would aid in classifying and performing computations in these theories. Until recently no ${\cal N}=3$ SCFTs were known, but recently a class of such theories was constructed by García-Etxebarria and Regalado in [7]. Their construction uses a generalization of orientifolds in string theory.
Orientifold 3-planes (the generalization to other dimensions is straightforward) are defined in type IIB string theory as planes in space-time, such that in the transverse space $y\in\mathbb{R}^{6}$ to these orientifolds, there is an identification between the points $y$ and $(-y)$, but with opposite orientations for strings (or, equivalently, with an opposite value for the $B_{2}$ and $C_{2}$ 2-form potentials of type IIB). This means that we identify configurations related by a $\mathbb{Z}_{2}$ symmetry that involves a spatial reflection in the transverse $SO(6)$, and also a transformation $(-I)$ in the $SL(2,\mathbb{Z})$ S-duality group of type IIB string theory. This breaks half of the supersymmetry, preserving a four dimensional ${\cal N}=4$ supersymmetry. In particular, putting $N$ D3-branes on the orientifold (these do not break any extra supersymmetries) gives at low energies four dimensional ${\cal N}=4$ SCFTs.
In [7] this was generalized to identifying configurations related by a $\mathbb{Z}_{k}$ symmetry, that acts both by a $(2\pi/k)$ rotation in the three transverse coordinates in $\mathbb{C}^{3}=\mathbb{R}^{6}$, and by an element of $SL(2,\mathbb{Z})$ whose $k$’th power is the identity. We will call the fixed planes of such transformations S-folds111This term, first coined in [8], generalizes the term T-folds that is used to describe identifications by elements of the T-duality group. S-folds of string theory involving S-duality twists together with shifts along a circle were studied, for instance, in [9, 10, 11, 12, 13], and similar S-folds were studied in the 4d $\mathcal{N}{=}\,4$ SYM theory in [14, 15, 16].; for $k=2$ they are the same as the usual orientifolds. Viewing $SL(2,\mathbb{Z})$ as the modular group of a torus, the $\mathbb{Z}_{k}$ S-duality transformation may be viewed as a rotation of the torus by an angle $(2\pi/k)$; such a rotation maps the torus to itself if and only if $k=3,4,6$ and its modular parameter is $\tau=e^{2i\pi/k}$, so S-folds of this type exist only for these values of $k$ and $\tau$. It is natural to define such an identification using F-theory [17], in which the $SL(2,\mathbb{Z})$ S-duality group is described as adding an extra zero-size torus whose modular parameter is the coupling constant $\tau$ of type IIB string theory; in this language the S-folds of [7] are the same as F-theory on $(\mathbb{C}^{3}\times T^{2})/\mathbb{Z}_{k}$. This specific $\mathbb{Z}_{k}$ identification preserves a four dimensional ${\cal N}=3$ supersymmetry. So, putting $N$ D3-branes on the S-fold gives at low energies theories with ${\cal N}=3$ superconformal symmetry. Note that D3-branes sitting on the S-fold are invariant under all the transformations discussed above, and, in particular, the D3-brane charge of the S-fold is well-defined.
In this note we analyze further the theories constructed in [7]. We focus on asking what are the extra parameters associated with S-folds, analogous to discrete torsion for orbifolds and orientifolds (namely, to non-trivial configurations of the various $p$-forms of type IIB string theory in the presence of the S-fold). S-folds with different parameters can carry different D3-brane charges. Upon putting $N$ D3-branes on the S-fold, these discrete parameters label different ${\cal N}=3$ SCFTs. Our main result is a classification of these extra parameters (following a preliminary discussion in [7]); we show that there are two variants of S-folds with $k=3,4$ and just a single variant with $k=6$. Each variant leads to different ${\cal N}=3$ SCFTs, with different central charges and chiral operators.
In some cases discrete global symmetries play an interesting role. In a specific type of orientifold (the $O3^{-}$ plane), the theory of $N$ D3-branes on the orientifold is an $\mathrm{O}(2N)$ gauge theory, and this may be viewed as an $\mathrm{SO}(2N)$ gauge theory in which a discrete global $\mathbb{Z}_{2}$ symmetry is gauged. Both the $\mathrm{SO}(2N)$ and the $\mathrm{O}(2N)$ theories exist as ${\cal N}=4$ SCFTs, and a few of their properties are different. We will see that a similar phenomenon happens also for D3-branes on S-folds, and discuss its realization in the AdS/CFT dual of these theories, in terms of discrete gauge symmetries in the bulk. This dual is obviously given by F-theory on $AdS_{5}\times(S^{5}\times T^{2})/\mathbb{Z}_{k}$.
We begin in section 2 with a naive analysis of the possible discrete parameters, by considering the possible discrete identifications on the moduli space of $N$ D3-branes, and the effects of discrete symmetries. In section 3 we discuss the holographic duals of the ${\cal N}=3$ SCFTs, and the realization of the discrete symmetries there. We show that holography suggests that only some of the possibilities found in the naive analysis are consistent. The compactification of F-theory on $(\mathbb{C}^{3}\times T^{2})/\mathbb{Z}_{k}$ on a circle gives M-theory on $(\mathbb{C}^{3}\times T^{2})/\mathbb{Z}_{k}$, with each S-fold splitting into several $\mathbb{C}^{4}/\mathbb{Z}_{l}$ singularities in M-theory. In section 4 we show that the consistency of this reduction implies that indeed only the possibilities found in section 3 are consistent. In section 5 we discuss the fact that for specific values of $k$ and $N$ some variants of the ${\cal N}=3$ theories have enhanced ${\cal N}=4$ supersymmetry, with gauge groups $SU(3)$, $SO(5)$ and $G_{2}$, and check the consistency of this. This provides new brane constructions and new (strongly coupled) AdS duals for these specific ${\cal N}=4$ SCFTs.
In this paper we discuss only the S-folds which give rise to four dimensional ${\cal N}=3$ supersymmetric theories. It would be interesting to study S-folds that preserve different amounts of supersymmetry, and that have different dimensions. A particularly interesting case, which should have many similarities to our discussion, is S-folds preserving four dimensional ${\cal N}=2$ supersymmetry. We leave the study of these theories to the future.
2 Preliminary analysis
In this paper we study the S-folds introduced by
García-Etxebarria and Regalado in [7], which are equivalent to F-theory on $(\mathbb{C}^{3}\times T^{2})/\mathbb{Z}_{k}$ for $k=3,4,6$, generalizing the standard orientifold 3-planes that arise for $k=2$.
Just as the orientifold 3-planes have four variants, $O3^{-}$, $\widetilde{O3}{}^{-}$, $O3^{+}$ and $\widetilde{O3}{}^{+}$, that differ by discrete fluxes, we expect that these new S-folds could also have a few variants.
In this section we perform a preliminary analysis of the possible variants,
from the viewpoint of the moduli space of the $\mathcal{N}{=}\,3$ superconformal field theories realized on $N$ D3-branes probing these S-folds. In the following sections we will discuss additional constraints on the possible variants.
2.1 Three general properties of ${\cal N}=3$ SCFTs
We start from the following three properties of theories of this type, that are generally believed to hold:
•
The conformal anomalies (central charges) $a$ and $c$ are equal in any $\mathcal{N}{=}\,3$ SCFT [5]:
$$a=c.$$
(2.1)
The value of $a=c$ determines also all anomalies of the $SU(3)_{R}\times U(1)_{R}$ symmetries in these theories.
•
The geometry of the gravitational dual of $N$ D3-branes sitting on a singularity involving a $\mathbb{Z}_{k}$ identification of their transverse space, and carrying $\epsilon$ units of D3-brane charge, is AdS${}_{5}\times S^{5}/\mathbb{Z}_{k}$, with $(N+\epsilon)$ units of 5-form flux. This implies that the large $N$ central charge of the corresponding CFTs is
$$a\sim c\sim k(N+\epsilon)^{2}/4+O(N^{0}).$$
(2.2)
•
In any $\mathcal{N}{=}\,2$ theory, the Coulomb branch operators (chiral operators whose expectation values label the Coulomb branch of the moduli space, namely the space of vacuum expectation values of the scalars in the vector multiplets) form a ring generated by $n$ operators, whose expectation values are all independent without any relation. Furthermore, up to a caveat mentioned below, there is a relation
$$2a-c=\sum_{i=1}^{n}(2\Delta_{i}-1)/4,$$
(2.3)
where $\Delta_{i}$ are the scaling dimensions of the generators of the Coulomb branch operators.
The property (2.2) can be derived easily following the analysis of [18, 19, 20].
Essentially, the curvature of the AdS space is supported by the total energy of the five-form field, and the volume of the $\mathbb{Z}_{k}$ quotient is $1/k$ of the volume before the quotient, changing the value of Newton’s constant on $AdS_{5}$ by a factor of $k$.
The equation (2.3), originally conjectured in [21], was given a derivation that applies to a large subclass of $\mathcal{N}{=}\,2$ theories in [22] (though it is not clear that this subclass includes the theories we discuss here). More precisely, this relation applies only to gauge theories whose gauge group has no disconnected parts.
To illustrate this, consider the $\mathcal{N}{=}\,4$ super Yang-Mills theories with gauge groups $\mathrm{U}(1)$ and $\mathrm{O}(2)$.
They have the same central charges $2a-c=1/4$, coming from a single free vector multiplet, but the gauge-invariant Coulomb branch operator has dimension $1$ for the former and $2$ for the latter,
so that property 3 only holds for the former. This is because the scalars in the vector multiplet change sign under the disconnected component of the $\mathrm{O}(2)$ gauge group. The $\mathrm{O}(2)$ theory is obtained by gauging a $\mathbb{Z}_{2}$ global symmetry of the $\mathrm{U}(1)=\mathrm{SO}(2)$ theory, where the $\mathbb{Z}_{2}$ symmetry is generated by the quotient $\mathrm{O}(2)/\mathrm{SO}(2)$; this operation does not change the central charges.
This means that the $\mathrm{O}(2)$ theory includes a $\mathbb{Z}_{2}$ gauge theory, implying that it has a non-trivial $\mathbb{Z}_{2}$ 2-form global symmetry in the sense of [23], as we further discuss below.
We assume in the following that the relation (2.3) holds whenever the theory does not have any non-trivial 2-form global symmetry, even if the theory does not have a gauge theory description.
2.2 D3-branes on S-folds and their discrete symmetries
Let us now see what the three properties recalled above tell us about the properties of S-folds.
Let us probe the $\mathbb{Z}_{k}$ S-fold by $N$ D3-branes. The moduli space of ${\cal N}=3$ theories is described by the expectation values of scalars in ${\cal N}=3$ vector multiplets, which are identical to ${\cal N}=4$ vector multiplets; viewing an ${\cal N}=3$ theory as an ${\cal N}=2$ theory, these decompose into a vector multiplet and a hypermultiplet.
Let us choose a specific $\mathcal{N}{=}\,2$ subgroup of the $\mathcal{N}{=}\,3$ symmetry, and consider the component of the moduli space which is a Coulomb branch from the point of view of this $\mathcal{N}{=}\,2$ subgroup. This implies that all the D3-branes
lie on a particular $\mathbb{C}/\mathbb{Z}_{k}$ within $\mathbb{C}^{3}/\mathbb{Z}_{k}$.
Denote by $z_{i}$ $(i=1,\ldots,N)$ the positions on $\mathbb{C}$ of these D3-branes.
Since the D3-branes are identical objects, the identifications that are imposed on the moduli space by the geometry are
$$\displaystyle z_{i}$$
$$\displaystyle\leftrightarrow z_{j},$$
$$\displaystyle\text{all other $z_{a}$ fixed},$$
$$\displaystyle z_{i}$$
$$\displaystyle\mapsto\gamma z_{i},$$
$$\displaystyle\text{all other $z_{j}$ fixed},$$
(2.4)
where $\gamma\equiv\exp(2\pi\mathrm{i}/k)$. For the purpose of the moduli space identifications we can ignore the additional S-duality transformation.
From these identifications we can see that the gauge-invariant Coulomb branch operators are the symmetric polynomials of $z_{i}{}^{k}$, and their generators are ($\sum_{i=1}^{N}z_{i}{}^{jk}$) for $j=1,\cdots,N$, with dimensions
$$k,2k,\cdots,Nk.$$
(2.5)
From (2.1) and (2.3), this naively implies that
$$4a=4c=kN^{2}+(k-1)N.$$
(2.6)
Applying (2.2), we find that the D3-brane charge of the $\mathbb{Z}_{k}$ S-fold is $\epsilon=(k-1)/(2k)$.
As discussed above, we know that even for $k=2$ this is not the whole story, due to the possibility of discrete torsion and discrete symmetries.
Let us review in detail this $k=2$ case, in a language that will be useful for our later analysis.
Among the known O3-planes, the above analysis only applies to $\widetilde{O3}{}^{-}$, $O3^{+}$ and $\widetilde{O3}{}^{+}$, for which $\epsilon=(k-1)/(2k)=+1/4$ is the correct value.
When the orientifold is $O3^{-}$, the $\mathrm{O}(2N)$ gauge group arising on $N$ D3-branes can be viewed as a $\mathbb{Z}_{2}$ gauging of the $\mathrm{SO}(2N)$ gauge theory, and thus, to compute $a$ and $c$, we need to use the $\mathrm{SO}(2N)$ theory instead of the $\mathrm{O}(2N)$ theory. In the $\mathrm{SO}(2N)$ theory
the residual gauge symmetries on the Coulomb branch, acting on the eigenvalues of a matrix in the adjoint representation of $\mathrm{SO}(2N)$, are generated by
$$\displaystyle z_{i}$$
$$\displaystyle\leftrightarrow z_{j},$$
$$\displaystyle\text{all other $z_{a}$ fixed},$$
$$\displaystyle(z_{i},z_{j})$$
$$\displaystyle\mapsto(-z_{i},-z_{j}),$$
$$\displaystyle\text{all other $z_{a}$ fixed}.$$
(2.7)
The dimensions of the independent gauge-invariant operators are then
$$2,4,\cdots,2(N-1);N,$$
(2.8)
where the last invariant is the Coulomb branch expression for the Pfaffian of an adjoint matrix of $SO(2N)$,
$z_{1}z_{2}\ldots z_{N}$.
From this list we can compute $a$ and $c$ again using (2.1) and (2.3) and find
$$4a=4c=2N^{2}-N,$$
(2.9)
from which we find $\epsilon=-1/4$ from (2.2).
This is indeed the correct value for $O3^{-}$.
The $\mathrm{SO}(2N)$ theory has a discrete $\mathbb{Z}_{2}$ global symmetry, corresponding to gauge transformations by elements of $\mathrm{O}(2N)$ which are not elements of $\mathrm{SO}(2N)$. Note that in the $\mathrm{SO}(2N)$ theory these are global symmetries rather than gauge symmetries. On the moduli space one of these transformations acts as $z_{1}\to-z_{1}$, and with this extra identification the group generated by (2.7) becomes the same as the group (2.4) acting on the D3-branes. In particular, this identification projects out the Pfaffian operator of dimension $N$, such that after it we obtain the Coulomb branch operators with dimensions (2.5). So, for this specific orientifold plane, the theory on the D3-branes is described by a $\mathbb{Z}_{2}$ quotient of some ‘parent’ theory which has a different group of identifications (2.7), and correspondingly different Coulomb branch operators. Only after gauging a $\mathbb{Z}_{2}$ global symmetry of this ‘parent’ theory do we get the theory of the D3-branes on the $O3^{-}$ orientifold.
We stress here that this gauging of $\mathbb{Z}_{2}$ symmetry does not change the central charges.
It is natural to ask how we could know directly from the theory of the D3-branes on the $O3^{-}$ plane that its central charges are not the same as the naive ones, because it arises as a $\mathbb{Z}_{2}$ gauging of some other theory. As discussed in [24, 23], when we have a discrete $\mathbb{Z}_{p}$ global symmetry, we have local operators that transform under this symmetry, as well as 3-plane operators that describe domain walls separating vacua that differ by a $\mathbb{Z}_{p}$ transformation. When we gauge the $\mathbb{Z}_{p}$ global symmetry, these local and 3-plane operators disappear from the spectrum. Instead we obtain new 2-plane operators (that may be viewed as worldvolumes of strings), characterized by having a $\mathbb{Z}_{p}$ gauge transformation when we go around them. These 2-plane operators are charged under a 2-form $\mathbb{Z}_{p}$ global symmetry in the language of [23]. So whenever we have a theory with a 2-form $\mathbb{Z}_{p}$ global symmetry, it is natural to expect that it arises by gauging a $\mathbb{Z}_{p}$ global symmetry of some ‘parent’ theory. And indeed, the analysis of [23] implies that this ‘parent’ theory can be obtained by gauging the 2-form $\mathbb{Z}_{p}$ global symmetry. Thus, whenever we have a theory with a $\mathbb{Z}_{p}$ 2-form global symmetry, we expect that its central charges would not be given by (2.3), but rather by those of its ‘parent’ theory222Gauging a discrete $\mathbb{Z}_{p}$ global symmetry does not change the dynamics on $\mathbb{R}^{4}$, but it does change the spectrum of local and non-local operators..
Suggested by this analysis for $k=2$, we expect that also for $k=3,4,6$, different versions of S-folds will be characterized by different 2-form $\mathbb{Z}_{p}\subset\mathbb{Z}_{k}$ global symmetries for the corresponding theories on the D3-branes, that will imply that these theories arise from ‘parent’ theories with $\mathbb{Z}_{p}$ global symmetries. The analysis above suggests that the identifications on the moduli spaces of these ‘parent’ theories should be subgroups of the group generated by (2.4), such that the ring of invariants is polynomial without any relation, and such that adding an extra $\mathbb{Z}_{p}$ generator produces the group (2.4).
Luckily such groups are already classified and are known as Shephard-Todd complex reflection groups; the fact that the ring of invariants is polynomial without any relation if and only if the group acting on $\mathbb{C}^{N}$ is a complex reflection group is known as the theorem of Chevalley, Shephard and Todd.333See [25] for a recent use of Shephard-Todd complex reflection groups in four-dimensional $\mathcal{N}{=}\,2$ theories. For more on complex reflection groups, see [26].
In our case, the available groups are known as $G(N,p,k)$, and are generated by the following elements:
$$\displaystyle z_{i}$$
$$\displaystyle\leftrightarrow z_{j},$$
$$\displaystyle\text{all other $z_{a}$ fixed},$$
$$\displaystyle(z_{i},z_{j})$$
$$\displaystyle\mapsto(\gamma z_{i},\gamma^{-1}z_{j}),$$
$$\displaystyle\text{all other $z_{a}$ fixed},$$
$$\displaystyle z_{i}$$
$$\displaystyle\mapsto\gamma^{p}z_{i},$$
$$\displaystyle\text{all other $z_{j}$ fixed},$$
(2.10)
where $p$ is a divisor of $k$. By adding another $\mathbb{Z}_{p}$ generator acting as $z_{i}\mapsto\gamma z_{i}$, all these groups become the groups (2.4).
Denoting $\ell=k/p$, we see that if there is a ‘parent’ theory with the identifications (2.10) on its moduli space, then its gauge-invariant Coulomb branch operators would be generated by the symmetric polynomials of $z_{i}^{k}$ and by $(z_{1}z_{2}\cdots z_{N})^{\ell}$, and therefore the dimensions of the Coulomb branch generators would be given by
$$k,2k,\ldots,(N-1)k;N\ell.$$
(2.11)
From (2.1) and (2.3) we then find that the central charges of the ‘parent’ theory, and also of its $\mathbb{Z}_{p}$ quotient that would describe the theory of $N$ D3-branes on the corresponding S-fold, are given by
$$4a=4c=kN^{2}+(2\ell-k-1)N.$$
(2.12)
Note that even though the $\mathbb{Z}_{p}$-gaugings of the theories with different values of $\ell$ all have the same moduli space, they are distinct theories with different central charges. When $p$ is not prime, one can also gauge subgroups of $\mathbb{Z}_{p}$, giving rise to additional theories, which again are not equivalent to the theories for other values of $\ell$ (even though they may have the same moduli space).
Therefore, it seems at this stage, that each $\mathbb{Z}_{k}$ S-fold with $k=2,3,4,6$ can have variants labeled by $\ell$ which is an integer dividing $k$.
The D3-brane charges $\epsilon_{k,\ell}$ of these S-folds can be easily found using (2.2) and (2.12):
$$\begin{array}[]{c||c|c|c|c|ccc}&\ell=1&\ell=2&\ell=3&\ell=4&\ell=6\\
\hline\cr\hline\cr k=2&\pagecolor{black!10}-1/4&\pagecolor{black!10}+1/4&&\\
k=3&\pagecolor{black!10}-1/3&&\pagecolor{black!10}+1/3&&\\
k=4&\pagecolor{black!10}-3/8&-1/8&&\pagecolor{black!10}+3/8&\\
k=6&\pagecolor{black!10}-5/12&-1/4&-1/12&&+5/12\end{array}$$
(2.13)
Note that just from the analysis above it is far from clear that S-folds and ${\cal N}=3$ theories corresponding to such identifications actually exist. However, we expect any S-fold to fall into one of these categories.
In the next two sections, we will see that the S-fold variants that really exist are only those shaded in the table (2.13) above, first by carefully studying the holographic dual and then by comparing with M-theory. Note also that there could be more than one theory with the same identifications. For $k=2$ there are three orientifold-types with the same $\ell=2$ identifications, though they all give rise to the same theory on D3-branes because they are all related by S-duality. We do not know the corresponding situation in our case. We also cannot rule out the existence of ${\cal N}=3$ theories having Coulomb branch operators (2.11) also for the values of $\ell$ and $k$ that do not come from S-folds. It would be interesting to shed further light on these questions, perhaps by a conformal bootstrap analysis of ${\cal N}=3$ SCFTs, or by analyzing the two dimensional chiral rings of the corresponding theories [27, 28].
3 Holographic dual
3.1 AdS duals of CFTs with discrete symmetries
Theories of quantum gravity are not expected to have any global symmetries (see, for instance, [24]). There are very strong arguments that this is the case for continuous symmetries, and it is believed to be true also for discrete symmetries. In the AdS/CFT correspondence, this implies that any global symmetry in the CFT should come from a gauge symmetry in anti-de Sitter (AdS) space.
For continuous global symmetries it is known that indeed they come from gauge fields in the bulk, with a boundary condition that sets their field strength to zero on the boundary. In some cases (like AdS${}_{4}$ [29]) there is also another consistent boundary condition in which the field strength does not go to zero on the boundary, and its boundary value becomes a gauge field in the CFT. Changing the boundary condition may be interpreted as gauging the global symmetry in the CFT (in cases where this gauging still gives a CFT).
In this section we discuss the analogous statements for discrete symmetries. We focus on the $AdS_{5}\leftrightarrow$ CFT${}_{4}$ case, but the generalization to other dimensions is straightforward. The discussion here is a special case of the discussion in appendix B of [23], generalized to five dimensions, but as far as we know its implications for the AdS/CFT correspondence were not explicitly written down before (see also [30]).
Consider a $\mathbb{Z}_{k}$ gauge symmetry on $AdS_{5}$ (thought of as a sector of a full theory of quantum gravity on $AdS_{5}$, that is dual to a 4d CFT). A universal way to describe such a gauge symmetry in five space-time dimensions is by a topological theory of a 1-form $A$ and a 3-form $C$, with an action
$${\cal L}={{ik}\over{2\pi}}A\wedge dC.$$
(3.1)
$A$ can be thought of as the gauge field for a $\mathbb{Z}_{k}$ symmetry; for example, the action above arises from a $U(1)$ gauge
symmetry that is spontaneously broken to $\mathbb{Z}_{k}$.
The forms $A$ and $C$ are both gauge fields, whose field strengths $dA$ and $dC$ vanish by the equations of motion. There are gauge transformations that shift the integrals of $A$ and $C$ over closed cycles by one. A gauge symmetry is just a redundancy in our description, but an invariant property of this theory is that it has line and 3-surface operators, given by $e^{i\oint A}$ and $e^{i\oint C}$, with $A$ and $C$ integrated over closed cycles. And, it has a 1-form $\mathbb{Z}_{k}$ global symmetry that multiplies the line operators $e^{i\oint A}$ by $e^{2\pi i/k}$, and a similar 3-form global symmetry.
When we put such a theory on $AdS_{5}$, we need to choose boundary conditions; the possible boundary conditions for such topological gauge theories were discussed in [23]. The variational principle implies that we need to set to zero $A\wedge C$ along the boundary. If we set to zero $C$ along the boundary, then the boundary value of $A$ gives a gauge field in the dual CFT, corresponding to a $\mathbb{Z}_{k}$ gauge symmetry in this CFT. With this boundary condition line operators in the bulk are allowed to approach the boundary and to become line operators in the CFT, while 3-surface operators cannot approach the boundary, but can end on the boundary, giving 2-surface operators in the CFT. This is as expected for a $\mathbb{Z}_{k}$ gauge symmetry in four space-time dimensions.
On the other hand, if we set $A$ to zero on the boundary, we obtain a $\mathbb{Z}_{k}$ global symmetry in the dual CFT. The line operators ending on the boundary now give local operators in the CFT, and the 1-form global symmetry in the bulk becomes a standard $\mathbb{Z}_{k}$ global symmetry in the CFT, under which these local operators are charged. The 3-surface operators going to the boundary give 3-surface operators in the CFT, which are domain walls between different vacua related by $\mathbb{Z}_{k}$.
Thus, whenever we have a $\mathbb{Z}_{k}$ gauge symmetry on $AdS_{5}$, there are two natural boundary conditions. One of them gives a conformal field theory with a $\mathbb{Z}_{k}$ global symmetry, and the other gives a conformal field theory with a $\mathbb{Z}_{k}$ gauge symmetry (and a $\mathbb{Z}_{k}$ 2-form global symmetry). The second theory is related to the first one by gauging its $\mathbb{Z}_{k}$ global symmetry, and similarly the first one arises from the second by gauging its $\mathbb{Z}_{k}$ 2-form global symmetry.
When $k$ is not prime, there are also additional possible boundary conditions, corresponding to gauging subgroups of $\mathbb{Z}_{k}$.
In the context of our discussion in the previous section, this implies that the ‘parent’ theories and their $\mathbb{Z}_{k}$-gaugings should arise from the same holographic dual, just with different boundary conditions for the $\mathbb{Z}_{k}$ gauge fields.
We will see in Sec. 3.4 how the $\mathbb{Z}_{k}$ gauge fields of (3.1) arise in F-theory on $AdS_{5}\times(S^{5}\times T^{2})/\mathbb{Z}_{k}$ from integrals of the type IIB five-form field $F_{5}$ on discrete cycles.
Note that all this applies already to the $AdS_{5}\times S^{5}/\mathbb{Z}_{2}$ case discussed in [31]; typically in that case only the option of having a global $\mathbb{Z}_{2}$ symmetry, leading to the $\mathrm{SO}(2N)$ gauge theory, is discussed.
Finally, note that a very similar story occurs already in type IIB string theory on $AdS_{5}\times S^{5}$, which includes a topological sector in the bulk corresponding to a 1-form $\mathbb{Z}_{N}$ gauge symmetry, with an action ${iN\over 2\pi}B_{2}\wedge dC_{2}$ (where $B_{2}$ and $C_{2}$ are the 2-form fields of type IIB string theory) [32, 33]; this topological theory was discussed in section 6 of [23]. In this case both simple boundary conditions give rise to a 1-form $\mathbb{Z}_{N}$ global symmetry in the dual CFT, and the resulting theories are $SU(N)$ and $SU(N)/\mathbb{Z}_{N}$ gauge theories [34, 35, 23]. In this specific case there is also an option of coupling these theories to a continuous $U(1)$ gauge symmetry, leading to a $U(N)$ theory [36, 37]. One can also quantize the topological theory in ways that do not lead to local field theories [38].
3.2 Discrete torsion
In [31], Witten showed how to characterize the variants of O3-planes by studying the discrete torsion on $S^{5}/\mathbb{Z}_{2}$.
In this section we generalize this to $S^{5}/\mathbb{Z}_{k}$ S-folds, for $k=3,4,6$.
For $k=2$, the discrete torsion of the NSNS and of the RR three-form field strengths takes values in $H^{3}(S^{5}/\mathbb{Z}_{2},\tilde{\mathbb{Z}})$, where the tilde over $\mathbb{Z}$ means that the coefficient system is multiplied by $(-1)$ when we go around the $\mathbb{Z}_{2}$ torsion 3-cycle of $S^{5}/\mathbb{Z}_{2}$.
For $k=3,4,6$, we have a $\mathbb{Z}_{k}$ torsion 3-cycle in $S^{5}/\mathbb{Z}_{k}$, and we have
an action of an element $\rho\in\mathrm{SL}(2,\mathbb{Z})$ on the NS-NS and R-R field strengths in $\mathbb{Z}\oplus\mathbb{Z}$
when we go around the $\mathbb{Z}_{k}$ torsion cycle of $S^{5}/\mathbb{Z}_{k}$. We can choose a specific form for this
$\rho$, given, say, by $\rho=\begin{pmatrix}-1&0\\
0&-1\end{pmatrix}$ for $k=2$, $\rho=\begin{pmatrix}-1&-1\\
1&0\end{pmatrix}$ for $k=3$, $\rho=\begin{pmatrix}0&-1\\
1&0\end{pmatrix}$ for $k=4$, and $\rho=\begin{pmatrix}0&-1\\
1&1\end{pmatrix}$ for $k=6$. Note that the eigenvalues of these matrices are $\gamma$ and $\gamma^{-1}$.
The discrete torsion is then given by $H^{3}(S^{5}/\mathbb{Z}_{k},(\mathbb{Z}\oplus\mathbb{Z})_{\rho})$, and its
computation is standard in mathematics444The method to compute cohomologies with twisted coefficients is explained in e.g. Hatcher [39] Chapter 3.H ; the cell decomposition of generalized lens spaces, of which $S^{5}/\mathbb{Z}_{k}$ is one, is given in the same book, Example 2.43. Section 5.2.1 of Davis-Kirk [40] was also quite helpful. .
In general, $H^{*}(S^{2n-1}/\mathbb{Z}_{k},A)$ where $A$ is a $\mathbb{Z}_{k}$-module is given by the cohomology of the complex
$$C^{0}\xrightarrow{1-t}C^{1}\xrightarrow{1+t+\cdots+t^{k-1}}C^{2}\xrightarrow{1-t}\cdots\xrightarrow{1+t+\cdots+t^{k-1}}C^{2n-2}\xrightarrow{1-t}C^{2n-1}$$
(3.2)
where all $C^{i}\simeq A$, $t$ is the generator of $\mathbb{Z}_{k}$, and the differential $d$ is alternately given by the multiplication by $1-t$ or by $1+t+\cdots+t^{k-1}$.
It is easy to see that $d^{2}=(1-t)(1+t+\cdots+t^{k-1})=1-t^{k}=0$.
When $k=2$, $t$ is just the multiplication by $-1$. Then $1+t=0$ and $1-t=2$, from which we conclude $H^{3}(S^{5}/\mathbb{Z}_{2},(\mathbb{Z}\oplus\mathbb{Z})_{\rho})=\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$, reproducing four types of O3-planes.
When $k=3,4,6$, the action $\rho$ of the generator of $\mathbb{Z}_{k}$ on $\mathbb{Z}\oplus\mathbb{Z}$ obeys
$1+\rho+\cdots+\rho^{k-1}=0$, and $\det(1-\rho)=3,2,1$ for $k=3,4,6$, respectively, and therefore
$$H^{3}(S^{5}/\mathbb{Z}_{k},(\mathbb{Z}\oplus\mathbb{Z})_{\rho})=\begin{cases}\mathbb{Z}_{3}&(k=3),\\
\mathbb{Z}_{2}&(k=4),\\
\mathbb{Z}_{1}&(k=6).\end{cases}$$
(3.3)
This gives the discrete torsion groups arising from the 3-form fields of type IIB string theory on these S-folds.
For $k=3$, we have three different possibilities, but
two non-trivial elements of $\mathbb{Z}_{3}$ are related by conjugation in $\mathrm{SL}(2,\mathbb{Z})$, so up to S-duality transformations there are just two types of S-folds with $k=3$ (in the same sense that up to S-duality there are just two types of O3-planes that give different theories for the D3-branes on them).
For $k=4$, the cohomology is $\mathbb{Z}_{2}$, and therefore we expect two types of S-folds.
Similarly, for $k=6$, there is only one type of S-fold.
3.3 Generalized Pfaffians
In the holographic duals of the S-folds, the discrete torsion described above corresponds to discrete 3-form fluxes on the $S^{3}/\mathbb{Z}_{k}$ discrete 3-cycle in $S^{5}/\mathbb{Z}_{k}$, and it affects the spectrum of wrapped branes on this 3-cycle. We can use this to match the discussion of the previous subsection to our analysis of Section 2.
For $k=2$ this was analyzed in [31]. There is a $\mathbb{Z}_{2}$-torsion three-cycle of the form $S^{3}/\mathbb{Z}_{2}$ within $S^{5}/\mathbb{Z}_{2}$,
and only when the discrete torsion is zero, i.e. when the 3-plane is $O3^{-}$, we can wrap a D3-brane on this cycle.
By analyzing the properties of this wrapped D3-brane we find that it corresponds to a dimension $N$ operator in the dual CFT,
which can be naturally identified with the Pfaffian operator, that only exists in the $\mathrm{SO}(2N)$ theory but not in $\mathrm{SO}(2N+1)$ or $\mathrm{Sp}(N)$ theories. Note that, according to the discussions above, this operator exists in the ‘parent’ $\mathrm{SO}(2N)$ theory, but not after we gauge the $\mathbb{Z}_{2}$ to get the $\mathrm{O}(2N)$ theory. So in the AdS dual it exists when we choose the boundary condition for the $\mathbb{Z}_{2}$ gauge theory in the bulk that gives a $\mathbb{Z}_{2}$ global symmetry, corresponding to the ‘parent’ theory, but not for the other boundary condition, that corresponds to the theory on the D3-branes.
The obstruction to wrap D3-branes on $S^{3}/\mathbb{Z}_{2}$ can be understood as follows.
The NSNS 3-form flux $G$ can in general be in a non-trivial cohomological class.
But when pulled-back to the worldvolume of a single D3-brane, $G$ is the exterior derivative of a gauge-invariant object $B-F$, where $F$ is the gauge field on the D3-brane,
and therefore the cohomology class $[G]$ should be trivial. The argument for the RR flux is the S-dual of this.
For all $k=3$, $4$, $6$, there is a $\mathbb{Z}_{k}$-torsion three-cycle in $S^{5}/\mathbb{Z}_{k}$ of the form $S^{3}/\mathbb{Z}_{k}$.
When the discrete torsion is zero, there is no obstruction to wrapping a D3-brane on this cycle.
The scaling dimension of this wrapped D3-brane can be easily found to be $kN/k=N$.
This matches the scaling dimensions we found for the $\ell=1$ variants of Section 2. So we identify
the S-fold with no discrete torsion with the $\ell=1$ case (either the ‘parent’ theory, or its $\mathbb{Z}_{k}$ gauging that gives the theory on the D3-branes, depending on the boundary conditions).
For $k=6$, there is nothing more to discuss, since $H^{3}(S^{5}/\mathbb{Z}_{k},(\mathbb{Z}\oplus\mathbb{Z})_{\rho})$ itself is trivial, so we do not find any variant except $\ell=1$.
For $k=3$, when the discrete torsion in $H^{3}(S^{5}/\mathbb{Z}_{k},(\mathbb{Z}\oplus\mathbb{Z})_{\rho})=\mathbb{Z}_{3}$ is non-trivial, we cannot wrap a D3-brane on this cycle. So
this should correspond to the $\ell=3$ variant (with no discrete symmetries).
The most subtle is the $k=4$ case, when the discrete torsion is given by an element in $H^{3}(S^{5}/\mathbb{Z}_{k},(\mathbb{Z}\oplus\mathbb{Z})_{\rho})=\mathbb{Z}_{2}$. We claim
that the non-trivial element of this discrete torsion gives the $\ell=4$ variant of Section 2, not the $\ell=2$ variant.
To see this, it is instructive to recall why in the case $k=2$ we can wrap two D3-branes on $S^{3}/\mathbb{Z}_{2}$ even with the discrete torsion.
Note that it is not just that for $N$ D3-branes wrapping on the same cycle, the triviality of $N\cdot[G]$ suffices. For one thing, if we have two D3-branes wrapping on the same locus with at least $\mathrm{U}(1)\times\mathrm{U}(1)$ unbroken, then $B-F_{1}$ and $B-F_{2}$ are both gauge-invariant (where $F_{1,2}$ are the $\mathrm{U}(1)$ gauge fields coming from the first and the second Chan-Paton factor), and therefore $[G]$ still needs to be zero.
To wrap two D3-branes on $S^{3}/\mathbb{Z}_{2}$ with discrete torsion consistently, one needs to require that the Chan-Paton indices 1,2 are interchanged when we go around the $\mathbb{Z}_{2}$ cycle. This means that in fact there is a single connected D3-brane of the shape $S^{3}$, which is wrapped on $S^{3}/\mathbb{Z}_{2}$ using a 2:1 quotient map. In this particular setting, we know how to judge the consistency of wrapping: on $S^{3}$, $(B-F)$ is a gauge-invariant object, so $[G]$ should vanish there. What needs to be checked is then to pull back the spacetime $G$ using the map
$$S^{3}\to S^{3}/\mathbb{Z}_{2}\to S^{5}/\mathbb{Z}_{2},$$
(3.4)
where the first is the 2:1 projection and the second is the embedding. The result is zero, from the trivial fact that $H^{3}(S^{3},\mathbb{Z})=\mathbb{Z}$ does not have non-zero two-torsion elements.
Thus we see that we can wrap two D3-branes in this way.
Now, let us come back to the $k=4$ case, and try to wrap two D3-branes on $S^{3}/\mathbb{Z}_{4}$ with discrete torsion, which should naively be possible since the discrete torsion lies in $\mathbb{Z}_{2}$. Again, if the Chan-Paton structure is trivial, we cannot wrap them. If we try to do the analogue of the $k=2$ case above, we can try wrapping two D3-branes such that the Chan-Paton indices 1 and 2 are exchanged. Just as above, this is equivalent to wrapping one D3-brane on $S^{3}/\mathbb{Z}_{2}$, covering $S^{3}/\mathbb{Z}_{4}$ with a 2:1 quotient map.
To test the consistency of the wrapping, one needs to pull back the spacetime obstruction on $S^{5}/\mathbb{Z}_{4}$ via
$$S^{3}/\mathbb{Z}_{2}\to S^{3}/\mathbb{Z}_{4}\to S^{5}/\mathbb{Z}_{4}.$$
(3.5)
One finds that the pull-back is non-trivial,
showing that this embedding is inconsistent.555In general, the pull-back map from $S^{n}/\mathbb{Z}_{ab}$ to $S^{n}/\mathbb{Z}_{b}$ is a multiplication map by $1+t+t^{2}+...+t^{a-1}$, where $t$ is the generator of the $\mathbb{Z}_{a}$ action that divides $S^{n}/\mathbb{Z}_{b}$ to $S^{n}/\mathbb{Z}_{ab}$.
It is clear that we can wrap four D3-branes on the discrete cycle, so this implies that the theory with this discrete torsion should be the $\ell=4$ variant discussed in Section 2, and that we cannot have S-folds with the $\ell=2$ variant.
This does not prove, we admit, that there are no other non-Abelian configurations on two D3-branes on $S^{3}/\mathbb{Z}_{4}$ that still allow the wrapping.
But we will see that the choice $\ell=4$ for this discrete torsion variant matches with the M-theory computation below.
3.4 Realization of the $\mathbb{Z}_{k}$ gauge theory in F-theory
After these discussions we can understand how the $\mathbb{Z}_{k}$ gauge theory of Section 3.1 arises in F-theory on $S^{5}/\mathbb{Z}_{k}$, when the discrete torsion in $H^{3}(S^{5}/\mathbb{Z}_{k},(\mathbb{Z}\oplus\mathbb{Z})_{\rho})$ is zero.
Recall that $H_{n}(S^{5}/\mathbb{Z}_{k},\mathbb{Z})$ is $\mathbb{Z}_{k}$ for $n=1$ and $3$, and its generator is $S^{1}/\mathbb{Z}_{k}$ and $S^{3}/\mathbb{Z}_{k}$, respectively.
We can wrap a D3-brane on these cycles to obtain a world 3-cycle and a worldline on $AdS_{5}$.
(Note that the latter is the generalized Pfaffian particle discussed above.)
Clearly, they both carry $\mathbb{Z}_{k}$ charges.
We are then naturally led to identify the worldline as coupling to the 1-form $A$ and the world 3-cycle to the 3-form $C$ in (3.1).
For this identification to make sense, it should be the case that if we rotate the generalized Pfaffian around an $S^{1}$ that has a unit linking number with the world 3-cycle of the D3-brane wrapped on $S^{1}/\mathbb{Z}_{k}$,
there should be a non-trivial $\mathbb{Z}_{k}$ holonomy of unit strength.
This can be seen as follows.
A D3-brane wrapped on $S^{1}/\mathbb{Z}_{k}$ creates an $F_{5}$ flux given by the Poincaré dual of its worldvolume.
In this case this is given by $s\otimes c$, where $s$ is the generator of $H^{1}(S^{1},\mathbb{Z})$ where this $S^{1}$ is linked in $AdS_{5}$ around the worldvolume of the D3-brane, and $c$ is the generator of $H^{4}(S^{5}/\mathbb{Z}_{k},\mathbb{Z})=\mathbb{Z}_{k}$.
Now, we use the cohomology long exact sequence associated to the short exact sequence $0\to\mathbb{Z}\to\mathbb{R}\to\mathrm{U}(1)\to 0$ to conclude that there is a natural isomorphism
$$H^{3}(S^{5}/\mathbb{Z}_{k},\mathrm{U}(1))\simeq H^{4}(S^{5}/\mathbb{Z}_{k},\mathbb{Z})=\mathbb{Z}_{k}.$$
(3.6)
In physics terms, this means that the discrete $\mathbb{Z}_{k}$ field strength (which is $\mathbb{Z}$-valued) with four legs along $S^{5}/\mathbb{Z}_{k}$ can be naturally identified with the discrete $\mathbb{Z}_{k}$ holonomy (which is $\mathrm{U}(1)$-valued) with three legs along $S^{5}/\mathbb{Z}_{k}$.
From this we see that we have a holonomy of the type IIB $C_{4}$ field given by the element
$$s\otimes c\in H^{1}(S^{1},\mathbb{Z})\otimes H^{3}(S^{5}/\mathbb{Z}_{k},\mathrm{U}(1)),$$
(3.7)
which can be naturally integrated on the cycle $S^{1}\times S^{3}/\mathbb{Z}_{k}$ to give $\exp(2\pi i/k)$.
This means that the generalized Pfaffian particle wrapped on $S^{3}/\mathbb{Z}_{k}$, when carried around the $S^{1}$ linking the worldvolume of the D3-brane wrapped on $S^{1}/\mathbb{Z}_{k}$, experiences this holonomy.
This is indeed the behavior we expect for the objects charged under the $\mathbb{Z}_{k}$ 1-form $A$ and the $\mathbb{Z}_{k}$ 3-form $C$ in (3.1).
4 Comparison to M-theory
In this section, we consider M-theory configurations on $(\mathbb{C}^{3}\times T^{2})/\mathbb{Z}_{k}$. We know that we obtain such configurations from any of our S-folds on a circle, using the standard relation between type IIB theory on a circle and M-theory on a torus. However, the opposite is not true, since when we translate some configuration of discrete fluxes in M-theory on $(\mathbb{C}^{3}\times T^{2})/\mathbb{Z}_{k}$ to F-theory on a circle, we could also have some non-trivial action of the shift around the F-theory circle, corresponding to a ‘shift-S-fold’
where there is a rotation on the transverse $\mathbb{C}^{3}$ when we go around the compactified $S^{1}$.
By analyzing all possible discrete charges in M-theory and translating them to F-theory,
we learn about all possible variants of S-folds.
4.1 Discrete fluxes in M-theory on $(\mathbb{C}^{3}\times T^{2})/\mathbb{Z}_{k}$
Let us start by analyzing the possible discrete fluxes in M-theory, which come from 4-form fluxes.
For a given $k$, $(\mathbb{C}^{3}\times T^{2})/\mathbb{Z}_{k}$ has several fixed points of the form $\mathbb{C}^{4}/\mathbb{Z}_{\ell_{i}}$,
each of which has an associated $H^{4}(S^{7}/\mathbb{Z}_{\ell_{i}},\mathbb{Z})=\mathbb{Z}_{\ell_{i}}$ in M-theory [41].
From this viewpoint, the possible discrete charges are given by the orbifold actions on the fixed points,
$$\bigoplus H^{3}(\mathbb{C}^{4}/\mathbb{Z}_{\ell_{i}},\mathbb{Z})=\begin{cases}\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}&(k=2),\\
\mathbb{Z}_{3}\oplus\mathbb{Z}_{3}\oplus\mathbb{Z}_{3}&(k=3),\\
\mathbb{Z}_{4}\oplus\mathbb{Z}_{4}\oplus\mathbb{Z}_{2}&(k=4),\\
\mathbb{Z}_{6}\oplus\mathbb{Z}_{3}\oplus\mathbb{Z}_{2}&(k=6).\end{cases}$$
(4.1)
We can alternatively measure the same charges by considering $H^{4}$ of the ‘asymptotic infinity’ of $(\mathbb{C}^{3}\times T^{2})/\mathbb{Z}_{k}$, which has the form $(S^{5}\times T^{2})/\mathbb{Z}_{k}$.
This is a $T^{2}$ bundle over $S^{5}/\mathbb{Z}_{k}$, and as such, one can apply the Leray-Serre spectral sequence666For an introduction on the Leray-Serre spectral sequence, see e.g. [42].
In our case, the computation goes as follows.
The second page $E^{p,q}_{2}$ of the spectral sequence is given by $E^{p,q}_{2}=H^{p}(S^{5}/\mathbb{Z}_{k},H^{q}(T^{2},\mathbb{Z}))$, with the differentials $d_{2}^{p,q}:E^{p,q}_{2}\to E^{p+2,q-1}_{2}$. The third page is given by $E^{p,q}_{3}=\mathrm{Ker}\ d^{p,q}_{3}/\mathrm{Im}\ d^{p-2,q+1}$, and has the differentials $d_{3}^{p,q}:E^{p,q}_{3}\to E^{p+3,q-2}_{3}$.
The second page and the part relevant for us of the third page are then given by
$\begin{array}[]{c|cccccc}2&\mathbb{Z}&0&\mathbb{Z}_{k}&0&\mathbb{Z}_{k}&\mathbb{Z}\\
1&0&X&0&X&0&X\\
0&\mathbb{Z}&0&\mathbb{Z}_{k}&0&\mathbb{Z}_{k}&\mathbb{Z}\\
\hline\cr&0&1&2&3&4&5\end{array},\qquad\begin{array}[]{c|cccccc}2&&0&\mathbb{Z}_{k}&&&\\
1&&&&X&&\\
0&&&&&\mathbb{Z}_{k}&\mathbb{Z}\\
\hline\cr&0&1&2&3&4&5\end{array}$
where $X=\mathbb{Z}_{2}\oplus\mathbb{Z}_{2},\mathbb{Z}_{3},\mathbb{Z}_{2},\mathbb{Z}_{1}$ for $k=2,3,4,6$, respectively.
From this we conclude that in fact $F^{p,q}/F^{p+1,q-1}=E^{p,q}_{\infty}\simeq E^{p,q}_{2}$ for $p+q=4$., that says that it has the filtration
$$H^{4}((S^{5}\times T^{2})/\mathbb{Z}_{k},\mathbb{Z})=F^{2,2}\supset F^{3,1}\supset F^{4,0},$$
(4.2)
where
$$\displaystyle F^{2,2}/F^{3,1}$$
$$\displaystyle=H^{2}(S^{5}/\mathbb{Z}_{k},H^{2}(T^{2},\mathbb{Z}))=H^{2}(S^{5}/\mathbb{Z}_{k},\mathbb{Z}),$$
(4.3)
$$\displaystyle F^{3,1}/F^{4,0}$$
$$\displaystyle=H^{3}(S^{5}/\mathbb{Z}_{k},H^{1}(T^{2},\mathbb{Z}))=H^{3}(S^{5}/\mathbb{Z}_{k},(\mathbb{Z}\oplus\mathbb{Z})_{\rho}),$$
(4.4)
$$\displaystyle F^{4,0}$$
$$\displaystyle=H^{4}(S^{5}/\mathbb{Z}_{k},H^{0}(T^{2},\mathbb{Z}))=H^{4}(S^{5}/\mathbb{Z}_{k},\mathbb{Z}).$$
(4.5)
The different subgroups here correspond in a sense to the ‘number of legs along the base $S^{5}/\mathbb{Z}_{k}$ and the fiber $T^{2}$’.
As we are taking the $\mathbb{Z}$-valued cohomology, we only have a filtration and it is not guaranteed that the group (4.2) is a direct sum of (4.3)–(4.5).
We can easily compute (4.3)–(4.5) to obtain
$$F^{2,2}/F^{3,1}=\mathbb{Z}_{k},\quad F^{3,1}/F^{4,0}=\begin{cases}\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}&(k=2)\\
\mathbb{Z}_{3}&(k=3)\\
\mathbb{Z}_{2}&(k=4)\\
\mathbb{Z}_{1}&(k=6)\end{cases},\quad F^{4,0}=\mathbb{Z}_{k}.$$
(4.6)
From the standard F-theory / M-theory mapping, we see that each piece (4.3)–(4.5)
has the following effect in the F-theory language:
•
The piece $F^{2,2}/F^{3,1}$ becomes a component of the metric of F-theory on $S^{1}$.
Concretely, it specifies the amount of the rotation on the transverse space while we go around $S^{1}$.
Therefore, the system is a plain $S^{1}$ compactification if the flux in this piece is zero, whereas it is a shift S-fold if it is non-zero.
•
The piece $F^{3,1}/F^{4,0}$ becomes the discrete RR and NSNS 3-form fluxes around the S-fold.
Indeed, the coefficient system $H^{1}(T^{2},\mathbb{Z})$ of the fiber is exactly the one $(\mathbb{Z}\oplus\mathbb{Z})_{\rho}$ that we discussed in the previous section, and there is a natural isomorphism of $F^{3,1}/F^{4,0}$ with the discrete torsion (3.3).
•
The piece $F^{4,0}$ becomes an $F_{5}$ flux having one leg along the $S^{1}$. As discussed in Sec. 3.4, we have a natural identification $H^{4}(S^{5}/\mathbb{Z}_{k},\mathbb{Z})=H^{3}(S^{5}/\mathbb{Z}_{k},\mathrm{U}(1))$.
So, this piece can also be regarded as specifying a $C_{4}$ holonomy having one leg along the $S^{1}$.
Note that
$$(F^{2,2}/F^{3,1})\oplus(F^{3,1}/F^{4,0})\oplus F^{4,0}=\begin{cases}\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}&(k=2),\\
\mathbb{Z}_{3}\oplus\mathbb{Z}_{3}\oplus\mathbb{Z}_{3}&(k=3),\\
\mathbb{Z}_{4}\oplus\mathbb{Z}_{4}\oplus\mathbb{Z}_{2}&(k=4),\\
\mathbb{Z}_{6}\oplus\mathbb{Z}_{3}\oplus\mathbb{Z}_{2}&(k=6),\end{cases}$$
(4.7)
and we see that (4.7) and (4.1) are the same as abstract groups.
What remains is to figure out the precise mapping between the two.
This is a purely geometrical question since it is just the relation of the $H^{4}$ of $(\mathbb{C}^{3}\times T^{2})/\mathbb{Z}_{k}$ computed at the origin and at the asymptotic infinity.
As we have not yet done this computation, we will use guesswork, and then verify that it leads to consistent results for the M2-brane and D3-brane charges of the various singularities.
The $k=2$ case can be worked out using the known properties of the orientifolds. In this case the map
$$a:\bigoplus H^{3}(\mathbb{C}^{4}/\mathbb{Z}_{\ell_{i}})=(\mathbb{Z}_{2})^{4}\to F^{2,2}/F^{3,1}=\mathbb{Z}_{2}$$
(4.8)
is given by the sum of the four $\mathbb{Z}_{2}$’s, and the map
$$b:F^{4,0}=\mathbb{Z}_{2}\to\bigoplus H^{3}(\mathbb{C}^{4}/\mathbb{Z}_{\ell_{i}})=(\mathbb{Z}_{2})^{4}$$
(4.9)
is the diagonal embedding.
Note that $a\circ b$ is indeed a zero map.
We assume that a similar relation holds also in the other cases. Namely, we assume that
$$a:\bigoplus H^{3}(\mathbb{C}^{4}/\mathbb{Z}_{\ell_{i}})\to F^{2,2}/F^{3,1}=\mathbb{Z}_{k}$$
(4.10)
is the ‘sum’ and that
$$b:F^{4,0}=\mathbb{Z}_{k}\to\bigoplus H^{3}(\mathbb{C}^{4}/\mathbb{Z}_{\ell_{i}})$$
(4.11)
is the ‘diagonal embedding’. For $k=4$ and $k=6$, we need to use natural maps such as $\mathbb{Z}_{2}\to\mathbb{Z}_{6}$ and $\mathbb{Z}_{6}\to\mathbb{Z}_{2}$. We just choose a multiplication by 3 in the former, and the mod-2 map in the latter, and similarly for $k=4$.
We see that the composition $a\circ b$ is indeed a zero map, which gives a small check of our assumptions.
4.2 M2-brane and D3-brane charges
Using the discussions above, let us test our identification by working out the M2-brane and D3-brane charges of our various configurations.
We use the formula of [43] for the charge of $\mathbb{C}^{4}/\mathbb{Z}_{k}$ orbifolds with discrete torsion,
$$\text{(M2-brane charge)}=-\frac{1}{24}(k-\frac{1}{k})+\frac{\ell(k-\ell)}{2k}.$$
(4.12)
The cases we use are
$$\begin{array}[]{cr|ccccccc}&&\ell=0&\ell=1&\ell=2&\ell=3&\ell=4&\ell=5\\
\hline\cr k=2:&-\frac{1}{16}+&0&\frac{1}{4}\\[10.00002pt]
k=3:&-\frac{1}{9}+&0&\frac{1}{3}&\frac{1}{3}\\[10.00002pt]
k=4:&-\frac{5}{32}+&0&\frac{3}{8}&\frac{1}{2}&\frac{3}{8}\\[10.00002pt]
k=6:&-\frac{35}{144}+&0&\frac{5}{12}&\frac{2}{3}&\frac{3}{4}&\frac{2}{3}&\frac{5}{12}\end{array}.$$
(4.13)
A long list of tables analyzing all possible cases follows.
The end result is that every possible M-theory configuration can be interpreted as a plain $S^{1}$ compactification or a shift-S-fold on $S^{1}$ of precisely the S-folds that we discussed in the previous section (the types shaded in (2.13)),
namely the S-folds with
$$(k,\ell)=(2,1),(2,2),(3,1),(3,3),(4,1),(4,4),\ \text{and}\ (6,1).$$
(4.14)
In all cases we compute the D3-brane charge using the sum of the M2-brane charges of the orbifold singularities (4.12) and successfully compare it with our expectation (2.13).
4.2.1 $k=2$
Let us denote the discrete charges of the four fixed points as elements in $\begin{pmatrix}\mathbb{Z}_{2}&\mathbb{Z}_{2}\\
\mathbb{Z}_{2}&\mathbb{Z}_{2}\end{pmatrix}$. The piece $F_{4,0}=\mathbb{Z}_{2}$ is generated by $\begin{pmatrix}0&0\\
0&0\end{pmatrix}$ and $\begin{pmatrix}1&1\\
1&1\end{pmatrix}$. The piece $F^{2,2}/F^{3,1}$ is given by the sum of the four entries.
In the interpretations below, we consider the columns of the matrices to correspond to O2-planes, when we interpret our
M-theory configuration in type IIA by shrinking one cycle of the torus (as we can do for $k=2$).
Of course everything should work out correctly in this $k=2$ case, and has been already worked out in [44, 45].
We reproduce the analysis here since it is a useful warm-up for $k=3,4,6$.
Shift $=0/2\in F^{2,2}/F^{3,1}$:
$$\begin{array}[]{c|ccccccc}\hline\cr\text{label}&&&\begin{pmatrix}0&0\\
0&0\end{pmatrix}&\begin{pmatrix}1&1\\
1&1\end{pmatrix}\\
\text{IIA}&&&O2{}^{-}+O2{}^{-}&\widetilde{O2}{}^{-}+\widetilde{O2}{}^{-}\\
\text{\#D3}&&&-\frac{1}{4}&-\frac{1}{4}+1\\
\hline\cr\text{label}&\begin{pmatrix}0&1\\
0&1\end{pmatrix}&\begin{pmatrix}1&0\\
1&0\end{pmatrix}&&&\begin{pmatrix}0&0\\
1&1\end{pmatrix}&\begin{pmatrix}1&1\\
0&0\end{pmatrix}\\
\text{IIA}&O2{}^{-}+\widetilde{O2}{}^{-}&\widetilde{O2}{}^{-}+O2{}^{-}&&&O2{}^{+}+O2{}^{+}&\widetilde{O2}{}^{+}+\widetilde{O2}{}^{+}\\
\text{\#D3}&+\frac{1}{4}&+\frac{1}{4}&&&+\frac{1}{4}&+\frac{1}{4}\\
\hline\cr\text{label}&&&\begin{pmatrix}0&1\\
1&0\end{pmatrix}&\begin{pmatrix}1&0\\
0&1\end{pmatrix}\\
\text{IIA}&&&O2{}^{+}+\widetilde{O2}{}^{+}&\widetilde{O2}{}^{+}+{O2}{}^{+}\\
\text{\#D3}&&&+\frac{1}{4}&+\frac{1}{4}\\
\hline\cr\end{array}$$
The first two come from $O3{}^{-}$, but the latter of the two has one additional mobile D3-brane stuck at the origin, due to a non-trivial Wilson line around $S^{1}$ in the component of $\mathrm{O}(2N)$ disconnected from the identity.
The rest all come from the other three O3-planes wrapped on $S^{1}$.
Shift $=1/2\in F^{2,2}/F^{3,1}$:
$$\begin{array}[]{c|ccccccc}\hline\cr\text{label}&&&\begin{pmatrix}1&0\\
0&0\end{pmatrix}&\begin{pmatrix}0&1\\
1&1\end{pmatrix}\\
\text{IIA}&&&\widetilde{O2}{}^{+}+O2{}^{-}&{O2}{}^{-}+\widetilde{O2}{}^{-}\\
\text{\#D3}&&&0&0+\frac{1}{2}\\
\hline\cr\text{label}&\begin{pmatrix}1&1\\
0&1\end{pmatrix}&\begin{pmatrix}0&0\\
1&0\end{pmatrix}&&&\begin{pmatrix}1&0\\
1&1\end{pmatrix}&\begin{pmatrix}0&1\\
0&0\end{pmatrix}\\
\text{IIA}&\widetilde{O2}{}^{+}+\widetilde{O2}{}^{-}&{O2}{}^{+}+O2{}^{-}&&&\widetilde{O2}{}^{-}+O2{}^{+}&{O2}{}^{-}+\widetilde{O2}{}^{+}\\
\text{\#D3}&0+\frac{1}{2}&0&&&0&0+\frac{1}{2}\\
\hline\cr\text{label}&&&\begin{pmatrix}1&1\\
1&0\end{pmatrix}&\begin{pmatrix}0&0\\
0&1\end{pmatrix}\\
\text{IIA}&&&\widetilde{O2}{}^{-}+\widetilde{O2}{}^{+}&{O2}{}^{-}+{O2}{}^{+}\\
\text{\#D3}&&&0+\frac{1}{2}&0\\
\hline\cr\end{array}$$
They are
$\mathbb{Z}_{2}$ shift-orientifolds around $S^{1}$.
More precisely, the geometry is a $\mathbb{C}^{3}$ fibration over $S^{1}$ such that when we go around $S^{1}$, we have a multiplication by $(-1)$ on $\mathbb{C}^{3}$.
The ones with D3-brane charge $0$ are an empty shift-orientifold, and the ones with charge $0+1/2$ have one D3-brane wrapped around $S^{1}$.
Note that the charge $1/2$ we are seeing here reflects the fact that the $T^{2}$ of M-theory is fibered over $\mathbb{C}^{3}/\mathbb{Z}_{2}$.
In the type IIB frame, a fiber over a particular point on $S^{1}$ is $\mathbb{C}^{3}$, and therefore the D3-brane charge is 1 if we integrate over the asymptotic infinity of this $\mathbb{C}^{3}$.
4.2.2 $k=3$
Let us denote the discrete charges of the three fixed points as elements in $(\mathbb{Z}_{3},\mathbb{Z}_{3},\mathbb{Z}_{3})$. The piece $F_{4,0}=\mathbb{Z}_{3}$ consists of $(0,0,0)$, $(1,1,1)$, $(2,2,2)=-(1,1,1)$.
The piece $F^{2,2}/F^{3,1}=\mathbb{Z}_{3}$ corresponding to the shift is given by the sum of the entries.
There are 27 choices in total.
Shift $=0/3\in F^{2,2}/F^{3,1}$:
$$\begin{array}[]{c|ccc}\hline\cr\text{label}&(0,0,0)&(1,1,1)&(2,2,2)\\
\text{\#D3}&-\frac{1}{3}&-\frac{1}{3}+1&-\frac{1}{3}+1\\
\hline\cr\text{label}&(0,1,2)&(1,2,0)&(2,0,1)\\
\text{\#D3}&+\frac{1}{3}&+\frac{1}{3}&+\frac{1}{3}\\
\hline\cr\text{label}&(0,2,1)&(1,0,2)&(2,1,0)\\
\text{\#D3}&+\frac{1}{3}&+\frac{1}{3}&+\frac{1}{3}\\
\hline\cr\end{array}$$
Among the first three, the first entry has the right D3-brane charge to be the $(k=3,\ell=1)$ S-fold, see (2.13). The other two have one more mobile D3-brane, stuck at the origin through a non-trivial $\mathbb{Z}_{3}$ background holonomy around $S^{1}$.
The others all have the right D3-charge to be the $(k=3,\ell=3)$ S-fold, see (2.13) again.
The second three and the third three have opposite $\mathbb{Z}_{3}$ charge characterizing the type of the S-fold.
Shift $=1/3\in F^{2,2}/F^{3,1}$:
$$\begin{array}[]{c|ccc}\hline\cr\text{label}&(1,0,0)&(2,1,1)&(0,2,2)\\
\text{\#D3}&0&\frac{2}{3}&\frac{1}{3}\\
\hline\cr\text{label}&(1,1,2)&(2,2,0)&(0,0,1)\\
\text{\#D3}&\frac{2}{3}&\frac{1}{3}&0\\
\hline\cr\text{label}&(1,2,1)&(2,0,2)&(0,1,0)\\
\text{\#D3}&\frac{2}{3}&\frac{1}{3}&0\\
\hline\cr\end{array}$$
They are the $\mathbb{Z}_{3}$ shift-S-folds, with the rotation angle $2\pi/3$, of the flat background on $S^{1}$, with 0, 1, 2 D3-brane(s) wrapped around at the origin.
Shift $=2/3\in F^{2,2}/F^{3,1}$:
$$\begin{array}[]{c|ccc}\hline\cr\text{label}&(2,0,0)&(0,1,1)&(1,2,2)\\
\text{\#D3}&0&\frac{2}{3}&\frac{1}{3}\\
\hline\cr\text{label}&(2,1,2)&(0,2,0)&(1,0,1)\\
\text{\#D3}&\frac{2}{3}&0&\frac{1}{3}\\
\hline\cr\text{label}&(2,2,1)&(0,0,2)&(1,1,0)\\
\text{\#D3}&\frac{2}{3}&0&\frac{1}{3}\\
\hline\cr\end{array}$$
They are again the $\mathbb{Z}_{3}$ shift-S-folds, but with the rotation angle $4\pi/3$, of the flat background on $S^{1}$, with 0, 1, 2 mobile D3-brane(s) wrapped around at the origin.
4.2.3 $k=4$
We denote the charges at the fixed points as elements in $(\mathbb{Z}_{4},\mathbb{Z}_{4};\mathbb{Z}_{2})$.
The piece $F_{4,0}=\mathbb{Z}_{4}$ consists of $(0,0;0)$, $(1,1;1)$, $(2,2;0)$, $(3,3;1)$.
Shift $=0/4\in F^{2,2}/F^{3,1}$:
$$\begin{array}[]{c|cccc}\hline\cr\text{label}&(0,0;0)&(1,1;1)&(2,2;0)&(3,3;1)\\
\text{\#D3}&-\frac{3}{8}&-\frac{3}{8}+1&-\frac{3}{8}+1&-\frac{3}{8}+1\\
\hline\cr\text{label}&(0,2;1)&(1,3;0)&(2,0;1)&(3,1;0)\\
\text{\#D3}&+\frac{3}{8}&+\frac{3}{8}&+\frac{3}{8}&+\frac{3}{8}\\
\hline\cr\end{array}$$
Among the first four, the first has the right D3-brane charge to be the $(k=4,\ell=1)$ S-fold, see (2.13).
The other three have one more mobile D3-brane, stuck at the origin through non-trivial $\mathbb{Z}_{4}$ background holonomy around $S^{1}$.
All the second four have the right D3-charge to be the $(k=4,\ell=4)$ S-fold, see (2.13).
Shift $=1/4\in F^{2,2}/F^{3,1}$:
$$\begin{array}[]{c|cccc}\hline\cr\text{label}&(0,1;0)&(1,2;1)&(2,3;0)&(3,0;1)\\
\text{\#D3}&0&+\frac{3}{4}&+\frac{1}{2}&+\frac{1}{4}\\
\hline\cr\text{label}&(0,3;1)&(1,0;0)&(2,1;1)&(3,2;0)\\
\text{\#D3}&+\frac{1}{4}&0&+\frac{3}{4}&+\frac{1}{2}\\
\hline\cr\end{array}$$
These are $\mathbb{Z}_{4}$ shift-S-folds of flat space, with rotation angle $\pi/2$,
and with $0$, $1$, $2$ or $3$ mobile D3-brane(s) around $S^{1}$.
Shift $=2/4\in F^{2,2}/F^{3,1}$:
$$\begin{array}[]{c|cccc}\hline\cr\text{label}&(0,0;1)&(1,1;0)&(2,2;1)&(3,3;0)\\
\text{\#D3}&-\frac{1}{8}&-\frac{1}{8}+\frac{1}{2}&-\frac{1}{8}+1&-\frac{1}{8}+\frac{1}{2}\\
\hline\cr\text{label}&(0,2;0)&(1,3;1)&(2,0;0)&(3,1;1)\\
\text{\#D3}&+\frac{1}{8}&+\frac{1}{8}+\frac{1}{2}&+\frac{1}{8}&+\frac{1}{8}+\frac{1}{2}.\\
\hline\cr\end{array}$$
These are the $\mathbb{Z}_{2}$ shift-S-folds on $S^{1}$ of the O3-planes.
Among the first four, the first has the right D3-brane charge to be the $\mathbb{Z}_{2}$ shift-S-fold of $O3{}^{-}$.
Recall that the base of the M-theory configuration is $\mathbb{C}^{3}/\mathbb{Z}_{4}$, but the in the type IIB description, the fiber at a particular point on $S^{1}$ is $\mathbb{C}^{3}/\mathbb{Z}_{2}$. The D3-brane charge of $O3{}^{-}$ is $-1/4$, and therefore the charge as seen from the M-theory configuration is $1/2\cdot(-1/4)=-1/8$.
The others have one or two additional mobile D3-brane(s) trapped at the origin.
Among the second four, the first and the third have the right D3-brane charge to be the $\mathbb{Z}_{2}$ shift-S-fold of $\widetilde{O3}{}^{+}$.
The second and the fourth have one additional mobile D3-brane trapped at the origin.
Note that we cannot take the $\mathbb{Z}_{2}$ shift S-fold of $\widetilde{O3}{}^{-}$ or $O3{}^{+}$,
since they are exchanged by this $\mathbb{Z}_{2}$ operation.
Shift $=3/4\in F^{2,2}/F^{3,1}$:
$$\begin{array}[]{c|cccc}\hline\cr\text{label}&(0,1;1)&(1,2;0)&(2,3;1)&(3,0;0)\\
\text{\#D3}&+\frac{1}{4}&+\frac{1}{2}&+\frac{3}{4}&0\\
\hline\cr\text{label}&(0,3;0)&(1,0;1)&(2,1;0)&(3,2;1)\\
\text{\#D3}&0&+\frac{1}{4}&+\frac{1}{2}&+\frac{3}{4}.\\
\hline\cr\end{array}$$
These are $\mathbb{Z}_{4}$ shift-S-folds of flat space, with rotation angle $3\pi/2$,
and with $0$, $1$, $2$ or $3$ mobile D3-brane(s) around $S^{1}$.
4.2.4 $k=6$
We denote the charges at the fixed points as elements in $(\mathbb{Z}_{6},\mathbb{Z}_{3},\mathbb{Z}_{2})$.
The piece $F_{4,0}=\mathbb{Z}_{6}$ consists of $(0,0,0)$, $(1,1,1)$, $(2,2,0)$, $(3,0,1)$, $(4,1,0)$, $(5,2,1)$.
Shift $=0/6\in F^{2,2}/F^{3,1}$:
$$\begin{array}[]{c|cccccc}\hline\cr\text{label}&(0,0,0)&(1,1,1)&(2,2,0)&(3,0,1)&(4,1,0)&(5,2,1)\\
\text{\#D3}&-\frac{5}{12}&-\frac{5}{12}+1&-\frac{5}{12}+1&-\frac{5}{12}+1&-\frac{5}{12}+1&-\frac{5}{12}+1\\
\hline\cr\end{array}$$
The first has the right D3-brane charge to be the $(k=6,\ell=1)$ S-fold, see (2.13).
The other five have one more mobile D3-brane, stuck at the origin through a non-trivial $\mathbb{Z}_{6}$ background holonomy around $S^{1}$.
Shift $=1/6\in F^{2,2}/F^{3,1}$:
$$\begin{array}[]{c|cccccc}\hline\cr\text{label}&(1,0,0)&(2,1,1)&(3,2,0)&(4,0,1)&(5,1,0)&(0,2,1)\\
\text{\#D3}&0&\frac{5}{6}&\frac{2}{3}&\frac{1}{2}&\frac{1}{3}&\frac{1}{6}\\
\hline\cr\end{array}$$
This is the $\mathbb{Z}_{6}$ shift-S-fold of flat space, with zero to five D3-branes stuck at the origin. The rotation angle is $2\pi/6$.
Shift $=2/6\in F^{2,2}/F^{3,1}$:
$$\begin{array}[]{c|cccccc}\hline\cr\text{label}&(2,0,0)&(3,1,1)&(4,2,0)&(5,0,1)&(0,1,0)&(1,2,1)\\
\text{\#D3}&-\frac{1}{12}+\frac{1}{3}&-\frac{1}{12}+1&-\frac{1}{12}+\frac{2}{3}&-\frac{1}{12}+\frac{1}{3}&-\frac{1}{12}&-\frac{1}{12}+\frac{2}{3}\\
\hline\cr\end{array}$$
The fifth has the correct charge to be the $\mathbb{Z}_{3}$ shift-S-fold of the standard $O3{}^{-}$-plane.
Since the $O3{}^{-}$-plane itself has an identification by the angle $\pi$, its $\mathbb{Z}_{3}$ quotient involves the rotation by $\pi/3$ of the transverse space.
Note also that the $O3{}^{-}$-plane has the D3-brane charge $-1/4$, therefore we see $1/3\cdot(-1/4)=-1/12$ in M-theory.
The others have one or two additional D3-brane(s) at the origin.
Shift $=3/6\in F^{2,2}/F^{3,1}$:
$$\begin{array}[]{c|cccccc}\hline\cr\text{label}&(3,0,0)&(4,1,1)&(5,2,0)&(0,0,1)&(1,1,0)&(2,2,1)\\
\text{\#D3}&-\frac{1}{6}+\frac{1}{2}&-\frac{1}{6}+1&-\frac{1}{6}+\frac{1}{2}&-\frac{1}{6}&-\frac{1}{6}+\frac{1}{2}&-\frac{1}{6}+1\\
\hline\cr\end{array}$$
The fourth has the correct charge to be the $\mathbb{Z}_{2}$ shift-S-fold of the $(k=3,\ell=1)$ S-fold. The others have one or two additional mobile D3-brane(s) on top.
Shift $=4/6\in F^{2,2}/F^{3,1}$:
$$\begin{array}[]{c|cccccc}\hline\cr\text{label}&(4,0,0)&(5,1,1)&(0,2,0)&(1,0,1)&(2,1,0)&(3,2,1)\\
\text{\#D3}&-\frac{1}{12}+\frac{1}{3}&-\frac{1}{12}+\frac{2}{3}&-\frac{1}{12}&-\frac{1}{12}+\frac{1}{3}&-\frac{1}{12}+\frac{2}{3}&-\frac{1}{12}+1\\
\hline\cr\end{array}$$
The third has the correct charge to be the $\mathbb{Z}_{3}$ shift-S-fold, of rotation angle $2\pi/3$, of the standard $O3{}^{-}$-plane. The others have one or two additional D3-brane(s) at the origin.
Shift $=5/6\in F^{2,2}/F^{3,1}$:
$$\begin{array}[]{c|cccccc}\hline\cr\text{label}&(5,0,0)&(0,1,1)&(1,2,0)&(2,0,1)&(3,1,0)&(4,2,1)\\
\text{\#D3}&0&\frac{1}{6}&\frac{1}{3}&\frac{1}{2}&\frac{2}{3}&\frac{5}{6}\\
\hline\cr\end{array}$$
This is the $\mathbb{Z}_{6}$ shift-S-fold of flat space, with zero to five D3-branes stuck at the origin. The rotation angle is $2\pi\cdot 5/6$.
Comments:
Note that in the cases with shift $2,4\in\mathbb{Z}_{6}$, we only find $\mathbb{Z}_{3}$ shift-S-folds of ${O3}{}^{-}$, but we
do not have $\mathbb{Z}_{3}$ shift-S-folds of $\widetilde{O3}{}^{-}$, $O3{}^{+}$ and $\widetilde{O3}{}^{+}$. This is as it should be, because these three types of O3-planes are permuted by the $\mathbb{Z}_{3}$ action.
Similarly, there is no $\mathbb{Z}_{2}$ shift-orientifold of the $(k=3,\ell=3)$ orientifold, since the $\mathbb{Z}_{2}$ action exchanges the two subtly different versions that we denoted by $(1,1,1)$ and (2,2,2) in Sec. 4.2.2.
5 Special cases and $\mathcal{N}{=}\,4$ enhancement
So far, we saw that $N$ D3-branes probing various variants of the S-folds give rise to $\mathcal{N}{=}\,3$ superconformal field theories characterized by
$$(k,\ell)=(3,1),(3,3),(4,1),(4,4),(6,1).$$
(5.1)
When $\ell<k$ we expect to have both ‘parent’ theories and their discrete gaugings. In this section we discuss some interesting special cases, including cases where the ${\cal N}=3$ supersymmetry is enhanced to ${\cal N}=4$.
As discussed in [5], an enhancement of supersymmetry to $\mathcal{N}{=}\,4$ occurs if and only if there is a Coulomb branch operator of dimension 1 or 2, since $\mathcal{N}{=}\,3$ supersymmetry then dictates the presence of extra supercharges.
The dimensions of the Coulomb branch operators of our ‘parent’ $\mathcal{N}{=}\,3$ theories were given in (2.11).
For $k=2$ we always have such an enhancement, but for $k=3,4,6$ we see that it happens just for $\ell=1$ and $N=1,2$.
Note that the theory with the lowest central charge which does not have any enhancement is the $N=1$, $k=\ell=3$ theory, whose only Coulomb branch operator has dimension 3. The central charges of this theory are the same as those of five vector multiplets.
Since the Coulomb branch operators of $\mathcal{N}{=}\,3$ theories must be integers as shown in Section 3.1 of [28], then this must be the ‘minimal’ $\mathcal{N}{=}\,3$ SCFT, assuming the general validity of the formula (2.3). It would be interesting to test this by a superconformal bootstrap analysis, generalizing the ones in [46, 47].
Going back to theories with $\mathcal{N}{=}\,4$ supersymmetry, the case $N=1$ is rather trivial: we just have a Coulomb branch operator of dimension one, so the moduli space is just $\mathbb{C}^{3}$, and we get the $\mathcal{N}{=}\,4$ super Yang-Mills theory with gauge group $\mathrm{U}(1)$.
So let us discuss the $N=2$ cases.
The spectrum of the Coulomb branch operators of the ‘parent’ theory is given by
$$\begin{cases}2,3&(k=3),\\
2,4&(k=4),\\
2,6&(k=6).\end{cases}$$
(5.2)
These spectra agree with those of an $\mathcal{N}{=}\,4$ super Yang-Mills theory with gauge group $\mathrm{SU}(3)$, $\mathrm{SO}(5)$ and $G_{2}$, respectively.
Below we give evidence that indeed, the ‘parents’ of these S-fold configurations give rise to these $\mathcal{N}{=}\,4$ super Yang-Mills theories, realized in a somewhat unusual manner. Note that ${\cal N}=4$ theories always have an exactly marginal deformation, sitting in the same multiplet as the dimension two Coulomb branch operator, and our conjectured relation implies that for our $N=2$ theories this is the gauge coupling of these ${\cal N}=4$ gauge theories. Our discussion in the previous sections implies that in the AdS dual of these ‘parent’ theories, this marginal deformation corresponds to a scalar field coming from a D3-brane wrapped on the torsion 3-cycle; for this specific case this wrapped D3-brane gives rise to a massless field.
Again let us limit ourselves to the points on the moduli space corresponding to a Coulomb branch from the point of view of an ${\cal N}=2$ description of our SCFTs. In the $N=2,\ell=1$ theories, this subspace is parameterized by $z_{1,2}\in\mathbb{C}$, with
the gauge symmetry (2.10)
$$(z_{1},z_{2})\mapsto(\gamma^{n}z_{2},\gamma^{-n}z_{1}),$$
(5.3)
where $\gamma=e^{2\pi i/k}$ and $n$ is any integer.
The charges in a basis that is natural from this point of view can be written as $(e_{1},m_{1};e_{2},m_{2})$.
Naively, one would expect the electric charges $e_{1},e_{2}$ to correspond to electric charges of the corresponding ${\cal N}=4$ theory, but this cannot be the case because of the non-trivial $Sp(4,\mathbb{Z})$ action on these charges, induced by the $\mathrm{SL}(2,\mathbb{Z})$ transformation that accompanies the identification (5.3). So instead we consider the
rank-2 sublattice containing charges of the form $(Q;\overline{Q})$ where we regard $(e,m)\in\mathbb{Z}\oplus\mathbb{Z}$ as a complex number $Q=e+m\gamma$.
Two charges from this sublattice are local with respect to each other. To see this, note that given $Q=e+m\gamma$ and $Q^{\prime}=e^{\prime}+m^{\prime}\gamma$, their Dirac pairing is
$$em^{\prime}-e^{\prime}m=(\overline{Q}Q^{\prime}-\overline{Q}{}^{\prime}Q)/(\gamma-\overline{\gamma}).$$
(5.4)
Then the Dirac pairing between the two charges $(Q;\overline{Q})$ and $(Q^{\prime};\overline{Q}{}^{\prime})$ is clearly zero.
This means that we can take these charges to be the “electric charges” in the $\mathcal{N}{=}\,4$ description. We provide some consistency checks for this below.
Now, note that the $\mathrm{SL}(2,\mathbb{Z})$ element associated to the $\mathbb{Z}_{k}$ orbifold then acts on $Q$ just by multiplication by $\gamma$.
Then the gauge transformation (5.3) acts on this variable $Q$ as
$$Q\mapsto\gamma^{n}\overline{Q},$$
(5.5)
which is a reflection of the complex plane along the line $e^{\pi in/k}\mathbb{R}$.
This makes it clear that the group generated by (5.3) for $k=3,4,6$ is the Weyl group of $\mathrm{SU}(3)$, $\mathrm{SO}(5)$ and $G_{2}$, respectively.
Let us test our identification by looking at half-BPS particles. There’s no string connecting $z_{1}$ and $\gamma^{n}z_{1}$, since we know nothing happens when $z_{1}=0$ for $\ell=1$. So there are only strings connecting $\gamma^{n}z_{1}$ and $\gamma^{m}z_{2}$.
Using (5.3) we can always restrict $z_{1}$ to have a phase between 0 and $2\pi/k$.
Then we just have to consider all $(p,q)$-strings connecting $z_{1}$ and $\gamma^{n}z_{2}$ for $n=0,\ldots,k-1$.
Since the IIB coupling constant is $\tau=\gamma$, the central charges of half-BPS particles are given by
$$(p+q\gamma)(z_{1}-\gamma^{n}z_{2}).$$
(5.6)
We conjectured that “electric” states have the charge $(Q;\overline{Q})$, and then their central charges are given by $Qz_{1}-\overline{Q}z_{2}$. Comparing with (5.6), we see that “electric” objects have the following $Q$:
•
For $k=3$, $Q=\omega^{n}$ and $Q=(1+\omega)\omega^{n}=-\omega^{n-1}$ where $\omega=e^{2\pi i/3}$,
•
For $k=4$, $Q=i^{n}$ and $Q=(1+i)i^{n}$,
•
For $k=6$, $Q=\gamma^{n}$ and $Q=(1+\gamma)\gamma^{n}$.
Clearly they can be identified with the roots of $\mathrm{SU}(3)$, $\mathrm{SO}(5)$, and $G_{2}$, respectively.
The metric on the moduli space is also correctly mapped to that on the Cartan subalgebra of these groups.
The original metric is $dz_{1}d\overline{z}_{1}+dz_{2}d\overline{z}_{2}$ on $\mathbb{C}^{2}$,
and we choose a real subspace $\mathbb{R}^{2}$ of the form $Qz_{1}-\overline{Q}z_{2}$.
Then, two vectors $Qz_{1}-\overline{Q}z_{2}$ and $Q^{\prime}z_{1}-\overline{Q}^{\prime}z_{2}$ in $\mathbb{R}^{2}$ have the induced inner product $(2\mathrm{Re}(Q\overline{Q}^{\prime}))$.
Using this, we can easily check that the vectors listed above have the same inner products as the root vectors of $\mathrm{SU}(3)$, $\mathrm{SO}(5)$, and $G_{2}$, in the normalization that the short roots have length squared 2.
Finally, recall that the dyons of $\mathcal{N}{=}\,4$ SYM have central charges of the form
$$(p+q\tau_{YM})(\alpha_{s}\cdot\phi),\qquad(p+q\frac{\tau_{YM}}{r})(\alpha_{l}\cdot\phi),$$
(5.7)
where $\alpha_{s,l}$ are short and long roots, and $r$ is the length squared of the long roots divided by that of the short roots.
We can check that the spectrum (5.6) can be matched with (5.7) with the identification of the roots given above, if we take $\tau_{YM}=-1/(1+\gamma)$, uniformly for $k=3,4,6$.
As a further check, note that for $k=6$ this is exactly the value of $\tau_{YM}$ for which the $G_{2}$ ${\cal N}=4$ theory has a discrete $\mathbb{Z}_{6}$ symmetry [48].
Acknowledgements
The authors would like to thank I. García-Etxebarria, D. Harlow, Z. Komargodski, T. Nishinaka, H. Ooguri and N. Seiberg for useful discussions.
The work of OA is supported in part by an Israel Science Foundation center for excellence grant, by the I-CORE program of the Planning and Budgeting Committee and the Israel Science Foundation (grant number 1937/12), by the Minerva foundation with funding from the Federal German Ministry for Education and Research, by a Henri Gutwirth award from the Henri Gutwirth Fund for the Promotion of Research, and by the ISF within the ISF-UGC joint research program framework (grant no. 1200/14). OA is the Samuel Sebba Professorial Chair of Pure and Applied Physics.
The work of YT is partially supported in part by JSPS Grant-in-Aid for Scientific Research No. 25870159,
and by WPI Initiative, MEXT, Japan at IPMU, the University of Tokyo.
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MIT-CTP/4956
\@open@row |
11institutetext: Université Laval, Québec, Canada 22institutetext: Depix, Montréal, Canada
https://lvsn.github.io/EditableIndoorLight/
Editable indoor lighting estimation
Henrique Weber
11
0000-0001-7103-7220
Mathieu Garon
22
0000-0003-1811-4156
Jean-François Lalonde
11
0000-0002-6583-2364
Abstract
We present a method for estimating lighting from a single perspective image of an indoor scene. Previous methods for predicting indoor illumination usually focus on either simple, parametric lighting that lack realism, or on richer representations that are difficult or even impossible to understand or modify after prediction. We propose a pipeline that estimates a parametric light that is easy to edit and allows renderings with strong shadows, alongside with a non-parametric texture with high-frequency information necessary for realistic rendering of specular objects. Once estimated, the predictions obtained with our model are interpretable and can easily be modified by an artist/user with a few mouse clicks. Quantitative and qualitative results show that our approach makes indoor lighting estimation easier to handle by a casual user, while still producing competitive results.
Keywords: lighting estimation, virtual object insertion, HDR
1 Introduction
Mixing virtual content realistically with real imagery is required in an increasing range of applications, from special effects to image editing and augmented reality (AR). This has created the need for capturing the lighting conditions of a scene with ever increasing accuracy and flexibility. In his seminal work, Debevec [6] suggested to capture the lighting conditions with a high dynamic range light probe. While it has been improved over the years, this technique, dubbed image-based lighting, is still at the heart of lighting capture for high end special effects in movies nowadays111See https://www.fxguide.com/fxfeatured/the-definitive-weta-digital-guide-to-ibl/.. Since the democratization of virtual object insertion for consumer image editing and AR, capturing light conditions with light probes restricts non professional users to have access to the scene and to use specialized equipment. To circumvent those limitations, approaches for automatically estimating the lighting conditions directly from images have been proposed.
In this line of work, the trend has been to estimate more and more complex lighting representations. This is exemplified by works such as Lighthouse [25], which propose to learn a multi-scale volumetric representation from an input stereo pair. Similarly, Li et al. [19] learn a dense 2D grid of spherical gaussians over the image plane. Wang et al. [27] propose to learn a 3D volume of similar spherical gaussians. While these lighting representations have been shown to yield realistic and spatially-varying relighting results, they have the unfortunate downside of being hard to understand: they do not lend themselves to being easily editable by a user. This quickly becomes a source of limitation when erroneous automatic results need to be corrected for improved accuracy or when creative freedom is required.
In this work, we depart from this trend and propose a simple, interpretable, and editable lighting representation (fig. 1).
But what does it mean for a lighting representation to be editable? We argue that an editable lighting representation must: 1) disentangle various components of illumination; 2) allow an intuitive control over those components; and, of course, 3) enable realistic relighting results.
Existing lighting representations in the literature do not possess all three properties. Environment maps [11, 24, 17] can be rotated but they compound light sources and environment textures together such that one cannot, say, easily increase the intensity of the light source without affecting everything else.
Rotating the environment map inevitably rotates the entire scene, turning walls into ceilings, etc., when changing the elevation.
Dense and/or volumetric representations [12, 19, 25, 27] are composed of 2D (or 3D) grids containing hundreds of parameters, which would have to be modified in a consistent way to achieve the desired result, an unachievable task for most. Parametric representations [10] model individual light sources with a few intuitive parameters, which can be modified independently of the others, but cannot generate realistic reflections.
Our proposed representation is the first to offer all three desired properties and is composed of two parts: 1) a parametric light source for modeling shading in high dynamic range; and 2) a non-parametric texture to generate realistic reflections off of shiny objects.
Our representation builds on the hypothesis (which we validate) that most indoor scenes can accurately be modeled by a single, dominant directional light source. We model this in high dynamic range with a parametric representation [10] that explicitly models the light source intensity, size, and 3D position. This representation is intuitive and can easily be edited by a user simply by moving the light source around in 3D.
This light source is complemented with a spatially-varying environment map texture, mapped onto a coarse 3D representation of the indoor scene. For this, we rely on a layout estimation network, which estimates a cuboid-like model of the scene from the input image. In addition, we also use a texture estimation network, whose output is conditioned on a combination of the input image, the scene layout and the parametric lighting representation. By explicitly tying the appearance of the environment texture with the position of the parametric light source, modifying the light source parameters (e.g. moving around the light) will automatically adjust the environment in a realistic fashion.
While our representation is significantly simplified, we find that it offers several advantages over the previous approaches. First, it renders both realistic shading (due to the high dynamic range of the estimated parametric light) and reflections (due to the estimated environment map texture). Second, it can efficiently be trained on real images, thereby alleviating any domain gap that typically arise when approaches need synthetic imagery for training [25, 19, 27]. Third—and perhaps most importantly—it is interpretable and editable. Since all automatic approaches are bound to make mistakes, it is of paramount importance in many scenarios that their output be adjustable by a user. By modifying the light parameters and/or the scene layout using simple user interfaces, our approach bridges the gap between realism and editability for lighting estimation.
2 Related work
For succinctness, we focus on single-image indoor lighting estimation methods in the section below, and refer the reader to the recent survey on deep models for lighting estimation for a broader overview [8].
2.0.1 Lighting estimation
Gardner et al. [11] proposed the first deep learning-based lighting estimation method for indoor scenes, and predicted an HDR environment map (equirectangular image) from a single image. This representation was also used in [17] for both indoors and outdoors, in [24] to take into account the object insertion position, in [23] which presented a real-time on-device approach, in [22] for scene decomposition, and in [3] which exploited the front and back cameras in current mobile devices. Finally, [28] propose to learn the space of indoor lighting using environment maps on single objects.
Other works explored alternative representations, such as spherical harmonics [12, 20, 34] that are useful for real-time rendering but are typically unsuitable for modeling high-frequency lighting (such as bright light sources) and are not ideal for non diffuse object rendering. [10] proposed to estimate a set of 3 parametric lights, which can easily be edited. However, that representation cannot generate realistic reflections. EMlight [33] propose a more expressive model by predicting gaussians on a spherical model. Similar to us, GMlight [31] back-projects the spherical gaussians to an estimated 3D model of the scene. This is further extended in [1] by the use of graph neural networks, and in [32] through the use of spherical wavelets dubbed “needlets”.
Recently, methods have attempted to learn volumetric lighting representations from images. Of note, Lighthouse [25] learns multi-scale volumetric lighting from a stereo pair, [19] predicts a dense 2D grid of spherical gaussians which is further extended into a 3D volumetric representation by Wang et al. [27]. While these yield convincing spatially-varying results, these representations cannot easily be interacted by a user.
2.0.2 Scene decomposition
Holistic scene decomposition [2] is deeply tied to lighting estimation as both are required to invert the image formation process. Li et al. [19] proposes to extract the scene geometry and the lighting simultaneously. Similarly, [7] extract only the geometry of the scene by estimating the normal and depth of the scene. These geometric representations are however non-parametric and thus difficult to edit or comprehend.
[16] proposes a simplified parametric model where a room layout is recovered in the camera field of view. Similarly, [35] presents a method to estimate the layout given a panoramic image of an indoor scene. We use the method of [16] to estimate a panoramic layout given a perspective image, thus providing a simple cuboid representation that allows for spatially varying textured lighting representation.
3 Editable indoor lighting representation
We begin by presenting our hybrid parametric/non-parametric lighting representation which aims at bridging the gap between realism and editability. We also show how that representation can be fitted to high dynamic range panoramas to obtain a training dataset, and conclude by presenting how it can be used for virtual object relighting.
3.1 Lighting representation
Our proposed light representation, shown in fig. 2, is composed of two main components: an HDR parametric light source $\mathbf{p}$; and an LDR textured cuboid $\mathcal{C}$.
3.1.1 Light source
As in [10], the light source parameters $\mathbf{p}$ are defined as
$$\mathbf{p}=\{\mathbf{l},d,s,\mathbf{c},\mathbf{a}\}\,,$$
(1)
where $\mathbf{l}\in\mathbb{R}^{3}$ is a unit vector specifying the light direction in XYZ coordinates, $d$ is the distance in meters, $s$ the radius (in meters), $\mathbf{c},\mathbf{a}\in\mathbb{R}^{3}$ are the light source and ambient colors in RGB, respectively. Here, $\mathbf{l}$, $d$ and $s$ are defined with respect to the camera. In contrast with [10], we use a single light source.
3.1.2 Textured cuboid
The cuboid $\mathcal{C}=\{\mathbf{T},\mathbf{L}\}$ is represented by a texture $\mathbf{T}\in\mathbb{R}^{2H\times H\times 3}$, which is an RGB spherical image of resolution $2H\times H$ stored in equirectangular (latitude-longitude) format, and a scene layout $\mathbf{L}\in\mathbb{R}^{2H\times H}$. The layout is a binary image of the same resolution, also in equirectangular format, indicating the intersections of the main planar surfaces in the room (walls, floor, ceiling) as an edge map [9].
3.2 Ground truth dataset
The ground truth is derived from the Laval Indoor HDR Dataset [11], which contains 2,100 HDR panoramas (with approximate depth labels from [10]). We extract $\mathbf{p}$ and $\mathcal{C}$ from each panorama using the following procedure. First, the HDR panorama is clipped to LDR (we re-expose such that the 90th-percentile is 0.8 then clip to [0, 1]) and directly used as the texture $\mathbf{T}$. Then the intersection between the main surfaces are manually labelled to define the layout $\mathbf{L}$. Lastly, we extract a dominant parametric light source from the HDR panorama. In order to determine the main light source, the $N=5$ brightest individual light sources are first detected using the region-growing procedure in [10]. A test scene (9 diffuse spheres arranged in a $3\times 3$ grid on a diffuse ground plane, seen from top as in fig. 4b) is rendered with each light source independently by masking out all other pixels—the brightest render determines the strongest light source.
An initial estimate of the light parameters $\mathbf{p}$ are obtained by the following. The distance $d$ is approximated by using the average depth of the region, direction $\mathbf{l}$ as the region centroid, the angular size from the major and minor axes of an ellipse fitted to the same region. Finally, the light color $\mathbf{c}$ and ambient term $\mathbf{a}$ are initialized with a least-squares fit to a rendering of the test scene using the HDR panorama. From the initial parameters, $\mathbf{p}$ is further refined:
$$\mathbf{p}^{*}=\operatorname*{arg\,min}_{\mathbf{p}}||\mathcal{R}(\mathbf{p})-\mathcal{R}(\tilde{\mathbf{P}})||_{2}\,.$$
(2)
$\mathcal{R}(x)$ is a differentiable rendering operator (implemented with Redner [18]) that renders a test scene using $\mathbf{p}$. The optimization is performed using gradient descent with Adam [15]. Finally, the texture map $\mathbf{T}$ is rescaled with the estimated ambient term $\mathbf{a}^{*}$ to ensure that the texture yields the same average RGB color.
3.3 Virtual object rendering
To render a virtual object using our lighting representation, we employ the Cycles rendering engine222Available within Blender at https://www.blender.org.. A scene, as shown in fig. 2, is composed of a 3D emissive sphere for the parametric light $\mathbf{p}$ and the textured cuboid mesh $\mathcal{C}$. The cuboid mesh is derived by detecting the cuboid corners from the layout using high pass filters. We use the following geometric constraints to simplify the back-projection of the scene corners to 3D. First, the shape is limited to a cuboid, meaning that opposing faces are parallel. Second, the panorama layouts were trained using a camera elevation of $0^{\circ}$ (pointing at the horizon) and height of 1.6 meter above the ground. Using these constraints, the bottom corners can easily be projected on the ground plane, and the top corners can be used to compute the ceiling height (averaged from the 4 corners). A texture map can then be computed using every planar surfaces of the cuboid. Finally, the parametric light and the texture are rendered in two rendering passes. After rendering, the relit virtual object can be composited into the image using differential rendering [6].
4 Approach
Our approach, illustrated in fig. 3, is composed of three main networks: light, layout, and texture which are combined together to estimate our light representation (c.f., sec. 3) from an image. We assume that the layout of the input image is available, in practice this is obtained with an off-the-shelf solution [30].
4.0.1 Light network
A “light” network is trained to learn the mapping from input image $\mathbf{I}\in\mathbb{R}^{128\times 128\times 3}$ to estimated lighting parameters $\mathbf{p}$ (sec. 3) using a similar approach to [10]. Specifically, the light network is composed of a headless DenseNet-121 encoder [14] to produce a 2048-dimensional latent vector, followed by a fully-connected layer (512 units), and ultimately with an output layer producing the light source parameters $\mathbf{p}$.
The light network is trained on light parameters fitted on panoramas from the Laval Indoor HDR Dataset [11] using the procedure described in sec. 3.2. To generate the input image from the panorama, we follow [11] and extract rectified crops from the HDR panoramas. The resulting images are converted to LDR by re-exposing to make the median intensity equal to 0.45, clipping to 1, and applying a $\gamma=\nicefrac{{1}}{{2.4}}$ tonemapping. The same exposure factor is subsequently applied to the color $\mathbf{c}$ and ambient $a$ light parameters to ensure consistency. Note that the training process is significantly simplified compared to [11] as the network predicts only a single set of parameters.
We employ individual loss functions on each of the parameters independently: L2 for direction $\mathbf{l}$, depth $d$, size $s$, and ambient color $a$, and L1 for light color $\mathbf{c}$. In addition, we also employ an angular loss for both the ambient and light colors $a$ and $\mathbf{c}$ to enforce color consistency. The weights for each term were obtained through a Bayesian optimization on the validation set (see supp. mat.).
4.0.2 Layout network
The mapping from the input RGB image $\mathbf{I}$ and its layout (obtained with [30]) to the estimated scene layout $\hat{\mathbf{L}}$ (sec. 3) is learned by the “layout” network whose architecture is that of pix2pixHD [26]. Both inputs are concatenated channel-wise. The layout network is trained on both the Laval and the Zillow Indoor Dataset [5], which contains 67,448 LDR indoor panoramas of 1575 unfurnished residences along with their scene layouts. To train the network, a combination of GAN, feature matching and perceptual losses are employed [26]. The same default weights as in [26] are used in training.
4.0.3 Texture network
Finally, the estimated environment texture $\hat{\mathbf{T}}$ is predicted by a “texture” network whose architecture is also that of pix2pixHD [26]. It accepts as input a channel-wise concatenation of three images: the input RGB image $\mathbf{I}$, the estimated light parameters $\hat{\mathbf{p}}$ projected in an equirectangular format, and the estimated scene layout $\hat{\mathbf{L}}$. The equirectangular images are vertically concatenated to the input image. Note that the $\hat{\mathbf{p}}$ projection is performed using a subset of all parameters (direction $\mathbf{l}$ and size $s$ only).
The texture network is also trained on both Laval and Zillow datasets. To obtain the required light source position from the Zillow dataset, we detect the largest connected component whose intensity is above the 98th percentile over the upper half of the panorama. To convert the Laval HDR panoramas to LDR, first a scale factor is found such as the crop taken from that panorama has its 90th percentile mapped to 0.8. This scale factor is then applied to the panorama such as its scale matches the one of the crop. The texture network is trained with the same combination of losses as the layout network.
5 Experiments
5.1 Validation of our 1-light approximation
We test our hypothesis that most indoor scenes are well-approximated by a single dominant light source with an ambient term. We render a scene with the ground truth environment map, and compare it with the renders obtained from the parametric lighting optimization procedure described in sec. 3. Fig. 4a shows the cumulative distribution of the contribution of the strongest light with respect to the entire lighting of the scene. Note that the strongest light source contributes to more than 95%/80%/60% of the total lighting for 25%/50%/75% of the images in our test set. Fig. 4b shows example images for each of these scenarios. We find that even if we expect indoor scenes to have multiple light sources, the vast majority can accurately be represented by a single dominant light.
5.2 Light estimation comparison
We now evaluate our method and compare it with recent state-of-the-art light estimation approaches. We first validate that our model performs better on quantitative metrics evaluated on physic-based renders of a scene using a test set provided by [11]. For each of the 224 panoramas in the test split, we extract 10 images using the same sampling distribution as in [11], for a total of 2,240 images for evaluation. We also show renders in various scenes to demonstrate how our solution is visually more appealing.
5.2.1 Quantitative comparison
To evaluate the lighting estimates, we render a test scene composed of an array of spheres viewed from above (sec. 3) and compute error metrics on the resulting rendering when compared to the ground truth obtained with the original HDR panorama. We report RMSE, si-RMSE [13], PSNR, and RGB angular error [17]. We also compute the FID444Implementation taken from https://pypi.org/project/pytorch-fid/. on the resulting environment maps to evaluate the realism of reflections (similar to [23]).
We evaluate against the following works.
First, two versions of [10] are compared: the original (3) where 3 light sources are estimated, and a version (1) trained to predict a single parametric light. Second, we also compare to Lighthouse [25], which expects a stereo pair as input. As a substitute, we generate a second image with a small baseline using Synsin [29] (visual inspection confirmed this yields reasonable results). For [12], we select the coordinates of the image center for the object position. For [23], we implemented their proposed “Cluster ID loss” and tonemapping (eq. 1 in [23]) but used pix2pixHD as backbone. Finally, we also compare against [33]. Results for each metrics are reported in tab. 1, which shows that despite our model being simple, it achieves the best score in every metric. We argue that it is because of its simplicity that we can achieve competitive results. Our approach can be trained on real data (as opposed to [12, 25] and does not require an elaborate 2-stage training process (compared to [10]).
We also demonstrate a significantly lower FID score than other methods thus bridging the gap between representation realism and HDR accuracy.
5.2.2 Qualitative comparison
We also present qualitative results in fig. 5 where predictions are rendered on 3 spheres with varying reflectance properties (diffuse, mirror, glossy). In addition, a tonemapped equirectangular view of the estimated light representation is provided under each render. We show an example from each error percentiles according to the RMSE metric. Our proposed method is perceptually better on the mirror spheres as other methods do not model high frequency details from the scene. We also notice accurate shadow and shading from all the spheres. We show objects realistically composited into photographs in fig. 6. Note how the reflections of the virtual objects and their cast shadows on the ground plane perceptually match the input photograph. Finally, we also compare against [27] in fig. 7.
5.3 Ablation study on input layout
One may consider that requiring the image layout as input may make our method sensitive to its estimation. To show this is not the case, we perform an experiment where we provide a black layout as input to the layout network (equivalent to no layout estimate). As can be seen in fig. 8, providing a black layout as input simply results in a different layout prediction where the texture still remains coherent with the RGB input and estimated light direction. The FID of the generated panoramas with no input layout is 88.68 (compared to 89.58 from tab. 1), showing that this essentially has no impact.
5.4 Ablation study on the texture network
We also tested different configurations for the texture network in order to validate our design choices. More specifically, we trained the texture network providing as input: (1) only the RGB crop (FID of 167.39), (2) RGB crop and parametric light (FID of 97.13), and (3) RGB crop and layout (FID of 151.04). In contrast, our full approach obtained an FID of 89.57 (see tab. 1).
6 Editing the estimated lighting
Because of its intuitive nature, it is simple and natural for a user to edit our estimated lighting representation, should the estimate not perfectly match the background image or simply for artistic purposes. Fig. 9 shows that our approach simultaneously disentangles various components of illumination, allows an intuitive control over those components, and enables realistic relighting results. First, fig. 9a shows that a user can rotate the light source about its azimuth angle. Note how the estimated texture (second row) is consistent with the desired light position (third row), while preserving the same overall structure. The renders (first row) exhibit realistic reflections and shadows that correspond to the desired lighting directions. A similar behaviour can be observed in figs 9b and 9c when the elevation angle and size are modified, respectively. In fig. 9d, we show that it is also possible to edit the scene layout and obtain an estimated texture map $\hat{\mathbf{T}}$ that is consistent with the users request. We also show results of compositing virtual objects directly into a scene in fig. 6. As shown in fig. 1, realistic rendering results can intuitively be edited to achieve the desired look.
To the best of our knowledge, the only other method which allows intuitive editing of indoor lighting estimate is that of Gardner et al. [10]. Unfortunately, realistic renders are limited to diffuse objects and cannot be extended to reflective objects as shown in fig. 5.
7 Discussion
This paper proposes a lighting estimation approach which produces an intuitive, user-editable lighting representation given a single indoor input image. By explicitly representing the dominant light source using a parametric model, and the ambient environment map using a textured cuboid, our approach bridges the gap between generating realistic shading (produced by HDR light sources) and reflections (produced by textured environment maps) on rendered virtual objects. We demonstrate, through extensive experiments, that our approach provides competitive quantitative performance when compared to recent lighting estimation techniques. In particular, when compared to the only other approach which can be user-edited [10], our approach yields significant improved results.
Limitations and future work
While our proposed approach estimates a 3D representation of the surrounding lighting environment, it does not reason about light occlusions in the scene as opposed to other techniques such as [12, 19, 27]. Incorporating these higher-order interactions while maintaining interpretability and editability of the output representation is an interesting direction for future research. In addition, the estimated environment textures were shown to produce realistic reflections on shiny objects, but a close inspection reveals that they are low resolution and contain some visual artifacts. It is likely that more recent image-to-image translation architectures [4, 21] could be used to improve realism.
Acknowledgements
This research was supported by MITACS and the NSERC grant RGPIN-2020-04799. The authors thank Pascal Audet for his help.
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A high-granularity calorimeter insert based on SiPM-on-tile technology
at the future Electron-Ion Collider
Miguel Arratia
[email protected]
Kenneth Barish
Liam Blanchard
Huan Z. Huang
Zhongling Ji
Bishnu Karki
Owen Long
Ryan Milton
Ananya Paul
Sebouh J. Paul
Sean Preins
Barak Schmookler
Oleg Tsai
Zhiwan Xu
Department of Physics and Astronomy, University of California, Riverside, CA 92521, USA
Department of Physics and Astronomy, University of California, Los Angeles, CA 90095, USA
Abstract
We present a design for a high-granularity calorimeter insert for future experiments at the Electron-Ion Collider (EIC). The sampling-calorimeter design uses scintillator tiles read out with silicon photomultipliers. It maximizes coverage close to the beampipe, while solving challenges arising from the beam-crossing angle and mechanical integration. It yields a compensated response that is linear over the energy range of interest for the EIC. Its energy resolution meets the requirements set in the EIC Yellow Report even with a basic reconstruction algorithm. Moreover, this detector will provide 5D shower data (position, energy, and time), which can be exploited with machine-learning techniques. This detector concept has the potential to unleash the power of imaging calorimetry at the EIC to enable measurements at extreme kinematics in electron-proton and electron-nucleus collisions.
Contents
1 Introduction
2 Proposed Design
2.1 External constraints
2.2 Sampling calorimeter layout
2.3 Scintillator cells
2.4 Silicon photo-multipliers
2.5 SiPM-carrying boards and backplane board
2.6 Accessibility for maintenance
2.7 Geometric acceptance
2.8 Total calorimeter thickness
3 Simulation and performance
3.1 Geometry and simulation setup with DD4HEP
3.2 Shower-shape examples
3.3 Hit distributions
3.4 Linearity for hadrons and electrons and $e/h$ ratio
3.5 Non-Gaussian tails due to leakage
3.6 Single-hadron energy resolution
3.7 Pseudorapidity dependence of performance
4 Summary and conclusions
1 Introduction
One of the key requirements for detectors at the future Electron-Ion Collider (EIC) is to have tracking and full calorimetry with 2$\pi$ azimuthal coverage over a large range in pseudorapidity—nominally $|\eta|<4.0$ (1). This would ensure a “$4\pi$, general-purpose detector” that could deliver the original EIC scientific goals (2) and much beyond.
While both the ATHENA (3) and ECCE (4) detector designs contemplate close-to-full coverage with full calorimetry, the specifics on how to implement it in the forward region remain undefined. Covering the region $3<|\eta|<4$ is challenging due to the EIC beam-crossing angle, which is 25 mrad (1).
As illustrated in Fig. 1, the beampipe envelope crosses the region where the forward hadronic calorimeter (HCal) would be located at an angle that approximately corresponds to the proton direction. Any detector covering this region needs to simultaneously fill a complex volume and keep clearance from the beampipe, while fitting the other calorimeters without needing additional support structures.
In addition, the detector in the $3<\eta<4$ region needs to be well matched to the particle densities and energies expected at the EIC. At the highest energy setting of 18 GeV electron beam and 275 GeV proton beam, the jet spectra reach close to the proton-beam energy with a high rate at nominal luminosity (1). Furthermore, single-particle measurements up to 60 GeV are set as a requirement (1). Given the large energies involved, an HCal becomes crucial since poor tracking performance is expected at forward angles (5; 1).
Instrumenting the $3<\eta<4$ region demands a relatively small detector area ($\approx 0.6\times 0.6$ m${}^{2}$ for a detector located at $z=3.8$ m) but also high granularity to yield a reasonable $\eta$ resolution and disentangle nearby particles. Small cells can also tessellate the complex geometry near the beampipe in an effective manner.
Good angular coverage and resolution would be beneficial for measurements of jets and their substructure, which will enable a wide variety of studies at the EIC, and inclusive deep-inelastic scattering at high-$x$ (1; 6).
The concept of high-granularity calorimetry has been developed for the last two decades or so (7), driven by the CALICE collaboration, and motivated by the particle-flow paradigm (8). More recently, this approach has been adopted for the CMS High-Granularity Calorimeter (HGCAL) upgrade (9), and is envisioned for experiments at several proposed $e^{+}e^{-}$ colliders such as ILC (10), CLIC (11), and CEPC (12; 13; 14).
Extensive R$\&$D efforts (7) and rapid progress in silicon-photomultiplier (SiPM) technology (15) have culminated in the “SiPM-on-tile” concept for hadronic calorimetry. In this approach, the active elements of the calorimeter are small scintillator tiles ($\approx$ 3$\times$ 3 cm${}^{2}$) air coupled to a SiPM at their center (16; 17), without using wavelength-shifting fibers that were used in earlier designs (18; 19; 20; 21).
The advent of high-granularity calorimeters has boosted studies of machine-learning algorithms (22) that could exploit the benefits of “imaging calorimetry” for a variety of tasks (23; 24; 25; 26; 25; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37).
Some early proposals for EIC detectors, such as TOPSIDE, considered imaging calorimetry over the entire solid angle (38; 1), including hadronic calorimetry similar to the CALICE design, although those options were not further pursued111In particular, they were not in any of the official EIC proposals (3; 4; 39). On the other hand, an imaging electromagnetic calorimeter based on silicon-pixel sensors was included in the ATHENA proposal (3). Imaging calorimetry with silicon sensors is also envisioned for the zero-degree calorimeters.. Instead, the forward HCals for both the ATHENA (3) and ECCE (40) proposals included a more traditional steel-scintillator design with longitudinal granularity that varies between 4 and 7 segments, which is similar to that used in the ZEUS (41) and H1 (42) detectors at HERA.
The SiPM-on-tile technology represents an attractive option to instrument the $3<\eta<4$ region at the EIC for various reasons. First, the expected energy range for jets largely overlaps with that of proposed $e^{+}e^{-}$ Higgs factories, so the vast CALICE literature and test-beam data (18; 19; 20; 43; 44; 45; 46; 47; 48; 49; 50; 51; 52; 53; 54) could inform designs and validate simulations. Second, the radiation level expected at the EIC is orders of magnitude smaller than the LHC so constraints from SiPM radiation damage are not a limiting factor. Third, the forward geometry limits the detector volume, numbers of channels, and cost, while still covering a significant range in $\eta$–$\phi$ space.
In this paper, we present a concept for a high-granularity calorimeter insert (HG-CALI) based on the SiPM-on-tile technology that achieves excellent acceptance near the beampipe, while satisfying the mechanical constraints imposed by integration with the rest of the detector. It is designed to be compensated ($e/h\approx 1$) and meets the requirements set in the EIC Yellow Report (1) with a basic reconstruction algorithm. Moreover, it will open up potential for improvements based on machine-learning techniques that exploit the 5D measurement of showers. It would be complemented with a high-granularity electromagnetic calorimeter insert based on tungsten powder/scintillating fiber technology (55), which will be described in a separate publication.
2 Proposed Design
2.1 External constraints
Two main factors constrain the design of the proposed calorimeter insert. First, it must integrate into the larger calorimeter endcap without additional support structures that would create dead areas. Second, it must surround the beampipe within some minimal acceptable clearance in order to maximize its coverage.
In the ATHENA design (3), the forward HCal design is based on the technology previously used in the STAR experiment (56; 57) and consists of alternating layers of steel absorbers (consisting of $96\times 98\times 20$ mm blocks) and scintillator tiles with a thickness of 3 mm. The absorber blocks of both the endcap and the insert are held in place using dowel pins which go into the holes on the top of each absorber block, through holes of the 1.9 mm thick “master plates”, and then into the holes in the blocks above them. This is illustrated in Fig. 2, which shows the high-granularity insert (described in more detail in the following sections) surrounded by the full HCal endcap (left), and zoomed-in with some of the adjacent structures within the endcap (right).
The endcap HCal is designed to open up by splitting in half vertically for maintenance, as it will rest on a support structure with a rail system (1). Therefore, the insert is designed to split into two parts along with the endcap.
The current design for the beampipe (58) is conically shaped and has an outer radius of 11.15 cm at the upstream face of the HCal ($z=380$ cm) and 13.40 cm at the front face of the last layer of the HCal ($z=498$ cm). Furthermore, the beampipe axis is at a small angle ($-$24.3 mrad) relative to the lab-frame $z$ axis, which is defined parallel to the solenoidal magnetic field axis and to the electron beampipe. The proton beam axis is at $-$25 mrad relative to the electron axis. Distances and angles defined with respect to the proton axis are labelled with an asterisk superscript throughout this paper. We assume a clearance of about 4 cm along the entire volume of the HCal insert.
Based on these requirements, the proposed HG-CALI will cover the equivalent area of 6$\times$6 blocks of the HCal endcap, or approximately 60$\times$60 cm. However, since the beampipe is offset in the negative $x$ direction, the part of the insert on the left side when viewed from upstream (positive $x$) covers the width of two HCal endcap blocks, plus the gaps between them, (19.6 cm), while the right side (negative $x$) of the insert covers the width of four HCal endcap blocks and the gaps between them (39.6 cm). Together with the 4 mm gap between the two halves of the HCal, and the 4 mm of readout backplanes on either side, the total width of the proposed insert is 60.4 cm.
To account for the beampipe, we include a hole in each layer of the insert, as shown in the left side of Fig. 3.
The shape of this hole is circular for the most part. However, the right side of the insert is shaped like a capital letter D, with flat extensions on the top and bottom to ensure clearance even while the calorimeter is sliding open.
The radii and position of the centers of the circles vary from layer to layer, as shown in the top image of Fig. 4, because of the conical shape and tilt (respectively) of the beampipe. This approach leads to an optimal layout, in the sense that it covers all the available volume, being limited only by the clearance from the beampipe. The dimensions of the insert are shown in the bottom image of Fig. 4.
Although we have shown a specific way to implement the HG-CALI with the ATHENA HCal design (3), this could be adapted to fit the ECCE HCal (40) design, which also uses a modular assembly structure based on rectilinear towers that run parallel to the $z$ axis. The HG-CALI could also be incorporated into the second EIC detector (59); for example, it is also compatible with the CORE design (39). It could also be used in future experiments at the Chinese EicC, which considers a crossing-angle of 50 mrad (60).
2.2 Sampling calorimeter layout
The HG-CALI consists of alternating absorber blocks and scintillator assemblies. Each scintillator assembly consist of a plastic scintillator sandwiched between a 2 mm cover on the upstream side of it and a printed-circuit board (PCB) with SiPMs for readout on the downstream side. It has 50 layers, each of which is 2.34 cm thick, followed by a downstream absorber-only layer. Figure 5 shows the longitudinal dimensions of the absorbers and the different components of the scintillator assemblies.
The HG-CALI is divided into two sets of layers with different absorber materials. In the first, tungsten is used; in the second, high-strength steel is used. The use of tungsten is motivated for two reasons: first, it allows for a compensated design; second, its lower radiation length and nuclear-interaction lengths compared to steel ($X_{0}=$ 0.35 cm, $\lambda_{n}$= 9.95 cm for W and $X_{0}=$ 1.76 cm, $\lambda_{n}$= 16.8 cm for Fe) helps in reducing leakage of the core of hadronic showers to the beampipe.
The use of magnetic steel is required to contain magnetic-field flux. The EIC poses a strict requirement on fringe fields, mainly driven by the need of accurate polarization of the beams. The amount of steel required in this region is still under study and no formal requirement has been established. We have assumed a total thickness of 33.81 cm of steel (the last 21 layers) for the simulations presented in Sec. 3.
2.3 Scintillator cells
A single scintillator “megatile” is used for each half-layer, which has a slightly different shape to match the corresponding absorber shape. The scintillator panels are tessellated with hexagonal cells defined by grooves.
Each cell will have a dimple in its center above the SiPMs that are hosted on the PCB. The grooves are filled with a mixture of epoxy and titanium dioxide pigment, following procedures similar to the STAR electromagnetic calorimeter (61). An alternative could be to paint the cells’ edges with reflective paint, such as Saint Gobain BC-621. The latter approach was used in the CMS HCAL (62).
To fully tessellate each layer, some of the cells on the edges will be cropped or extended. This is not necessarily an issue given that the design allows for a cell-to-cell calibration based on minimum-ionizing particles.
The megatile is covered on the front and back with reflective foil like the VM2000 enhanced specular reflector (ESR) to increase the light yield and to improve uniformity across the cell area. The ESR foil on the downstream side of the megatiles will include holes for the dimples in each cell. The ESR foil will not be wrapped to facilitate assembly; instead, the edges of the scintillator megatiles will be painted.
The dimples in the center of each cell will have the same dimensions used in the CMS HGCAL test beam (63), which define a spherical dome with a radius of curvature of 3.8 mm and a depth of 1.6 mm, as shown in Fig. 6. The dimples improve signal uniformity across the area of the cell by alleviating hot-spot effects near the SiPM (16; 17).
The megatile concept has several advantages over a design in which each tile is wrapped individually, such as the ones used in the CALICE prototype (20) and the CMS HGCAL (9), especially for a small and complicated geometry such as that of HG-CALI. The megatile design would simplify assembly and reduce cell-to-cell variations caused by wrapping with ESR foil (64).
Previous experience with calorimeters that use megatiles such as the STAR electromagnetic calorimeter (61) suggest that optical cross-talk can be reduced to about 2$\%$ when using a mixture of epoxy and titanium dioxide to fill the grooves that define the cells. This level of optical crosstalk is acceptable for this application, given that hadronic showers encompass a large number of hits over a wide area.
The megatile concept provides us with the flexibility to define the transverse granularity of the detector, and thus optimize it. We envision that the area of these cells will vary from layer to layer, for example as shown in Fig. 7, with finer granularity in the furthest upstream layers and coarser granularity in the downstream. The granularity is yet to be optimized, but we estimate that a reasonable choice would yield a total of less than 5k channels.
Test-beam results by the CMS HGCAL collaboration (63) showed that the light yield of 3 mm thick scintillator cells (Eljen
Technology EJ-200) is about 32 photo-electrons for 9 cm${}^{2}$ cells; about 26 photo-electrons for 12 cm${}^{2}$ cells, and about 20 photoelectrons for 36 cm${}^{2}$ cells.
2.4 Silicon photo-multipliers
Both the CALICE and CMS collaborations use 1.3$\times$1.3 mm${}^{2}$ SiPMs (63; 9). Larger-area SIPMs were considered, e.g., by Jiang et al. in the context of R$\&$D for the CEPC (65). Unlike the applications for ILC or CEPC, which will use millions of channels, or the CMS HGCAL, which uses about 200k channels, our design will have just a few thousand channels. Therefore, the total cost of SiPM would still be reasonable even if we choose models with larger areas.
We anticipate that larger area SiPMs will make the calorimeter more robust against non-uniformity created by SiPM misalignment (64), as well as saturation effects. The photoelectron yield increases linearly with the area of the SiPM, so choosing a 9 mm${}^{2}$ SiPM could increase the light yield by roughly a factor of 4.5 with respect to the CMS HGCAL (63).
As we will show in Sec. 3, the maximum hit energy reaches $\approx 200$ MeV or 250 MIPs. We anticipate an operation for 30 photo-electrons per MIP for the $3\times 3$ cm${}^{2}$ cells. This high light yield would create a buffer to cover for the reduced signal-to-noise ratio after radiation damage.
The Hamamatsu model S13360-3025PE has 14.4k pixels, which translates into a dynamic range of more than 400 MIPs, so no saturation effect is expected for $3\times 3$ cm${}^{2}$ cells.
The neutron fluence expected at the EIC near the insert region will not exceed $10^{12}$ neutrons/cm${}^{2}$ per year from the electron-proton collisions at the top luminosity $(10^{34}$ cm${}^{-2}$s${}^{-1})$ (1). The corresponding ionizing radiation dose is 25 Gy per year.
The expected SiPM radiation damage would be classified as “modest” in Garutti’s review (66) on the topic. It would mainly lead to increased dark current that can affect the ability to identify single-photon peaks. This level of damage can partially be recovered with annealing (66), which could be done after each run (as explained in Sec. 2.6).
For reference, CMS uses SiPM-on-tile technology for regions where the total dose is less than 3000 Gy and fluences less than 8$\times$10${}^{13}$ $n_{\mathrm{eq}}$/cm${}^{2}$, although the SiPMs will be operated at $-$30 C to reduce the impact of the increased noise (9).
The STAR experiment at RHIC is currently operating a steel-scintillator calorimeter readout with SiPMs (56) that is expected to receive fluences that are similar to those expected at the EIC. Detailed data on radiation-induced noise taken with STAR will guide future iterations of the HG-CALI design.
2.5 SiPM-carrying boards and backplane board
The SiPMs will be hosted in a PCB with no active elements. The PCB will also host LEDs next to each SiPM in the dimple area to be used for monitoring and calibration purposes, similarly to the CMS HGCAL design (9). The SiPM-carrying board will host a high-density connector to connect to the backplane board. The SiPMs will not be actively cooled.
On either side of the insert are 4 mm PCB backplanes used for readout, which the PCB for each scintillator assemblies connect into like a traditional crate. Figure 8 shows the location of the PCB backplanes.
The SiPM-carrying boards will be plugged in and unplugged one-at-a-time during assembly and maintenance. The backplane board will also be used to power the SiPMs and control the LED system for calibration and monitoring.
2.6 Accessibility for maintenance
The scintillator assemblies could be removed for maintenance by sliding them out laterally from between the absorbers when the HCal is opened, as illustrated in Fig. 9. This will be possible because the interaction point will have a rail system that will be integrated to the support structure (1).
This feature ensures that the SiPMs boards could be annealed at high-temperature after each run to minimize radiation damage, and that SiPMs and scintillator tiles could be replaced.
2.7 Geometric acceptance
The overall effects of the EIC crossing angle and other beam parameters are discussed in Ref. (67). Events in the lab frame, where the beams have a non-zero crossing angle, can be Lorentz transformed into a frame where the beams are anti-parallel and have energies very close to those in the lab frame. This “minimally-transformed, collinear frame” is a natural choice to extract many physics observables and the accompanying particles’ angular distributions.
In the hadron- (electron-) going direction, scattering angles defined in the lab frame relative to the hadron (electron) beam will be equivalent to the scattering angle in the minimally-transformed collinear frame (68). In particular, for a detector in the hadron-going direction, its angular acceptance should be defined relative to the hadron beam.
To estimate the geometrical acceptance of the insert, we determined where cones of constant $\eta^{*}$ (that is, pseudorapidity with respect to the $z^{*}$ axis – the proton beam axis) would intersect the detector.
Figure 10 shows projections of these $\eta^{*}$ cones superimposed on a $yz^{*}$ (vertical vs. longitudinal) cross-section of the insert. Within this slice, the $\eta^{*}=4.0$ cone skims the inner edge of the detector, while those at lower $\eta^{*}$ hit the face of the detector.
To determine the geometrical acceptance over the full azimuthal range we projected these cones of constant $\eta^{*}$ onto the faces of the scintillators planes of the insert, resulting in elliptical rings of constant $\eta^{*}$. An example is shown in the inset of Fig. 11 for $\eta^{*}$=3.7, which covers the full azimuthal acceptance on all layers of the detector.
We show in Fig. 11 the cross sections of these cones for several values of $\eta^{*}$ at the first layer (top left panel), one of the middle layers (top right panel), and the last layer (bottom left panel).
While there is some acceptance up to $\eta^{*}=4.0$, this is limited to one side of the detector, and it barely skims the first layer, causing a degradation of the signal, as we quantify in Sec. 3.
Since the final detector design may differ from the baseline design presented in this work, we discuss possible alternatives: with a clearance parameter of 3 cm instead of 4 cm, the entire $\eta^{*}=3.8$ ring fits entirely within the first scintillator layer. If the clearance parameter were increased to 5 cm, the full $\eta^{*}=3.7$ ring would still be within the acceptance of the detector, but it would come very close to the edge, resulting in some degradation.
The relevant scale to compare various clearance options has to be the effective radiation length of the insert, which in our design is minimized using mostly tungsten as an absorber. This is because the relevant metric is whether or not the core of the hadronic showers can be well contained, as we describe more in Sec. 3.
Given that even seemingly small differences of clearance definitions can have a large impact in extending the $\eta^{*}$ coverage of the detector, careful engineering design and studies are required to minimize the clearance as much as possible.
2.8 Total calorimeter thickness
We determined the amount of material (in nuclear interaction lengths, $\lambda_{n}$) that a particle travelling in a straight line would pass through in the HG-CALI and the beampipe as modeled in Ref. (58), as a function of its polar angle $\theta^{*}$ with respect to the $z^{*}$ axis. We averaged this with respect to the azimuthal angle $\phi^{*}$ and show the results in the left panel of Fig. 12.
The HG-CALI covers polar angles from about 2${}^{\circ}$ to 6${}^{\circ}$, with up to about 7 nuclear interaction lengths. Between 3${}^{\circ}$ and 6${}^{\circ}$, there is overlap between the HG-CALI and the HCal endcap. The HCal endcap has a somewhat lower number of nuclear interaction lengths (about 6) than the insert. There is a small amount of material traversed by particles passing through the sides of the beampipe (about 1$\lambda_{n}$), which may allow the detection of showering at very small $\theta^{*}$.
We show in the right panel of Fig. 12 the dependence of the amount of material traversed in the HCal insert on $\phi^{*}$ and $\eta^{*}$. The amount of material is nearly uniform in $\phi^{*}$ in the $3.5\leq\eta^{*}\leq 3.8$ range. Since the insert’s outer shape is (approximately) a square, the value of $\eta^{*}$ at the edge of the insert depends non-trivially on $\phi^{*}$. At about $3.0<\eta^{*}<3.3$ (depending on $\phi^{*}$) particles may skim the edge of the insert, creating a signal in both the insert and the HCal. Particles with $3.3<\eta^{*}<3.9$ encounter at least two nuclear interaction lengths of material in the insert at all azimuthal angles.
The HG-CALI can complement the HCal endcap at angles where the effective thickness is not maximal by providing a measurement of the shower-start position and incident-particle angle, both of which will be better measured in the insert.
3 Simulation and performance
3.1 Geometry and simulation setup with DD4HEP
We used the DD4HEP framework (69) to define the geometry and ran the Geant4 framework (70) to evaluate the performance of the HG-CALI. The version of Geant4 used is 10.7.p03.
Table 1 shows the materials, thicknesses, and numbers of interaction lengths for each component of a single layer of the insert (see Fig. 5). The previously described layer design is followed: 30 layers W/Sc, 20 layers steel/Sc, and 1 final layer of steel.
For simplicity, the granularity of the insert was assumed to be 3$\times$3 cm${}^{2}$ for all the layers. Additionally, the model did not include the dimples and grooves in the scintillator layers, nor the SiPMs in the PCB. The model included a realistic beampipe geometry and material description (58).
We also included in the simulation an HCal model similar to the STAR/ATHENA design (56; 3), assuming a granularity of 10$\times$10 cm${}^{2}$, 20 mm thick absorbers, and 3 mm thick scintillators.
The solenoidal magnetic field is expected to be low in the forward region relevant for the insert, but is not precisely known at this time so it was not included in the simulation.
For the simulation of the particle shower in the material, we used the FTFP_BERT_HP physics list of Geant4, which well describes the CALICE test-beam data (51), including the time evolution of hadronic showers in both tungsten and steel absorbers (49). To model saturation effects in the scintillator response we applied Birks’ law (71) with a constant 0.126 mm/MeV.
The particle origin position was assumed to be at $z=0$ and the first layer of the HCal insert was placed at $z=380$ mm.
No noise was included in the simulation, as a realistic implementation is not available at this time, but its effect was accounted for with an energy threshold as described in the next section. For the digitization step, a 12-bit dynamic range for the ADC and maximum value set to 200 MeV.
Each cell was assigned a timestamp dictated by the particle with the earliest time of arrival to the cell. The effect of the time resolution was included as a cell-by-cell smearing with a Gaussian with $\mu=0$, $\sigma=$1 ns. Unlike in Ref. (51), no SiPM-saturation effect was simulated since this effect is expected to be negligible with the model chosen as baseline222For reference, the SiPM used in the CALICE W-AHCAL prototype contained about 1k channels whereas the one used in our design has 14.4k channels..
The hits were also required to have $E_{\mathrm{hit}}>0.1E_{\mathrm{MIP}}$, where $E_{\mathrm{MIP}}$ is the most-probable energy deposited by a minimum-ionizing particle (MIP), which is about 0.8 MeV as obtained from simulations of high-energy muons. This energy cut was assumed to reduce the noise to a negligible level, as it will correspond to a cut of about 3 photoelectrons. Our selection was more relaxed than the CALICE studies (20) that used an older generation of SiPMs with significantly larger noise, after-pulsing, and cross-talk.
We used a time cut of $t<50$ ns, where $t=0$ is defined when the generated particle hit the front of the insert. As we will show in Sec. 3.4, this selection yields a compensated response with a basic reconstruction algorithm. Much larger integration times might be possible given the EIC bunch structure and absence of pileup. Such time information can be exploited with advanced algorithms, such as those proposed in Refs. (72; 29; 73).
3.2 Shower-shape examples
Figure 13 shows some example shower shapes for a 50 GeV $\pi^{-}$ with $\eta^{*}=3.7$ and a random $\phi^{*}$. As expected, the $\pi^{-}$ showers can be roughly described as having a narrow core from electromagnetic sub-showers and a halo from hadronic sub-showers. The core typically contains cells with large energy deposits, whereas the halo is composed of mainly low-energy cells. The latter are dominated by the copious production of neutrons produced in tungsten. Similar observations have been made with CALICE prototypes (7).
The hadronic-shower shapes, although irregular, provide a hint on the performance of the insert. The production of neutrons, which drives the hadronic halo, occurs over wide angles. Some of the neutrons will be emitted in the direction of the beampipe hole, but often these appear in the other side of the calorimeter instead of being lost altogether. On the other hand, the core of the shower is electromagnetic in nature so its size is driven by the radiation length, which is short in tungsten.
3.3 Hit distributions
Figure 14 shows the hit-energy and multiplicity distributions for a 50 GeV $\pi^{-}$ with $\eta^{*}=3.7$. We show the distributions separately for all hits above the 0.1 MIP threshold, all hits that have more energy than the average of all the hits in their respective showers, and those with more than 5 MIPs of energy.
The average hit energy was found to be about 1.6 MeV, with 17% of the hits containing energy above 1.6 MeV and 7.6% containing energy above 5 MIPs (4 MeV).
The maximum hit energy was about 200 MeV. The hits-per-shower distribution shows a broad peak around 275 hits, whereas the hits above average show a narrower distribution centered around 50 hits, and those above 5 MIPs show an average of 20 hits.
3.4 Linearity for hadrons and electrons and $e/h$ ratio
Figure 15 shows the energy distributions of showers at $\eta^{*}$= 3.7 both for $\pi^{-}$ and $e^{-}$ with various energies. The total reconstructed energy was computed as:
$$E_{total}=\frac{E_{Insert}}{SF_{Insert}}+\frac{E_{HCal}}{SF_{HCal}}$$
(1)
where $E_{Insert}$ and $E_{HCal}$ are the sum of hit energies in the insert and HCal respectively, and $SF_{Insert}$ and $SF_{HCal}$ are the sampling fractions for the insert and HCal, respectively. The sampling fraction is computed by taking the ratio of the output energy to the input energy in the calorimeter. The sampling fraction is 0.9$\%$ for the insert and 2.2$\%$ for the HCal.
The response for pions in the insert is mostly Gaussian but it shows a low-energy tail, which is much less pronounced with the combined measurement using the HCal insert and the HCal (this is quantified in Sec. 3.5). This suggests that the low-energy tail is due to leakage in the transverse direction from the HCal insert towards the HCal.
The response for electrons is mostly Gaussian. This also suggests that that the electromagnetic core of the hadronic showers is well contained due to short radiation length of tungsten.
Figure 16 shows the mean energy deposited in the HCal insert for electrons and pions, along with the $e/h$ ratio, which is found to be about 0.97 on average.
3.5 Non-Gaussian tails due to leakage
Leakage is quantified as the fraction of events in which the shower energy is below $\mu-3\sigma$, where $\mu$ and $\sigma$ are obtained from the Gaussian fit to the distribution. Figure 17 shows the leakage for $\pi^{-}$ and $e^{-}$ at different energies and $\eta^{*}$=3.7. The leakage in the insert alone for $\pi^{-}$ is about 10% for the entire energy range but relatively small (about 1% or lower) for electrons. After considering both the HCal and the HCal insert, the leakage for $\pi^{-}$ showers drops to about $1\%$, suggesting most of the leakage at $\eta^{*}$=3.7 is in the transverse direction with a minimal amount of it leaking through the beampipe. This indicates that neutrons, which are produced over a large solid angle, traverse the hole and are measured rather than being lost in the small solid angle of the beampipe exit (as illustrated in Fig. 13).
3.6 Single-hadron energy resolution
Figure 18 shows that the resolution, defined as ratio of the width ($\sigma$) to the mean value of the fitted energy distribution, for charged pions at $\eta^{*}$=3.7. The HCal insert meets the requirements set in the EIC Yellow Report (1), even beyond the pseudorapidity range for which it was set.
The expected resolution for the HG-CALI is similar to the CALICE test-beam data (51). We have also implemented the CALICE geometry in DD4HEP for both the steel and tungsten prototypes and were able to reproduce the linearity and the resolutions reported in Ref. (51) to within 10$\%$, indicating that the similarity that we see in Fig. 18 is not coincidental.
3.7 Pseudorapidity dependence of performance
Figure 19 shows the distribution of shower energies at a constant pion energy of 50 GeV but with different $\eta^{*}$. The measured distributions using both the HCal insert and HCal can be well described by a Gaussian fit, but the fit quality deteriorates with increasing $\eta^{*}$. For $\eta^{*}>$3.9, the fraction of events in the low-energy tail increases due to losses to the beampipe hole. This stems from the worse geometrical acceptance of the HCal insert here (see Sec. 2.7).
Figure 20 shows the leakage for 50 GeV $\pi^{-}$ and $e^{-}$ at different $\eta^{*}$. Taking both the HCal endcap and the HCal insert into account removes most of the leakage, which is in the transverse direction and can be recovered, as shown by solid markers. At the high $\eta^{*}$ the leakage increases sharply but is still only about 5$\%$ at $\eta^{*}=3.9$ for $\pi^{-}$ and about 2$\%$ for $e^{-}$.
Figure 21 shows the energy resolutions obtained with the HCal insert and the HCal as a function of $\eta^{*}$ for 50 GeV $\pi^{-}$. The resolution obtained with HG-CALI only is constant at about $9\%$ over the region $3.4<\eta^{*}<3.8$ and it degrades at lower and higher $\eta^{*}$. Including the HCal improves the resolution only at low $\eta^{*}$ but not at high $\eta^{*}$. The horizontal line represents the EIC Yellow Report requirement (1), which is defined up to $\eta=3.5$, beyond which only “degraded performance” is specified. The HG-CALI resolution meets the YR requirement up to $\eta^{*}=3.8$, and gets slightly worse at the highest $\eta^{*}$ region due to the poorer acceptance as discussed in Sec. 2.7.
4 Summary and conclusions
We have presented a design for a high-granularity calorimeter insert (HG-CALI) for the forward hadronic calorimeter of the EIC detector. Its sampling design is based on the CALICE “SiPM-on-tile” concept and uses scintillator megatiles that are grooved to define hexagonal cells. Each layer is distinctly shaped to adapt to the conical beampipe shape that accommodates a beam-crossing angle of 25 mrad.
The HG-CALI represents the first concrete proposal to instrument the $3<\eta^{*}<4$ range, which is challenging due to the EIC crossing angle. It offers an optimal acceptance coverage that is limited only by the clearance from the beampipe. It thus represents a way to maximize acceptance up to nearly $\eta^{*}=4.0$, which is a key requirement set in the EIC Yellow Report (1).
The HG-CALI is designed to integrate well with the rest of the endcap calorimeters and will enable easy access to the scintillator layers and SiPMs for maintenance and upgrades. Such a feature will enable frequent annealing of the SiPMs, which will keep radiation damage within manageable levels.
The energy resolution of the HG-CALI meets the EIC Yellow Report requirements even with a basic reconstruction algorithm. Moreover, this device will provide 5D shower information (energy, time, and 3D position) that can be exploited with machine-learning techniques.
The HG-CALI will complement the forward HCal in the region $3.0<\eta^{*}<3.4$ or so by providing an accurate estimate of the shower start position and incident angle over some range in azimuth. Such information could be used with machine-learning approaches to account for shower fluctuations and thus improve the energy resolution of the combined system.
This design will be complemented by a high-granularity electromagnetic calorimeter insert based on tungsten-powder/scintillating fiber with SiPM readout. The complete insert system will be compensated, yielding the prospect of precise and accurate measurements.
The HG-CALI has the potential to unleash the promise of imaging calorimetry at the EIC, which could support a rich physics program centered on forward jets and their substructure in polarized electron-proton and electron-nucleus collisions.
Model availability
The Google Sketchup model, as well as the CAD file in .step format, for the proposed detector can be found in Ref. (74).
The DD4HEP model can be found in Ref. (75).
Acknowledgements
We thank members of the EIC Detector-1 proto-collaboration for useful discussions about the insert, as well as help from their Software Team on matters related to the DD4Hep framework. We thank Oskar Hartbrich for pointing out the importance of the timing cuts and the choice of resolution metric to reproduce CALICE results. We also thank Gerard Visser for feedback on the readout scheme. We also thank members of the California EIC consortium for their feedback on our design. This work was supported by MRPI program of the University of California Office of the President, award number 00010100. M.A, A.P and S.P acknowledge support from DOE grant award number DE-SC0022324. M.A, K.B, and B.K acknowledge support from DOE grant award number DE-SC0022355.
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Application of belief functions to medical image segmentation: A review
Ling Huang
Su Ruan
Heudiasyc, CNRS, Université de technologie de Compiègne, France
Quantif, LITIS, University of Rouen Normandy, France
Abstract
The investigation of uncertainty is of major importance in risk-critical applications, such as medical image segmentation. Belief function theory, a formal framework for uncertainty analysis and multiple evidence fusion, has made significant contributions to medical image segmentation, especially since the development of deep learning. In this paper, we provide an introduction to the topic of medical image segmentation methods using belief function theory. We classify the methods according to the fusion step and explain how information with uncertainty or imprecision is modeled and fused with belief function theory. In addition, we discuss the challenges and limitations of present belief function-based medical image segmentation and propose orientations for future research. Future research could investigate both belief function theory and deep learning to achieve more promising and reliable segmentation results.
keywords:
Belief function theory, Medical image segmentation, Uncertainty quantification, Information fusion, Evidence fusion, Reliability, Deep learning
1 Introduction
The precise delineation of the target lesion on the medical image is essential for optimizing disease treatment. In clinical routine, this segmentation is performed manually by physicians and contains shortcomings. First, the pixel or voxel-level segmentation process is time-consuming, especially with 3D images. Second, the segmentation performances are limited by the quality of medical images, the difficulty of the disease, and the domain knowledge of the experts. Thus, physicians always diagnose a disease by summarising multiple sources of information. The advances in medical imaging machines and technology now allow us to obtain medical images in several modalities, such as Magnetic Resonance Imaging (MRI)/Positron Emission Tomography(PET), multi-sequence MRI, or PET/ Computed Tomography (CT). The different modalities provide different information about cancer and other abnormalities in the human body. Combining these modalities makes it possible to segment the organ and lesion region in a reliable way.
Former medical image segmentation approaches batenburg2009adaptive ; kimmel2003fast ; onoma2014segmentation ; salvador2004determining ; suykens1999least ; StroblBias ; kleinbaum1996logistic focus on low-level feature analysis, e.g., gray and textual features, or hand-crafted features. Those approaches have limitations in segmentation accuracy, efficiency, and reliability, which make it a great gap between experimental performance and clinical application. More recently, the success of deep learning in the medical domain brought a lot of contributions to medical image segmentation tasks ronnebergerconvolutional ; myronenko20183d ; isensee2018nnu ; bahdanau2014neural ; vaswani2017attention ; carion2020end ; han2021transformer . It first solves the segmentation efficiency problem that enables large-scale medical image segmentation and then contributes greatly to its accuracy.
Despite the excellent performance of deep learning-based medical image segmentation methods, the doubt about the reliability of the segmentation results remains hullermeier2021aleatoric , which has been the obstacle between experimental performance and the application of therapeutic decision-making for complex oncological cases. A reliable segmentation model should be well calibrated with accuracy and confidence to be an identity function $y=x$. Figure 1 shows an example of an over-confidence segmentation model that outputs a result with high segmentation accuracy and low confidence. Therefore, a trustworthy representation of uncertainty is desirable and should be considered a key feature of any deep learning method, especially in safety-critical application domains, e.g., medical image segmentation. In general, deep models have two sources of uncertainty: aleatoric uncertainty and epistemic uncertainty hora1996aleatory ; der2009aleatory . Aleatoric uncertainty refers to the notion of randomness, i.e., the variability in an experiment’s outcome due to inherently random effects. In contrast, epistemic uncertainty refers to uncertainty caused by a lack of knowledge (ignorance) about the best model, i.e., the ignorance of the learning algorithm or decision-maker. As opposed to uncertainty caused by randomness, uncertainty caused by ignorance can be reduced based on additional information or the design of a suitable learning algorithm.
The medical image segmentation, the uncertainty can be summarized into three levels lakshminarayanan2017simple : pixel/voxel-level uncertainty, instance-level uncertainty, and subject-level uncertainty. Pixel/voxel-level uncertainty is useful for interaction with physicians by providing additional guidance for correction segmentation results. Instance-level is the uncertainty aggregate by pixel/voxel-level uncertainty, which can be used to reduce the false discovery rate. Subject-level uncertainty offers information on whether the segmentation model is reliable. Early approaches to quantify the segmentation uncertainty quantification were based on Bayesian theory hinton1993keeping ; mackay1992practical .
The popularity of deep segmentation models has revived the research on model uncertainty estimation and has given rise to specific methods such as variational dropout gal2016dropout ; tran2018bayesian , and model ensembles lakshminarayanan2017simple ; rupprecht2017learning . While the probabilistic segmentation models capture knowledge in terms of a single probability distribution and can not distinguish between aleatoric and epistemic uncertainty, which limits the further study of uncertainty and reliability. In this paper, we focus on a different uncertainty quantification approach based on belief function theory (BFT) dempster1967upper ; shafer1976mathematical ; denoeux20b with the specific focus on medical image segmentation. It can model the epistemic uncertainty directly and enable researchers further explore the possibility of improving the model’s reliability based on the existing uncertainty. Researchers from the medical image segmentation community have been actively involved in the research of BFT in handling uncertain information modeling, and fusion and have been shown to be very promising for the segmentation task for imperfect medical images lian2018joint ; ghesu2021quantifying ; huang2021belief .
The rest of the paper will provide an overview of the BFT-based medical image segmentation methods. Recent review work for medical image segmentation focus on deep learning-based methods razzak2018deep ; hesamian2019deep ; liu2021review , with a specific focus on deep feature extraction framework siddique2021u , multimodal information fusion zhou2019review , embracing imperfect dataset ramesh2021review , etc. This paper will focus on discussing:
(1)
how imperfect medical images can be modeled by assigning them the degree of belief and uncertainty directly,
(2)
how multiple sources of evidence (with conflict) can be fused by BFT,
(3)
how the above advantages can be merged with the popular deep learning methods to get an accurate and reliable deep segmentation model.
To the best of our knowledge, this is the first review that summarizes the existing medical image segmentation methods with uncertainty quantification and multiple evidence fusion. We hope this review will raise the community’s awareness of the existing solutions for imperfect medical image data segmentation and further inspire researchers to explore the possibility of taking the benefit of both BFT and deep learning to make automatic segmentation methods reliable and interpretable.
We organize this paper as follows: Section 2 summarizes the development of existing medical image segmentation methods and their limitations. Section 3 introduces the fundamentals of BFT. Section 4 introduces the BFT-based methods to model uncertainty, with Sections 4.1 and 4.2 present supervised and unsupervised methods to model uncertainty, respectively. Section 5 gives an overview of BFT-based medical image segmentation methods with Sections 5.1 and 5.2 present the BFT-based medical image segmentation with, respectively, single and multiple classifiers or clustering algorithms. Section 6 concludes this review and give some potential research directions.
Search criterion. To identify related contributions, we mainly retrieved papers containing “medical image segmentation” and “belief function theory” or “Dempster-Shafer theory” or “evidence theory” in the title or abstract from IEEE, Springer, PubMed, Google Scholar, and ScienceDirect. Additionally, conference proceedings for NIPS, CVPR, ECCV, ICCV, MICCAI, and ISBI were searched based on the titles of papers. Papers that do not primarily focus on medical image segmentation problems were excluded.
2 Medical image segmentation
Medical image segmentation involves the extraction of regions of interest (ROI) from 2D/3D image data (e.g., pathological or optical imaging with color mages, MRI, PET, and CT scans. The main goal of segmenting medical images is to identify areas of cancer and other abnormalities in the human body, for example, brain tissue and tumor segmentation, the interior of the human body (such as lung, spinal canal, and vertebrae), skin and cell lesion segmentation with optical imaging with color. Previously, segmenting medical images was time-consuming for physicians, while recent advances in machine learning techniques, especially deep learning, make it easier to perform routine tasks.
2.1 Traditional approaches to medical images segmentation
Early image segmentation methods used the information provided by the image itself, e.g., gray, textual, contrast and histogram features, and segmenting ROI based threshold batenburg2009adaptive , edge detection kimmel2003fast , graph partitioning onoma2014segmentation , and clustering salvador2004determining . More recently, researchers are interested in hard-crafted features, e.g., Scale Invariant Feature Transform (SIFT) lowe1999object , Features from Accelerated Segment Test (FAST) rosten2006machine and Geometric hashing mian2006three , and segmenting ROI using machine learning-based methods such as support vector machine (SVM) suykens1999least , random forest (RF) StroblBias and logistic regression (LR) kleinbaum1996logistic . Those methods have attracted great interest for a while, but their accuracy cannot meet clinical application requirements.
2.2 Deep learning-based methods
Long et al. long2015fully were the first authors to show that a fully convolutional network (FCN) could be trained end-to-end for semantic segmentation, exceeding the state-of-the-art when the paper was published in 2015. UNet ronnebergerconvolutional , a successful modification and extension of FCN, has become the most popular model for medical image segmentation in recent years. Based on UNet, research for deep learning-based medical image segmentation can be summarized in two major directions: feature representation and model optimization.
To better represent input information, well-designed DNNs with encoder-decoder architecture (e.g., 3D UNetcciccek20163d , Res-UNet xiao2018weighted , Dense-UNet guan2019fully , MultiResUNet ibtehaz2020multiresunet , Deep3D-UNet zhu2018anatomynet , V-Netmilletari2016v , nnUNetisensee2018nnu ) were proposed and have achieved good performance. Recently, the popular ideas e.g., residual connection he2016deep , attention mechanism bahdanau2014neural and transformer mechanism vaswani2017attention ; carion2020end ; han2021transformer , have achieved promising performance with DNNs, which also lead to the contributions in medical image domain, e.g., Residual-UNet ronnebergerconvolutional ; xiao2018weighted , Attention UNet oktay2018attention ; trebing2021smaat and Transformer-UNet cao2021swin ; hatamizadeh2022unetr ; hatamizadeh2022swin .
As for the optimization of the DNN-based medical image segmentation models, researchers in the medical image segmentation domain tend to use Dice Loss or a combination of dice loss and cross entropy as a total loss milletari2016v instead of the classical loss functions (e.g., Cross-Entropy Loss, Mean Square Loss) to handle the unbalanced label distribution problem, with more focuses on mining the foreground area. Based on Dice loss, some variants such as GeneralizedDiceLoss sudre2017generalised , TverskyLoss salehi2017tversky and ContrastiveLoss chen2020simple , have been proposed to further solve the unbalanced label problem.
Dice score, Sensitivity, and Precision are the most commonly used evaluation criteria to assess the quality of deep medical image segmentation methods, which are defined as follows:
$$\textsf{Dice}(P,T)=\frac{2\times TP}{FP+2\times TP+FN},$$
(1)
$$\textsf{Sensitivity}(P,T)=\frac{TP}{TP+FN},$$
(2)
$$\textsf{Precision}(P,T)=\frac{TP}{TP+FP},$$
(3)
where $TP$, $FP$, and $FN$ denote the numbers of true positive, false positive, and false negative voxels, respectively. (See Figure 2). The Dice score is a global measure of segmentation performance. Sensitivity is the proportion, among actual tumor voxels, of voxels correctly predicted as tumors. Precision is the proportion, among predicted tumor voxels, of voxels that actually belong to the tumor region; it is, thus, an estimate of the probability that the model is correct when it predicts that a voxel is in a lymphoma region. These two criteria, thus, have to be considered jointly. A more comprehensive introduction to the evaluation criteria for medical image segmentation can be found in taha2015metrics .
2.3 The limitations
The above research focuses on improving the accuracy of segmentation performance under the assumption of adequate and perfect input information and accurate and appropriate prior knowledge. While in reality, especially in the medical image segmentation domain, both the input information and prior knowledge are imperfect and contain a degree of uncertainty. Figure 3a shows the example of a information with uncertainty. Taking a brain tumor segmentation task as an example. Let $X$ be the type of tumor of a voxel, and $\Omega=\{ED,ET,NCR,Others\}$, corresponding to the possibilities: edema, enhancing tumor, necrotic core, and others. Assuming that a specialist provides the information $X\in\{ED,ET\}$, while there is a probability $p=0.1$ that the information is unreliable. How to represent this situation by a probability function becomes a challenging problem. Another situation is if we have multiple information sources containing uncertainty, i.e., Figure 3b; how can we model that kind of uncertainty and fusion the evidence based on that? Furthermore, if the multiple information sources are in conflict and contain uncertainty as well, i.e., Figure 3c, it is difficult to represent and summarize that information by probabilistic models. Thanks to BFT, these challenges can be addressed by designing new frameworks for modeling, reasoning, and fusing imperfect (uncertain, imprecise) information. Next section we will give a brief introduction to BFT.
3 Fundamentals of belief function theory
BFT is a generalization of Bayesian theory, but it is more flexible than the Bayesian approach and suitable for weaker conditions sun2018multi , i.e., imperfect (uncertain, imprecise, partial) information. Figure 4 shows the difference between a probabilistic model and a BFT model when applied to a three-class classification task ($\Omega=\{a,b,c\}$). For input $x$, the probabilistic model outputs the probability that $x$ belongs to classes a, b, and c as 0.4, 0.5, and 0.1, respectively. While the BFT model can generate the belief that $x$ belongs to any subset of $\Omega$, e.g., $\{a,b\}$, $\{b,c\}$. Compared with the probabilistic model, the BFT model has more degrees of freedom to represent its uncertainty directly, i.e., it models the uncertainty into the subset of $\Omega$ and the ignorance set $\Omega$.
In the past decades, BFT has generated considerable interests and has had a great success in diverse fields, including uncertain reasoning smets1990combination ; yager1987dempster ; dubois1988representation ; denoeux2008conjunctive ; xiao2020generalization , classification denoeux1995k ; denoeux2000neural ; xiao2022negation and clustering denoeux2004evclus ; masson2009recm , etc. It was first originated by Dempster dempster1967upper in the context of statistic inference in 1968 and was formalized by Shafer shafer1976mathematical as a theory of evidence in 1976. In 1986, Dubois and Parde proposed an approach to the computerized processing of uncertainty dubois2012possibility . In 1978, Yager proposed a new combination rule of the belief function framework yager1987dempster . In 1990, BFT was further popularized and developed by Smets smets1990combination as the ’Transferable Belief Model’ with pignistic transformation for decision making. Since then, booming developments have been made. More detailed information about the development of BFT in 40 years can be found in denzux201640 .
We will first briefly introduce the basic notions of BFT in Section 3.1, which includes evidence representation (mass functions, belief, and plausibility functions). Second, we introduce Dempster’s combination rule to explain the operations of multiple sources of evidence in Section 3.2. Third, we introduce a discounting operation for unreliable sources in Section 3.3. Fourth, we introduce some commonly used decision-making methods in Section 3.4.
3.1 Representation of evidence
Let $\Omega=\{\omega_{1},\omega_{2},...,\omega_{C}\}$ be a finite set of all possible hypotheses about some problem, called a frame of discernment. Evidence about a variable $\omega$ taking values in $\Omega$ can be represented by a mass function $m$, from the power set $2^{\Omega}$ to $[0,1]$, such that
$$\displaystyle\sum_{A\subseteq\Omega}m(A)=1,$$
(4a)
$$\displaystyle m(\emptyset)=0.$$
(4b)
which can also be called basic belief assignment (BBA). The BBA methods used in medical image segmentation to generate mass functions will be introduced in Section 4 Each subset $A\subseteq\Omega$ and $m(A)$ is called a focal set of $m$. The total ignorance of the problem can be represented as $m(\Omega)$. If all focal sets are singletons, then $m$ is said to be Bayesian. It is equivalent to a probability distribution. The information provided by a mass function $m$ can be represented by a belief function $Bel$ or a plausibility function $Pl$ from $2^{\Omega}$ to $[0,1]$, respectively, as
$$Bel(A)=\sum_{B\subseteq A}m(B)$$
(5)
and
$$Pl(A)=\sum_{B\cap A\neq\emptyset}m(B)=1-Bel(\bar{A}),$$
(6)
for all $A\subseteq\Omega$. The quantity $Bel(A)$ can be interpreted as a degree of support to $A$, while $Pl(A)$ can be interpreted as a measure of lack of support given to the complement of $A$ (or we can say $Pl(A)$ represents the belief that might be committed to $A$).
3.2 Dempster’s rule
In BFT, the belief about a certain item is elaborated by aggregating different belief functions over the same frame of discernment. Given two mass functions $m_{1}$ and $m_{2}$ derived from two independent items of evidence, the final belief that supports $A$ can be obtained by combining $m_{1}$ and $m_{2}$ with Dempster’s rule shafer1976mathematical , which is defined as
$$(m_{1}\oplus m_{2})(A)=\frac{1}{1-\kappa}\sum_{B\cap D=A}m_{1}(B)m_{2}(D),$$
(7)
for all $A\subseteq\Omega,A\neq\emptyset$, and $(m_{1}\oplus m_{2})(\emptyset)=0$. The coefficient $\kappa$ is the degree of conflict between $m_{1}$ and $m_{2}$, it is defined as
$$\kappa=\sum_{B\cap D=\emptyset}m_{1}(B)m_{2}(D).$$
(8)
3.3 Discounting
In (8), if $m_{1}$ and $m_{2}$ are logically contradictory and, consequently, then we can not use Dempster’s rule to combine the two pieces of evidence. Discounting strategies thus can be used to combine evidence under high contradiction shafer1976mathematical ; mercier2008refined ; denoeux2019new . Let $m$ be a mass function on $\Omega$ and $\xi$ a coefficient in $[0,1]$. The discounting operation shafer1976mathematical with discount rate $\xi$ transforms $m$ into a weaker, less informative mass function defined as follows:
$${}^{\xi}m(A)=(1-\xi)m(A),\quad\forall A\subset\Omega,$$
(9a)
$${}^{\xi}m(\Omega)=(1-\xi)m(\Omega)+\xi,$$
(9b)
coefficient $1-\xi$ represents the degree of belief that the source $m$ is reliable. In the medical domain, the discounting function is widely used in multimodal evidence fusion by considering source information reliability and discounts their outputs using these precalculated reliability factors.
3.4 Decision-making
After combining all the available evidence in the form of a mass function, it is necessary to decide the belief to be assigned into focal sets. In this section, we introduce some classical BFT-based decision-making methods.
Upper and lower expected utilities
Let $u$ be a utility function. The lower and upper expectations of $u$ with respect to $m$ are defined, respectively, as the averages of the minima and the maxima of $u$ within each focal set of $m$:
$$\displaystyle\underline{E}_{m}(u)=\sum_{A\subseteq\Omega}m(A)\min_{\omega\in A}u(\omega),$$
(10a)
$$\displaystyle\overline{E}_{m}(u)=\sum_{A\subseteq\Omega}m(A)\max_{\omega\in A}u(\omega).$$
(10b)
When $m$ is Bayesian, $\underline{E}_{m}(u)=\overline{E}_{m}(u)$. If $m$ is logical with focal set $A$, then $\underline{E}_{m}(u)$ and $\overline{E}_{m}(u)$ are, respectively, the belief and plausibility functions as we mentioned in Section 3.1. Both lower or upper expectations can be chosen for the final decision according to the given task.
Pignistic criterion
In 1990, Smets proposed a pignistic transformation smets1990combination that each mass of belief is distributed equally among the elements for all, which is defined as
$$BetP(\omega)=\sum_{\omega\in A}\frac{m(A)}{\mid A\mid},\quad\forall\omega\in\Omega,$$
(11)
where $\left|A\right|$ denotes the cardinality of $A\subseteq\Omega$.
Besides the above methods, there are various decision-making methods proposed for BFT, such as Generalized OWA criterion yager1992decision , Generalized minimax regretYAGER2004109 , Generalized divergence xiao2022generalized , etc. More details about decision-making with BFT can be found in the review paper denoeux2019decision .
4 BBA methods to generate mass functions
To model segmentation uncertainty, the first step is to generate mass functions. In this section, we introduce the BBA methods applied to medical image segmentation. Figure 5 is an overview of BBA methods. In general, those methods can be separated into supervised and unsupervised methods according to if the annotations are used.
4.1 Supervised BBA methods
The supervised BBA methods can be classified into two categories. One is the likelihood-based methods, such as Shafer’s model shafer1976mathematical and Appriou’s model appriou1999multisensor ; appriou200501 . The other is the distance-based methods, such as evidential KNN denoeux1995k , evidential neural network classifier denoeux2000neural and Radial basis function network denoeux2019decision . It should be noted that the distance-based BBA methods are easy to be merged with popular deep segmentation models and have shown promising results huang2021belief ; huang2022lymphoma .
4.1.1 The likelihood-based BBA methods
Shafer’s model
In shafer1976mathematical , Shafer proposed a likelihood-based evidential model to calculate mass functions. Assuming that conditional density functions $f(x\mid\omega_{c})$ are known, then the conditional likelihood associated with the pattern $X$ is defined by $\ell(\omega_{c}\mid x)=f(x\mid\omega_{c})$. The mass functions are defined according to the knowledge of all hypotheses $\omega_{1},...,\omega_{C}$. Firstly, the plausibility of a simple hypothesis $\omega_{c}$ is proportional to its likelihood. The plausibility is thus given by
$$Pl(\{\omega_{c}\})=\hbar\cdot\ell(\omega_{c}\mid x),\quad\forall\omega_{c}\in\Omega,$$
(12)
where $\hbar$ is a normalization factor with $\hbar=1\max_{\omega\in\Omega}\ell(\omega|x)$. The plausibility of a set $A$ is thus given by
$$Pl(A)=\hbar\cdot\underset{\omega_{c}\in A}{\max}\ell(\omega_{c}\mid x).$$
(13)
Appriou’s model
Same as Shafer’s one, Appriou appriou1999multisensor ; appriou200501 proposed two likelihood-based models to calculate mass functions with the frame of discernment $\Omega=\{\omega_{c},\neg{\omega_{c}}\}$. For the first model, the mass functions are defined by
$$\displaystyle m(\{\omega_{c}\})=0,$$
(14a)
$$\displaystyle m(\{\neg{\omega_{c}}\})=\alpha_{c}({1-\hbar\cdot\ell(\omega_{c}\mid x)}),$$
(14b)
$$\displaystyle m(\Omega)=1-\alpha_{c}{(1-\hbar\cdot\ell(\omega_{c}\mid x))},$$
(14c)
where $\alpha_{c}$ is a reliability factor depending on the hypothesis $\omega_{c}$ and on the source information. The second model is defined as
$$\displaystyle m(\{\omega_{c}\})=\alpha_{c}\cdot\hbar\cdot\ell(\omega_{c}\mid x)/{(1+\hbar\cdot\ell(\omega_{c}\mid x))},$$
(15a)
$$\displaystyle m(\{\neg{\omega_{c}}\})=\alpha_{c}/{(1+\hbar\cdot\ell(\omega_{c}\mid x))},$$
(15b)
$$\displaystyle m(\Omega)=1-\alpha_{c}.$$
(15c)
4.1.2 The distance-based BBA methods
Evidential KNN (EKNN) rule
In denoeux1995k , Denoeux proposed a distance-based KNN classifier for classification tasks. Let $N_{K}(x)$ denote the set of the $K$ nearest neighbors of $x$ in learning set $Z$. Each $x_{i}\in N_{K}(x)$ is considered as a piece of evidence regarding the class label of $x$. The strength of evidence decreases with the distance between $x$ and $x_{i}$. The evidence of $(x_{i},y_{i})$ support class $c$ is represented by
$$\displaystyle m_{i}(\{\omega_{c}\})=\varphi_{c}(d_{i})y_{ic},\quad 1\leq c\leq C,$$
(16a)
$$\displaystyle m_{i}(\Omega)=1-\varphi_{c}(d_{i}),$$
(16b)
where $d_{i}$ is the distance between $x$ and $x_{i}$, which can be Euclidean or other distance function; and $y_{ic}=1$ if $y_{i}=\omega_{c}$ and $y_{ic}=0$ otherwise. The parameter $\varphi_{c}$ is a decreasing function with
$$\varphi_{c}(d)=\alpha\exp(-\gamma d^{2}),$$
(17)
where $\alpha$ and $\gamma$ are two tuning parameters. The evidence of the $K$ nearest neighbors of $x$ is fused by Dempster’s rule:
$$m=\bigoplus_{x_{i}\in N_{K}(x)}m_{i}.$$
(18)
The final decision is made according to maximum plausibility. The detailed optimization of these parameters is described in zouhal1998evidence . Based on the first work, Denoeux et al. proposed contextual discounting evidential KNN denoeux2019new with partially supervised learning to address the annotation limitation problem.
Evidential neural network (ENN)
The success of machine learning encouraged the exploration of applying belief function theory with learning methods. In denoeux2000neural , Denœux proposed an ENN classifier in which mass functions are computed based on distances to prototypes.
The ENN classifier is composed of an input layer of $H$ neurons, two hidden layers, and an output layer. The first input layer is composed of $I$ units, whose weights vectors are prototypes $p_{1},\ldots,p_{I}$ in input space. The activation of unit $i$ in the prototype layer is
$$s_{i}=\alpha_{i}\exp(-\gamma_{i}d_{i}^{2}),$$
(19)
where $d_{i}=\left\|x-p_{i}\right\|$ is the Euclidean distance between input vector $x$ and prototype $p_{i}$, $\gamma_{i}>0$ is a scale parameter, and $\alpha_{i}\in[0,1]$ is an additional parameter. The second hidden layer computes mass functions $m_{i}$ representing the evidence of each prototype $p_{i}$, using the following equations:
$$\displaystyle m_{i}(\{\omega_{c}\})$$
$$\displaystyle=u_{ic}s_{i},\quad c=1,...,C$$
(20a)
$$\displaystyle m_{i}(\Omega)$$
$$\displaystyle=1-s_{i},$$
(20b)
where $u_{ic}$ is the membership degree of prototype $i$ to class $\omega_{c}$, and $\sum_{c=1}^{C}u_{ic}=1$. Finally, the third layer combines the $I$ mass functions $m_{1},\ldots,m_{I}$ using Dempster’s rule. The output mass function $m=\bigoplus_{i=1}^{I}m_{i}$ is a discounted Bayesian mass function that summarizes the evidence of the $I$ prototypes.
Radial basis function (RBF) network
In huang2022lymphoma , Huang et al. confirmed the RBF network can be an alternative approach to ENN that is based on the aggregation of weights of evidence. Consider an RBF network with $I$ prototype (hidden) units. The activation of hidden unit $i$ is
$$s_{i}=\exp(-\gamma_{i}d_{i}^{2}),$$
(21)
where, $d_{i}=\left\|\boldsymbol{x}-\boldsymbol{p}_{i}\right\|$ is the Euclidean distance between input vector $\boldsymbol{x}$ and prototype $\boldsymbol{p}_{i}$, and $\gamma_{i}>0$ is a scale parameter. Here we only show an example of binary classification task $C=2$ and $\Omega=\{\omega_{1},\omega_{2}\}$ (The multi-class example can be found in denoeux19d ). Let $v_{i}$ be the weight of the connection between hidden unit $i$ and the output unit, and let $w_{i}=s_{i}v_{i}$ be the product of the output of unit $i$ and weight $v_{i}$. The quantities $w_{i}$ can be interpreted as weights of evidence for class $\omega_{1}$ or $\omega_{2}$, depending on the sign of $v_{i}$.
To each prototype $i$ can be associated with the following simple mass function:
$$m_{i}=\{\omega_{1}\}^{w_{i}^{+}}\oplus\{\omega_{2}\}^{w_{i}^{-}},$$
where $w_{i}^{+}$ and $w_{i}^{-}$ denote, respectively, the positive and negative parts of $w_{i}$. Combining the evidence of all prototypes in favor of $\omega_{1}$ or $\omega_{2}$ by Dempster’s rule, we get the mass function $m=\bigoplus_{i=1}^{I}m_{i}$ that has the following expression:
$$\displaystyle m(\{\omega_{1}\})$$
$$\displaystyle=\frac{[1-\exp(-w^{+})]\exp(-w^{-})}{1-\kappa},$$
(22a)
$$\displaystyle m(\{\omega_{2}\})$$
$$\displaystyle=\frac{[1-\exp(-w^{-})]\exp(-w^{+})}{1-\kappa},$$
(22b)
$$\displaystyle m(\Omega)$$
$$\displaystyle=\frac{\exp(-w^{+}-w^{-})}{1-\kappa}=\frac{\exp(-\sum_{i=1}^{I}|w_{i}|)}{1-\kappa},$$
(22c)
where $\kappa$ is the degree of conflict between $\{\omega_{1}\}^{w^{+}}$ and $\{\omega_{2}\}^{w^{-}}$ with
$$\kappa=[1-\exp(-w^{+})][1-\exp(-w^{-})].$$
(23)
4.2 Unsupervised BBA methods
For unsupervised BBA methods, the goal is to find an algorithm to generate mass functions without any label information. In earlier BBA studies, Fuzzy C-means (FCM) dunn1973fuzzy was the most popular algorithm used to generate membership values (MVs). Based on MVs, the authors can obtain mass functions according to some domain knowledge, e.g., threshold zhu2002automatic , or user-specific parameters ghasemi2012brain . Sigmoid and the one-side Gaussian function can also be used to generate MVs safranek1990evidence . Credal partition denoeux2004evclus , the extension of fuzzy partition, enables us to generate mass functions directly masson2008ecm . Besides the two popular BBA methods, mass functions can also be generated from the Gaussian distribution of the input to the cluster center chen2012manifold .
4.2.1 MVs-based BBA methods
FCM
Considering that there are some BFT-based methods that use FCM to generate MVs, thus we introduce it here to offer a basic view for readers. With FCM, any $\mathbf{x}$ has a set of coefficients giving the degree of being in the $k$-th cluster $w_{k}(x)$. The centroid of a cluster is the mean of all points, weighted by their degree of belonging to the cluster,
$$c_{k}={{\sum_{x}{w_{k}(x)}^{m}x}\over{\sum_{x}{w_{k}(x)}^{m}}},$$
(24)
where $m$ is the hyper-parameter that controls how fuzzy the cluster will be. The higher it is, the fuzzier. Given a finite set of data, the FCM algorithm returns a list of cluster centres $P=\{c_{1},...,c_{C}\}$ and a partition matrix $W=\omega_{ij},i=1,...N,j=1,...,C$,
$$w_{{ij}}={\frac{1}{\sum_{{k=1}}^{{C}}\left({\frac{\left\|{\mathbf{x}}_{i}-{\mathbf{c}}_{j}\right\|}{\left\|{\mathbf{x}}_{i}-{\mathbf{c}}_{c}\right\|}}\right)^{{{\frac{2}{m-1}}}}}},$$
(25)
where $w_{ij}$, is the the degree of $\mathbf{x}_{i}$ belongs to cluster $\mathbf{c}_{j}$. The objective function is defined as
$$\underset{P}{arg\max}\sum_{i=1}^{N}\sum_{j=1}^{C}w_{ij}^{m}\left\|\mathbf{x}_{i}-\mathbf{c}_{j}\right\|.$$
(26)
Zhu’s model
In zhu2002automatic , Zhu et al. proposed a method to determine mass functions using FCM and neighborhood information. Mass of simple hypothesis $\{\omega_{c}\}$ is directly obtained from the filtered membership functions $\upsilon_{c}(g)$ of the gray level $g(x,y)$ to cluster $c$ as $m(\{\omega_{c}\})=\upsilon_{c}(g)$, indicates that for a given gray level, the piece of evidence of belonging to the cluster $c$ is directly given by its degree of membership to the same cluster. If there is a high ambiguity in assigning a gray level $g(x,y)$ to cluster $c$ or $c+1$, that is, $\left|\upsilon_{c}(g)-\upsilon_{c+1}(g)\right|<\varepsilon$, where $\varepsilon$ is a thresholding value, then a double hypothesis is formed. The value of the threshold $\varepsilon$ is chosen following the applications in question. The authors suggested fixing $\varepsilon$ at 0.1. Once the double hypotheses are formed, their associated mass is calculated according to the following formula:
$$m(\{\omega_{c},\omega_{c+1}\})=\frac{S[\upsilon_{c}(g),\upsilon_{c+1}(g)]}{2S_{max}},$$
(27)
where $S$ represents a surface of a triangle and $S_{max}$ is the maximum of ambiguity. The surface of such a triangle depends
both on the degrees of the membership functions of $g(x,y)$ to clusters $c$ and $c+1$ and on the conflicts between these MVs. Figure 6a shows how the triangle is constructed and how the mass of double hypotheses $\{\omega_{c},\omega_{c+1}\}$ is derived from the surface of the triangle. The vertical axis of Figure 6a represents the MVs. The surface of the two dotted triangles represents two so-called triangular membership functions corresponding to classes $c$ and $c+1$. The two triangles are isosceles and have the same length for their bases. The heights of the triangles are equal to $m(\{\omega_{c}\})$ and $m(\{\omega_{c+1}\})$, respectively. The overlapping surface $S$ of the two triangles represents the MV to the double hypothesis $\{\omega_{c},\omega_{c+1}\}$. Therefore, the mass value attributed to the double hypothesis $\{\omega_{c},\omega_{c+1}\}$ can be directly calculated from the surface $S$. Figure 6b shows the condition of the maximum ambiguity case.
Ratio MV (RMV) transformation
In ghasemi2012brain , Ghasemi et al. proposed a ratio membership value transformation method to calculate mass functions. The FCM algorithm was first used to generate MVs $f_{\omega_{c}}$ for each pixel. Then the MVs are used to build the mass functions. For this purpose, the three ratios of the available MVs are calculated, corresponding to three situations: no-uncertainty (NU), semi-uncertainty (SU), and perfect-uncertainty (PU). First, PU is a critical situation in which the RMVs are smaller than $\alpha$, then the mass function is calculated as $m(\{\omega_{1}\})=m(\{\omega_{2}\})=m(\Omega)=(f_{\omega_{1}}+f_{\omega_{2}})/3$. Second, two thresholds $\alpha$ and $\beta$ with $\alpha=1.5$ and $\beta=3$ are selected to control the boundary between SU and PU, and between NU and SU, separately. For example, with
$$f_{\omega_{1}}=0.18,\\
f_{\omega_{2}}=0.81,\\
RMV=f_{\omega_{1}}/f_{\omega_{2}}=4.5,RMV>\beta,$$
the two MVs fall in the NU category. If
$$f_{\omega_{1}}=0.25,\\
f_{\omega_{2}}=0.65,\\
RMV=f_{\omega_{1}}/f_{\omega_{2}}=2.6,\\
\alpha<RMV<\beta,$$
the two MVs are in the SU category. The mass functions are calculated using
$$\displaystyle m(\{\omega_{1}\})=f_{\omega_{1}}-\frac{\lambda_{\omega_{1},\omega_{2}}}{2},$$
(28a)
$$\displaystyle m(\{\omega_{2}\})=f_{\omega_{2}}-\frac{\lambda_{\omega_{1},\omega_{2}}}{2},$$
(28b)
$$\displaystyle m(\Omega)=\lambda_{\omega_{1},\omega_{2}},$$
(28c)
where $\lambda$ is an uncertainty distance value that defined as $\lambda_{\omega_{1},\omega_{2}}=\frac{\left|f_{\omega_{1}}-f_{\omega_{2}}\right|}{\beta-\alpha}$.
4.2.2 Evidential C-means (ECM)
In denoeux2004evclus , Denoeux et al. proposed an evidential clustering algorithm that extends the notion of fuzzy partition with credal partition, which extends the existing concepts of hard, fuzzy (probabilistic), and possibilistic partition by allocating each object a ’mass of belief,’ not only to single clusters but also to any subsets of $\Omega=\{\omega_{1},...,\omega_{C}\}$.
Credal partition
Assuming there is a collection of five objects for two classes, mass functions for each source are given in Table 1. They represent different situations: the class of source 1 indicates strong evidence that the class of source 1 does not lie in any focal set; the class of source 2 is completely unknown, and the class of source 3 is known with certainty; the cases of sources 4 and 5 correspond to situations of partial knowledge ($m_{5}$ is Bayesian). We can then assign basic beliefs to each partition with the given objective functions with the credal partition.
Evidential C-Means (ECM)
Based on the credal partition, another BBA method, ECM masson2008ecm , was introduced to generate mass functions. In ECM, a cluster is represented by a prototype $p_{c}$. For each non-empty set $A_{j}\subseteq\Omega$, a prototype $\bar{p_{j}}$ is defined as the center of mass of the prototypes $p_{c}$ such that $\omega_{c}\in A_{j}$. Then the non-empty focal set is defined as $F=\{A_{1},...,A_{f}\}\subseteq 2^{\Omega}\setminus\left\{\emptyset\right\}$. Deriving a credal partition from object data implies determining, for each object $x_{i}$, the quantities $m_{i}j=m_{i}(A_{j}),A_{i}\neq\emptyset,A_{j}\subseteq\Omega$. In such a way that $m_{i}j$ is low (resp. high) when the distance between $x_{i}$ and the focal set $\bar{p_{j}}$ is high (resp. low). The distance between an object and any nonempty subset of $\Omega$ has thus to be defined by
$$d_{ij}^{2}=\left\|x_{i}-\bar{p_{j}}\right\|^{2}.$$
(29)
4.2.3 Gaussian distribution (GD)-based model
Besides the FCM-based and credal partition-based BBA method, the mass functions can also be generated from GD chen2012manifold .
The mass of simple hypotheses $\{\omega_{c}\}$ can be obtained from the assumption of GD according to the information $x_{i}$ of a pixel from an input image to cluster $c$ as follows:
$$m(\{\omega_{c}\})=\frac{1}{\sigma_{c}\sqrt{2\pi}}\exp{\frac{-(x_{i}-\mu_{c})^{2}}{2\sigma_{c}^{2}}},$$
(30)
where $\mu_{c}$ and $\sigma^{2}_{c}$ represent, respectively, the mean and the variance of the cluster $c$, which can be estimated by
$$\mu_{c}=\frac{1}{n_{c}}\sum_{i=1}^{n_{c}}x_{i},$$
(31)
$$\sigma_{c}^{2}=\frac{1}{n_{c}}\sum_{i=1}^{n_{c}}(x_{i}-\mu_{c})^{2},$$
(32)
where $n_{c}$ is the number of pixels in the cluster $c$. The mass of double hypotheses involves several sets $\{\omega_{1},\omega_{2},...,\omega_{T}\}$ is determined as
$$m(\{\omega_{1},\omega_{2},...,\omega_{T}\})=\frac{1}{\sigma_{t}\sqrt{2\pi}}\exp{\frac{-(x_{i}-\mu_{t})^{2}}{2\sigma_{t}^{2}}},$$
(33)
where $\mu_{t}=\frac{1}{T}\sum_{i=1}^{T}\mu_{i}$, $\sigma_{t}=\max(\sigma_{1},\sigma_{2},...,\sigma_{T}),2\leq T\leq C$, $C$ is the number of clusters.
4.2.4 Binary frames of Discernment (BFOD)
Under the assumption that the membership value is available, Safranek et al. introduced a BFOD-based BBA method safranek1990evidence to transform membership value into mass functions. The BFOD is constructed as $\Omega=\{\omega,\neg\omega\}$ with a function $cf(\nu)$, $cf(\nu)\in\left[0,1\right]$ that indicates confidence factor. Sigmoid and the one-sided Gaussian function were the most appropriate functions for $cf(\nu)$ reported by the authors. Once a confidence value is obtained, the transformation for mass functions can be accomplished by defining appropriate transfer functions with parameters A and B.
$$\displaystyle m(\{\omega\})=\frac{B}{1-A}cf(\nu)-\frac{AB}{1-A},$$
(34a)
$$\displaystyle m(\{\neg\omega\})=\frac{-B}{1-A}cf(\nu)+B,$$
(34b)
$$\displaystyle m(\Omega)=1-m(\{\omega\})-m(\{\neg\omega\}),$$
(34c)
where $A$ and $B$ are user-specific parameters. The left-hand side stays clamped at zero when the right-hand side goes negative. The parameter $A$ is the confidence-factor axis intercept of the curve that depicts the dependence of $m(\{\omega\})$ on confidence factors, and $B$ is the maximum support value assigned to $m(\{\omega\})$ or $m(\{\neg\omega\})$.
4.3 Discussion
If segmentation labels are available, we suggest using the supervised BBA methods to generate mass functions, especially the distance-based BBA methods, because the distance-based BBA methods can be merged easily with the popular deep learning methods and construct a learnable end-to-end deep segmentation model. The detailed merging example can be found in huang2021evidential , where the authors first apply the idea of merging the ENN or RBF to the deep medical image segmentation model by mapping feature vectors into mass functions instead of using Softmax to map them into probabilities.
If the segmentation labels are not available or only partially available, researchers can still generate mass functions using the unsupervised BBA methods, e.g., using ECM for tumor segmentation lian2018unsupervised . To our best knowledge, there is no research that works on merging unsupervised BBA methods into deep segmentation models even though some studies already make the end-to-end neural network-based evidential clustering model possible, e.g., denoeux2021nn . We believe it is a good choice for researchers who work on unlabeled datasets and wish to directly quantify segmentation uncertainty, they can consider applying unsupervised BBA methods to deep segmentation models.
5 BFT-based medical image segmentation methods
To summarize the BFT-based medical image segmentation methods, we can either classify them by the input modality of the images or by the specific clinical application. Figure 7a shows the proportion of types of medical images applied in the segmentation task and Figure 7b the proportions of application in the medical image segmentation task.
We presented the common BBA methods to generate mass functions for medical images in Section 4, the most interesting way to analyze those methods is to classify them according to the stage at which they fuse mass functions. Figure 8 gives an overview of the BFT-based medical image segmentation methods where the fusion operation is performed.
We first classify them by the number of classifiers/clusterers, i.e., if the fusion of mass functions is performed at the classifiers/clusterers level. Then we classify them by the number of input modalities, i.e., if the fusion of mass functions is performed at the modality level. It should be noted that the medical image segmentation with single modal input and single classifier/clusterer can also have a fusion operation, which is performed at the pixel/voxel level. Figure 9 gives the proportions of the BFT-based segmentation methods if multiple classifiers/clusterers or multimodal medical images are used. 76% of the methods use a single classifier or cluster. Among these methods, 18% take single modal medical images as input, and 58% take multimodal medical images as input. The remaining 24% of the methods use several classifiers or clusterers Among those methods, 21% take single modal medical images as input, and 3% take multimodal medical images as inputs. Sections 5.1 and 5.2 will go for more details for the fusion operations and discuss their performances.
5.1 BFT-based medical image segmentation with a single classifier or clusterer
The medical image segmentation methods we introduced in this section focus on using a single classifier or clusterer. To simplify the introduction, we classify them into single modal and multimodal inputs and introduce them in Sections 5.1.1 and 5.1.2, respectively.
5.1.1 Single modal evidence fusion
Figure 10 shows the framework of single modal evidence fusion with a single classifier (we only take the classifier as an example in the rest of our paper to simplify the explanation). The inputs of the framework are single-modal medical images. The segmentation process comprises three steps: masses calculation (including feature extraction and BBA), feature-level masses fusion, and decision-making. Since decision-making is not the focus of this paper, we will not go into details about it and readers can refer to Ref. denoeux2019decision for a recent review of decision methods based on belief functions.
1)
First, the mass calculation step assigns each pixel/voxel $K$ mass functions based on the evidence of $K$ different BBA methods, or $K$ kinds of input features, or $K$ nearest neighbors, or $K$ prototype centers.
2)
Dempster’s rule is then used for each pixel/voxel to fuse the feature-level mass functions.
3)
Finally, decision-making is done to obtain the final segmentation results.
The feature extraction method here is used to generate MVs (corresponding to traditional medical image segmentation methods) or deep features (corresponding to deep learning-based medical image segmentation methods). The feature extraction method could be intensity-based methods such as threshold intensity, probabilistic-based methods such as SVM, or fuzzy set-based methods such as FCM, etc. The BBA methods are various, therefore we introduce them in specific tasks as follows. In general, the methods introduced in this section focus on feature-level evidence fusion. Table 2 shows the segmentation methods with a single modal input and classifier/cluster that focus on feature-level evidence fusion.
In SuhKnowledge , Suh et al. developed a knowledge-based endocardial and epicardial boundary detection and segmentation system with cardiac MR image sequences. Pixel and location information were mapped into mass functions by Shafer’s model shafer1976mathematical . The two source mass functions were fused by using Dempster’s rule. Experiments were applied to cardiac short-axis images and obtained an excellent success rate ($>90\%$). However, this work only focused on cardiac boundary detection and did not discuss the details of the heart. In 1509843 , Vauclin et al. proposed a BFT-based lung and spinal canal segmentation model. The k-means clustering algorithm was first used to perform a pre-segmentation. Then a 3D filter exploits the results of the pre-segmentation to compute the MVs from spatial neighbors using Shafer’s model. Segmentation results showed the credal partition permits the reduction of the connection risks between the trachea and the lung when they are very close and between the left and right lungs at the anterior or posterior junctions.
In gerig2000exploring , Gerig et al. presented a method for automatic segmentation and characterization of object changes in time series of 3D MR images. A set of MVs was derived from time series according to brightness changes. BFOD transformation safranek1990evidence was used here to map the obtained features into mass functions. Then the set of evidence was combined by using Dempster’s rule. Experiments visually compared with results from alternative segmentation methods revealed an excellent sensitivity and specificity performance in the brain lesion region. The author also pointed out that better performance could be obtained with multimodal and multiple time-series evidence fusion. Simulation experimental results showed that about 80% of the implanted voxels could be detected for most generated lesions.
In vannoorenberghe2006belief , Vannoorenberghe et al. presented a BFT-based thoracic image segmentation method. First, a K-means algorithm performed coarse segmentation on the original CT images. Second, the EKNN rule denoeux1995k was applied by considering spatial information and calculating feature-level mass functions. The authors claimed that using the motioned segmentation scheme leads to a complementary approach combining region segmentation and edge detection. Experimental results showed promising results on 2D and 3D CT images for lung segmentation.
In derraz2013globally , Derraz et al. proposed an active contours (AC)-based chan2000active global segmentation method for vector-valued image that incorporated both probability and mass functions. All features issued from the vector-valued image were integrated with inside/outside descriptors to drive the segmentation results by maximizing the Maximum-Likelihood distance between foreground and background. Appriou’s second model appriou200501 was used to calculate the imprecision caused by low contrast and noise between inside and outside descriptors issued from the multiple channels. Then the fast algorithm based on Split Bregman goldstein2009split was used for final segmentation by forming a fast and accurate minimization algorithm for the Total Variation (TV) problem. Experiments were conducted on color biomedical images (eosinophil, lymphocyte, eosinophil, monocyte, and neutrophil cell mohamed2012enhanced ) and have around 6% improvements by using F-score on five image groups. In the same year, Derraz et al. proposed a new segmentation method derraz2013image based on Active Contours (AC) for the vector-valued image that incorporates Bhattacharyya distance michailovich2007image . The only difference is that the authors calculate the probability functions by Bhattacharyya distance instead of the Maximum-Likelihood distance in this paper. The performance of the proposed algorithm was demonstrated on the retinopathy dataset quellec2008optimal niemeijer2009retinopathy and with an increase of 3% in F-score compared with the best-performed methods.
In lian2017spatial , Lian et al. introduced a tumor delineation method in fluorodeoxyglucose positron emission tomography (FDG-PET) images by using spatial ECM lelandais2014fusion with adaptive distance metric. The authors proposed the adaptive distance metric to extract the most valuable features, and spatial ECM was used to calculate mass functions. Compared with ECM and spatial ECM, the proposed method showed a 14% and 10% increase in Dice score when evaluated on the FDG-PET images of non-small cell lung cancer (NSCLC) patients, showing good performance.
In huang2021evidential , Huang et al. proposed a 3D PET/CT lymphoma segmentation framework with BFT and deep learning. This is the first work that applied BFT with a deep neural network for medical image segmentation. In this paper, the PET and CT images were concatenated as a signal modal input method and transferred into UNet to get high-level semantic features. Then the ENN classifier was used to map high-level semantic features into mass functions by fusing the contribution of $K$ prototypes. Moreover, the segmentation uncertainty was considered in this paper with an uncertainty loss during training. The reported quantitative and qualitative results showed that the proposal outperforms the state-of-the-art methods. Based on the first work, Huang et al. verified the similarity of RBF and ENN in uncertainty quantification and merged them with the deep neural network (UNet) for lymphoma segmentation huang2022lymphoma . The segmentation performance confirmed that RBF is an alternative approach of ENN denoeux19d to act as an evidential classifier and showed that the proposal outperforms the baseline method UNet and the state-of-the-art both in accuracy and reliability.
Before 2020, the BFT-based medical image segmentation methods with single modal medical used low-level image features, e.g., grayscale and shape features, to generate mass functions, which limits the segmentation accuracy. Moreover, None of them discussed the reliability of the segmentation results. Huang et al. first merge BFT with a deep segmentation model (UNet) and learn the representation of mass functions with some learning algorithms huang2021evidential . Based on this, Huang et al further discuss the relationship between segmentation accuracy and reliability in huang2022lymphoma , which takes a new direction to study reliable medical image segmentation methods and bridge the gap between experimental results and clinical application.
5.1.2 Multimodal evidence fusion
Single-modality medical images often do not contain enough information to present the information about the disease and are often tainted with uncertainty. In addition to feature-level evidence fusion, the fusion of multimodal evidence is also important to achieve accurate and reliable medical image segmentation performance. Approaches for modality-level evidence fusion can be summarized into three main categories according to the way they calculate the evidence: probabilistic-based fusion, fuzzy set-based fusion, and BFT-based fusion. The development of convolution neural networks (CNNs) further contributes to the probabilistic-based fusion methods zhou2019review , which can be summarized into image-level fusion (e.g., data concatenation peiris2021volumetric ), feature-level fusion (e.g., attention mechanism concatenation zhou2020fusion zhou2022tri ), and decision-level fusion (e.g., model ensembles kamnitsas2017ensembles ). While none of those methods consider the conflict of source evidence, i.e., the modality-level uncertainty is not well studied, which limits the reliability and explainability of the performance.
This section focus on the BFT-based segmentation methods with modality-level evidence fusion. Figure 11 shows an example framework of multimodal evidence fusion with a single classifier. We separate the segmentation process into four steps: masses calculation, feature-level evidence fusion (alternative), modality-level evidence fusion, and decision-making. Compared with single-modal evidence fusion 5.2.1, multimodal evidence fusion focuses here not only on feature-level but also on modality-level evidence fusion (It should be noted that feature-level evidence fusion is not necessary in this case). multimodal evidence fusion is the most popular application for BFT in the medical image segmentation domain. Therefore we classify those methods according to their input modal for better analysis. Table 3 summarizes the segmentation methods with multimodal inputs and a single classifier/cluster with the main focus on modality-level evidence fusion.
Fusion of multimodal MR images
In bloch1996some , Bloch first proposed a BFT-based dual-echo MR pathological brains tissue segmentation model with uncertainty and imprecision quantification. The author assigned mass functions based on a reasoning approach that uses grey-level histograms provided by each image to choose focal elements. After defining mass functions, Dempster’s rule combined the mass from dual-echo MR images for each pixel. Based on the first work with BFT, in bloch2008defining , Bloch proposed to use fuzzy mathematical morphology bloch1995fuzzy , i.e., erosion and dilation, to generate mass functions by introducing imprecision in the probability functions and estimating compound hypotheses. Then Dempster’s rule is used to fuse mass functions from multimodal images. It should be noted that this paper is under the strong assumption that it is possible to represent imprecision by a fuzzy set, also called the structuring element. Application examples on dual-echo MR image fusion showed that the fuzzy mathematical morphology operations could represent either spatial domain imprecision or feature space imprecision (i.e., grey levels features). The visualized brain tissue segmentation results yield the robustness of the proposed method.
As we motioned in Section 4.2.1, Zhu et al. proposed modeling mass functions in BFT using FCM and spatial neighborhood information for image segmentation. The visualized segmentation results on MR brain images showed that the fusion-based segmented regions are relatively homogeneous, enabling accurate measurement of brain tissue volumes compared with single modal input MR image input.
In ghasemi2012brain , Ghasemi et al. presented a brain tissue segmentation approach based on FCM and BFT. The authors used the FCM to model two different input features: pixel intensity and spatial information, as membership values (MVs). Then for each pixel, the RMV transformation ghasemi2012brain was used to map MVs into mass functions. Last, the authors used Dempster’s rule to fuse intensity-based and spatial-based mass functions to get final segmentation results. Compared with FCM, the authors reported an increase in Dice and accuracy. As an extension of ghasemi2012brain , Ghasemi et al. proposed an unsupervised brain tumor segmentation method that modeled pixel intensity and spatial information into mass functions with RMV transformation and fused the two mass functions with Dempster’s rule in ghasemi2013novel .
In Rui2013Lesion , Wang et al. proposed a lesion segmentation method for infarction and cytotoxic brain edema. The authors used a method similar to Zhu’s model to define simple and double hypotheses. FCM bezdek1984fcm was used first to construct the mass functions of simple hypotheses $\{a\}$ and $\{b\}$. Then the mass function of double hypotheses was defined by
$m(\{a,b\})=\frac{1}{4}\times\frac{min(m(\{a\}),m(\{b\}))}{m(\{a\})+m(\{b\})}$. Then the authors used Dempster’s rule for modality-level evidence fusion. The results showed that infarction and cytotoxic brain edema around the infarction could be well segmented by final segmentation.
In makni2014introducing , Makni et al. introduced a multi-parametric MR image segmentation model by using spatial neighborhood in ECM for prostate zonal anatomy. The authors extended ECM with neighboring voxels information to map multi-parametric MR images into mass functions. Then the authors used prior knowledge related to defects in the acquisition system to reduce uncertainty and imprecision. Finally, the authors used Dempster’s rule to fuse the mass functions from the multi-parametric MR images. The method achieved good performance on prostate multi-parametric MR image segmentation.
In tavakoli2018brain , Tavakoli et al. proposed a segmentation method based on the evidential fusion of different modalities (T1, T2, and Proton density (PD)) for brain tissue. The authors used FCM to get MVs and used the RMV transformation ghasemi2012brain to transform the clustering MVs into mass functions. The authors first formed the belief structure for each modal image and used Dempster’s rule to fuse the three modalities’ mass functions of T1, T2, and PD. Compared with FCM, this proposal achieved a 5% improvement in the Dice score. Based on Tavakoli’s method tavakoli2018brain , Lima et al. proposed a modified brain tissue segmentation method in lima2019modified . The authors tested their method with four modality inputs: T1, T2, PD, and Flair. The reported results outperformed both the baseline method FCM and Tavakoli’s method with four modality evidence fusions.
In xiao2017vascular , Xiao et al. proposed an automatic vascular segmentation algorithm, which combines the grayscale and shape features of blood vessels and extracts 3D vascular structures from the head phase-contrast MR angiography dataset. First, grayscale and shape features are mapped into mass functions by using the GD-based method. Second, a new reconstructed vascular image was established according to the fusion of vascular grayscale and shape features by Dempster’s rule. Third, a segmentation ratio coefficient was proposed to control the segmentation result according to the noise distribution characteristic. Experiment results showed that vascular structures could be detected when both grayscale and shape features are robust. Compared with traditional grayscale feature-based or shape feature-based methods, the proposal showed better performance in segmentation accuracy with the decreased over-segmentation and under-segmentation ratios by fusing two sources of information.
Since Bloch’s early work fully demonstrated the advantages of BFT in modeling uncertain and imprecision, introducing partial or global ignorance, and fusing conflicting evidence in a multimodal MR images segmentation task, researchers in this domain have gone further to explore the advantages of BFT in multi MR image fusion. Among these research works, FCM is the most popular clustering algorithm to map input information into MVs. Ration MV transformation or Zhu’s model is usually used to generate mass functions from MVs. Apart from these two-step methods, the GD-based model can also be used to generate mass functions directly.
Fusion of RGB channels
In vannoorenberghe1999dempster , Vannoorenberghe et al. pointed out that taking R, G, and B channels as three independent information sources can be limited and weak for medical image tasks and proposed a color image segmentation method based on BFT. They calculated the degree of evidence by mapping R, G, and B channel’s intensity into mass functions using the Gaussian distribution information (similar to GD-based model chen2012manifold ) with an additional discounting operation). Then three pieces of evidence were fused with Dempster’s rule. The proposed segmentation method was applied to biomedical images to detect skin cancer (melanoma). Experiments showed a significant part of the lesion is correctly extracted from the safe skin. The segmentation performance is limited by feature representation, e.g., some regions correspond to pixels that cannot be classified as either the safe skin or the lesion because only the pixel-level feature is insufficient for hard-example segmentation.
In chaabane2009relevance Chaabane et al. proposed a color medical image segmentation method based on fusion operation. Compared to vannoorenberghe1999dempster , the authors first modeled the probability of each region belongs an image by a Gaussian distribution, then used Appriou’s second model to map probability into mass functions. Dempster’s rule then combines three color channel evidence. Compared with single-channel segmentation results, the fused results achieved a 10% increase in segmentation sensitivity.
Different from the methods vannoorenberghe1999dempster and chaabane2009relevance that decompose color images into R, G, B channels, Harrabi et al. harrabi2012color presented a color image segmentation method that represents the color image with 6 color spaces (RGB, HSV, YIQ, XYZ, LAB, LUV). The segmentation operation is based on multi-level thresholding and evidence fusion techniques. First, the authors identify the most significant peaks of the histogram by multi-level thresholding with the two-stage Otsu optimization approach. Second, the GD-based model was used to calculate the mass functions for each color space. Then the authors used Dempster’s rule to combine six sources of information. Compared with single color spaces, such as RGB and HSV, the fused result of six color spaces has a great increase in segmentation sensitivity, for example, an increase of 4% and 7% in RGB and HSV, respectively.
In trabelsi2015skin , Trabelsi et al. applied BFT in optical imaging with color to improve skin lesion segmentation performance. The authors decomposed color images into R, G, and B channels and applied the FCM method on each channel to get probability functions for pixel $x$ in each color space. In this paper, the author takes the probability functions as mass functions and calculates the orthogonal sum of the probability functions for the three-channel images. Even experiments showed about 10% improvements in segmentation accuracy compared with single-channel results, further study could be done on this work to generate uncertainty from probability functions to get achieve performance.
In general, the BFT-based RGB medical image segmentation approaches are used to generate mass functions by possibility distributions, e.g., Gaussian distribution and Possibility C-means distribution, and fuse them by Dempster’s rule. Though the authors claimed they could get better performance compared with single color input, the segmentation performance is limited by features because gray-scale and intensity features are not robust and efficient in representing image information. Further research could take both feature extraction and uncertainty quantification into consideration, e.g., a deep feature extraction model with an evidential classifier, to improve the performance.
Fusion of PET/CT
In lelandais2012segmentation , Lelandais et al. proposed a BFT-based multi-trace PET images segmentation method to segment biological target volumes. First, the authors used a modified FCM algorithm with the discounting algorithm to determine mass functions. The modification integrated a disjunctive combination of neighboring voxels inside the iterative process. Second, the operation of reduction of imperfect data was conducted by fusing neighboring voxels using Dempster’s rule. Based on this first work, Lelandais et al. proposed an ECM-based fusion model lelandais2014fusion for biological target volume segmentation with multi-tracer PET images. The segmentation method introduced in this paper is similar to the one introduced in makni2014introducing with a different application.
In derraz2015joint , Derraz et al. proposed a multimodal tumor segmentation framework for PET and CT inputs. Different from Lelandais et.al.’s work that uses ECM or optimized ECM to generate mass functions directly, the authors get mass functions in two steps. They first proposed a NonLocal Active Contours (NLAC) based variational segmentation framework to get probability results. Then, similar to the authors’ previous work derraz2013globally ; derraz2013image , they used Appriou’s second model appriou200501 to map MVs into mass functions. Then Dempster’s rule is used to fuse the mass functions from PET and CT modalities. The framework was evaluated on a lung tumor segmentation task. Compared with the state-of-the-art methods, this framework yielded the highest Dice score for tumor segmentation.
Based on Lelandais et al.’s work, Lian et al. proposed a tumor segmentation method lian2017accurate by using Spatial ECM lelandais2014fusion with Adaptive Distance Metric lian2017spatial in FDG-PET images with the guidance of CT images. Based on the first work, Lian et al. proposed a co-segmentation method of lung tumor segmentation lian2018joint ; lian2018unsupervised . They take PET and CT as independent inputs and use ECM to generate mass functions. At the same time, an adaptive distance metric is used to quantify clustering distortions and spatial relationships during the evidential clustering procedure. Then Dempster’s rule is used to fuse mass functions from PET and CT modalities. The quantitative and qualitative evaluation results showed superior performance compared with single modal segmentation results with an increase of 1% and 58% in PET and CT in Dice scores, respectively.
ECM is the most common BBA method to generate mass functions for PET/CT medical image segmentation approaches. Similar to the BFT-based RGB medical image segmentation methods, the segmentation performance here is limited by feature extraction methods. Further research could build on the study of deep feature representation and combine ECM with deep neural networks to learn mass feature representation. A good example of applying the evidential cluster in a neural network can be found in denoeux2021nn .
5.2 BFT-based medical image segmentation with several classifiers or clusterers
It is normal to see that two or more physicians cooperate together for disease diagnosis, which can minimize the impact of physicians’ misjudgments. Therefore, combining the results from multiple decision mechanisms as well as addressing the conflicts is critical to achieving a more reliable diagnosis. This section introduces the BFT-based medical image segmentation methods with several classifiers or clusterers. We follow the same way in Section 5.1 to separate them into a single modal and multimodal evidence fusion and introduce them in Sections 5.2.1 and 5.2.2, respectively.
5.2.1 Single modal evidence fusion
Compared with the methods presented in Section 5.2.1, the methods introduced in this section focus on feature-level (alternative) and classifier-level evidence fusion, which aims to minimize the impact of misjudgments caused by the model’s inner shortcomings. Figure 12 shows an example of a medical image segmentation framework with single modal input and several classifiers. We separate the segmentation process into four steps: masses calculation, feature-level evidence fusion (alternative), classifier-level evidence fusion, and decision-making. Table 4 summarizes the segmentation methods with single modal inputs and several single classifiers/clusterers with the main focus on classifier/clusterer level evidence fusion.
In capelle2002segmentation , Capelle et al. proposed a segmentation scheme for MR images based on BFT. The authors followed the same idea of the Evidential K-NN rule denoeux1995k to map image features into mass functions. Then, the authors applied the evidential fusion process to classify the voxels. Based on the first work, Capelle et al. proposed an evidential segmentation scheme of multimodal MR images for brain tumor detection in capelle2004evidential . This work focused on analyzing different evidential modeling techniques and on the influence of the introduction of spatial information to find the most effective brain tumor segmentation method. Three different BFT-based models: the distance-based BFT model (EKNN) denoeux2000neural , the likelihood function-based BFT method (Shafer’s model shafer1976mathematical , and Appriou’s first model (appriou1999multisensor ) were used to model information, and Dempster’s rule is used to fuse the three mass functions. This study concluded that neighborhood information increases the evidence of class membership for each voxel, thus making the final decision more reliable. Experimental results showed better segmentation performance compared with the state-of-the-art methods when the paper was published.
In taleb2002information , Taleb-Ahmed proposed a segmentation method of MR sequences of vertebrae in the form of images of their multiple slices with BFT. The authors used three different classifiers to calculate three kinds of mass functions. Firstly, the authors used grey-level intensity and standard deviation information to calculate two pixel-level mass functions with two fixed thresholds. Then the distance between two matching contours of consecutive slices ($P$ and $Q$) is used to calculate contour-level mass functions with the following definition:
$$\displaystyle m(\{S_{PQ}\})=1-e^{-\eta\left|d(P_{i},Q_{i})-\beta\right|},\quad d(P_{i},Q_{i})\in[\rho..\beta],$$
(35a)
$$\displaystyle m(\overline{\{S_{PQ}\}})=1-e^{-\eta\left|d(P_{i},Q_{i})-\beta\right|},\quad d(P_{i},Q_{i})\in[\beta..\propto],$$
(35b)
$$\displaystyle m(\Omega)=e^{-\eta\left|d(P_{i},Q_{i})-\beta\right|},\quad\forall d(P_{i},Q_{i}),$$
(35c)
where $P_{i}$ and $Q_{i}$ are two matching points of the slices $P$ and $Q$, $d(P_{i},Q_{i})$ is corresponding distance; $\Omega=\{S_{PQ},\overline{S_{PQ}}\}$, $S_{PQ}$ means two points $P_{i}$ and $Q_{i}$ are both belonging to cortical osseous. Parameter $\beta$ represents the tolerance which the expert associates with the value $d(P_{i},Q_{i})$, $\rho$ is the inter-slice distance and $\eta$ makes it possible to tolerate a greater inaccuracy in the geometrical resemblance of two consecutive contours. Then Dempster’s rule was used to combine the three mass functions for final segmentation.
In barhoumi2007collaborative , Barhoumi et al. introduced a new collaborative computer-aided diagnosis system for skin malignancy tracking. First, two diagnosis opinions were produced by perceptron neural network classification and content-based image retrieval (CBIR) schemes. Then Dempster’s rule was used to fuse the two opinions to achieve a final malignancy segmentation. Simulation results showed that the proposed BFT-based combination could generate accurate diagnosis results. The frame of discernment in this paper is composed of two singletons, and the mass functions in this paper are said to be Bayesian.
In guan2011study , Guan et al. proposed a human brain tissue segmentation method with BFT. The authors first used Markov random forest (MRF) li2009markov for spatial information segmentation and then used a Two-Dimensional Histogram (TDH) method of fuzzy clustering duan2008multi to get a vague segmentation. Then a redundancy image was generated, representing the difference between the MRF and TDH methods. Then Zhu’s model zhu2002automatic was used to calculate mass functions. Finally, Dempster’s rule fused the two segmentation results and the generated redundancy image to handle controversial pixels. The visual segmentation results showed that this method has higher segmentation accuracy compared with the state-of-the-art.
As discussed in Section 3.2, Dempster’s rule becomes numerically unstable when combining highly conflicting mass functions. In this condition, the fused results will be improper, and any tiny changes in mass functions can bring sharp changes to the fusion results. Researchers in the medical domain has also awarded this limitation. In ketout2012improved , Ketout et al. proposed a modified mass computation method to break this limitation and apply their proposal for endocardial contour detection. First, the output of each active contour model (ACM) kass1988snakes was regarded as mass functions. Second, a threshold was proposed to check if the evidence $m$ conflicts with others. If there was conflict, a modifying operation was used on the conflicting evidence $m$: $m=m+1/l$ with modified pace $1/l$. The parameter $l$ is related to the number of hypothesis $W$ in $\Omega$, generally, $l=2W$. Then the results of edge set-based segmentation li2005level and region set-based segmentation chan2001active were fused by using the improved BFT xin2005improved to get a more accurate contour of the left ventricle. Experimental results showed that the fused contour is closer to the ground truth than the contour from the edge or region. And false detection of the two contours was suppressed in the resulting image by rejecting the conflicting events by the fusion algorithm. Meanwhile, the proposed method could detect the edges of the endocardial borders even in low-contrast regions.
In wen2018improved , Wen et al. proposed an improved MR image segmentation method based on FCM and BFT. First, the authors did a image-level fusion for two modality images $A$ and $B$ with the function $F(x,y)=w_{1}\times g_{A}(x,y)+w_{2}\times g_{B}(x,y)$, where $x$ and $y$ are image pixels and $w_{1}$ and $w_{2}$ are the weighted coefficients which are used to adjust the importance of different images for the final fusion result with $w_{1}+w_{2}=1$. Second, the authors calculated the MV by FCM and calculated the mass functions of simple and double hypotheses by Zhu’s model zhu2002automatic without the neighboring pixel information. Third, the authors generated another mass functions by weighting those of its neighboring pixels with the GD-based model and used Zhu’s model zhu2002automatic again to construct simple and double hypotheses mass functions. Finally, the authors used Dempster’s rule to complete the fusion of the two mass functions to achieve the final segmentation. Compared with the FCM-based method, the proposed method can better decrease the conflict in multiple sources to achieve easy convergence and significant improvement by using Dempster’s rule for classifier-level evidence fusion.
Besides, with the development of CNNs, the research community used Dempster’s rule for the fusion of multiple CNN classifiers. In george2020breast , George et al. proposed a breast cancer detection system using transfer learning and BFT. This first work first applied BFT in multiple evidence fusion with deep learning. Image features were represented in high-level features using a convolutional neural network for feature extraction, such as ResNet-18, ResNet-50, ResNet-101, GoogleNet, and AlexNet. A patch-based feature extraction strategy was used to avoid wrong segmentation of the boundaries and provide features with good discriminative power for classification. The extracted features were classified into benign and malignant classes using a support vector machine (SVM). A discounting operation was applied to transfer probability-based segmentation maps into mass functions. The discounted outputs from the different CNN-SVM frameworks are now combined using Dempster’s rule. This work takes advantage of deep learning and BFT and has achieved good performance. Compared with majority voting-based fusion methods, BFT-based fusion showed superior segmentation accuracy. Compared with a single classifier, such as ResNet-101, the fused framework achieves the accuracy of sensitivity, specificity, and AUC with an increase of 1%, 0.5%, 3%, and 2%, respectively. Also, the authors compared their results with the state-of-the-art method and achieved comparable segmentation accuracy.
Apart from using BFT to combine the discounted probabilities from the CNN classifiers george2020breast , another solution is to construct a deep evidential segmentation framework directly. In huang2021belief , Huang et al. proposed a BFT-based semi-supervised learning framework (EUNet) for brain tumor segmentation. This work applied BFT in a deep neural network to quantify segmentation uncertainty directly. During training, two kinds of evidence were obtained: the segmentation probability functions and mass functions generated by UNet and EUNet, respectively. Dempster’s rule was then used to fuse the two pieces of evidence. Experimental results showed that the proposal has better performance than state-of-the-art methods. It achieved around 10%, 4%, and 2% increase in Dice score, Positive predictive value, and sensitivity, respectively, compared with the baseline (UNet). Moreover, the authors showed how BFT exceeds probability-based methods in addressing uncertainty boundary problems. This is the first work embeds BFT into CNN and achieves an end-to-end deep evidential segmentation model.
The approaches introduced in this section use several classifiers/clusterers to generate different mass functions and fuse them by Dempster’s rule. Among those approaches, George et al. george2020breast first applied Dempster’s rule to combine the discounted probabilities from different deep segmentation models. Huang et al. huang2021belief first merged ENN with UNet to construct an end-to-end segmentation model and fuse two kinds of evidence by Dempster’s rule. Compared to George et al.’s approach george2020breast , Huang et al.’s approach huang2021belief can generate mass functions directly from a deep segmentation model, which is more promising.
5.2.2 Multimodal evidence fusion
Figure 13 shows an example of a medical image segmentation framework with several classifiers and multimodal inputs, which is more complex than the frameworks we introduced in Sections 5.2.2, 5.2.2, and 5.1.1. The segmentation process comprises five steps: masses calculation, feature-level evidence fusion (alternative), classifier-level evidence fusion, modality-level evidence fusion, and decision-making. Pixels at the same position from different modalities are transferred into different classifiers and different BBA methods to get pixel-level mass functions. Dempster’s rule is used first to fuse feature-level evidence fusion, then to fuse classifier-level evidence, and last to fuse modality-level evidence. Table 5 summarizes the segmentation methods with multimodal inputs and several classifiers/clusterers focusing on classifier/clusterer fusion and modality-level evidence fusion
In gautier2000belief , Gautier et al. proposed a method for helping physicians monitor the spinal column’s diseases with multimodal MR images. At first, an initial segmentation was applied with active contour lai1994deformable . Then several mass functions were obtained from experts’ opinions on constructing the frame of discernment. Thus, the mass functions here are human-defined. Finally, Dempster’s rule was then used to fuse the mass functions from different experts. The proposal got the most reliable segmentation maps with the experimental illustration results when the paper was published.
Based on the first work of multimodal medical image fusion with a single cluster chaabane2009relevance , Chaabane et al. presented another BFT-based segmentation method with several clustererschaabane2011new . First, possibilistic C-means clustering bezdek2013pattern was used on R, G, and B channels to get three MVs. Then the MVs were mapped into the mass functions of simple and double hypotheses using Zhu’s model zhu2002automatic . Dempster’s rule is used first to fuse three mass functions from three corresponding color spaces. Based on the initial segmentation results, another mass function was calculated associated with each pixel and its neighboring pixels for each color space. Finally, the authors used Dempster’s rule again to fuse the two mass functions from two corresponding clusterers. Experimental segmentation performance in cell images highlights the effectiveness of the proposed method. Compared with the FCM-based segmentation method, the proposal increased by 15% in sensitivity segmentation.
In lajili2018two , Lajili et al. proposed a two-step evidential fusion approach for breast region segmentation. The first evidential segmentation results were obtained by a grey-scale-based K-means clustering method. A sigmoid function was used to define the mass functions for each class $C_{x}(x=1...,k)$:
$$\displaystyle m_{z\in C_{x}}(\{Br\})=\frac{1}{1+\exp^{2-x}},$$
(36a)
$$\displaystyle m_{z\in C_{x}}(\{Ba\})=\frac{1}{1+\exp^{-2-x}},$$
(36b)
$$\displaystyle m_{z\in C_{x}}(\Omega)=1-m(\{Br\})-m(\{Ba\}),$$
(36c)
where $x$ denotes the index of the class $C_{x}$ for pixel $z$, $Br$ represents breast, and $Ba$ represents background. For local-homogeneity-based segmentation, the authors modeled the uncertainty with a threshold value $\alpha$, by defining $m(\{Br\})=1-\alpha$, $m(\{Ba\})=0$, $m(\{Br,Ba\})=\alpha$, where $\alpha$ represents the belief mass affected by the uncertainty on the membership of the pixel $z$. A final fusion strategy with Dempster’s rule was applied to combine evidence from the two mass functions. Experiments were conducted on two breast datasets suckling1994mammographic bowyer1996digital . The proposed segmentation approach yielded more accurate results than the best-performed method. It extracted the breast region with correctness equal to 0.97, which is 9% higher than the best-performed method.
In huang2021deep , Huang et al. proposed to take PET and CT as two modal inputs and use to UNet model to segment lymphoma separately. Then the two segmentation masks are fused by Dempster’s rule. Though this is the first work that applied BFT in multimodal evidence fusion with several deep segmentation models, the shortcoming of this work is that only probability functions other than mass functions are used for evidence fusion. To improve this first work, Huang et al. proposed a multimodal evidence fusion framework with contextual discounting for brain tumor segmentation huang2022contextual . In this work. With four modules of feature extraction and one module of evidential segmentation, the method assigns to each voxel a mass function. In addition, a contextual discounting layer is designed to take into account the reliability of the method when classifying different classes. Finally, the Dempster rule is used to combine the discounted evidence to obtain a final segmentation. The authors also claimed that this method could be used together with any state-of-the-art segmentation module to improve the final performance.
There are only a few studies that take multimodal medical images as independent inputs and use independent classifiers to generate mass functions. The performance of this kind of approach is limited by the representation of image features and the ability to quantify the model’s uncertainty. Huang et al. huang2022contextual first merge ENN with UNet for the fusion of multimodal MR images with contextual discounting. It enables the model to generate a learned reliability metric from input modalities during different segmentation tasks, which is good potential to make the results more explainable.
5.3 Discussion
The choice of using single modal or multimodal depends on the dataset. Generally, the more source data we have, the more reliable segmentation results we will get. The choice of a single or several classifiers/clusterers depends on the limitation of the computation source and the requirement of computation efficiency.
Considering the main advantages of using multiple classifiers for multimodal medical images, as shown in the work of huang2022contextual , a potential research direction based on this approach could be promising.
Prior to 2020, BFT-based medical image segmentation methods limited segmentation accuracy due to the use of low-level image features, such as grayscale and shape features, to generate mass functions. Moreover, none of them considered the segmentation reliability. Since the application of deep learning in medical image segmentation has been very successful, the use of BFT in deep neural networks should be a promising research direction, in particular, to quantify the uncertainty and reliability of the segmentation results.
6 Conclusion and future work
6.1 Conclusion
The main challenges we have for medical image segmentation tasks are: 1) imperfect medical image resolution, 2) noisy ground truth, and 3) unreliable segmentation models due to imperfect data and ground truths. This leads to epistemic segmentation uncertainty and has been the gap between experimental performance and clinical application for a while. For example, traditional medical image segmentation methods are limited by the use of low-level image features for decision-making kimmel2003fast ; salvador2004determining ; onoma2014segmentation , leading to poor segmentation accuracy. The success of machine learning, especially deep learning, has made promising contributions to segmentation accuracy thanks to its powerful ability in high-level semantic feature representation, such as UNet ronnebergerconvolutional and its variants trebing2021smaat ; cao2021swin ; hatamizadeh2022unetr . In contrast, a clinically applicable deep segmentation model depends not on the segmentation accuracy but also the reliability, which is critical for the high-risk task.
The systematic and comprehensive discussion of uncertainty and reliability of machine learning methods started in 2020 hullermeier2021aleatoric , which also attracted researchers’ attention in the medical image segmentation domain to study both accurate and reliable segmentation methods kwon2020uncertainty ; mehrtash2020confidence ; ghoshal2021estimating ; zhou2022trustworthy . This review provides a comprehensive overview of the BFT-based medical image segmentation methods. It first shows how imperfect medical image could be modeled by the degree of belief and ignorance(uncertainty) by mass functions; and then shows how unreliable or conflicting source information can be modeled to reach reliable fusion results. The application of segmentation methods covers a range of segmenting different human tissues (i.e., heart, lung) and tumors (i.e., brain, breast, and skin). In particular, we first briefly introduce medical image segmentation methods in Section 2 by introducing the traditional and deep learning medical segmentation methods and by discussing their shortcomings and limitations. Then we presented the history and the fundamental knowledge of BFT in Section 3. Section 4 comprehensively introduced the BBA methods used to represent uncertainty (generate mass functions) in medical image segmentation tasks. Sections 5 introduced the BFT-based medical image segmentation methods in detail by grouping them according to the number of input modalities and classifiers used to generate mass functions.
In certain cases, the BFT-based methods could surpass the probability-based methods by modeling information more effectively (with uncertainty or ignorance quantification in focal sets) and fusing multiple evidence in different stages, such ad feature-level, modality-level, and classifier-level evidence fusion. We hope this review could provide an intuitive understanding of BFT-based techniques that significantly contribute to medical image segmentation, especially for uncertain/imprecise multimodal medical image segmentation.
6.2 Future work
Although medical image segmentation with belief function theory has advantages in expressing segmentation uncertainty and improving the reliability of the performance, limitations of existing methods remain and can be further investigated.
First, most of the existing BFT-based medical image segmentation methods still stay at the stage of using low-level features and do not fully use the advantages of deep learning. Applying BFT in deep segmentation networks should be a potential research direction for getting both accurate and reliable segmentation models, particularly in medical image segmentation because of the availability of medical knowledge a priori easily modeled by mass functions. Ref george2020breast is the first successful application of Dempster’s rule for the fusion of different CNNs. Ref huang2022lymphoma confirmed the similarity of ENN and RBF when acting as an evidential classifier and integrated both the two classifiers within a deep segmentation network to improve the segmentation accuracy and reliability. We believe that more promising results will be obtained by cooperating BFT with the existing powerful deep medical image segmentation models trebing2021smaat ; cao2021swin ; hatamizadeh2022unetr .
Second, even though the BFT-based fusion methods have considered the conflict of multiple sources of evidence, there is a problem in that the multimodal medical images may have different reliability when segmenting different regions. For example, for the BRATS brain tumor dataset 111http://braintumorsegmentation.org/, the domain knowledge from the physicians points that areas describe the enhancing tumor with visually faint and enhancement on T1Gd MR images, and edema is defined by the abnormal hyperintense signal envelope on the FLAIR modality baid2021rsna . The multimodal medical image fusion operation may fail if some source information is unreliable or partially reliable when doing different segmentation tasks. In huang2022contextual , Huang et al. proposed a new method to learn reliability coefficients conditionally on different segmentation tasks in a deep neural network. Another interesting future work, therefore, is to obtain task-specific reliability, as well as to explain the results with deep evidential neural networks using contextual discounting for multimodal or cross-modal medical image segmentation tasks.
Third, acquiring big labeled training data is particularly challenging for medical image segmentation tasks, which has become the bottleneck of learning-based segmentation performance. The successful application of using unsupervised BBA methods for medical image segmentation points out another new direction to address the short annotation problem. As far as we know, there is no research work on the study of unsupervised BBA methods in a deep learning network. Evidential Clustering with neural network (NN-EVCLUS) denoeux21b and EKNN with partially supervised learning denoeux2019new are two successful applications of generating mass functions with unsupervised and semi-supervised learning. These works provide insights into how to use unsupervised/semi-supervised learning to quantify segmentation uncertainty with unannotated or partially annotated data sets.
We hope this review could increase awareness of the challenges of existing BFT-based medical image segmentation methods and call for future contributions to bridge the gap between experimental performance and clinical application, as well as achieve accurate, reliable, and explainable deep segmentation models.
Acknowledgements
The authors like to thank Thierry Denoeux for helpful remarks and comments on the content of this paper. We are also grateful for the insight suggestions made by the reviewers. The first author received funding from the China Scholarship Council (No.201808331005).
This work was carried out in the framework of the Labex MS2T, which was funded by the French Government through the program “Investments for the future” managed by the National Agency for Research (Reference ANR-11-IDEX-0004-02)
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Braided diagram groups and local similarity groups
Daniel S. Farley
Department of Mathematics and Statistics
Miami University
Oxford, OH 45056 U.S.A.
[email protected]
and
Bruce Hughes
Department of Mathematics
Vanderbilt University
Nashville, TN 37240 U.S.A.
[email protected]
(Date:: November 22, 2020)
Abstract.
Hughes defined a class of groups that act as local similarities on compact ultrametric spaces. Guba and Sapir had previously defined braided diagram groups over semigroup presentations. The two classes of groups share some common characteristics: both act properly by isometries on CAT(0) cubical complexes, and certain
groups in both classes have type $F_{\infty}$, for instance.
Here we clarify the relationship between these families of groups: the braided diagram groups over tree-like semigroup presentations are precisely
the groups that act on compact ultrametric spaces via small similarity structures. The proof can be considered a generalization of the proof that Thompson’s group $V$
is a braided diagram group over a tree-like semigroup presentation.
We also prove that certain additional groups, such as the Houghton groups
$H_{n}$, and $QAut(T_{2,c})$, lie in both classes.
The second-named author was supported in part by NSF Grant DMS–0504176.
Contents
1 Introduction
2 Braided diagram groups
3 Groups defined by finite similarity structures
4 Braided diagram groups and groups determined by finite similarity structures
2000 Math. Subject Class. Primary 20F65, 54E45
Contents
1 Introduction
2 Braided diagram groups
3 Groups defined by finite similarity structures
4 Braided diagram groups and groups determined by finite similarity structures
1. Introduction
In [7], Hughes described a class of groups that act as homeomorphisms on compact ultrametric spaces. Fix a compact ultrametric space $X$.
The essence of the idea
was to associate to $X$ a finite similarity structure, which is a function that associates to each ordered pair of balls $B_{1},B_{2}\subseteq X$ a finite set ${\rm Sim}_{X}(B_{1},B_{2})$ of surjective similarities from $B_{1}$ to $B_{2}$. (A similarity is a map that stretches or contracts distances by a fixed constant.) The finite
sets ${\rm Sim}_{X}(B_{1},B_{2})$ are assumed to have certain desirable closure properties (such as closure under composition). A homeomorphism $h:X\rightarrow X$ is said to be locally determined by ${\rm Sim}_{X}$ if each $x\in X$ has a ball neighborhood $B$ with the property that $h(B)$
is a ball and
the restriction of $h$ to $B$ agrees with one of the local similarities $\sigma\in{\rm Sim}_{X}(B,h(B))$. The collection of all homeomorphisms that are locally determined by
${\rm Sim}_{X}$ forms a group under composition. We will call such a group an FSS group (finite similarity structure group) for short. Hughes [7] proved
that each FSS group has the Haagerup property, and even acts properly on a CAT(0) cubical complex. In [5], the authors described a class of FSS groups that
have type $F_{\infty}$. That class includes Thompson’s group $V$, and the main theorem of [5] is best understood as a generalization of [2],
where Brown originally showed that $V$ has type $F_{\infty}$.
In earlier work, Guba and Sapir [6] had sketched a theory of braided diagram groups over semigroup presentations, and proved that Thompson’s group $V$
is a braided diagram group over the semigroup presentation $\langle x\mid x=x^{2}\rangle$. Farley [4] showed that braided diagram groups over semigroup presentations
act properly on CAT(0) cubical complexes.
The class $\mathcal{F}$ of FSS groups and the class $\mathcal{B}$ of braided diagram groups therefore have a common origin, as generalizations of Thompson’s group $V$. Both classes also share other features in common (as noted above). It is therefore natural to wonder to what extent the two classes are the same. The main goal of this note is to prove Theorem 4.12, which says that the FSS groups determined by small similarity structures (Definition 4.6)
are precisely the same as the braided diagram groups determined by tree-like semigroup presentations (Definition 4.1). It is even possible that Theorem 4.12 describes the precise
extent of the overlap between $\mathcal{F}$ and $\mathcal{B}$, but we do not know how to prove this.
We include all relevant definitions, and our treatment is fairly self-contained as a result. A precise definition of braided diagram groups is given in
Section 2, the precise definition of FSS groups appears in Section 3, and the main theorem is proved in Section
4. Along the way, we give additional examples in the class $\mathcal{F}\cap\mathcal{B}$, including the Houghton groups $H_{n}$ and
a certain group $QAut(T_{2,c})$ of quasi-automorphisms of the infinite binary tree. (These are Examples 4.3 and 4.4, respectively.)
This note has been adapted from the longer preprint [5]. The first part of the latter preprint (including roughly the first six sections) will be published elsewhere. The first author would like to thank the organizers of the Durham Symposium (August 2013) for the opportunity to speak. Example 4.3 first appeared as part of the
first author’s lecture. The idea of Example 4.4 occurred to the first author after listening to Collin Bleak’s lecture at the Symposium.
2. Braided diagram groups
In this section, we will recall the definition of braided diagram groups over semigroup presentations.
Note that the theory of braided diagram groups was first sketched by Guba and Sapir [6]. A more extended introduction to braided diagram groups
appears in [4].
Definition 2.1.
Let $\Sigma$ be a set, called an alphabet. The free semigroup on $\Sigma$, denoted $\Sigma^{+}$, is
the collection of all positive non-empty strings formed from $\Sigma$, i.e.,
$$\Sigma^{+}=\{u_{1}u_{2}\ldots u_{n}\mid n\in\mathbb{N},u_{i}\in\Sigma\text{ %
for }i\in\{1,\ldots,n\}\}.$$
The free monoid on $\Sigma$, denoted $\Sigma^{\ast}$, is the union $\Sigma^{+}\cup\{1\}$, where $1$ denotes the
empty string. (Here we assume that $1\not\in\Sigma$ to avoid ambiguity.) The operations in $\Sigma^{+}$ and $\Sigma^{\ast}$ are concatenation.
We write $w_{1}\equiv w_{2}$ if $w_{1}$ and $w_{2}$ are equal as words in $\Sigma^{\ast}$.
Definition 2.2.
A semigroup presentation $\mathcal{P}=\langle\Sigma\mid\mathcal{R}\rangle$ consists of an alphabet $\Sigma$ and
a set $\mathcal{R}\subseteq\Sigma^{+}\times\Sigma^{+}$. The elements of $\mathcal{R}$ are called relations.
Remark 2.3.
A relation $(w_{1},w_{2})\in\mathcal{R}$ can be viewed as an equality between the words $w_{1}$ and $w_{2}$.
We use ordered pairs to describe these equalities because we will occasionally want to make a distinction between the
left and right sides of a relation.
A semigroup presentation $\mathcal{P}$ determines a semigroup $S_{\mathcal{P}}$, just as a group presentation determines a group. We will, however,
make essentially no use of this semigroup $S_{\mathcal{P}}$. Our interest is in braided diagrams over $\mathcal{P}$ (see below).
Definition 2.4.
(Braided Semigroup Diagrams)
A frame is a homeomorphic copy of $\partial([0,1]^{2})=(\{0,1\}\times[0,1])\cup([0,1]\times\{0,1\})$. A frame
has a top side, $(0,1)\times\{1\}$, a bottom side, $(0,1)\times\{0\}$, and left and right sides,
$\{0\}\times[0,1]$ and $\{1\}\times[0,1]$, respectively. The top and bottom of a frame have obvious left to right orderings.
A transistor is a homeomorphic copy of $[0,1]^{2}$. A transistor has top, bottom, left, and right sides, just as a frame does. The
top and bottom of a transistor also have obvious left to right orderings.
A wire is a homeomorphic copy of $[0,1]$. Each wire has a bottom $0$ and a top $1$.
Let $\mathcal{P}=\langle\Sigma\mid\mathcal{R}\rangle$ be a semigroup presentation. Let $\mathcal{T}(\Delta)$ be a finite
(possibly empty) set of transistors. Let $\mathcal{W}(\Delta)$ be a finite, nonempty set of wires. We let $F(\Delta)=\partial([0,1]^{2})$ be
a frame. We let $\ell_{\Delta}:\mathcal{W}(\Delta)\rightarrow\Sigma$ be an arbitrary function, called the labelling function.
For each wire $W\in\mathcal{W}(\Delta)$, we choose a point $t(W)$ on the bottom of a transistor, or on the top of the frame, and a point $b(W)$
on the top of a transistor, or on the bottom of the frame. The points $t(W)$ and $b(W)$ are called the top and bottom contacts of $W$, respectively.
We attach the top of each wire $W$ to $t(W)$ and the bottom of $W$ to $b(W)$. The resulting topological space $\Delta$ is called a braided diagram over
$\mathcal{P}$ if the following additional conditions are satisfied:
(1)
If $W_{i}$, $W_{j}\in\mathcal{W}(\Delta)$, $t(W_{i})=t(W_{j})$ only if $W_{i}=W_{j}$, and $b(W_{i})=b(W_{j})$ only if $W_{i}=W_{j}$.
In other words, the disjoint union of all of the wires maps injectively into the quotient.
(2)
We consider the top of some transistor $T\in\mathcal{T}(\Delta)$. Reading from left to right, we find contacts
$$b(W_{i_{1}}),b(W_{i_{2}}),\ldots,b(W_{i_{n}}),$$
where $n\geq 0$. The word $\ell_{t}(T)=\ell(W_{i_{1}})\ell(W_{i_{2}})\ldots\ell(W_{i_{n}})$ is called the top label of $T$.
Similarly, reading from left to right along the bottom of $T$, we find contacts
$$t(W_{j_{1}}),t(W_{j_{2}}),\ldots,t(W_{j_{m}}),$$
where $m\geq 0$. The word $\ell_{b}(T)=\ell(W_{j_{1}})\ell(W_{j_{2}})\ldots\ell(W_{j_{m}})$ is called the bottom label of $T$.
We require that, for any $T\in\mathcal{T}(\Delta)$, either $(\ell_{t}(T),\ell_{b}(T))\in\mathcal{R}$ or $(\ell_{b}(T),\ell_{t}(T))\in\mathcal{R}$.
(We emphasize that it is not sufficient for $\ell_{t}(T)$ to be equivalent to $\ell_{b}(T)$ modulo the relation $\sim$ determined by $\mathcal{R}$. Note also that this condition implies that
$\ell_{b}(T)$ and $\ell_{t}(T)$ are both non-empty, since $\mathcal{P}$ is a semigroup presentation. In particular, each transistor has wires attached to its top and bottom faces.)
(3)
We define a relation $\preceq$ on $\mathcal{T}(\Delta)$ as follows. Write $T_{1}\preceq T_{2}$ if there is some wire $W$ such that
$t(W)\in T_{2}$ and $b(W)\in T_{1}$. We require that the transitive closure $\dot{\preceq}$ of $\preceq$ be a strict partial order on $\mathcal{T}(\Delta)$.
Definition 2.5.
Let $\Delta$ be a braided diagram over $\mathcal{P}$. Reading from left to right across the top of the frame $F(\Delta)$,
we find contacts
$$t(W_{i_{1}}),t(W_{i_{2}}),\ldots,t(W_{i_{n}}),$$
for some $n\geq 1$. The word $\ell(W_{i_{1}})\ell(W_{i_{2}})\ldots\ell(W_{i_{n}})=\ell_{t}(\Delta)$
is called the top label of $\Delta$. We can similarly define the bottom label of $\Delta$, $\ell_{b}(\Delta)$.
We say that $\Delta$ is a braided $(\ell_{t}(\Delta),\ell_{b}(\Delta))$-diagram over $\mathcal{P}$.
Remark 2.6.
One should note that braided diagrams, despite the name, are not truly braided. In fact, two braided diagrams are equivalent
(see Definition 2.10) if there is a certain type of marked homeomorphism between them. Equivalence therefore does not depend
on any embedding into a larger space. Braided diagram groups (as defined in Theorem 2.13) also seem to have little in common
with Artin’s braid groups.
Example 2.7.
Let $\mathcal{P}=\langle a,b,c\mid ab=ba,ac=ca,bc=cb\rangle$. Figure 1 shows an example of a braided $(aabc,acba)$-diagram
over the semigroup presentation $\mathcal{P}$. The frame is the box formed by the dashed line. The wires that appear to cross in the figure do not
really touch, and it is unnecessary to specify which wire passes over the other one. See Remark 2.6.
Definition 2.8.
(Concatenation of braided diagrams)
Let $\Delta_{1}$ and $\Delta_{2}$ be braided diagrams over $\mathcal{P}$. We suppose that $\Delta_{1}$
is a $(w_{1},w_{2})$-diagram and $\Delta_{2}$ is a $(w_{2},w_{3})$-diagram. We can multiply $\Delta_{1}$ and $\Delta_{2}$ by stacking them. More explicitly,
we remove the bottom of the frame of $\Delta_{1}$ and the top of the frame of $\Delta_{2}$, and then glue together the wires in order from left to right. This gluing is compatible with the labeling of the wires, since the bottom label of $\Delta_{1}$ is the same as the top label of $\Delta_{2}$. The result
is a braided diagram $\Delta_{1}\circ\Delta_{2}$, called the concatenation of $\Delta_{1}$ and $\Delta_{2}$.
Definition 2.9.
(Dipoles)
Let $\Delta$ be a braided semigroup diagram over $\mathcal{P}$. We say that the transistors $T_{1},T_{2}\in\mathcal{T}(\Delta)$, $T_{1}\,\preceq\,T_{2}$,
form a dipole if:
(1)
the bottom label of $T_{1}$ is the same as the top label of $T_{2}$, and
(2)
there are wires $W_{i_{1}},W_{i_{2}},\ldots,W_{i_{n}}(n\geq 1)$ such that the bottom contacts $T_{2}$, read from left to right, are
precisely
$$t(W_{i_{1}}),t(W_{i_{2}}),\ldots,t(W_{i_{n}})$$
and the top contacts of $T_{1}$, read from left to right, are precisely
$$b(W_{i_{1}}),b(W_{i_{2}}),\ldots,b(W_{i_{n}}).$$
Define a new braided diagram as follows. Remove the transistors $T_{1}$ and $T_{2}$ and all of the wires $W_{i_{1}},\ldots,W_{i_{n}}$ connecting the top
of $T_{1}$ to the bottom of $T_{2}$. Let $W_{j_{1}},\ldots,W_{j_{m}}$ be the wires attached (in that order)
to the top of $T_{2}$, and let $W_{k_{1}},\ldots,W_{k_{m}}$ be the wires attached to the bottom of $T_{1}$. We glue the bottom of
$W_{j_{\ell}}$ to the top of $W_{k_{\ell}}$. There is a natural well-defined labelling function on the resulting wires, since
$\ell(W_{j_{\ell}})=\ell(W_{k_{\ell}})$ by our assumptions. We say that the new diagram $\Delta^{\prime}$ is obtained from $\Delta$ by reducing the dipole
$(T_{1},T_{2})$. The inverse operation is called inserting a dipole.
Definition 2.10.
(Equivalent Diagrams)
We say that two diagrams $\Delta_{1}$, $\Delta_{2}$ are equivalent if there is a homeomorphism
$\phi:\Delta_{1}\rightarrow\Delta_{2}$ that preserves the labels on the wires, restricts to a homeomorphism
$\phi_{|}:F(\Delta_{1})\rightarrow F(\Delta_{2})$, preserves the tops and bottoms of the transistors and frame, and preserves
the left to right orientations on the transistors and the frame. We write $\Delta_{1}\equiv\Delta_{2}$.
Definition 2.11.
(Equivalent Modulo Dipoles; Reduced Diagram)
We say that $\Delta$ and $\Delta^{\prime}$ are equivalent modulo dipoles
if there is a sequence $\Delta\equiv\Delta_{1}\equiv\Delta_{2}\equiv\ldots\equiv\Delta_{n}\equiv\Delta%
^{\prime}$,
where $\Delta_{i+1}$ is obtained from $\Delta_{i}$ by either inserting or removing a dipole, for $i\in\{1,\ldots,n-1\}$. We write $\Delta=\Delta^{\prime}$.
(The relation of equivalence modulo dipoles is indeed an equivalence relation – see [4].)
A braided diagram $\Delta$ over a semigroup presentation is called reduced if it contains no dipoles. Each equivalence class modulo dipoles
contains a unique reduced diagram [4].
Example 2.12.
In Figure 2, we have two braided diagrams over the semigroup presentation $\mathcal{P}=\langle a,b,c\mid ab=ba,ac=ca,bc=cb\rangle$.
The two rightmost transistors in the diagram on the left form a dipole, and the diagram on the right is the result of reducing that dipole.
Theorem 2.13.
[4]
Let $\mathcal{P}=\langle\Sigma\mid\mathcal{R}\rangle$ be a semigroup presentation, and let $w\in\Sigma^{+}$. We let $D_{b}(\mathcal{P},w)$ denote the set of equivalence classes of braided $(w,w)$-diagrams modulo dipoles. The operation of concatenation induces a well-defined group operation on $D_{b}(\mathcal{P},w)$.
This group $D_{b}(\mathcal{P},w)$ is called the braided diagram group over $\mathcal{P}$ based at $w$.
3. Groups defined by finite similarity structures
3.1. Review of ultrametric spaces and finite similarity structures
We now give a quick review of finite similarity structures on compact ultrametric spaces, as defined in Hughes [7]. Most of this subsection is taken directly from
[5].
Definition 3.1.
An ultrametric space is a metric space $(X,d)$ such that $d(x,y)\leq\max\{d(x,z),d(z,y)\}$ for all
$x,y,z\in X$.
Lemma 3.2.
Let $X$ be an ultrametric space.
(1)
Let $B_{r}(x)$ be an open metric ball in $X$. If $y\in B_{r}(x)$, then $B_{r}(x)=B_{r}(y)$.
(2)
If $B_{1}$ and $B_{2}$ are open metric balls in $X$, then either the balls are disjoint, or one is contained in the other.
(3)
Every open ball in $X$ is a closed set, and every closed ball in $X$ is an open set.
(4)
If $X$ is compact, then each open ball $B$ is contained in at most finitely many distinct open balls of $X$.
(5)
If $X$ is compact, then each open ball in $X$ is a closed ball (possibly of a different radius), and each closed ball is an open ball.
(6)
If $X$ is compact and $x$ is not an isolated point, then each open ball $B_{r}(x)$ is partitioned by its maximal proper open subballs, which are finite in number.
∎
Convention 3.3.
We assume for the rest of the section that $X$ is a compact ultrametric space. By Lemma 3.2(5), open balls are closed balls, and closed balls are open balls,
so we can refer to both simply as balls, and we will follow this practice from now on.
Definition 3.4.
Let $f:X\rightarrow Y$ be a function between metric spaces. We say that $f$ is a similarity if there is a constant $C>0$ such that
$d_{Y}(f(x_{1}),f(x_{2}))=Cd_{X}(x_{1},x_{2})$, for all $x_{1}$ and $x_{2}$ in $X$.
Definition 3.5.
A
finite similarity structure for $X$ is a function
${\rm Sim}_{X}$ that assigns to
each ordered pair $B_{1},B_{2}$ of balls in $X$
a (possibly empty)
set ${\rm Sim}_{X}(B_{1},B_{2})$ of surjective similarities
$B_{1}\to B_{2}$ such that whenever
$B_{1},B_{2},B_{3}$ are balls in $X$, the following properties
hold:
(1)
(Finiteness) ${\rm Sim}_{X}(B_{1},B_{2})$ is a finite set.
(2)
(Identities) $\mathrm{id}_{B_{1}}\in{\rm Sim}_{X}(B_{1},B_{1})$.
(3)
(Inverses) If $h\in{\rm Sim}_{X}(B_{1},B_{2})$, then $h^{-1}\in{\rm Sim}_{X}(B_{2},B_{1})$.
(4)
(Compositions)
If $h_{1}\in{\rm Sim}_{X}(B_{1},B_{2})$ and $h_{2}\in{\rm Sim}_{X}(B_{2},B_{3})$, then
$h_{2}h_{1}\in{\rm Sim}_{X}(B_{1},B_{3})$.
(5)
(Restrictions)
If $h\in{\rm Sim}_{X}(B_{1},B_{2})$ and $B_{3}\subseteq B_{1}$,
then
$$h|B_{3}\in{\rm Sim}_{X}(B_{3},h(B_{3})).$$
Definition 3.6.
A homeomorphism $h\colon X\to X$ is locally determined by ${\rm Sim}_{X}$ provided that for every $x\in X$, there exists a ball
$B^{\prime}$ in $X$ such that $x\in B^{\prime}$, $h(B^{\prime})$ is a ball in $X$, and $h|B^{\prime}\in{\rm Sim}(B^{\prime},h(B^{\prime}))$.
Definition 3.7.
The finite similarity structure
(FSS) group
$\Gamma({\rm Sim}_{X})$ is the set of all homeomorphisms $h\colon X\to X$ such that $h$ is locally determined by
${\rm Sim}_{X}$.
Remark 3.8.
The fact that $\Gamma({\rm Sim}_{X})$ is a group under composition is due to Hughes [7].
3.2. A description of the homeomorphisms determined by a similarity structure
In this subsection, we offer a somewhat simpler description
of the elements in the groups $\Gamma({\rm Sim}_{X})$ (Proposition 3.11), which shows that elements $\gamma\in\Gamma({\rm Sim}_{X})$ can be described in a manner reminiscent of the tree pair representatives for elements in Thompson’s group $V$ (see [3]).
Definition 3.9.
We define the standard partitions of $X$ inductively as follows.
(1)
$\{X\}$ is a standard partition.
(2)
If $\mathcal{P}=\{\widehat{B}_{1},\ldots,\widehat{B}_{n}\}$ is a standard partition, and $\{B_{1},\ldots,B_{m}\}$ is the partition of $\widehat{B}_{i}$ into maximal proper subballs, then
$(\mathcal{P}-\{\widehat{B}_{i}\})\cup\{B_{1},\ldots,B_{m}\}$ is also a standard partition.
Clearly, each standard partition is a partition of $X$ into balls.
Lemma 3.10.
Every partition $\mathcal{P}$ of $X$ into balls is standard.
Proof.
We prove this by induction on $|\mathcal{P}|$. It is clearly true if $|\mathcal{P}|=1$. We note that compactness implies that each
partition $\mathcal{P}$ of $X$ into balls must be finite.
For an arbitrary ball $B\subseteq X$, we define the depth of $B$, denoted $d(B)$, to be the number of distinct balls
of $X$ that contain $B$. (This definition is similar to Definition 3.19 from [5].) We note that $d(B)$ is a positive integer by Lemma 3.2(4), and $d(X)=1$.
Now we suppose that a partition $\mathcal{P}$ is given to us. We assume inductively that all partitions with smaller numbers of
balls are standard. By finiteness of $\mathcal{P}$,
there is some ball $B$ having maximum depth $m$, where we can assume that $m\geq 2$. Let $\widehat{B}$ denote the ball
containing $B$ as a maximal proper subball. Clearly, $d(\widehat{B})=m-1$. We let $\{B_{0},\ldots,B_{k}\}$ be the collection
of maximal proper subballs of $\widehat{B}$, where $B=B_{0}$ and $k\geq 1$.
We claim that $\{B_{0},B_{1},B_{2},\ldots,B_{k}\}\subseteq\mathcal{P}$. Choose $x\in B_{i}$. Our assumptions
imply that $x$ is in some ball $B^{\prime}$ of $\mathcal{P}$ such that $d(B^{\prime})\leq m$. The only such balls are $B_{i}$, $\widehat{B}$, and
any balls that contain $\widehat{B}$. (This uses an appeal to Lemma 3.2(2).) Since $\widehat{B}\cap B_{0}\neq\emptyset$ and $B_{0}=B\in\mathcal{P}$,
the only possibility is that $B^{\prime}=B_{i}$, since $\mathcal{P}$ is a partition. This proves that $\{B_{0},\ldots,B_{k}\}\subseteq\mathcal{P}$.
Now we consider the partition $\mathcal{P}^{\prime}=(\mathcal{P}-\{B_{0},\ldots,B_{k}\})\cup\{\widehat{B}\}$.
This partition is standard by the induction hypothesis, and it follows directly that $\mathcal{P}$ itself is standard.
∎
Proposition 3.11.
Let ${\rm Sim}_{X}$ be a finite similarity structure on $X$, and let $\gamma\in\Gamma({\rm Sim}_{X})$. There exist standard partitions $\mathcal{P}_{1}=\{B_{1},\ldots,B_{n}\}$ and $\mathcal{P}_{2}$ of $X$, a bijection $\phi:\mathcal{P}_{1}\rightarrow\mathcal{P}_{2}$, and elements $\sigma_{i}\in{\rm Sim}(B_{i},\phi(B_{i}))$
such that $\gamma_{\mid B_{i}}=\sigma_{i}$, for $i=1,\ldots,n$.
Moreover, we can arrange that the balls $B_{i}$ are maximal in the sense that if $B\subseteq X$ and $\gamma(B)$ are balls such
that $\gamma_{\mid B}\in{\rm Sim}_{X}(B,\gamma(B))$, then $B\subseteq B_{i}$, for some $i\in\{1,\ldots,n\}$.
Proof.
Since $\gamma$ is locally determined by ${\rm Sim}_{X}$, we can find an open cover of $X$ by balls such that the restriction of $\gamma$ to each ball is a local similarity in the
${\rm Sim}_{X}$-structure. By compactness of $X$, we can pass to a finite subcover. An application of Lemma 3.2(2) allows us to pass to a subcover that is also
a partition. We call this partition $\mathcal{P}_{1}$. We can then set $\mathcal{P}_{2}=\gamma(\mathcal{P}_{1})$. Both partitions are standard by Lemma 3.10.
The final statement is essentially Lemma 3.7 from [7].
∎
4. Braided diagram groups and groups determined by finite similarity structures
4.1. Braided diagram groups over tree-like semigroup presentations
Definition 4.1.
A semigroup presentation $\mathcal{P}=\langle\Sigma\mid\mathcal{R}\rangle$ is tree-like if,
(1)
every relation $(w_{1},w_{2})\in\mathcal{R}$ satisfies $|w_{1}|=1$ and $|w_{2}|>1$;
(2)
if $(a,w_{1}),(a,w_{2})\in\mathcal{R}$, then $w_{1}\equiv w_{2}$.
Example 4.2.
The generalized Thompson’s groups $V_{d}$ are isomorphic to the braided diagram groups $D_{b}(\mathcal{P},x)$,
where $\mathcal{P}=\langle x\mid(x,x^{d})\rangle$ is a tree-like semigroup presentation.
This fact was already proved in [6] and [4], and it is also a consequence of Theorem 4.12.
Example 4.3.
Consider the graph $G_{n}$ made up from a disjoint union of $n$ rays: $G_{n}=\{1,\ldots,n\}\times[0,\infty)$.
We assume that each ray is given the standard CW-complex structure with a vertex at
each integer. We define the Houghton group $H_{n}$ to be the set of bijections $h$ of the vertices $G_{n}^{0}$ such that
(1)
$h$ preserves adjacency, with at most finitely many exceptions, and
(2)
$h$ preserves ends: that is, for each $i\in\{1,\ldots,n\}$, there are $i_{1},i_{2}\in\mathbb{N}$ such that $h(\{i\}\times[i_{1},\infty))=\{i\}\times[i_{2},\infty)$.
Ken Brown [2] showed that $H_{n}$ is a group of type $F_{n-1}$ but not of type $F_{n}$.
We will sketch a proof that each Houghton group $H_{n}$ is a braided diagram group over a tree-like semigroup presentation. (It follows, in particular, that each of these
groups is $\Gamma({\rm Sim}_{X})$, for an appropriate compact ultrametric space $X$ and finite similarity structure ${\rm Sim}_{X}$, by Theorem 4.12.)
For $n\geq 2$, consider the semigroup presentation
$$\mathcal{P}_{n}=\langle a,r,x_{1},\ldots,x_{n}\mid(r,x_{1}x_{2}x_{3}\ldots x_{%
n}),(x_{1},ax_{1}),(x_{2},ax_{2}),\ldots,(x_{n},ax_{n})\rangle.$$
Similarly, we can define $\mathcal{P}_{1}=\langle a,r\mid(r,ar)\rangle$.
We claim that $D_{b}(\mathcal{P}_{n},r)$ is isomorphic to $H_{n}$. We sketch the proof for $n\geq 2$; the proof in case $n=1$ is very similar.
The elements of $D_{b}(\mathcal{P}_{n},r)$ can be expressed in the form $\Delta_{2}\circ\Delta_{1}^{-1}$, where each transistor in $\Delta_{i}$ is “positive”; i.e., the top label of the transistor is the left side of a relation in $\mathcal{P}_{n}$, and the bottom label is the right side. (This is proved as part of the proof of Theorem 4.12.)
We can think of each diagram $\Delta_{i}$ as a recipe for separating $G_{n}$ into connected components. The wires running between $\Delta_{1}$ and $\Delta_{2}$
in the concatenation $\Delta_{2}\circ\Delta_{1}^{-1}$ describe how these connected components should be matched by the bijection $h\in H_{n}$. To put it more explicitly,
the relations represent the following operations:
(1)
the relation $(r,x_{1}x_{2}\ldots x_{n})$ describes the initial configuration $G_{n}$ of $n$ disjoint rays. The letters $x_{i}$ ($i\in\{1,\ldots,n\}$) represent the isomorphism types of the different rays.
The different subscripts prevent different ends of $G_{n}$ from being permuted nontrivially. (If we wish to remove the end-preserving condition above, we can simply replace the
$n$ distinct symbols $x_{1}$, $x_{2}$, $\ldots$, $x_{n}$ by the single symbol $x$.)
(2)
the relation $(x_{i},ax_{i})$ (for $i\in\{1,\ldots,n\}$) represents the action of breaking the initial vertex away from the ray of isomorphism type $x_{i}$. The initial vertex of the ray gets the label $a$, and the new ray retains the label $x_{i}$ (since it is of the same combinatorial type as the original ray, and the new ray is a permissible target for the original ray
under the action of the Houghton group). The letter $a$ thus represents a single floating vertex. Any two such vertices can be matched by an element of the Houghton group, which
is why we use a single label for all of these vertices.
We illustrate how an $(r,r)$-diagram over $\mathcal{P}_{2}$ represents an element of $H_{2}$. Figure 3 depicts an $(r,r)$-diagram $\Delta_{h}$ over $\mathcal{P}_{2}$.
This diagram $\Delta_{h}$ represents the element $h\in H_{2}$ that sends: $(2,n)$ to $(2,n-1)$, for each $n\geq 1$, $(1,n)$ to $(1,n+1)$ (for all $n$), and $(2,0)$ to $(1,0)$.
Note that the bottom portion of the diagram represents a subdivision of the domain, and the top portion represents a subdivision of the range.
It is straightforward to check that the indicated function does not change if we insert or remove dipoles.
Example 4.4.
The same principle can be used to exhibit the group $\Gamma=\mathrm{QAut}(\mathcal{T}_{2,c})$ as a braided diagram group over a tree-like
semigroup presentation. Here $\Gamma$ (as defined in [1]) is the group of self-bijections $h$ of the vertices of the infinite ordered rooted binary tree $T$ such that
(1)
$h$ preserves adjacency, with at most finitely many exceptions, and
(2)
$h$ preserves the left-right ordering of the edges incident with and below a given vertex, again with at most finitely many exceptions.
Consider the semigroup presentation $\mathcal{P}=\langle a,x\mid(x,xax)\rangle$. We claim that $D_{b}(\mathcal{P},x)$ is isomorphic to $\Gamma$. Much of the discussion from the previous example carries over identically.
We will simply indicate how the single relation allows us to simulate breaking the binary tree into pieces. (Such a dissection of $T$ would be represented by
a positive diagram, as above. The wires connecting the bottoms of the positive diagrams $\Delta_{1}$ and $\Delta_{2}$ would again represent how the resulting pieces are matched by
a bijection.)
The letter $x$ represents the isomorphism type of the binary tree $T$. The relation represents breaking the tree at the root. The result of this operation yields a floating vertex
(represented by the letter $a$), and two new rooted binary trees (both represented by $x$). The first $x$ in $xax$ represents the left branch, and the second $x$ represents the
right. The description of the isomorphism of $D_{b}(\mathcal{P},x)$ with $\Gamma$ now follows the general pattern of the previous example.
Remark 4.5.
Example 4.3 shows that the $F_{\infty}$ result of [5] cannot be extended to all groups determined by finite similarity structures (as defined in Section 3).
All of the groups in the above examples act properly on CAT(0) cubical complexes by a construction of [4].
We note also that the representation of the above groups as braided diagram groups suggests a method for producing embeddings into other groups, such as (perhaps most notably) Thompson’s
group $V$. For instance, the group from Example 4.4 can be embedded into $V$ as follows. Given a braided diagram $\Delta$ over $\mathcal{P}$, systematically
replace each $a$ label with an $x$. The result is a braided diagram over the semigroup presentation $\mathcal{P}^{\prime}=\langle x\mid(x,x^{3})\rangle$. The indicated function
$\phi:D_{b}(\mathcal{P},x)\rightarrow D_{b}(\mathcal{P}^{\prime},x)$ is
easily seen to be a homomorphism, and $\phi$ is injective since it sends reduced diagrams to reduced diagrams. We can now appeal to the fact that
$D_{b}(\mathcal{P}^{\prime},x)\cong V_{3}$ (the $3$-ary version of Thompson’s group $V$), and the latter group embeds in $V$ itself.
The group from Example 4.4 was previously known to embed in $V$ by a result of [1].
4.2. Groups determined by small similarity structures
Definition 4.6.
Let $X$ be a compact ultrametric space. We say that the finite similarity structure ${\rm Sim}_{X}$ is small
if, for every pair of balls $B_{1},B_{2}$ in $X$, $|{\rm Sim}_{X}(B_{1},B_{2})|\leq 1$.
Definition 4.7.
Let $X$ be a compact ultrametric space endowed with a small similarity structure ${\rm Sim}_{X}$. If $B\subseteq X$ is a ball in
$X$ that is not an isolated point, then a local ball order at $B$ is an assignment of a linear order $<$ to the set $\{\widehat{B}_{1},\ldots,\widehat{B}_{n}\}$
of maximal proper subballs of $B$. A ball order on $X$ is an assignment of such a linear order to each ball $B\subseteq X$ that is not a singleton. The ball order is
compatible with ${\rm Sim}_{X}$ if each $h\in{\rm Sim}_{X}(B_{1},B_{2})$ induces an order-preserving bijection of the maximal proper subballs of $B_{1}$ and $B_{2}$,
for all choices of $B_{1}$ and $B_{2}$.
Lemma 4.8.
Let $X$ be a compact ultrametric space endowed with a small similarity structure. There exists a ball order on $X$ that is compatible with ${\rm Sim}_{X}$.
Proof.
We recall a definition from [5].
Let $B\subseteq X$ be a metric ball. Let
$[B]=\{B^{\prime}\subseteq X\mid{\rm Sim}_{X}(B,B^{\prime})\neq\emptyset\}$; $[B]$ is called the ${\rm Sim}_{X}$-class of $B$.
From a given ${\rm Sim}_{X}$-class of balls, choose a ball $B$. If $B$ is not a singleton, then there exists a collection of maximal proper subballs
$B_{1}$, $\ldots$, $B_{n}$ of $B$. Choose a linear order on this collection of balls; without loss of generality, $B_{1}<B_{2}<\ldots<B_{n}$. If $B^{\prime}$ is another ball
in $[B]$, then we can let $h$ denote the unique element of ${\rm Sim}_{X}(B,B^{\prime})$. This $h$ carries the maximal proper subballs of $B$ into maximal proper subballs of $B^{\prime}$, and thereby
induces an order $h(B_{1})<h(B_{2})<\ldots<h(B_{n})$ on the maximal proper subballs of $B^{\prime}$. This procedure gives a local ball order to each ball $B^{\prime}\in[B]$.
We repeat this procedure for each ${\rm Sim}_{X}$-class of balls. The result is a ball order on $X$ that is compatible with ${\rm Sim}_{X}$.
∎
Remark 4.9.
A ball order on $X$ also determines a linear order on any given collection of pairwise disjoint balls in $X$. For let $\mathcal{C}$
be such a collection, and let $B_{1},B_{2}\in\mathcal{C}$. There is a unique smallest ball $B\subseteq X$ that contains
both $B_{1}$ and $B_{2}$, by Lemma 3.2(4). Let $\{\widehat{B}_{1},\ldots,\widehat{B}_{n}\}$
be the collection of maximal proper subballs of $B$. By minimality of $B$,
we must have that $B_{1}$ and $B_{2}$ are contained in distinct
maximal proper subballs of $B$; say $B_{1}\subseteq\widehat{B}_{1}$ and $B_{2}\subseteq\widehat{B}_{2}$. We write
$B_{1}<B_{2}$ if and only if $\widehat{B}_{1}<\widehat{B}_{2}$. This defines a linear order on $\mathcal{C}$.
The verification is straightforward.
Definition 4.10.
Let $X$ be a compact ultrametric space with a small similarity structure ${\rm Sim}_{X}$ and a compatible ball order.
Define a semigroup presentation $\mathcal{P}_{{\rm Sim}_{X}}=\langle\Sigma\mid\mathcal{R}\rangle$
as follows. Let
$$\Sigma=\{[B]\mid B\text{ is a ball in }X\}.$$
If $B\subseteq X$ is a ball, let $B_{1},\ldots,B_{n}$ be the maximal proper subballs
of $B$, listed in order. If $B$ is a point, then $n=0$. We set
$$\mathcal{R}=\{([B],[B_{1}][B_{2}]\ldots[B_{n}])\mid n\geq 1,B\text{ is a ball %
in }X\}.$$
Remark 4.11.
We note that $\mathcal{P}_{{\rm Sim}_{X}}$ will always be a tree-like semigroup presentation, for any choice of
compact ultrametric space $X$, small similarity structure ${\rm Sim}_{X}$, and compatible ball order.
4.3. The main theorem
Theorem 4.12.
If $X$ is a compact ultrametric space with a small similarity structure ${\rm Sim}_{X}$ and compatible ball order,
then
$$\Gamma({\rm Sim}_{X})\cong D_{b}(\mathcal{P}_{{\rm Sim}_{X}},[X]).$$
Conversely, if $\mathcal{P}=\langle\Sigma\mid\mathcal{R}\rangle$ is a tree-like semigroup presentation,
and $x\in\Sigma$, then there is a compact ultrametric space $X_{\mathcal{P}}$, a small
finite similarity structure ${\rm Sim}_{X_{\mathcal{P}}}$, and a compatible ball order such that
$$D_{b}(\mathcal{P},x)\cong\Gamma({\rm Sim}_{X_{\mathcal{P}}}).$$
Proof.
If $\gamma\in\Gamma({\rm Sim}_{X})$, then, by Proposition 3.11, there are standard
partitions $\mathcal{P}_{1}$, $\mathcal{P}_{2}$ of $X$ into balls,
and a bijection $\phi:\mathcal{P}_{1}\rightarrow\mathcal{P}_{2}$ such that, for any $B\in\mathcal{P}_{1}$,
$\gamma(B)=\phi(B)$ and $\gamma|_{B}\in{\rm Sim}_{X}(B,\gamma(B))$.
Since $|{\rm Sim}_{X}(B,\gamma(B))|\leq 1$, the triple $(\mathcal{P}_{1},\mathcal{P}_{2},\phi)$ determines $\gamma$ without
ambiguity. We call $(\mathcal{P}_{1},\mathcal{P}_{2},\phi)$ a defining triple for $\gamma$. Note that a given $\gamma$
will usually have many defining triples. Let $\mathcal{D}$ be the set of all defining triples, for $\gamma$ running over all of
$\Gamma({\rm Sim}_{X})$.
We will now define a map $\psi:\mathcal{D}\rightarrow D_{b}(\mathcal{P}_{{\rm Sim}_{X}},[X])$. To a partition $\mathcal{P}$
of $X$ into balls, we first assign a braided diagram $\Delta_{\mathcal{P}}$ over $\mathcal{P}_{{\rm Sim}_{X}}$. There is a transistor
$T_{B}\in\mathcal{T}(\Delta_{\mathcal{P}})$ for each ball $B$ which properly contains some ball of $\mathcal{P}$. There is a wire
$W_{B}\in\mathcal{W}(\Delta_{\mathcal{P}})$ for each ball $B$ which contains a ball of $\mathcal{P}$. The wires are attached as follows:
(1)
If $B=X$, then we attach the top of $W_{B}$ to the top of the frame.
If $B\neq X$, then the top of the wire $W_{B}$ is attached to the bottom of the transistor $T_{\widehat{B}}$, where $\widehat{B}$ is the (unique) ball
that contains $B$ as a maximal proper subball.
Moreover, we attach the wires in an “order-respecting” fashion. Thus, if $\widehat{B}$ is a ball properly containing balls of $\mathcal{P}$, we let
$B_{1},B_{2},\ldots,B_{n}$ be the collection of maximal proper subballs of $\widehat{B}$, listed in order. We attach the wires $W_{B_{1}},W_{B_{2}},\ldots,W_{B_{n}}$ so that $t(W_{B_{i}})$ is to the left of $t(W_{B_{j}})$ on the bottom of $T_{\widehat{B}}$ if $i<j$.
(2)
The bottom of the wire $W_{B}$ is attached to the top of $T_{B}$ if $B$ properly contains a ball of $\mathcal{P}$. If not (i.e., if $B\in\mathcal{P}$),
then we attach the bottom of $W_{B}$ to the bottom of the frame. We can arrange, moreover, that the wires are attached in an order-respecting manner to the
bottom of the frame. (Thus, if $B_{1}<B_{2}$ ($B_{1},B_{2}\in\mathcal{P}$), we have that $b(W_{B_{1}})$ is to the left of $b(W_{B_{2}})$.)
The labelling function $\ell:\mathcal{W}(\Delta_{\mathcal{P}})\rightarrow\Sigma$ sends $W_{B}$ to $[B]$. It is straightforward to check that the
resulting $\Delta_{\mathcal{P}}$ is a braided diagram over $\mathcal{P}_{{\rm Sim}_{X}}$. The top label of $\Delta_{\mathcal{P}}$ is $[X]$.
Given a bijection $\phi:\mathcal{P}_{1}\rightarrow\mathcal{P}_{2}$, where $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$ are partitions of $X$ into balls
and $[B]=[\phi(B)]$, we can define a braided diagram $\Delta_{\phi}$ over $\mathcal{P}_{{\rm Sim}_{X}}$ as follows. We let
$\mathcal{T}(\Delta_{\phi})=\emptyset$, and $\mathcal{W}(\Delta_{\phi})=\{W_{B}\mid B\in\mathcal{P}_{1}\}$.
We attach the top of each wire to the frame in such a way that $t(W_{B_{1}})$ is to the left of $t(W_{B_{2}})$ if $B_{1}<B_{2}$.
(Here $<$ refers to the ordering from Remark 4.9.) We attach the
bottom of each wire to the bottom of the frame in such a way that $b(W_{B_{1}})$ is to the left of $b(W_{B_{2}})$ if $\phi(B_{1})<\phi(B_{2})$.
Now, for a defining triple $(\mathcal{P}_{1},\mathcal{P}_{2},\phi)\in\mathcal{D}$, we set
$\psi((\mathcal{P}_{1},\mathcal{P}_{2},\phi))=\Delta_{\mathcal{P}_{2}}\circ%
\Delta_{\phi^{-1}}\circ\Delta_{\mathcal{P}_{1}}^{-1}\in\mathcal{D}_{b}(%
\mathcal{P}_{{\rm Sim}_{X}},[X])$.
We claim that any two defining triples $(\mathcal{P}_{1},\mathcal{P}_{2},\phi)$, $(\mathcal{P}_{1}^{\prime},\mathcal{P}_{2}^{\prime},\phi^{\prime})$
for a given $\gamma\in\Gamma({\rm Sim}_{X})$
have the same image
in $D_{b}(\mathcal{P}_{{\rm Sim}_{X}},[X])$, modulo dipoles. We begin by proving an intermediate statement. Let $(\mathcal{P}_{1},\mathcal{P}_{2},\phi)$
be a defining triple. Let $B\in\mathcal{P}_{1}$, and let $\widehat{B}_{1},\ldots,\widehat{B}_{n}$ be the collection of maximal proper subballs of $B$, listed
in order. We let $B^{\prime}=\phi(B)$ and let $\widehat{B}^{\prime}_{1},\ldots,\widehat{B}^{\prime}_{n}$ be the collection of maximal proper subballs of $B^{\prime}$. (Note that
$[B^{\prime}]=[B]$ by our assumptions, so both have the same number of maximal proper subballs, and in fact $[\widehat{B}_{i}]=[\widehat{B}^{\prime}_{i}]$ for $i=1,\ldots,n$,
since $\gamma|_{B}\in{\rm Sim}_{X}(B,B^{\prime})$ and the elements of ${\rm Sim}_{X}(B,B^{\prime})$ preserve order.) We set
$\widehat{\mathcal{P}}_{1}=(\mathcal{P}_{1}-\{B\})\cup\{\widehat{B}_{1},\ldots,%
\widehat{B}_{n}\}$,
$\widehat{\mathcal{P}}_{2}=(\mathcal{P}_{2}-\{B^{\prime}\})\cup\{\widehat{B}^{%
\prime}_{1},\ldots,\widehat{B}^{\prime}_{n}\}$,
and $\widehat{\phi}|_{\mathcal{P}_{1}-\{B\}}=\phi|_{\mathcal{P}_{1}-\{B\}}$,
$\widehat{\phi}(\widehat{B}_{i})=\widehat{B}^{\prime}_{i}$. We say that $(\widehat{\mathcal{P}}_{1},\widehat{\mathcal{P}}_{2},\widehat{\phi})$
is obtained from $(\mathcal{P}_{1},\mathcal{P}_{2},\phi)$ by subdivision at $(B,B^{\prime})$.
A straightforward argument shows that $\psi((\widehat{\mathcal{P}}_{1},\widehat{\mathcal{P}}_{2},\widehat{\phi}))$ is in fact obtained from
$\psi((\mathcal{P}_{1},\mathcal{P}_{2},\phi))$ by inserting a dipole. We omit the details, which rely on the fact that each
element of the ${\rm Sim}_{X}$-structure preserves the local ball order.
Now suppose that $(\mathcal{P}_{1},\mathcal{P}_{2},\phi)$ and $(\mathcal{P}^{\prime}_{1},\mathcal{P}^{\prime}_{2},\phi^{\prime})$ are defining triples
for the same element $\gamma\in\Gamma({\rm Sim}_{X})$. We can find a common refinement $\mathcal{P}^{\prime\prime}_{1}$ of $\mathcal{P}_{1}$ and $\mathcal{P}^{\prime}_{1}$.
Using the fact that all partitions of $X$ into balls are standard (Lemma
3.10),
we can pass from $(\mathcal{P}_{1},\mathcal{P}_{2},\phi)$ to $(\mathcal{P}^{\prime\prime}_{1},\widehat{P}_{2},\widehat{\phi})$ by repeated subdivision
(for some partition $\widehat{\mathcal{P}}_{2}$ of $X$ into balls and some bijection $\widehat{\phi}:\mathcal{P}^{\prime\prime}_{1}\rightarrow\widehat{\mathcal{P}}_%
{2}$).
Since subdivision does not change the values of $\psi$ modulo dipoles, $\psi((\mathcal{P}_{1},\mathcal{P}_{2},\phi))=\psi((\mathcal{P}^{\prime\prime}_%
{1},\widehat{\mathcal{P}}_{2},\widehat{\phi}))$ modulo dipoles. Similarly, we can subdivide
$(\mathcal{P}^{\prime}_{1},\mathcal{P}^{\prime}_{2},\phi^{\prime})$ repeatedly in order to obtain $(\mathcal{P}^{\prime\prime}_{1},\widehat{\mathcal{P}}^{\prime}_{2},\widehat{%
\phi}^{\prime})$, where
$\psi((\mathcal{P}^{\prime}_{1},\mathcal{P}^{\prime}_{2},\phi^{\prime}))=\psi((%
\mathcal{P}^{\prime\prime}_{1},\widehat{\mathcal{P}}^{\prime}_{2},\widehat{%
\phi}^{\prime}))$ modulo dipoles.
Both $(\mathcal{P}^{\prime\prime}_{1},\widehat{\mathcal{P}}^{\prime}_{2},\widehat{%
\phi}^{\prime})$
and $(\mathcal{P}^{\prime\prime}_{1},\widehat{\mathcal{P}}_{2},\widehat{\phi})$ are defining triples for $\gamma$, so we are forced to have
$\widehat{\phi}=\widehat{\phi}^{\prime}$ and $\widehat{\mathcal{P}}_{2}=\widehat{\mathcal{P}}^{\prime}_{2}$.
It follows that $\psi((\mathcal{P}_{1},\mathcal{P}_{2},\phi))=\psi((\mathcal{P}^{\prime}_{1},%
\mathcal{P}^{\prime}_{2},\phi^{\prime}))$, so $\psi$ induces a function
from $\Gamma({\rm Sim}_{X})$ to $D_{b}(\mathcal{P}_{{\rm Sim}_{X}},[X])$. We will call this function $\widehat{\psi}$.
Now we will show that $\widehat{\psi}:\Gamma({\rm Sim}_{X})\rightarrow D_{b}(\mathcal{P}_{{\rm Sim}_{%
X}},[X])$ is a homomorphism. Let $\gamma,\gamma^{\prime}\in\Gamma({\rm Sim}_{X})$. After subdividing as necessary, we
can choose defining triples $(\mathcal{P}_{1},\mathcal{P}_{2},\phi)$ and $(\mathcal{P}^{\prime}_{1},\mathcal{P}^{\prime}_{2},\phi^{\prime})$ for
$\gamma$ and $\gamma^{\prime}$ (respectively) in such a way that $\mathcal{P}_{2}=\mathcal{P}^{\prime}_{1}$. It follows easily that
$(\mathcal{P}_{1},\mathcal{P}^{\prime}_{2},\phi^{\prime}\phi)$ is a defining triple for $\gamma^{\prime}\gamma$. Therefore,
$\widehat{\psi}(\gamma^{\prime}\gamma)=\Delta_{\mathcal{P}^{\prime}_{2}}\circ%
\Delta_{(\phi^{\prime}\phi)^{-1}}\circ\Delta^{-1}_{\mathcal{P}_{1}}$.
Now
$$\displaystyle\widehat{\psi}(\gamma^{\prime})\circ\widehat{\psi}(\gamma)$$
$$\displaystyle=\Delta_{\mathcal{P}^{\prime}_{2}}\circ\Delta_{(\phi^{\prime})^{-%
1}}\circ\Delta^{-1}_{\mathcal{P}^{\prime}_{1}}\circ\Delta_{\mathcal{P}_{2}}%
\circ\Delta_{\phi^{-1}}\circ\Delta^{-1}_{\mathcal{P}_{1}}$$
$$\displaystyle=\Delta_{\mathcal{P}^{\prime}_{2}}\circ\Delta_{(\phi^{\prime})^{-%
1}}\circ\Delta_{\phi^{-1}}\circ\Delta^{-1}_{\mathcal{P}_{1}}$$
$$\displaystyle=\Delta_{\mathcal{P}^{\prime}_{2}}\circ\Delta_{(\phi^{\prime}\phi%
)^{-1}}\circ\Delta^{-1}_{\mathcal{P}_{1}}$$
Therefore, $\widehat{\psi}$ is a homomorphism.
We now show that $\widehat{\psi}:\Gamma({\rm Sim}_{X})\rightarrow D_{b}(\mathcal{P}_{{\rm Sim}_{%
X}},[X])$ is injective. Suppose that $\widehat{\psi}(\gamma)=1$.
Using the final statement of Proposition 3.11, we choose a defining triple $(\mathcal{P}_{1},\mathcal{P}_{2},\phi)$ for $\gamma$ with the property that, if $B\subseteq X$ is a ball, $\gamma(B)$ is a ball,
and $\gamma|_{B}\in{\rm Sim}_{X}(B,\gamma(B))$, then $B$ is contained in some ball of $\mathcal{P}_{1}$. We claim that
$\psi((\mathcal{P}_{1},\mathcal{P}_{2},\phi))$ is a reduced diagram. If there were a dipole $(T_{1},T_{2})$, then we would have
$T_{1}\in\mathcal{T}(\Delta_{\mathcal{P}_{1}}^{-1})$ and $T_{2}\in\mathcal{T}(\Delta_{\mathcal{P}_{2}})$, since it is impossible for $\Delta_{\mathcal{P}}$
to contain any dipoles, for any partition $\mathcal{P}$ of $X$ into balls. Thus $T_{1}=T_{B_{1}}$ and $T_{2}=T_{B_{2}}$, where $[B_{1}]=[B_{2}]$ and
the wires from the bottom of $T_{B_{2}}$ attach to the top of $T_{B_{1}}$, in order.
This means that, if $\widehat{B}_{1},\ldots,\widehat{B}_{n}$ are the maximal proper subballs of $B_{1}$,
and $\widehat{B}^{\prime}_{1},\ldots,\widehat{B}^{\prime}_{n}$ are the maximal proper subballs of $B_{2}$, then $\gamma(\widehat{B}_{i})=\widehat{B}^{\prime}_{i}$, where the latter
is a ball, and $\gamma|_{\widehat{B}_{i}}\in{\rm Sim}_{X}(\widehat{B}_{i},\widehat{B}^{\prime}%
_{i})$.
Now, since $[B_{1}]=[B_{2}]$, there is $h\in{\rm Sim}_{X}(B_{1},B_{2})$. Since ${\rm Sim}_{X}$ is closed under restrictions and $h$ preserves order,
we have $h_{i}\in{\rm Sim}_{X}(\widehat{B}_{i},\widehat{B}^{\prime}_{i})$ for $i=1,\ldots,n$, where $h_{i}=h|_{\widehat{B}_{i}}$. It follows that
$\gamma|_{\widehat{B}_{i}}=h_{i}$, so, in particular, $\gamma|_{B_{1}}=h$.
Since $B_{1}$ properly contains some ball in $\mathcal{P}_{1}$, this is a contradiction. Thus, $\psi((\mathcal{P}_{1},\mathcal{P}_{2},\phi))$
is reduced.
We claim that $\psi((\mathcal{P}_{1},\mathcal{P}_{2},\phi))$ contains no transistors (due to the condition $\widehat{\psi}(\gamma)=1$).
We have shown that $\psi((\mathcal{P}_{1},\mathcal{P}_{2},\phi))$ is a reduced diagram in the same class as the identity
$1\in D_{b}(\mathcal{P}_{{\rm Sim}_{X}},[X])$. The identity can be represented as the (unique) $([X],[X])$-diagram $\Delta_{1}$ with only a single wire,
$W_{X}$, and no transistors. We must have $\psi((\mathcal{P}_{1},\mathcal{P}_{2},\phi))\equiv\Delta_{1}$. Thus,
there is no ball that properly contains a ball of $\mathcal{P}_{1}$. It can only be that $\mathcal{P}_{1}=\{X\}$, so we must have
$\gamma\in{\rm Sim}_{X}(X,X)$. This forces $\gamma=1$, so $\widehat{\psi}$ is injective.
Finally we must show that $\widehat{\psi}:\Gamma({\rm Sim}_{X})\rightarrow D_{b}(\mathcal{P}_{{\rm Sim}_{%
X}},[X])$ is surjective. Let
$\Delta$ be a reduced $([X],[X])$-diagram over $\mathcal{P}_{{\rm Sim}_{X}}$. A transistor $T\in\mathcal{T}(\Delta)$ is called positive
if its top label is the left side of a relation in $\mathcal{P}_{{\rm Sim}_{X}}$, otherwise (i.e., if the top label is the right side of a relation
in $\mathcal{P}_{{\rm Sim}_{X}}$) the transistor $T$ is negative. It is easy to see that the sets of positive and negative transistors partition
$\mathcal{T}(\Delta)$. We claim that, if $\Delta$ is reduced, then we cannot have $T_{1}\dot{\preceq}T_{2}$ when $T_{1}$ is positive and $T_{2}$
is negative. If we had such $T_{1}\dot{\preceq}T_{2}$, then we could find $T^{\prime}_{1}\preceq T^{\prime}_{2}$, where $T^{\prime}_{1}$ is positive and $T^{\prime}_{2}$ is negative.
Since $T^{\prime}_{1}$ is positive, there is only one wire $W$ attached to the top of $T^{\prime}_{1}$. This wire must be attached to the bottom of $T^{\prime}_{2}$, since
$T^{\prime}_{1}\preceq T^{\prime}_{2}$, and it must be the only wire attached to the bottom of $T^{\prime}_{2}$, since $T^{\prime}_{2}$ is negative and $\mathcal{P}_{{\rm Sim}_{X}}$
is a tree-like semigroup presentation by Remark 4.11. Suppose that $\ell(w)=[B]$. By the definition of $\mathcal{P}_{{\rm Sim}_{X}}$, $[B]$ is the left
side of exactly one relation, namely $([B],[B_{1}][B_{2}]\ldots[B_{n}])$, where the $B_{i}$ are maximal proper subballs of $B$, listed in order. It follows
that the bottom label of $T^{\prime}_{1}$ is $[B_{1}][B_{2}]\ldots[B_{n}]$ and the top label of $T^{\prime}_{2}$ is $[B_{1}][B_{2}]\ldots[B_{n}]$. Therefore
$(T^{\prime}_{1},T^{\prime}_{2})$ is a dipole. This proves the claim.
A diagram over $\mathcal{P}_{{\rm Sim}_{X}}$ is positive if all of its transistors are positive, and negative if all of its transistors are negative.
We note that $\Delta$ is positive if and only if $\Delta^{-1}$ is negative, by the description of inverses in the proof of Theorem 2.13.
The above reasoning shows that any reduced $([X],[X])$-diagram over $\mathcal{P}_{{\rm Sim}_{X}}$ can be written
$\Delta=\Delta^{+}_{1}\circ\left(\Delta_{2}^{+}\right)^{-1}$, where $\Delta^{+}_{i}$ is a positive diagram for $i=1,2$.
We claim that any positive diagram $\Delta$ over $\mathcal{P}_{{\rm Sim}_{X}}$ with top label $[X]$ is $\Delta_{\mathcal{P}}$ (up to a reordering of the bottom
contacts), where $\mathcal{P}$ is some partition of $X$. There is a unique wire $W\in\mathcal{W}(\Delta)$ making a top contact with the frame. We call
this wire $W_{X}$. Note that its label is $[X]$ by our assumptions. The bottom contact of $W_{X}$ lies either on the bottom of the frame, or on top of some
transistor. In the first case, we have $\Delta=\Delta_{\mathcal{P}}$ for $\mathcal{P}=\{X\}$ and we are done. In the second, the bottom contact of
$W_{X}$ lies on top of some transistor $T$, which we call $T_{X}$. Since the top label of $T_{X}$ is $[X]$, the bottom label must be $[B_{1}]\ldots[B_{k}]$,
where $B_{1},\ldots,B_{k}$ are the maximal proper subballs of $X$. Thus there are wires $W_{1},\ldots,W_{k}$ attached to the bottom of $T_{X}$, and we
have $\ell(W_{i})=[B_{i}]$, for $i=1,\ldots,k$. We relabel each of the wires $W_{B_{1}},\ldots,W_{B_{k}}$, respectively. Note that
$\{B_{1},\ldots,B_{k}\}$ is a partition of $X$ into balls. We can continue in this way, inductively labelling each wire with a ball $B\subseteq X$.
If we let $\overline{B}_{1},\ldots,\overline{B}_{m}$ be the resulting labels of the wires which make bottom contacts with the frame, then
$\{\overline{B}_{1},\ldots,\overline{B}_{m}\}=\mathcal{P}$ is a partition of $X$ into balls, and $\Delta=\Delta_{\mathcal{P}}$ by construction,
up to a reordering of the bottom contacts.
We can now prove surjectivity of $\widehat{\psi}$. Let $\Delta\in D_{b}(\mathcal{P}_{{\rm Sim}_{X}},[X])$ be reduced. We can write
$\Delta=\Delta_{2}^{+}\circ\left(\Delta_{1}^{+}\right)^{-1}$, where $\Delta_{i}^{+}$ is positive, for $i=1,2$. It follows that
$\Delta_{i}^{+}=\Delta_{\mathcal{P}_{i}}\circ\sigma_{i}$, for $i=1,2$, where $\mathcal{P}_{i}$ is a partition of $X$ into balls and $\sigma_{i}$ is
diagram containing no transistors. Thus, $\Delta=\Delta_{\mathcal{P}_{2}}\circ\sigma_{2}\circ\sigma_{1}^{-1}\circ\Delta_%
{\mathcal{P}_{1}}^{-1}=\psi((\mathcal{P}_{1},\mathcal{P}_{2},\phi))$, where $\phi:\mathcal{P}_{1}\rightarrow\mathcal{P}_{2}$ is a bijection determined by
$\sigma_{2}\circ\sigma_{1}^{-1}$. Therefore, $\widehat{\psi}$ is surjective.
Now we must show that if $\mathcal{P}=\langle\Sigma\mid\mathcal{R}\rangle$ is a tree-like semigroup presentation, $x\in\Sigma$, then there
is a compact ultrametric space $X_{\mathcal{P}}$, a small similarity structure ${\rm Sim}_{X_{\mathcal{P}}}$, and a compatible ball order, such that
$D_{b}(\mathcal{P},x)\cong\Gamma({\rm Sim}_{X_{\mathcal{P}}})$.
Construct a labelled ordered simplicial tree $T_{(\mathcal{P},x)}$ as follows. Begin with a vertex $\ast$, the root, labelled by $x\in\Sigma$.
By the definition of tree-like semigroup presentation (Definition 4.1), there is at most one relation in $\mathcal{R}$ having
the word $x$ as its left side. Let us suppose first that $(x,x_{1}x_{2}\ldots x_{k})\in\mathcal{R}$, where $k\geq 2$.
We introduce $k$ children of the root, labelled $x_{1},\ldots,x_{k}$ (respectively), each connected to the root by an edge.
The children are ordered from left to right in such a way that we read the
word $x_{1}x_{2}\ldots x_{k}$ as we read the labels of the children from left to right. If, on the other hand, $x$ is not the left side of any relation
in $\mathcal{R}$, then the tree terminates – there is only the root. We continue similarly: if $x_{i}$ is the left side of some relation
$(x_{i},y_{1}y_{2}\ldots y_{m})\in\mathcal{R}$ ($m\geq 2$),
then this relation is unique and we introduce a labelled ordered collection of children, as above. If $x_{i}$ is not the left side of any relation
in $\mathcal{R}$, then $x_{i}$ has no children. This builds a labelled ordered tree $T_{(\mathcal{P},x)}$. We note that if a vertex $v\in T_{(\mathcal{P},x)}$ is labelled by $y\in\Sigma$, then the subcomplex $T_{v}\leq T_{(\mathcal{P},x)}$ spanned by $v$ and all of its descendants is isomorphic
to $T_{(\mathcal{P},y)}$, by a simplicial isomorphism which preserves the labelling and the order.
We let $\mathrm{Ends}(T_{(\mathcal{P},x)})$ denote the set of all edge-paths $p$ in $T_{(\mathcal{P},x)}$ such that:
i) $p$ is without backtracking; ii) $p$ begins at the root; iii) $p$ is either infinite, or $p$ terminates at a vertex without children.
We define a metric on $\mathrm{Ends}(T_{(\mathcal{P},x)})$ as follows. If $p,p^{\prime}\in\mathrm{Ends}(T_{(\mathcal{P},x)})$ and $p,p^{\prime}$ have exactly
$m$ edges in common, then we set $d(p,p^{\prime})=e^{-m}$. This metric makes $\mathrm{Ends}(T_{(\mathcal{P},x)})$ a compact ultrametric space, and a ball order is given
by the ordering of the tree. We can describe the
balls in $\mathrm{Ends}(T_{(\mathcal{P},x)})$ explicitly. Let $v$ be a vertex of $T_{(\mathcal{P},x)}$. We set $B_{v}=\{p\in\mathrm{Ends}(T_{(\mathcal{P},x)})\mid v\text{ lies on }p\}$. Every such set is a ball, and every ball in $\mathrm{Ends}(T_{(\mathcal{P},x)})$
has this form. We can now describe a finite similarity structure ${\rm Sim}_{X_{\mathcal{P}}}$ on $\mathrm{Ends}(T_{(\mathcal{P},x)})$. Let $B_{v}$ and $B_{v^{\prime}}$ be the balls
corresponding to the vertices $v,v^{\prime}\in T_{(\mathcal{P},x)}$. If $v$ and $v^{\prime}$ have different labels, then we set ${\rm Sim}_{X_{\mathcal{P}}}(B_{v},B_{v^{\prime}})=\emptyset$.
If $v$ and $v^{\prime}$ have the same label, say $y\in\Sigma$, then there is label- and order-preserving simplicial isomorphism $\psi:T_{v}\rightarrow T_{v^{\prime}}$.
Suppose that $p_{v}$ is the unique edge-path without backtracking connecting the root to $v$. Any point in $B_{v}$ can be expressed in the form
$p_{v}q$, where $q$ is an edge-path without backtracking in $T_{v}$. We let $\widehat{\psi}:B_{v}\rightarrow B_{v^{\prime}}$ be defined by the
rule $\widehat{\psi}(p_{v}q)=p_{v^{\prime}}\psi(q)$. The map $\widehat{\psi}$ is easily seen to be a surjective similarity.
We set ${\rm Sim}_{X_{\mathcal{P}}}(B_{v},B_{v^{\prime}})=\{\widehat{\psi}\}$. The resulting assignments give a small similarity structure ${\rm Sim}_{X_{\mathcal{P}}}$ on
the compact
ultrametric space $\mathrm{Ends}(T_{(\mathcal{P},x)})$ that is compatible with the ball order.
Now we can apply the first part of the theorem: setting $X_{(\mathcal{P},x)}=\mathrm{Ends}(T_{(\mathcal{P},x)})$, we
have $\Gamma({\rm Sim}_{X_{(\mathcal{P},x)}})\cong D_{b}(\mathcal{P}_{{\rm Sim}_{X_{%
(\mathcal{P},x)}}},[X_{(\mathcal{P},x)}])\cong D_{b}(\mathcal{P},x)$, where the first isomorphism follows from the forward direction of the theorem, and the second isomorphism follows from the canonical identification of the semigroup presentation $\mathcal{P}_{{\rm Sim}_{X_{(\mathcal{P},x)}}}$
with $\mathcal{P}$.
∎
References
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Daniel S. Farley and Bruce Hughes.
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arXiv:1206.2692.
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Groups Geom. Dyn., 3:299–315, 2009. |
Spatial properties of entangled photon pairs generated in nonlinear layered structures
Jan Peřina, Jr.
Palacký University, RCPTM, Joint Laboratory of
Optics, 17. listopadu 12, 771 46 Olomouc, Czech Republic
[email protected]
Abstract
A spatial quantum model of spontaneous parametric down-conversion
in nonlinear layered structures is developed expanding the
interacting vectorial fields into monochromatic plane waves. A
two-photon spectral amplitude depending on the signal- and
idler-field frequencies and propagation directions is used to
derive transverse profiles of the emitted fields as well as their
spatial correlations. Intensity spatial profiles and their spatial
correlations are mainly determined by the positions of
transmission peaks formed in these structures with photonic bands.
A method for geometry optimization of the structures with respect
to efficiency of the nonlinear process is suggested. Several
structures composed of GaN/AlN layers are analyzed as typical
examples. They allow the generation of photon pairs correlated in
several emission directions. Photon-pair generation rates
increasing better than the second power of the number of layers
can be reached. Also structures efficiently generated photon pairs
showing anti-bunching and anti-coalescence can be obtained. Three
reasons for splitting the correlated area in photonic-band-gap
structures are revealed: zig-zag movement of photons inside the
structure, spatial symmetry and polarization-dependent properties.
Also spectral splitting can be observed in these structures.
nonlinear photonic-band-gap structure, nonlinear layered
structure, entangled photon pair, correlated area
pacs: 42.65.Lm,42.70.Qs,42.50.Dv
I Introduction
The first temporal correlations between the signal and idler
photons generated in the process of spontaneous parametric
down-conversion (SPDC) were observed already more than thirty
years ago Hong et al. (1987). They manifested quantum entanglement in
the common state of both photons generated in one quantum event of
the spontaneous nonlinear process Mandel and Wolf (1995). Since then
the understanding of properties of such photon pairs has grown
enormously. Polarization properties and namely polarization
entanglement between the signal and idler photons originating in
tensorial character of the nonlinear susceptibility have attracted
the greatest attention from the very beginning. The reason lies in
the simplicity of such states described in the Hilbert space with
dimension $2\times 2$. This made these states easily
experimentally accessible. Despite their simplicity, such states
have allowed to reveal many substantial features of quantum
physics related to correlations between subsystems
Bou (2000). Violation of the Bell inequalities that
ruled out neoclassical theories Peřina et al. (1994), the effect of
collapse of a wave-function Peřina et al. (1994) and the ability of
teleportation of a quantum state Bouwmeester et al. (1997) belong to
the most important.
In the time domain, both photons occur in a very narrow temporal
window Hong et al. (1987) because they are emitted at one instant
after the annihilation of a pump photon. Moreover the conservation
law of energy dictates entanglement among monochromatic components
of the signal and idler fields Mandel and Wolf (1995). This
entanglement leads to a typical finite entanglement time between
the detection instants of the signal and idler photons observable
in a Hong-Ou-Mandel interferometer Hong et al. (1987) or
sum-frequency generation of the paired photons
Harris (2007); Brida et al. (2009). These properties have been
extensively studied namely in connection with the generation by
pulsed pump fields Keller and Rubin (1997); Peřina Jr.
et al. (1999) that allows
precise synchronization of photons from different photon pairs.
Spatial properties of photon pairs have attracted attention last.
Correlations in the transverse planes of the signal and idler
photons occur here due to geometric properties of a photon-pair
source and pump-beam spatial profile. They originate in the
necessity of reaching good spatial phase matching of the
interacting fields Mandel and Wolf (1995) that results in an efficient
nonlinear process. For example, the sum of wave vectors of the
signal and idler fields has to approximately give the wave vector
of the pump field to observe an efficient photon-pair generation
in an extended bulk crystal using collimated pumping. This results
in strong correlations in emission directions of the signal and
idler photons Joobeur et al. (1994, 1996); Vallone et al. (2007). These
correlations may even be exploited to ’transfer’ spatial
properties of the pump beam into spatial correlations of the
signal and idler beams Monken et al. (1998); Walborn et al. (2004). We may
observe analogy between spatial and spectral correlations
(entanglement) of two photons in a pair. Similarly as spectral
correlations may be tailored by a pump-field spectral profile,
spatial correlations may be controlled by a pump-beam spatial
profile. Also signal and idler electric-field phase variations in
the transverse plane belong to important characteristics of photon
pairs. They can be quantified in terms of eigenstates of the
angular orbital momentum operator Law and Eberly (2004). Under certain
conditions, entanglement between such states in the signal and
idler fields has been observed Mair et al. (2001); Oemrawsingh et al. (2005).
Experimentally, spatial correlations can be conveniently measured
by moving a fiber tip in the transverse plane, as it has been
done, e.g., in Molina-Terriza
et al. (2005). The use of an
intensified CCD camera has provided a more elegant way for the
experimental investigations of spatial correlations
Jost et al. (1998); Haderka et al. (2005). Spatial correlations between the
down-converted beams have also been extensively used for quantum
ghost imaging Rubin and Shih (2008). We note that spatial correlations
occur not only in spontaneous regime, they have been found also in
the regime with prevailing stimulated emission
Brambilla et al. (2004); Jedrkiewicz et al. (2004).
All these forms of entanglement are potentially interesting both
for fundamental physical experiments and practical applications
including metrology Migdall (1999), quantum cryptography
Bruß and Lütkenhaus (2000) and quantum-information processing
Bou (2000). In principle, all these forms of
entanglement can occur simultaneously depending on the source of
photon pairs. However, usual sources of photon pairs are
constructed such that only one or two forms of entanglement (e.g.,
polarization and spectral) are found and may efficiently be
experimentally exploited.
The effort to generate entangled photon pairs extended over as
many basis states as possible (and defined in several degrees of
freedom) and to modify ’the structure of entanglement’ belong
together with the effort to enhance quantum efficiency of the
generation process to the leitmotifs of the development in this
field. Bulk nonlinear crystals that were nearly exclusively used
at the beginning have been gradually replaced by more complex and
efficient nonlinear structures including poled nonlinear materials
Harris (2007); Kitaeva (2007); Svozilík and
Peřina Jr. (2009, 2010), nonlinear
waveguides URen et al. (2004); Spillane et al. (2007); Chen et al. (2009) and nonlinear
photonic structures Ber (2001). Nonlinear photonic
structures are extraordinarily interesting because they allow
efficient photon-pair generation owing to enhanced electric-field
amplitudes observed inside structures with photonic band-gaps
Yablonovitch (1987); John (1987) on one side, they also allow
relatively wide tailoring of properties of the emitted photon
pairs on the other side
Vamivakas et al. (2004); Centini et al. (2005); Peřina Jr.
et al. (2006). Structured
nonlinear fibers that rely on four-wave mixing
Li et al. (2005); Fulconis et al. (2005); Fan et al. (2005) represent a typical example. Or
waveguides with Bragg mirrors Abolghasem et al. (2009); Svozilík et al. (2011)
can be mentioned as perspective sources.
In the article, we consider systems composed of parallel nonlinear
layers. Back-scattering of the fields on the boundaries between
layers provides suitable conditions for the enhancement of
electric-field amplitudes under certain conditions Yeh (1988).
Moreover, spatial properties of photon pairs can be efficiently
tailored using parameters of these systems (e.g., the number of
layers). Nonlinear layered structures have already been studied in
the simplified geometry both in the framework of classical
Centini et al. (2005) and quantum Peřina Jr.
et al. (2006) descriptions.
Here, we generalize the quantum model present in
Peřina Jr.
et al. (2006) to the real spatial geometry including
vectorial character of the interacting fields. This allows us to
study transverse intensity profiles of the down-converted beams as
well as correlated areas of the signal and idler photons
considering typical layered structures made of GaN/AlN. We note
that GaN/AlN structures with random layers’ lengths have already
been studied as sources of photon pairs with ultra-narrow spectral
widths that originated in an optical analog of Anderson
localization of the down-converted fields
Peřina Jr.
et al. (2009a, b). It has also been shown that
GaN/AlN nonlinear layers allow the generation of photon-pair
states antisymmetric with respect to the exchange of the signal-
and idler-field frequencies that exhibit anti-bunching and
anti-coalescence Peřina Jr.
et al. (2007a).
The investigated layered structures as sources of photon pairs can
be compared with other photon-pair sources with respect to
intensity transverse profile, correlated area and efficiency of
photon-pair generation as follows. The vast majority of
photon-pair sources including wave-guiding structures and
nonlinear crystals with/withour poling are designed such that the
signal (and similarly the idler) photon is generated into one
compact and small emission area. In some sources, bulk crystals
are cut in such a way that the spatial symmetry allows to generate
photons around the whole cone surface Vallone et al. (2007). In this
case the generated state is entangled also in the wave vectors of
the signal and idler photons. As the Hilbert space corresponding
to this degree of freedom has a greater number of independent
states we obtain a higher-dimensional entangled state. In case of
layered structures, the signal (and idler) photon can even be
generated around several concentric cone surfaces depending on
complexity of the structure. This even enlarges the number of
independent states in the transverse area making the generated
state suitable for ’parallel processing’ of quantum information in
the transverse plane. As for correlated areas, they are described
by compact profiles (usually of an elliptical shape) in all common
photon-pair sources with cw pumping. Splitting of correlated areas
into several well-separated parts is a distinguished property of
nonlinear layered structures. Even three different mechanisms
leading to this splitting exist in layered structures: zig-zag
movement of photons inside the structure, spatial symmetry and
polarization-dependent properties. The last two mechanisms may in
principle occur also in other photon-pair sources. However,
enhancement of the fields’ amplitudes inside a structure is needed
to make them significant. Finally, periodically-poled nonlinear
crystals provide the greatest photon-pair fluxes. On the other
hand, wave-guiding structures including nonlinear planar
wave-guides and nonlinear structured fibers have the greatest
nonlinear conversion efficiencies due to the transverse
confinement of the interacting fields. However, they allow for
only moderate pumping intensities because of a possible material
damage and competing nonlinear processes. Nonlinear layered
structures lie in the middle in this comparison. They provide
greater nonlinear conversion efficiencies than bulk crystals owing
to enhanced electric-field amplitudes along the propagation
direction. Compared to wave-guiding structures, their nonlinear
conversion efficiencies are lower because the ’fields’
confinement’ is only in one dimension. On the other hand, similar
pumping intensities as those used for bulk crystals can be
applied.
The paper is organized as follows. A spatial vectorial quantum
model of SPDC in layered media is present in Sec. II and provides
quantities characterizing the emitted photon pairs. A systematic
method for designing efficient nonlinear structures is described
in Sec. III. Transverse intensity profiles and correlated areas
are discussed in Sec. IV using structures with different numbers
of layers. Conclusions are drawn in Sec. V.
II Spatial quantum model of spontaneous parametric
down-conversion
Nonlinear Hamiltonian $\hat{H}_{\rm int}$ characterizing SPDC in
a nonlinear medium of volume $\cal V$ at time $t$ can be
written in the form Mandel and Wolf (1995):
$$\displaystyle\hat{H}_{\rm int}(t)$$
$$\displaystyle=$$
$$\displaystyle\epsilon_{0}\int_{\cal V}d{\bf r}$$
(1)
$$\displaystyle{\bf d}({\bf r}):\left[{\bf E}_{p}^{(+)}({\bf r},t)\hat{\bf E}_{s%
}^{(-)}({\bf r},t)\hat{\bf E}_{i}^{(-)}({\bf r},t)+{\rm h.c.}\right].$$
In Eq. (1), ${\bf d}$ stands for a third-order tensor of
nonlinear coefficients and the symbol $:$ means shorthand of the
tensor ${\bf d}$ with respect to its three indices. A strong
pump field is described by the positive-frequency part ${\bf E}_{p}^{(+)}({\bf r},t)$ of its electric-field amplitude vector.
A signal [idler] field at single-photon level is characterized by
the negative-frequency part $\hat{\bf E}_{s}^{(-)}({\bf r},t)$
[$\hat{\bf E}_{s}^{(-)}({\bf r},t)$] of its electric-field
operator amplitude. Symbol $\epsilon_{0}$ denotes permittivity of
vacuum whereas ${\rm h.c.}$ stands for the Hermitian conjugate
term.
The positive-frequency amplitudes ${\bf E}_{m}^{(+)}({\bf r},t)$
of the interacting fields ($m=p,s,i$) can be in general
decomposed into plane waves with wave vectors ${\bf k}_{m}$ and
amplitudes ${\bf E}_{m}^{(+)}({\bf k}_{m})$:
$${\bf E}_{m}^{(+)}({\bf r},t)=\frac{1}{(\sqrt{2\pi})^{3}}\int d^{3}{\bf k}_{m}{%
\bf E}_{m}^{(+)}({\bf k}_{m})\exp(i{\bf k}_{m}{\bf r}-i\omega_{m}t);$$
(2)
$\omega_{m}$ is the frequency of field $m$ determined in
accordance with dispersion relations.
Considering the incident and un-depleted pump field, we assume
that its temporal spectrum ${\cal E}_{p}(\omega_{p})$ as well as
spatial spectrum ${\cal E}_{p}^{\rm tr}(k_{px},k_{py})$ in the
transverse plane are given. In this case, the decomposition of
amplitude ${\bf E}_{p}^{(+)}({\bf r},t)$ in Eq. (2) can be
rewritten as:
$$\displaystyle{\bf E}_{p}^{(+)}({\bf r},t)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{(\sqrt{2\pi}c)^{3}}\int_{-\pi/2}^{\pi/2}\sin(\vartheta_{%
p})d\vartheta_{p}\int_{-\pi/2}^{\pi/2}d\psi_{p}$$
(3)
$$\displaystyle\int_{0}^{\infty}\omega_{p}^{2}d\omega_{p}\;{\cal E}_{p}(\omega_{%
p}){\cal E}_{p}^{\rm tr}\left[k_{p,x}({\bf\Omega_{p}}),k_{p,y}({\bf\Omega_{p}}%
)\right]$$
$$\displaystyle\mbox{}\times\exp\left[ik_{p,x}({\bf\Omega_{p}})x+ik_{p,y}({\bf%
\Omega_{p}})y\right]\sum_{\alpha={\rm TE,TM}}{\bf E}_{p,\alpha}^{(+)}(z,{\bf%
\Omega_{p}})$$
$$\displaystyle\mbox{}\times\exp(-i\omega_{p}t);$$
using the vector ${\bf\Omega}_{p}\equiv(\omega_{p},\vartheta_{p},\psi_{p})$ of ’spherical coordinates’ $\omega_{p}$, $\vartheta_{p}$ and $\psi_{p}$. Speed of light in
vacuum is denoted as $c$. Assuming for simplicity air around the
structure the $x$ and $y$ components of wave vector ${\bf k}_{p}$ in front of the structure are given as:
$$\displaystyle k_{p,x}({\bf\Omega}_{p})$$
$$\displaystyle=$$
$$\displaystyle-\frac{\omega_{p}\sin(\psi_{p})\sin(\vartheta_{p})}{c},$$
$$\displaystyle k_{p,y}({\bf\Omega}_{p})$$
$$\displaystyle=$$
$$\displaystyle\frac{\omega_{p}\cos(\psi_{p})\sin(\vartheta_{p})}{c}.$$
(4)
The decomposition of pump-field amplitude ${\bf E}_{p}^{(+)}$
into TE and TM waves as given in Eq. (3) is done with
respect to the plane of incidence of the wave with wave vector ${\bf k}_{p}$ propagating through the layered structure (see
Fig. 1 for a scheme of the structure). We note that
projections of ${\bf k}$ vectors into the planes of boundaries
are conserved through the structure. The pump-field amplitudes ${\bf E}_{p,\alpha}^{(+)}$ introduced in Eq. (3) describe
field evolution along the $z$ axis where the field undergoes
back-scattering at the boundaries.
Considering a structure with $N$ layers and boundaries
perpendicular to the $z$ axis localized at positions $z_{n}$, $n=0,\ldots,N$, the pump-field amplitude ${\bf E}_{p,\alpha}^{(+)}$ can be written in the form (for more
details, see Peřina Jr.
et al. (2006)):
$$\displaystyle{\bf E}_{p,\alpha}^{(+)}(z,{\bf\Omega}_{p})$$
$$\displaystyle=$$
$$\displaystyle{\rm rect}_{-\infty,z_{0}}(z)\sum_{a=F,B}A_{p_{a},\alpha}^{(0)}({%
\bf\Omega}_{p})$$
(5)
$$\displaystyle\mbox{}\times{\bf e}_{p_{a},\alpha}^{(0)}({\bf\Omega}_{p})\exp%
\left[iK_{p_{a},z}^{(0)}({\bf\Omega}_{p})(z-z_{0})\right]$$
$$\displaystyle\mbox{}\hskip-8.535827pt+\sum_{l=1}^{N}{\rm rect}_{z_{l-1},z_{l}}%
(z)\sum_{a=F,B}A_{p_{a},\alpha}^{(l)}({\bf\Omega}_{p})$$
$$\displaystyle\mbox{}\hskip-28.452756pt\times{\bf e}_{p_{a},\alpha}^{(l)}({\bf%
\Omega}_{p})\exp\left[iK_{p_{a},z}^{(l)}({\bf\Omega}_{p})(z-z_{l-1})\right]$$
$$\displaystyle\mbox{}\hskip-8.535827pt+{\rm rect}_{z_{N},\infty}(z)\sum_{a=F,B}%
A_{p_{a},\alpha}^{(N+1)}({\bf\Omega}_{p})$$
$$\displaystyle\mbox{}\hskip-28.452756pt\times{\bf e}_{p_{a},\alpha}^{(N+1)}({%
\bf\Omega}_{p})\exp\left[iK_{p_{a},z}^{(N+1)}({\bf\Omega}_{p})(z-z_{N})\right];$$
$$\displaystyle\alpha={\rm TE,TM}.$$
The function ${\rm rect}_{z_{a},z_{b}}(z)$ equals one for $z_{a}\leq z\leq z_{b}$ and is zero otherwise. Symbols ${\bf e}^{(l)}_{p_{F},\alpha}$ and ${\bf e}^{(l)}_{p_{B},\alpha}$ denote
polarization vectors of $\alpha$ waves for forward- and
backward-propagating fields with respect to the $+z$ axis,
respectively.
The $z$ component $K_{p_{a},z}^{(l)}({\bf\Omega}_{p})$ of wave
vector in an $l$th layer belonging to wave $a$ with frequency
$\omega_{p}$ and propagating in direction ($\vartheta_{p},\psi_{p}$) in front of the structure can be expressed as:
$$K_{p_{a},z}^{(l)}({\bf\Omega}_{p})=\pm\frac{n_{p}^{(l)}(\omega_{p})\omega_{p}}%
{c}\cos(\vartheta_{p}^{(l)}),$$
(6)
where the sign $+$ ($-$) is appropriate for a forward-
(backward-) propagating wave. Index of refraction of the pump
field in an $l$th layer is denoted as $n_{p}^{(l)}$. As we
assume that the layered structure is surrounded by air, we have $n_{p}^{(0)}=n_{p}^{(N+1)}=1$. The propagation angle $\vartheta_{p}^{(l)}$ in the $l$th layer is then derived from
Snell’s law:
$$n_{p}^{(0)}\sin(\vartheta_{p}^{(0)})=n_{p}^{(l)}\sin(\vartheta_{p}^{(l)}),%
\hskip 14.226378ptl=1,\ldots,N+1;$$
(7)
$\vartheta_{p}^{(0)}\equiv\vartheta_{p}$.
The coefficients $A_{p_{F},\alpha}^{(l)}({\bf\Omega}_{p})$ and $A_{p_{B},\alpha}^{(l)}({\bf\Omega}_{p})$ introduced in Eq. (5)
determine amplitudes of $\alpha$ waves with frequency $\omega_{p}$ propagating forward and backward, respectively, in the ($\vartheta_{p},\psi_{p}$) direction. These coefficients are derived
from Fresnel’s relations at the boundaries using, e.g., the
transfer-matrix formalism Yeh (1988). Values of the
coefficients $A_{p_{F},\alpha}^{(0)}({\bf\Omega}_{p})$ and $A_{p_{B},\alpha}^{(N+1)}({\bf\Omega}_{p})$ for $\alpha={\rm TE,TM}$ characterize the pump fields incident on the structure from
both sides and represent the boundary conditions. The
transfer-matrix formalism has been elaborated in detail in
Peřina Jr.
et al. (2006) for layered structures. The coefficients $A^{(l)}$ of the fields with wave vectors lying in the plane of
incidence given by an angle $\psi_{p}$ are given by Eq. (23) of
Ref. Peřina Jr.
et al. (2006).
The signal and idler fields are at single-photon level in the
spontaneous process and so they have to be treated using quantum
theory Mandel and Wolf (1995). Their electric-field operator amplitudes
in the layered structure can be decomposed into plane waves using
Eq. (1) and subsequently described in analogy to the pump
field. Using the ’spherical coordinates’ $\omega_{m}$ $,\vartheta_{m}$ and $\psi_{m}$ and defining ${\bf\Omega}_{m}=(\omega_{m},\vartheta_{m},\psi_{m})$, $m=s,i$, we can express the
positive-frequency part $\hat{E}_{m}^{(+)}$ of the
electric-field operator amplitude of mode $m$ as:
$$\displaystyle\hat{\bf E}_{m}^{(+)}({\bf r},t)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{c^{3}}\int_{-\pi/2}^{\pi/2}\sin(\vartheta_{m})d\vartheta%
_{m}\int_{-\pi/2}^{\pi/2}d\psi_{m}$$
(8)
$$\displaystyle\int_{0}^{\infty}\omega_{m}^{2}d\omega_{m}\exp\left[ik_{m,x}({\bf%
\Omega}_{m})x+ik_{m,y}({\bf\Omega}_{m})y\right]$$
$$\displaystyle\mbox{}\times\sum_{\alpha={\rm TE,TM}}\sqrt{\frac{\hbar\omega_{m}%
}{16\pi^{3}\epsilon_{0}}}\;\hat{\bf a}_{m,\alpha}(z,{\bf\Omega}_{m})\exp(-i%
\omega_{m}t);$$
$\hbar$ stands for the reduced Planck constant and $\hat{\bf E}_{m}^{(-)}=\hat{\bf E}_{m}^{(+)\dagger}$ . The expression $\sqrt{\hbar\omega_{m}/(16\pi^{3}\epsilon_{0})}$ gives an
electric-field amplitude per one photon with energy $\hbar\omega_{m}$ propagating at speed $c$. The $x$ and $y$
components of wave vectors ${\bf k}_{m}$ of the signal and idler
fields ($m=s,i$) are defined outside the structure:
$$\displaystyle k_{m,x}({\bf\Omega}_{m})$$
$$\displaystyle=$$
$$\displaystyle-\frac{\omega_{m}\sin(\psi_{m})\sin(\vartheta_{m})}{c},$$
$$\displaystyle k_{m,y}({\bf\Omega}_{m})$$
$$\displaystyle=$$
$$\displaystyle\frac{\omega_{m}\cos(\psi_{m})\sin(\vartheta_{m})}{c}.$$
(9)
The operator amplitudes $\hat{\bf a}_{m,\alpha}(z,{\bf\Omega}_{m})$ introduced in Eq. (8) can be derived in the considered
layered structure in the form:
$$\displaystyle\hat{\bf a}_{m,\alpha}(z,{\bf\Omega}_{m})$$
$$\displaystyle=$$
$$\displaystyle{\rm rect}_{-\infty,z_{0}}(z)\sum_{a=F,B}\hat{a}_{m_{a},\alpha}^{%
(0)}({\bf\Omega}_{m})$$
(10)
$$\displaystyle\mbox{}\times{\bf e}_{m_{a},\alpha}^{(0)}({\bf\Omega}_{m})\exp%
\left[iK_{m_{a},z}^{(0)}({\bf\Omega}_{m})(z-z_{0})\right]$$
$$\displaystyle\mbox{}\hskip-8.535827pt+\sum_{l=1}^{N}{\rm rect}_{z_{l-1},z_{l}}%
(z)\sum_{a=F,B}\hat{a}_{m_{a},\alpha}^{(l)}({\bf\Omega}_{m})$$
$$\displaystyle\mbox{}\hskip-28.452756pt\times{\bf e}_{m_{a},\alpha}^{(l)}\exp%
\left[iK_{m_{a},z}^{(l)}({\bf\Omega}_{m})(z-z_{l-1})\right]$$
$$\displaystyle\mbox{}\hskip-8.535827pt+{\rm rect}_{z_{N},\infty}(z)\sum_{a=F,B}%
\hat{a}_{m_{a},\alpha}^{(N+1)}({\bf\Omega}_{m})$$
$$\displaystyle\mbox{}\hskip-28.452756pt\times{\bf e}_{m_{a},\alpha}^{(N+1)}({%
\bf\Omega}_{m})\exp\left[iK_{m_{a},z}^{(N+1)}({\bf\Omega}_{m})(z-z_{N})\right];$$
$$\displaystyle m=s,i;\hskip 14.226378pt\alpha={\rm TE,TM}.$$
Here, the symbols ${\bf e}^{(l)}_{m_{F},\alpha}$ and ${\bf e}^{(l)}_{m_{B},\alpha}$ stand for polarization vectors of $\alpha$ wave of field $m$ propagating forward and backward,
respectively. The annihilation operators $\hat{a}_{m_{a},\alpha}^{(l)}({\bf\Omega}_{m})$ are defined at the
end of an $l$th layer for the $\alpha$ wave with frequency $\omega_{m}$ of field $m$ propagating along the direction ($\vartheta_{m},\psi_{m}$) either forward ($a=F$) or backward ($a=B$).
The $z$ component $K_{m_{a},z}^{(l)}({\bf\Omega}_{m})$ of wave
vector in the $l$th layer is determined as:
$$\displaystyle K_{m_{a},z}^{(l)}({\bf\Omega}_{m})=\pm\frac{n_{m}^{(l)}(\omega_{%
m})\omega_{m}}{c}\cos(\vartheta_{m}^{(l)}),$$
$$\displaystyle m=s,i;\hskip 14.226378pta=F,B.$$
(11)
The sign $+$ ($-$) stands for a forward- (backward-)
propagating wave and $n_{m}^{(l)}$ gives index of refraction of
field $m$ in the $l$th layer. Snell’s law provides the
propagation angles $\vartheta_{m}^{(l)}$ needed in the $l$th
layer:
$$\displaystyle n_{m}^{(l)}(\omega_{m})\sin(\vartheta_{m}^{(l)})=n_{m}^{(N+1)}(%
\omega_{m})\sin(\vartheta_{m}^{(N+1)}),$$
$$\displaystyle \hskip 2.845276ptl=0,\ldots,N;$$
(12)
$\vartheta_{m}^{(N+1)}\equiv\vartheta_{m}$.
The operators $\hat{a}_{m_{a},\alpha}^{(l)}({\bf\Omega}_{m})$ for $l=0,\ldots,N+1$ and fixed $m$, $\alpha$, $\omega_{m}$, $\vartheta_{m}$, and $\psi_{m}$ are connected by unitary
transformations at boundaries (Fresnel’s relations) and free-field
propagation transformations. This means that the usual boson
commutation relations obeyed by the incident fields are
’transferred’ through the structure and the only nonzero
commutation relations Vogel et al. (2001) are the following ones:
$$\displaystyle[\hat{a}_{m_{a},\alpha}^{(l)}({\bf\Omega}_{m}),\hat{a}_{{m^{%
\prime}}_{a^{\prime}},\alpha^{\prime}}^{(l^{\prime})\dagger}({\bf\Omega}^{%
\prime}_{m^{\prime}})]$$
$$\displaystyle=$$
$$\displaystyle\frac{c^{2}}{|\sin(\vartheta_{m})|\omega_{m}^{2}}\delta_{m,m^{%
\prime}}\delta_{a,a^{\prime}}$$
(13)
$$\displaystyle\delta_{\alpha,\alpha^{\prime}}\delta_{l,l^{\prime}}\delta(\omega%
_{m}-\omega^{\prime}_{m^{\prime}})\delta(\vartheta_{m}-\vartheta^{\prime}_{m^{%
\prime}})\delta(\psi_{m}-\psi^{\prime}_{m^{\prime}}).$$
The transfer-matrix formalism allows to express the operators $\hat{a}_{m_{a},\alpha}^{(l)}({\bf\Omega}_{m})$ in terms of the
operators $\hat{a}_{m_{F},\alpha}^{(N+1)}({\bf\Omega}_{m})$ and $\hat{a}_{m_{B},\alpha}^{(0)}({\bf\Omega}_{m})$ describing the
outgoing fields. The appropriate relations valid for fields with
wave vectors lying in the plane of incidence given by an angle $\psi_{m}$ are given in Eq. (21) of Ref. Peřina Jr.
et al. (2006).
The generation of a photon pair in the process of SPDC governed by
the Hamiltonian $\hat{H}_{\rm int}$ in Eq. (1) is
described by a perturbation solution of the Schrödinger
equation to the first order at time $t\rightarrow\infty$
assuming an incident vacuum state $|{\rm vac}\rangle$ in the
signal and idler fields at time $t\rightarrow-\infty$. The
resulting state $|\psi\rangle_{s,i}^{\rm out}$ can be derived in
the form:
$$\displaystyle|\psi\rangle_{s,i}^{\rm out}$$
$$\displaystyle=$$
$$\displaystyle|{\rm vac}\rangle-\frac{i}{2c^{8}}\sum_{l=1}^{N}\sum_{a,b,g=F,B}%
\sum_{\alpha,\beta,\gamma={\rm TE,TM}}$$
(14)
$$\displaystyle\prod_{m=p,s,i}\int_{-\pi/2}^{\pi/2}\sin(\vartheta_{m})d\vartheta%
_{m}\int_{-\pi/2}^{\pi/2}d\psi_{m}\int_{0}^{\infty}\omega_{m}^{2}d\omega_{m}$$
$$\displaystyle\sqrt{\omega_{s}\omega_{i}}\;{\cal E}_{p}(\omega_{p}){\cal E}_{p}%
^{\rm tr}(k_{p,x},k_{p,y})\delta(\omega_{p}-\omega_{s}-\omega_{i})$$
$$\displaystyle\mbox{}\times\delta\left[k_{p,x}-k_{s,x}-k_{i,x}\right]\delta%
\left[k_{p,y}-k_{s,y}-k_{i,y}\right]$$
$$\displaystyle\mbox{}\times{\bf d}^{(l)}:{\bf e}_{p_{a},\alpha}^{(l)}({\bf%
\Omega}_{m}){\bf e}_{s_{b},\beta}^{(l)*}({\bf\Omega}_{s}){\bf e}_{i_{g},\gamma%
}^{(l)*}({\bf\Omega}_{i})$$
$$\displaystyle\mbox{}\times\exp\left[\frac{i}{2}\Delta K_{p_{a}s_{b}i_{g},z}^{(%
l)}({\bf\Omega}_{p},{\bf\Omega}_{s},{\bf\Omega}_{i})L_{l}\right]$$
$$\displaystyle\mbox{}\times L_{l}{\rm sinc}\left[\frac{1}{2}\Delta K_{p_{a}s_{b%
}i_{g},z}^{(l)}({\bf\Omega}_{p},{\bf\Omega}_{s},{\bf\Omega}_{i})L_{l}\right]$$
$$\displaystyle\mbox{}\times A_{p_{a},\alpha}^{(l)}({\bf\Omega}_{m})\hat{a}_{s_{%
b},\beta}^{(l)\dagger}({\bf\Omega}_{s})\hat{a}_{i_{g},\gamma}^{(l)\dagger}({%
\bf\Omega}_{i})|{\rm vac}\rangle.$$
The functions $\Delta K_{p_{a}s_{b}i_{g},z}^{(l)}({\bf\Omega}_{p},{\bf\Omega}_{s},{\bf\Omega}%
_{i})=K_{p_{a},z}^{(l)}({\bf\Omega}_{p})-K_{s_{b},z}^{(l)}({\bf\Omega}_{s})-K_%
{i_{g},z}^{(l)}({\bf\Omega}_{i})$ stand for phase mismatches in the $l$th layer. Symbol $L_{l}$
means the length of $l$th layer ($L_{l}=z_{l}-z_{l-1}$). The
transverse wave vectors $k_{m,x}({\bf\Omega}_{m})$ and $k_{m,y}({\bf\Omega}_{m})$ for $m=p,s,i$ are defined in
Eqs. (4) and (9) and characterize the fields outside
the structure. We note that the approach based on the solution of
Schrödinger equation does not take into account surface SPDC
that generates additional photon pairs
Peřina Jr.
et al. (2009c, d).
The phase-matching conditions in the transverse plane $xy$ are
described by two $\delta$ functions in Eq. (14), that
determine the emission direction ($\vartheta_{i},\psi_{i}$) of an
idler photon provided that the signal-photon emission direction ($\vartheta_{s},\psi_{s}$) is given and the pump-field is in the form
of a plane wave propagating in direction ($\vartheta_{p},\psi_{p}$).
Simple geometric considerations provide the following formulas:
$$\displaystyle\psi_{i}$$
$$\displaystyle=$$
$$\displaystyle\psi_{p}+\arctan\left[\frac{\omega_{s}\sin(\vartheta_{s})\sin(%
\psi_{p}-\psi_{s})}{\omega_{p}\sin(\vartheta_{p})-\omega_{s}\sin(\vartheta_{s}%
)\cos(\psi_{p}-\psi_{s})}\right],$$
$$\displaystyle\vartheta_{i}$$
$$\displaystyle=$$
$$\displaystyle\arcsin\left[\frac{\omega_{p}\sin(\vartheta_{p})}{\omega_{i}\cos(%
\psi_{p}-\psi_{i})}\right.$$
(15)
$$\displaystyle\mbox{}\hskip 56.905512pt\left.-\frac{\omega_{s}\cos(\psi_{p}-%
\psi_{s})}{\omega_{i}\cos(\psi_{p}-\psi_{i})}\sin(\vartheta_{s})\right].$$
If the pump beam is focused, the strict phase-matching conditions
in the transverse plane formulated in Eq. (15) are relaxed
and we arrive at correlation areas with finite spreads.
The expression in Eq. (14) for the state $|\psi\rangle_{si}^{\rm out}$ can be rewritten into the form
containing only the outgoing creation operators $\hat{a}_{m_{F},\beta}^{(N+1)\dagger}({\bf\Omega}_{m})$ and $\hat{a}_{m_{B},\beta}^{(0)\dagger}({\bf\Omega}_{m})$ for $m=s,i$
and $\beta={\rm TE,TM}$ using the formulas in Eqs. (21) and
(23) of Ref. Peřina Jr.
et al. (2006). The outgoing operators can be
finally transformed into the polarization basis of the detector
using a suitable unitary transformation. We assume that the
detection plane is perpendicular to the field propagation
direction ($\vartheta,\psi$) and its $s$-polarization
(denoted as $\perp$) is parallel to the horizontal plane $xz$;
$p$-polarization (denoted as $\|$) is defined by the
orthogonality conditions. Assuming field $m$ ($m=s,i$) at
frequency $\omega_{m}$ and propagating along the angles $\vartheta_{m}$ and $\psi_{m}$ the needed $\vartheta_{m}$- and $\psi_{m}$-dependent unitary transformation can be written in the
form:
$$\displaystyle\left[\matrix{\hat{a}_{m_{F},{\rm TE}}^{(N+1)}({\bf\Omega}_{m})%
\cr\hat{a}_{m_{F},{\rm TM}}^{(N+1)}({\bf\Omega}_{m})}\right]$$
$$\displaystyle=$$
$$\displaystyle\left[\matrix{\cos(\zeta_{m})&\sin(\zeta_{m})\cr-\sin(\zeta_{m})&%
\cos(\zeta_{m})}\right]\left[\matrix{\hat{a}_{m_{F},\perp}({\bf\Omega}_{m})\cr%
\hat{a}_{m_{F},\|}({\bf\Omega}_{m})}\right],$$
$$\displaystyle\left[\matrix{\hat{a}_{m_{B},{\rm TE}}^{(0)}({\bf\Omega}_{m})\cr%
\hat{a}_{m_{B},{\rm TM}}^{(0)}({\bf\Omega}_{m})}\right]$$
$$\displaystyle=$$
$$\displaystyle\left[\matrix{\cos(\zeta_{m})&\sin(\zeta_{m})\cr-\sin(\zeta_{m})&%
\cos(\zeta_{m})}\right]\left[\matrix{\hat{a}_{m_{B},\perp}({\bf\Omega}_{m})\cr%
\hat{a}_{m_{B},\|}({\bf\Omega}_{m})}\right],$$
$$\displaystyle\zeta_{m}(\vartheta_{m},\psi_{m})$$
$$\displaystyle=$$
$$\displaystyle\arccos\left[\frac{\cos(\psi_{m})}{\sqrt{1+\sin^{2}(\psi_{m})\tan%
^{2}(\theta_{m})}}\right]$$
(16)
$$\displaystyle\mbox{}\times{\rm sign}(\psi_{m}),\hskip 28.452756ptm=s,i.$$
The function ${\rm sign}$ gives the sign of its argument. The
newly introduced annihilation operators $\hat{a}_{m_{b},\alpha}({\bf\Omega}_{m})$, $m=s,i$, $b=F,B$, $\alpha=\|,\perp$, describe the signal and idler fields in the
polarization bases connected with the detectors.
The terms of state $|\psi\rangle^{\rm out}_{s,i}$ in
Eq. (14) describing the created photon pair can be
decomposed into four groups according to the propagation
directions of the signal and idler photons with respect to the $+z$ axis ($FF$, $FB$, $BF$, $BB$). Inside these groups
there occur four contributions that differ in signal- and
idler-photon polarization directions ($\|\,\|$, $\|\,\perp$, $\perp\,\|$, $\perp\,\perp$). Each contribution can be
written in the form:
$$\displaystyle|\psi\rangle_{s_{a},i_{b}}^{\alpha,\beta}({\bf r},t)$$
$$\displaystyle=$$
$$\displaystyle\prod_{m=s,i}\int_{-\pi/2}^{\pi/2}\sin(\vartheta_{m})d\vartheta_{%
m}\int_{-\pi/2}^{\pi/2}d\psi_{m}$$
(17)
$$\displaystyle\int_{0}^{\infty}d\omega_{m}\phi_{ab}^{\alpha,\beta}({\bf\Omega}_%
{s},{\bf\Omega}_{i})\hat{a}_{s_{a},\alpha}^{\dagger}({\bf\Omega}_{s})\hat{a}_{%
i_{b},\beta}^{\dagger}({\bf\Omega}_{i})|{\rm vac}\rangle$$
$$\displaystyle\mbox{}\hskip-65.441339pt\times\exp[-i({\bf k}_{s_{a}}^{\rm out}+%
{\bf k}_{i_{b}}^{\rm out}){\bf r}]\exp[i(\omega_{s}+\omega_{i})t];$$
$$\displaystyle a,b=F,B;\hskip 14.226378pt\alpha,\beta=\|,\perp.$$
The wave vectors ${\bf k}_{s_{a}}^{\rm out}$ and ${\bf k}_{i_{b}}^{\rm out}$ characterize free-field evolution of the
emitted signal and idler fields, respectively, outside the
structure. The functions $\phi_{ab}^{\alpha,\beta}({\bf\Omega}_{s},{\bf\Omega}_{i})$ introduced in Eq. (17)
characterize completely properties of the generated photon pair.
It gives the probability amplitude of having an $\alpha$-polarized signal photon at frequency $\omega_{s}$ propagating
along direction ($\vartheta_{s},\psi_{s}$) together with its $\beta$-polarized idler photon at frequency $\omega_{i}$
propagating along direction ($\vartheta_{i},\psi_{i}$) at the
output $ab$ of the structure.
Intensity spatial and spectral properties of a photon pair
Peřina Jr.
et al. (2006) can be conveniently derived from a density $n_{ab}^{\alpha,\beta}$ of the mean photon-pair numbers belonging
to the state $|\psi\rangle_{s_{a},i_{b}}^{\alpha,\beta}$. The
density $n_{ab}^{\alpha,\beta}$ is defined along the expression
$$\displaystyle n_{ab}^{\alpha,\beta}({\bf\Omega}_{s},{\bf\Omega}_{i})={}_{s_{a}%
,i_{b}}^{\alpha,\beta}\langle\psi|\hat{n}_{s_{a},\alpha}({\bf\Omega}_{s})\hat{%
n}_{i_{b},\beta}({\bf\Omega}_{i})|\psi\rangle_{s_{a},i_{b}}^{\alpha,\beta}.$$
(18)
The photon-number density operator $\hat{n}_{m_{a},\alpha}({\bf\Omega}_{m})$ is expressed as
$$\displaystyle\hat{n}_{m_{a},\alpha}({\bf\Omega}_{m})$$
$$\displaystyle=$$
$$\displaystyle\hat{a}_{m_{a},\alpha}^{\dagger}({\bf\Omega}_{m})\hat{a}_{m_{a},%
\alpha}({\bf\Omega}_{m}).$$
(19)
Using Eq. (17) the formula for density $n_{ab}^{\alpha,\beta}$ of mean photon-pair numbers in Eq.
(18) attains a simple form:
$$n_{ab}^{\alpha,\beta}({\bf\Omega}_{s},{\bf\Omega}_{i})=|\phi_{ab}^{\alpha,%
\beta}({\bf\Omega}_{s},{\bf\Omega}_{i})|^{2}.$$
(20)
A density $n_{s,ab}^{\alpha,\beta}({\bf\Omega}_{s})$ of mean
signal-photon numbers in the state $|\psi\rangle_{s_{a},i_{b}}^{\alpha,\beta}$ can easily be determined
from the density $n_{ab}^{\alpha,\beta}$ defined in
Eq. (18) and using the relation in Eq. (20):
$$\displaystyle n_{s,ab}^{\alpha,\beta}({\bf\Omega}_{s})$$
$$\displaystyle=$$
$$\displaystyle\int_{-\pi/2}^{\pi/2}\sin(\vartheta_{i})d\vartheta_{i}\int_{-\pi/%
2}^{\pi/2}d\psi_{i}\int_{0}^{\infty}d\omega_{i}$$
(21)
$$\displaystyle|\phi_{ab}^{\alpha,\beta}({\bf\Omega}_{s},{\bf\Omega}_{i})|^{2}.$$
If spectral resolution in detection of the signal-field transverse
profile is not available, a spatial density $n_{s,ab}^{{\rm tr},\alpha,\beta}(\vartheta_{s},\psi_{s})$ of mean signal-photon
numbers emitted in the direction ($\vartheta_{s},\psi_{s}$) is a
useful characteristic. It can be derived using the density $n_{s,ab}^{\alpha,\beta}$ of mean signal-photon numbers given in
Eq. (21):
$$\displaystyle n_{s,ab}^{{\rm tr},\alpha,\beta}(\vartheta_{s},\psi_{s})$$
$$\displaystyle=$$
$$\displaystyle\int_{0}^{\infty}d\omega_{s}n_{s,ab}^{\alpha,\beta}({\bf\Omega}_{%
s}).$$
(22)
The above densities can be analogously defined also for the idler
field.
Spatial correlations between the signal- and idler-field photon
numbers in their transverse planes can be quantified in terms of
the fourth-order correlation functions $n_{ab}^{{\rm cor},\alpha,\beta}(\vartheta_{s},\psi_{s},\vartheta_{i},\psi_{i})$ that
give joint densities of photon-pair numbers such that a signal
photon propagates along the ($\vartheta_{s},\psi_{s}$) direction and
the idler twin propagates along the ($\vartheta_{i},\psi_{i}$)
direction:
$$\displaystyle n_{ab}^{{\rm cor},\alpha,\beta}(\vartheta_{s},\psi_{s},\vartheta%
_{i},\psi_{i})$$
$$\displaystyle=$$
$$\displaystyle\int_{0}^{\infty}d\omega_{s}\int_{0}^{\infty}d\omega_{i}n_{ab}^{%
\alpha,\beta}({\bf\Omega}_{s},{\bf\Omega}_{i}).$$
Provided that the signal-photon propagation direction ($\vartheta_{s}^{0},\psi_{s}^{0}$) is given, the joint density $n_{ab}^{{\rm cor},\alpha,\beta}(\vartheta_{s}^{0},\psi_{s}^{0},\vartheta_{i},%
\psi_{i})$
remains a function of the idler-field emission angles $\vartheta_{i}$ and $\psi_{i}$ and its profile defines a correlated
area. The correlated area of an idler photon determines an area in
the idler-field transverse plane where an idler photon can be
expected provided that its signal twin has been detected in the ($\vartheta_{s}^{0},\psi_{s}^{0}$) direction.
Finally, an overall mean photon-pair number $N_{ab}^{\alpha,\beta}$ related to the state $|\psi\rangle_{s_{a},i_{b}}^{\alpha,\beta}$ can be found using the
following relation:
$$\displaystyle N_{ab}^{\alpha,\beta}$$
$$\displaystyle=$$
$$\displaystyle\prod_{m=s,i}\int_{-\pi/2}^{\pi/2}\sin(\vartheta_{m})d\vartheta_{%
m}\int_{-\pi/2}^{\pi/2}d\psi_{m}\int_{0}^{\infty}d\omega_{m}$$
(24)
$$\displaystyle n_{ab}^{\alpha,\beta}({\bf\Omega}_{s},{\bf\Omega}_{i}).$$
An important feature of layered structures is an increase of the
efficiency of nonlinear process due to the enhanced electric-field
amplitudes caused by interference along the $z$ axis. This
increase can be quantified with respect to a certain reference
structure which fully exploits the nonlinearity, but does not rely
on interference. In this reference structure, there occurs no
back-scattering of the propagating fields and also the nonlinear
process is assumed to be fully phase matched. The orientations of
nonlinear layers and polarizations of the interacting fields are
such that the most intense nonlinear effect occurs. This reference
structure provides an ideal photon pair with a signal photon
emitted in an arbitrary direction ($\vartheta_{s},\psi_{s}$) and an
idler photon in the corresponding ($\vartheta_{i},\psi_{i}$)
direction. Fixing $\vartheta_{s}$ and $\psi_{s}$, the emitted pair
can be described by the following output state $|\psi\rangle_{s,i}^{\rm ref}$ [compare Eq. (14)]:
$$\displaystyle|\psi\rangle_{s,i}^{\rm ref}$$
$$\displaystyle=$$
$$\displaystyle-\frac{i}{2c^{8}}\int_{0}^{\infty}\,\omega_{s}^{2}d\omega_{s}\int%
_{0}^{\infty}\,\omega_{i}^{2}d\omega_{i}\sqrt{\omega_{s}\omega_{i}}$$
(25)
$$\displaystyle\mbox{}{\cal E}_{p}(\omega_{s}+\omega_{i})\sum_{l=1}^{N}\max({\bf
d%
}^{(l)})L_{l}\hat{a}_{s}^{\dagger}(\omega_{s})\hat{a}_{i}^{\dagger}(\omega_{i}%
)|{\rm vac}\rangle,$$
where $\hat{a}_{s}^{\dagger}(\omega_{s})$ [$\hat{a}_{i}^{\dagger}(\omega_{i})$] stands for a signal- [idler-]
field creation operator outside the reference structure. The
function $\max$ used in Eq. (25) gives the maximum value
among the elements of tensor ${\bf d}^{(l)}$.
Using the reference structure a relative density $\eta_{s,ab}^{\alpha,\beta}({\bf\Omega}_{s})$ of mean signal-photon
numbers belonging to the state $|\psi\rangle_{s_{a},i_{b}}^{\alpha,\beta}$ can naturally be defined
as
$$\eta_{s,ab}^{\alpha,\beta}({\bf\Omega}_{s})=\frac{n_{s,ab}^{\alpha,\beta}({\bf%
\Omega}_{s})}{n_{s}^{\rm ref}(\omega_{s})},$$
(26)
where the signal-field photon-number density $n_{s,ab}^{\alpha,\beta}$ is written in Eq. (21). The
reference signal-field photon-number density $n_{s}^{\rm ref}$
characterizes the state $|\psi\rangle_{s,i}^{\rm ref}$ in
Eq. (25) and does not depend on the propagation angles $\vartheta_{s}$ and $\psi_{s}$.
Properties of photon pairs in the time domain are complementary to
those found in the spectral domain and belong to important
characteristics of photon pairs. We can mention signal- and
idler-field photon fluxes or coincidence-count interference
patterns in different kinds of interferometers as examples. These
photon-pair properties can be investigated, e.g., using the
formulas contained in Sec. IIC of Ref. Peřina Jr.
et al. (2006) even in
this spatial vectorial model.
Numerical calculations that follow are performed for a cw pump
field with a Gaussian transverse profile described as follows:
$$\displaystyle{\cal E}_{p}(\omega_{p})$$
$$\displaystyle=$$
$$\displaystyle\xi_{p}\delta(\omega_{p}-\omega_{p}^{0}),$$
$$\displaystyle{\cal E}_{p}^{\rm tr}(k_{x},k_{y})$$
$$\displaystyle=$$
$$\displaystyle\frac{r_{p}}{\sqrt{2\pi}}\exp\left[\frac{r^{2}_{p}(k_{x}^{2}+k_{y%
}^{2})}{4}\right].$$
(27)
In Eq. (27), $\xi_{p}$ gives an amplitude of the pump field
with the carrying frequency $\omega_{p}^{0}$; $r_{p}$ denotes the
width of amplitude transverse profile. Normalization of the
function ${\cal E}_{p}^{\rm tr}$ is such that $\int dk_{x}\int dk_{y}|{\cal E}_{p}^{\rm tr}(k_{x},k_{y})|^{2}=1$. In case of cw
pumping, there occur formulas in the above equations that contain
a formal expression $\delta^{2}(\omega)$. This expression has to
be replaced by the expression $2T/(2\pi)\delta(\omega)$, where
the detection interval extends over $(-T,T)$ and the limit $T\rightarrow\infty$ may be considered.
III Design of an efficient layered structure
Here we consider structures with odd numbers $N$ of layers made
of two kinds of materials. Layers of material $b$ of length $l_{b}$ are sandwiched by layers of material $a$ having lengths $l_{a}$. Also pumping at a defined carrying frequency $\omega_{p}^{0}$
and impinging on the structure at normal incidence is assumed. The
down-converted signal and idler fields are assumed to have nearly
degenerate frequencies.
We suggest a method for designing an efficient layered structure
from the point of view of, in general, three-mode nonlinear
interaction. It is based on two observations:
•
An efficient nonlinear process occurs provided that all three
nonlinearly interacting fields lie inside their transmission
peaks. This behavior originates in the fact that the
electric-field amplitudes of monochromatic fields with frequencies
in transmission peaks are enhanced inside the structure owing to
constructive interference of back-scattered light. It follows from
the band-gap theory that the closer the transmission peak to a
band gap, the greater the electric-field amplitudes.
•
As numerical calculations have revealed, the overlap integral over the
amplitudes of three interacting fields giving the strength of the
effective nonlinear interaction [see Eq. (1)] vanishes if
the signal- and idler-field amplitudes along the structure are the
same. This means that photon-pair states degenerate in
frequencies, emitted in symmetric directions and having the same
polarizations cannot be generated.
We note that these facts have been found crucial in designing
layered structures efficient for second-harmonic generation
Scalora et al. (1997). Considering a collinear interaction, the
requirements are even more strict because of only one propagation
direction. This requires specific approaches that rely, e.g., on
tuning the frequencies of transmission peaks by changing the index
of refraction of one type of the layers Scalora et al. (1997).
Alternatively, non-collinear second-harmonic generation has been
considered. In this case, there exists one free parameter (radial
emission angle) that can be varied in order to find an efficient
structure. This represents an equivalent problem to that
considered here and can be treated by the developed systematic
approach.
Returning back to the considered layered structures they are
characterized by three parameters: number $N$ of layers and
lengths $l_{a}$ and $l_{b}$ of these layers. A detailed inspection
has revealed that the number $N$ of layers significantly
determines the number of generated photon pairs as well as angular
extensions of the emission areas of photons in a pair (see
Fig. 3 bellow). The greater the number $N$ of layers
the greater the number of generated photon pairs and also the
smaller the angular extensions of emission areas. From practical
point of view, the number $N$ of layers is approximately fixed
considering these dependencies. Having the number $N$ of layers
fixed, there remain two adjustable parameters - layers’ lengths $l_{a}$ and $l_{b}$. However, these two lengths cannot be chosen
arbitrarily because the pump field at the carrying frequency $\omega_{p}^{0}$ has to be in a transmission peak.
Let us fix the lengths $l_{a}$ and $l_{b}$ (together with the
number $N$ of layers) for a moment and determine the spectral
intensity transmission $T(\omega)$ along the $+z$ direction
(of the pump-field propagation) using, e.g., the transfer-matrix
formalism Yeh (1988). Increasing the frequency $\omega$
there occur forbidden bands one following the other. The
difference in central frequencies of the neighbor forbidden bands
is roughly the same in accord with the band-gap theory. This may
be convenient for spectrally nearly degenerate SPDC provided that
the process of SPDC can be tuned such that the pump-field
frequency lies inside a transmission peak close to the second
forbidden band whereas the signal- and idler-field frequencies are
inside the transmission peaks near to the first forbidden band. We
note that all three interacting fields can accommodate themselves
either to the transmission peaks above or bellow the forbidden
bands. This helps to obey the quite strong requirements of the
three-field interaction by varying the signal-field emission angle
$\vartheta_{s}$. However, suitable conditions can only be revealed
numerically.
Now we return back to the problem in which the pump-field carrying
frequency $\omega_{p}^{0}$ is fixed. As follows from the
considerations of the previous paragraph, useful structures are
those that have the first upper or the first lower transmission
peak near to the second forbidden band at the pump-field carrying
frequency $\omega_{p}^{0}$. Inspection of the behavior of structures
with different layers’ lengths $l_{a}$ and $l_{b}$ has shown that
there exists a system of curves in the plane spanned by layers’
lengths $l_{a}$ and $l_{b}$ that provides the required
transmission peaks at the frequency $\omega_{p}^{0}$. One curve
corresponds to the first lower transmission peak of the second
forbidden band. Similarly, another curve is associated with the
first upper transmission peak of the second forbidden band. They
can be revealed as follows using the scaling property of
diffraction phenomena in optics.
We consider a fictitious dispersion-free structure with the
indexes of refraction appropriate for the pump-field frequency $\omega_{p}^{0}$ and define the corresponding optical lengths $l_{a}^{\rm opt}$ and $l_{b}^{\rm opt}$. It can be shown that the
ratio $L=l_{b}^{{\rm opt}}/l_{a}^{{\rm opt}}$ of optical lengths
represents a suitable variable for parametrization of these
curves. Suitable optical lengths $l_{a}^{\rm opt}$ and $l_{b}^{\rm opt}$ for a given value of the ratio $L$ can be revealed easily
using the scaling property of diffraction phenomena. We fix the
value of length $l_{a}^{\rm opt}$ to, e.g., $l_{a}^{{\rm opt},0}=\lambda_{p}^{0}/2=\pi c/\omega_{p}^{0}$. This gives the unit length of
diffraction phenomena that can be, in principle, chosen
arbitrarily. As the ratio $L$ is given, the optical length $l_{b}^{{\rm opt},0}$ is derived as $l_{b}^{{\rm opt},0}=Ll_{a}^{{\rm opt},0}$ and we can calculate the spectral intensity
transmission $T_{p}(\omega_{p})$ for this structure. We further
identify the frequency $\omega_{p}^{\rm max}$ of the first lower
(or upper) transmission peak of the second forbidden band. Then we
have to ’transfer’ the actual frequency $\omega_{p}^{\rm max}$ of
the transmission peak to the required frequency $\omega_{p}^{0}$
using the scaling property. The scaling property provides the
layers’ optical lengths in the form: $l_{a}^{\rm opt}=l_{a}^{{\rm opt},0}\omega_{p}^{0}/\omega_{p}^{\rm max}$ and $l_{b}^{\rm opt}=Ll_{a}^{\rm opt}$. We note that the obtained lengths differ for the
lower and the upper transmission peaks.
In the next step we move along the obtained two curves (for the
upper and the lower transmission peaks) parameterized by the ratio
$L$ and numerically analyze the structures. The maximum $\eta_{s}^{\rm max}$ of relative density $\eta_{s}(\omega_{s},\vartheta_{s},\psi_{s}^{0})$ of mean signal-photon
numbers taken over the signal-field frequency $\omega_{s}$ and
radial emission angle $\vartheta_{s}$ assuming the fixed
signal-field azimuthal emission angle $\psi_{s}^{0}$ has been found
a suitable quantity for monitoring efficiency of the nonlinear
process. The signal- and idler-field polarizations are assumed to
be fixed. The greater the value of maximum $\eta_{s}^{\rm max}$ of
relative density the closer the signal- and idler-field
transmission peaks to the first forbidden band. A curve giving the
dependence of maximum $\eta_{s}^{\rm max}$ of relative density on
the ratio $L$ is thus a good indicator for choosing suitable
layers’ lengths. We note that the curve depends on polarization
properties of the signal and idler fields as well as the azimuthal
signal-field emission angle $\psi_{s}^{0}$. It is also possible to
monitor another quantity in this procedure, e.g., the overall mean
photon-pair number $N$ given in Eq. (24). Or the density
of mean signal-photon numbers emitted for a fixed azimuthal
emission angle $\psi_{s}^{0}$ and determined as $\int_{-\pi/2}^{\pi/2}\sin(\vartheta_{s})d\vartheta_{s}n_{s}^{\rm tr}(\vartheta%
_{s},\psi_{s})$ using the formula in Eq. (22) may
be considered.
As an example, the dependence of maximum $\eta_{s}^{\perp,\|,\rm max}$ of relative density of two photons propagating along the $+z$ axis on the ratio $L$ for the first lower and the first
upper pump-field transmission peak of the second forbidden band of
the structure composed of 11 and 101 layers, respectively, is
plotted in Fig. 2 (for details, see below). The curve in
Fig. 2(a) appropriate for the structure with 11 layers is
continuous and rather flat. This means that there exists a whole
continuous set of structures giving an efficient nonlinear
process. On the other hand, efficient nonlinear structures with $N=101$ layers are found only in peaks of the curve in
Fig. 2(b). This is a consequence of complex interference
of back-scattered light inside these structures. We note that the
appropriate transmission peaks in all three interacting fields do
not necessarily exist for all values of the ratio $L$. Also the
greatest values of maximum $\eta_{s}^{\perp,\|,\rm max}$ of the
relative density are found for the values of ratio $L$ in a
certain restricted region. As the graph in Fig. 2(b)
shows, the greatest values of maximum $\eta_{s}^{\perp,\|,\rm max}$ occur around $L=0.5$.
The curves in Fig. 2 demonstrate a ’weak’ increase of
values of the maximum $\eta_{s}^{\perp,\|,\rm max}$ of relative
signal-field density with the increasing values of the number $N$ of layers. As the densities $n_{s}^{\rm ref}$ of the mean
signal-photon numbers of the reference structure are linearly
proportional to the second power of the length of nonlinear
material inside the structure, the densities $n_{s}$ of the mean
signal-photon numbers increase better than $N^{2}$. This is very
important, because the mean number of emitted photon pairs
increases less than the second power of the length of nonlinear
material in other structures producing photon pairs like bulk
crystals, periodically-poled crystals or wave-guiding structures
because of ’longitudinal’ phase-matching conditions. Nearly linear
dependence of the mean photon-pair numbers on the length of a
nonlinear structure is commonly observed.
IV Intensity transverse profiles and correlated areas
We consider three examples of layered structures made of GaN/AlN
that demonstrate typical features of the emitted photon pairs.
They generate photon pairs with different polarizations of the
signal and idler photons. They are designed in such a way that
three different mechanisms of splitting the correlated area are
observed. The pump field propagating at normal incidence at the
carrying wavelength $\lambda_{p}^{0}=400$ nm is assumed. The
signal and idler fields are emitted around the degenerate
wavelengths $\lambda_{s}^{0}=\lambda_{i}^{0}=800$ nm in
non-collinear geometry. The GaN and AlN layers are positioned such
that their optical axes are perpendicular to the boundaries. The
considered structures differ in the number $N$ of layers ($N=11$, 51 and 101). This results in different signal-field intensity
transverse profiles plotted in Fig. 3. Whereas the
shortest structure with $N=11$ layers has only one emission
area, the longest one with $N=101$ layers already forms five
emission rings. We note that transverse profiles of the
down-converted fields can also be efficiently tailored by
modifying the pump-beam profile Walborn et al. (2004).
The analyzed structures have certain properties in common. Namely,
they cannot generate proton pairs in nearly collinear geometries
because of the symmetry that cancels the overlap integral. As for
correlated areas, their azimuthal spreads $\Delta\psi_{i}$ depend
on the width $r_{p}$ of transverse pump-field profile
Molina-Terriza
et al. (2005); Hamar et al. (2010). The radial spreads $\Delta\vartheta_{i}$ depend also on the geometry of the layered
structure; the greater the number $N$ of layers, the smaller the
correlated area. In more detail, the considered structures behave
as follows.
IV.1 Structure with 11 layers
The structure is composed of 6 nonlinear GaN layers 90.14 nm long
and 5 linear AlN layers 74.92 nm long. Despite a relatively small
number of layers, photonic band gaps are already formed though the
bottoms of forbidden bands have intensity transmissions around
0.3. For $\psi_{s}=0$ deg, an efficient SPDC occurs for
polarizations (TE,TM,TE) and (TE,TE,TM) for the
(pump,signal,idler) fields due to the nonlinear coefficient $d(1,1,3)$. The fields are tuned to the first lower transmission
peaks of the first and second forbidden bands, so the efficiency
of SPDC is nearly optimal. The signal and idler transmission peaks
are spectrally broad and, as a consequence, we observe one large
emission area in the signal-field transverse plane that extends
from cca $\vartheta_{s}=20$ deg to 60 deg [see Fig. 3(a)].
The relative density $\eta_{s}$ of mean signal-photon numbers as
plotted in Fig. 4 reveals that the emitted photons have
nearly degenerate frequencies and relatively broad spectra. The
shape of relative density $\eta_{s}$ as shown in Fig. 4
is caused by a different radial dependence of intensity
transmissions $T_{\rm TE}$ and $T_{\rm TM}$. In the area
around $\psi_{s}=\pm 90$ deg there occurs no photon-pair
generation because of geometric reasons (the propagation of signal
and idler fields as TM waves is not supported). Compared to one
GaN layer $6\times 90.14$ nm long, the structure gives cca 2
times greater relative densities $\eta_{s}$.
The correlated area of an idler photon for a collimated pump beam
is composed of several ’islands’ localized around different
idler-field radial emission angles $\vartheta_{i}$ [see
Fig. 5(a)]. This breaking is caused by the zig-zag
movement of two photons inside the structure after being generated
in one layer. The comparison of correlated areas for collimated
and focused pump beams using, e.g., the graphs in
Fig. 5(a) and (b) reveals that the spread of correlated
area along the azimuthal emission angle $\psi_{i}$ depends
strongly on the amount of focusing the pump beam. On the other
hand, the shape of correlated area in the radial emission angle $\vartheta_{i}$ is blurred by a focused pump beam keeping the
overall spread of correlated area roughly the same.
IV.2 Structure with 51 layers
The second structure contains 26 nonlinear GaN layers 106.87 nm
long and 25 linear AlN layers 65.99 nm long. This structure has
already well-formed forbidden bands. As Fig. 3(b) shows
there occurs an efficient SPDC in two concentric rings in the
transverse plane. The structure is designed such that an efficient
nonlinear interaction for fields’ polarizations (TM,TM,TM) occurs
along the direction $\psi_{s}=0$ deg using the nonlinear
coefficient $d(2,2,3)$. The pump field lies in the first upper
transmission peak of the second forbidden band and the signal and
idler fields are in the first and second lower transmission peaks
of the first forbidden band. The relative density $\eta_{s}$ of
mean signal-photon numbers as plotted in Fig. 6
demonstrates that the signal-field (and also idler-field)
intensity spectra are composed of two symmetric peaks. There
occurs no photon-pair generation for degenerate signal- and
idler-field frequencies because of symmetry. This indicates the
generation of photon pairs in the state that is anti-symmetric
with respect to the exchange of the signal- and idler-field
frequencies. Anti-bunching of photons in a pair and
anti-coalescence (observed in a Hong-Ou-Mandel interferometer) are
the distinguished features of this state Peřina Jr.
et al. (2007a). The
graph in Fig. 3(b) reveals that an efficient SPDC occurs
also for azimuthal emission angles $\psi_{s}$ around $\pm 90$ deg. In this case, the signal photon is generated as TE wave and
the idler photon as TM wave or vice versa. The structure gives cca
50 times greater relative densities $\eta_{s}$ in comparison with
one GaN layer $26\times 106.87$ nm long.
As a consequence of two-peak structure of the signal- and
idler-field spectra there occurs splitting of the correlated area
of an idler photon, as documented in Fig. 7(a). The
overall correlated area is composed of two symmetric parts.
Because this splitting arises from symmetry, it survives even for
focused pump beams, as demonstrated in Fig. 7(b).
IV.3 Structure with 101 layers
The last structure is composed of 51 nonlinear GaN layers
106.42 nm long and 50 linear AlN layers 65.71 nm long. The pump
field lies in the first upper transmission peak of the second
forbidden band. A detailed comparison of the relative density $\eta_{s}$ of mean signal-photon numbers (see Fig. 8) with
intensity transmission spectra $T_{\rm TE}$ and $T_{\rm TM}$
valid for TE and TM waves, respectively, and obtained for $\psi=0$ deg (see Fig. 9) reveals the following. An efficient
SPDC occurs at intersections of $j$th and $(j+1)$th lower
transmissions peaks of the first forbidden band for $j=2,3,4,5,6$. This results in five concentric rings in the density $n_{s}^{\rm tr}$ of mean signal-photon numbers clearly visible in
Fig. 3(c). We note that the nonlinear coefficient $d(1,1,3)$ is exploited here and (TE,TM,TE) and (TE,TE,TM)
[(TM,TM,TM)] polarizations are suitable for the angles around $\psi_{s}=0$ deg [$\psi_{s}=\pm 90$ deg]. The relative density $\eta_{s}$ of mean signal-photon numbers indicates that the
signal-field intensity spectra are composed of two peaks of
different weights for $\psi_{s}=0$ deg. One peak is related to a
TE signal-field wave, whereas the second one arises from a TM
signal-field wave. Comparison with one GaN layer $51\times 106.42$ nm long reveals that the enhancement of optical fields inside
the structure results in an increase of the relative density $\eta_{s}$ cca 330 times.
A two-peak spectral structure leads to splitting of the correlated
area of the idler photon into two parts Peřina Jr.
et al. (2007b), as
shown in Fig. 10(a). However, two parts are not symmetric
in this case and they can also merge together provided that the
pump beam is sufficiently focused [see Fig. 10(b)]. The
distance between two parts in the idler-photon radial emission
angle $\vartheta_{i}$ increases with the increasing signal-photon
radial emission angle $\vartheta_{s}^{0}$. Whereas this distance
equals cca 0.6 deg for $\vartheta_{s}^{0}=23$ deg (the first
ring), it equals already cca 4 deg for $\vartheta_{s}^{0}=66$ deg
(the fifth ring). Idler photons found in different parts of the
correlated area differ in their polarizations.
The above investigated structures demonstrate main features of
photon pairs generated in nonlinear layered structures. These
structures allow to generate photon pairs entangled in
frequencies, polarizations as well as in emission directions.
Moreover, they allow tailoring of properties of photon pairs
varying namely the number of layers. A small amount of nonlinear
material inside them is compensated by an increase of
electric-field amplitudes originating in fields’ back-scattering.
V Conclusions
We have developed a spatial vectorial quantum model of spontaneous
parametric down-conversion in nonlinear layered structures. Photon
pairs generated in these structures can be entangled in
frequencies, polarizations as well as emission directions. Namely
entanglement in emission directions is important because it can be
efficiently tailored varying the number of layers. A method for
designing efficient layered structures has been suggested. Its
efficiency has been demonstrated suggesting three typical
structures. It has been shown that the number of generated
photon-pairs increases greater than the second power of the number
of layers. Signal-field intensity profiles and correlated areas in
the transverse plane have been analyzed in the designed
structures. Intensity profiles are typically composed of
concentric rings. The greater the number of layers, the greater
the number of rings. Correlated areas may be broken into several
parts because of three possible reasons: i) Zig-zag movement of
photons inside the structure, ii) Necessity to obey geometric
symmetries, and iii) Polarization-dependent properties. Splitting
of the correlated area arising from the geometric symmetry
survives even for a focused pump beam. Also states of photon pairs
showing anti-bunching and anti-coalescence can efficiently be
generated in layered structures. We believe that nonlinear layered
structures are potentially interesting namely as efficient sources
of photon-pair fields entangled in propagation directions which
might be useful, e.g., in ’parallel processing’ of quantum
information.
Acknowledgements.Support by projects IAA100100713 of GA AV ČR
and COST OC 09026, 1M06002 and Operational Program Research and
Development for Innovations - European Social Fund (project
CZ.1.05/2.1.00/03.0058) of the Ministry of Education of the Czech
Republic is acknowledged. The author thanks M. Centini for useful
discussions.
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Supersymmetric quantum mechanics on the lattice:
III. Simulations and algorithms
David Baumgartner and Urs Wenger
Albert Einstein Center for Fundamental Physics,
Institute for Theoretical Physics, University of Bern,
Sidlerstrasse 5, CH–3012 Bern, Switzerland
()
Abstract
In the fermion loop formulation the contributions to the partition
function naturally separate into topological equivalence classes
with a definite sign. This separation forms the basis for an
efficient fermion update algorithm using a fluctuating open fermion
string. It guarantees sufficient tunnelling between the topological
sectors, and hence provides a solution to the fermion sign problem
affecting systems with broken supersymmetry. Moreover, the algorithm
shows no critical slowing down even in the massless limit and can
hence handle the massless Goldstino mode emerging in the
supersymmetry broken phase. In this paper – the third in a series
of three – we present the details of the simulation algorithm and
demonstrate its efficiency by means of a few examples.
1 Introduction
The reformulation of supersymmetric quantum mechanics on the lattice
in terms of bosonic and fermionic bonds as derived in the first paper
of our series [1] provides a perfect setup for
Monte Carlo simulations. First of all, the reduction in complexity by
going from continuous to discrete variables is enormous. More
specifically though, expressing the Grassmann fields in terms of
fermionic bonds avoids the expensive calculation of the fermion
determinant and allows the use of special algorithms for which
critical slowing down is essentially absent
[2, 3] and simulations are possible even
in the massless limit [4]. This is of particular
importance for systems with broken supersymmetry, since the physics of
those is driven by the massless Goldstino mode. In the present paper
– the last in a series of three – we describe in detail such an
algorithm and demonstrate its efficiency. Since the model can be
solved exactly at finite lattice spacing by means of transfer
matrices, as discussed in the second paper of our series
[5], there is in principle no need for
numerical simulations. Hence, the present paper rather constitutes a
feasibility study to test the practicability and efficiency of the
proposed simulation algorithm for the quantum mechanical system in the
bond formulation. In that sense it also serves as a preparation for
the application of the algorithm, in particular the fermionic part, in
more complex situations, such as in supersymmetric Yang-Mills quantum
mechanics [6], in the ${\cal N}=1$ Wess-Zumino
model [7, 8, 9]
or in the supersymmetric nonlinear $O(N)$ sigma model
[10]. The advantage of the application of the
algorithm in the quantum mechanical model presented here is of course
the fact that the correctness of the algorithm can be crosschecked
with the exact results from the transfer matrix approach, and that the
algorithm can hence be validated in detail.
There is another rather pedagogical reason which motivates to consider
a new simulation algorithm for quantum mechanics in the bond
formulation. Often, simple quantum mechanical systems such as the
harmonic and anharmonic oscillator are used to introduce the path
integral approach. Similarly, the systems also provide a pedagogical
context in which various Monte Carlo simulation algorithms can be
illustrated and discussed, see for example [11] for an
early example. However, it turns out that the standard Metropolis
algorithms and even more advanced algorithms such as the
overrelaxation or heat bath algorithm become extremely inefficient
towards the continuum limit. This has to do with the usual critical
slowing down of the simulations towards that limit, and for the
anharmonic oscillator also with the suppressed tunnelling at small
lattice spacing. The algorithms presented here do not suffer from
these deficiencies, because they eliminate critical slowing down. In
addition, in the bond formulation the $\mathbb{Z}_{2}$-symmetry $\phi\rightarrow-\phi$ is exactly maintained for each bond configuration.
Last but not least, the numerical simulations presented here serve as
a test of the practicability of the solution of the fermion sign
problem proposed in [4] and discussed further in
the first paper of our series [1]. The solution
is based on two ingredients. Firstly, the lattice regulates the
vanishing Witten index and therefore also the sign problem. Secondly,
the fermion loop formulation provides a tool to handle the fluctuating
sign, because it naturally separates the contributions to the
partition function into topological equivalence classes, each
possessing a definite sign. Nevertheless, it is a priori not clear
whether the lattice artefacts and the statistical fluctuations can be
kept under sufficient control in a practical simulation. The
statistical fluctuations of the sign are essentially determined by the
amount of tunnelling between the topological sectors, i.e., between
the fermionic and bosonic vacuum. In order for the fermion update
algorithm to be a true solution to the sign problem, it must guarantee
a sufficiently efficient tunnelling rate. The results in this paper
demonstrate that this is indeed the case. Not surprisingly, the open
fermion string algorithm discussed here has proven to be extremely
successful in the ${\cal N}=1$ Wess-Zumino model
[9] in which the fermion sign problem is
prevailing.
Of course, supersymmetric quantum mechanics has already been simulated
on the lattice in various setups using standard algorithms, cf. for
example
[12, 13, 14, 15, 16, 17, 18, 19, 20]. However,
the bond formulation together with the simulation algorithm presented
here brings the numerical nonperturbative calculations to a new,
unprecedented level of accuracy. In that sense, the results presented
here and partly in [4] serve as a benchmark
against which new formulations or simulation algorithms can be tested.
The present paper is organised as follows. In Section
2 we construct in detail an algorithm designed for
updating the bosonic and fermionic bond configurations. The
discussion includes the explicit update steps and the derivation of
the corresponding acceptance ratios. Their evaluation requires the
calculation of site weight ratios which turn out to become numerically
unstable for large site occupation numbers. Therefore, in Section
3 we present a computational strategy which allows to
evaluate the ratios for arbitrarily large occupation numbers. In
Section 4, we then present the results obtained using
the proposed algorithm. The simulations are for the same
discretisation schemes and superpotentials we used in the previous two
papers [1, 5]. Since this
section is merely meant as a validation of the algorithm, the
discussion of the physics behind the results is kept short and we
refer to the exact results in [5] for a more
throrough discussion.
2 Simulation algorithm
We start our discussion from the partition function of supersymmetric
quantum mechanics on the lattice written as a sum over all allowed,
possibly constrained bond configurations $\mathcal{C}=\{n^{b}_{i}(x),n^{f}(x)\}$ in
the configuration space $\mathcal{Z}$,
$$Z=\sum_{\mathcal{C}\subset\mathcal{Z}}W_{F}(\mathcal{C})\,,$$
(1)
where the fermion number $F=0,1$ is determined by the fermionic bond
configuration $\{n^{f}(x)\}$ with $n^{f}(x)=F,\,\forall x$, and the weight $W_{F}(\mathcal{C})$
of a configuration is given by
$$W_{F}(\mathcal{C})=\prod_{x}\left(\prod_{i}\frac{w_{i}^{n^{b}_{i}(x)}}{n^{b}_{%
i}(x)!}\right)\prod_{x}Q_{F}(N(x))\,.$$
(2)
Here, $x$ denotes the sites of the lattice and $i$ labels the various
types of bosonic bonds $b_{i}$ with $i\in\{j\rightarrow k\,|\,j,k\in\mathbbm{N}\}$. The corresponding bosonic bond weights are denoted
by $w_{i}$ and $n_{i}^{b}(x)\in\mathbbm{N}_{0}$ is the occupation number of
the bond $b_{i}$ connecting the sites $x$ and $x+1$. The site weight
$Q_{F}$ depends on the site occupation number, i.e., the total number of
bosonic bonds connected to site $x$,
$$N(x)=\sum_{j,k}\left(j\cdot n^{b}_{j\rightarrow k}(x)+k\cdot n^{b}_{j%
\rightarrow k}(x-1)\right)$$
(3)
and is given by
$$Q_{F}(N)=\int_{-\infty}^{\infty}d\phi\ \phi^{N}\mathrm{e}^{-V(\phi)}M(\phi)^{1%
-F}\,.$$
(4)
In Section 3 we will discuss in detail the
computational strategy necessary to reliably evaluate ratios of these
integrals for arbitrary and possibly large site occupation
numbers. The type of bonds $b_{i}$, the weights $w_{i}$ as well as the
potential $V(\phi)$ and the monomer term $M(\phi)$ in eq.(4) depend on the specifics of the chosen discretisation and
the superpotential $P(\phi)$. We refer to the appendix of our first
paper [1] for a compilation of the
discretisations and superpotentials considered in our series.
As mentioned above, the bond configurations $\mathcal{C}=\{n^{b}_{i}(x),n^{f}(x)\}$
are possibly constrained. In particular we have the local fermionic
constraints
$$n^{f}(x-1)=n^{f}(x)\,$$
(5)
while the local bosonic constraints
$$N(x)=0\mod 2\,$$
(6)
may or may not be present depending on the bosonic symmetries of the system.
The challenge of updating constrained bond configurations lies
precisely in the difficulty to maintain the constraints while moving
efficiently through the configuration space $\mathcal{Z}$. In
[21] Prokof’ev and Svistunov proposed to extend the
constrained bosonic bond configuration space by introducing local
sources which explicitly violate the constraints. The so-called worm
algorithm then probes the extended configuration space by moving the
local violations around the lattice, thereby sampling directly
the bosonic correlation function corresponding to the sources
introduced. The contact with the original configuration space $\mathcal{Z}$ is
established when the violations annihilate each other, e.g. when
moving to the same site on the lattice, such that the bond
configuration fulfills again all constraints.
In [2] the idea has been extended to fermionic
systems expressed in terms of fermionic bonds. The fermionic
constraint in eq.(5) allows only either an
empty or a completely filled fermion bond configuration. The
difficulty for the direct application of the worm idea to the
fermionic system lies in the fact that the introduction of the fermionic
source term $\psi_{x}\overline{\psi}_{x}$ is incompatible with the presence of
the fermion loop at site $x$. A simple solution is to allow the
unphysical situation of the site $x$ being occupied by a propagating
fermion and two additional sources. Such a configuration violates the
Pauli exclusion principle and does not contribute to any physical
observable. In the Grassmann path integral such a configurations
indeed vanishes trivially.
In order to be more explicit it is necessary to introduce the bond
configuration spaces of bosonic and fermionic two-point correlation
functions, $\mathcal{G}^{b}_{F}$ and $\mathcal{G}^{f}$, respectively, following the notation
in our first paper [1]. Bond configurations in
$\mathcal{G}^{b}_{F}$ contribute to the non-normalised bosonic two-point function
according to
$$g^{b}_{F}(x_{1}-x_{2})\equiv\langle\!\langle\phi_{x_{1}}\phi_{x_{2}}\rangle\!%
\rangle_{F}=\sum_{\mathcal{C}\subset\mathcal{G}^{b}_{F}}\left(\prod_{x}\frac{Q%
_{F}(N(x)+\delta_{x,x_{1}}+\delta_{x,x_{2}})}{Q_{F}(N(x))}\right)\cdot W_{F}(%
\mathcal{C})\,,$$
(7)
while the configurations in $\mathcal{G}^{f}$ contribute to the non-normalised
fermionic two-point function as
$$g^{f}(x_{1}-x_{2})\equiv\langle\!\langle\psi_{x_{1}}\overline{\psi}_{x_{2}}%
\rangle\!\rangle=\sum_{\mathcal{C}\subset\mathcal{G}^{f}}\left[\prod_{x\in%
\phantom{\notin}\!\!\!\!\mathcal{F}}\frac{Q_{1}(N(x))}{Q_{0}(N(x))}\right]%
\cdot W_{0}(\mathcal{C})\,,$$
(8)
where $\cal{F}$ denotes the set of lattice sites belonging to the open
fermion string associated with the fermionic correlation function. The
key point of the bosonic and fermionic updating algorithm is that the
bond configurations for $g^{b}_{F}(0)$, $g^{f}(0)$ and $Z_{F}$ have identical
bond elements. As a consequence, statistics for $g^{b,f}$ and $Z$ can
be accummulated in the same Monte Carlo process. If the bosonic
constraints in eq.(6) are not present,
e.g. for superpotentials with broken supersymmetry, the equivalence of
bond configurations even extends to $g^{b}_{F}(x)$, i.e., $\mathcal{Z}_{F}=\mathcal{G}^{b}_{F}$. The movements from one configuration space to the other are
induced by introducing or removing bosonic or fermionic sources
according to the scheme given in figure 8 of our first paper
[1]. For convenience we reproduce it here in
figure 1.
In the following we will now discuss in detail the various updating
steps which establish explicitly the connection between the bond
configuration spaces $\mathcal{G}^{f},\mathcal{Z}_{F},\mathcal{G}^{b}_{F}$ and in addition move the
system within $\mathcal{G}^{f}$ and $\mathcal{G}^{b}_{F}$. The moves are generated by a Monte
Carlo process with probabilities given by the weights of the
configurations in eqs.(2), (7)
and (8). In particular, we derive the transition
probabilities $P_{X}(\mathcal{C}\rightarrow\mathcal{C}^{\prime})$ for the transition $X$
from bond configuration $\mathcal{C}$ to $\mathcal{C}^{\prime}$, which is then accepted by the
usual Metropolis prescription
$$P_{acc}(\mathcal{C}\rightarrow\mathcal{C}^{\prime})=\mathop{\rm min}\{1,P_{X}(%
\mathcal{C}\rightarrow\mathcal{C}^{\prime})\}.$$
(9)
In order to simplify the discussion we select the update from $\mathcal{Z}_{F}$
to $\mathcal{G}^{b}_{F}$ or $\mathcal{G}^{f}$ with equal probability which is balanced by
corresponding proposal probabilities to select between moving in
$\mathcal{G}^{f}$ and $\mathcal{G}^{b}_{F}$ or returning to $\mathcal{Z}_{F}$.
2.1 Updating the fermionic bond configuration
Here we discuss the various update steps which moves the system within
$\mathcal{G}^{f}$ and relate the bond configurations spaces $\mathcal{Z}_{0}$ and $\mathcal{Z}_{1}$ via
$\mathcal{G}^{f}$.
Moves within $\mathcal{G}^{f}$ are induced by shifting $\overline{\psi}$ by one lattice
spacing from site $x$ to site $x+1$, and vice versa, while keeping the
other source $\psi$ fixed. Such an update step is graphically
illustrated in figure 2 and is called ‘shift’
update step.
A shift in forward direction, $x\rightarrow x+1$, automatically
involves the removal of the fermionic bond $b^{f}(x)$, whereas a shift
in backward direction, $x+1\rightarrow x$, requires the addition of a
new fermionic bond $b^{f}(x)$. Both directions are proposed with equal
probability $1/2$ and are hence balanced against each other as long as
the new site does not coincide with the position of the source
$\psi$. The formula in eq.(8) provides us with the
acceptance ratios
$$\displaystyle P_{\mathrm{sh}}(x+1\rightarrow x)$$
$$\displaystyle=$$
$$\displaystyle\displaystyle\frac{Q_{1}(N(x))}{Q_{0}(N(x))}\,,$$
(10)
$$\displaystyle P_{\mathrm{sh}}(x\rightarrow x+1)$$
$$\displaystyle=$$
$$\displaystyle\displaystyle\frac{Q_{0}(N(x))}{Q_{1}(N(x))}\,.$$
(11)
Now let us consider the case when we propose to shift the source
$\overline{\psi}$ forward to the site $x$ where the source $\psi$ is present,
as depicted in the upper half of figure 3.
The forward shift update step $x-1\rightarrow x$ is balanced with the
backward shift update step $x\rightarrow x-1$. This backward shift,
however, is proposed with probability 1 instead of probability 1/2
since the shift of $\overline{\psi}$ from $x\rightarrow x+1$ would involve
the creation of an open fermion string around the entire lattice. The
asymmetry in the proposition probabilities is balanced by the choice
of the probability $p_{\mathrm{rm}}=1/2$ to remove the sources
$\psi\overline{\psi}$, such that we find the acceptance ratio for a shift in
backward direction $x\rightarrow x-1$ to be the same as in
eq.(10), namely
$$P_{\mathrm{sh}}(x\rightarrow x-1)=\frac{Q_{1}(N(x-1))}{Q_{0}(N(x-1))}.$$
(12)
The shift step is balanced with the corresponding one in forward
direction with acceptance ratio as given in eq.(11),
$$P_{\mathrm{sh}}(x-1\rightarrow x)=\displaystyle\frac{Q_{0}(N(x-1))}{Q_{1}(N(x-%
1))}$$
(13)
The step from $\mathcal{G}^{f}$ to $\mathcal{Z}_{0}$ and vice versa is induced by
introducing or removing a pair of fermionic sources $\psi\overline{\psi}$
at site $x$, respectively. It is called ‘put/remove’ update step and
is graphically illustrated in the lower half of figure
3.
The removal of the fermionic sources is suggested with probability
$p_{\mathrm{rm}}=1/2$ and is balanced on one side by the probability
to add bosonic sources, and on the other by the probability to shift
one of the sources and hence move within $\mathcal{C}^{f}$. Because the
‘put/remove’ update step does not alter the fermionic bond
configuration, we have $\mathcal{Z}_{0}=\mathcal{G}^{f}(0)$. On the other hand it adds or
removes a fermionic monomer term $M(\phi)$ at site $x$. The relative
weight of the configurations with or without this term is given by
$Q_{1}/Q_{0}$ and the acceptance ratios on a lattice with $L_{t}$ sites
become
$$\displaystyle P_{\mathrm{rm}}(x\rightarrow\varnothing)$$
$$\displaystyle=$$
$$\displaystyle\frac{2}{L_{t}}\displaystyle\frac{Q_{0}(N(x))}{Q_{1}(N(x))},$$
(14)
$$\displaystyle P_{\mathrm{put}}(\varnothing\rightarrow x)$$
$$\displaystyle=$$
$$\displaystyle\frac{L_{t}}{2}\displaystyle\frac{Q_{1}(N(x))}{Q_{0}(N(x))}\,.$$
(15)
The factor $L_{t}$ compensates for the proposition probability to choose
lattice site $x$ out of $L_{t}$ possibilities when putting the sources,
while the factor 2 compensates for the asymmetric shift proposal
probability when moving $\overline{\psi}$ from $x$ to $x-1$, since the
shift of $\overline{\psi}$ from $x$ to $x+1$ is not allowed.
Next we consider the shift update step for the case when the source
$\overline{\psi}$ at site $x+1$ is shifted backwards to site $x$ which is
already occupied by the sink $\psi$. The step is graphically
illustrated in the upper half of figure
4.
While the resulting fermion bond configuration is a valid one (it
belongs to $\mathcal{Z}_{1}$), the whole fermion configuration including the
source and the sink represents an unphysical situation, and in fact
does not contribute to any physical observable, as discussed
before. Therefore, such a backward shift from $\mathcal{G}^{f}$ to $\mathcal{Z}_{1}$,
essentially closing the open fermion string, automatically induces the
removal of the fermionic source and sink pair $\psi\overline{\psi}$ from site
$x$ as illustrated in the lower half of figure
4. Such a step is called a hybrid
‘shift/remove’ update step. Of course, the step is balanced with a
hybrid ‘put/shift’ update step when the additional fermionic sink and
source variables are put on a closed fermion loop at the site $x$. As
usual, the acceptance ratios for the hybrid update steps can be read
off from the weights of the configurations involved and yield
$$\displaystyle P_{\mathrm{sh/rm}}(x+1\rightarrow x\rightarrow\varnothing)$$
$$\displaystyle=$$
$$\displaystyle\frac{2}{L_{t}},$$
(16)
$$\displaystyle P_{\mathrm{put/sh}}(\varnothing\rightarrow x\rightarrow x+1)$$
$$\displaystyle=$$
$$\displaystyle\frac{L_{t}}{2}.$$
(17)
The factor $L_{t}$ compensates for the proposition probability to choose
the same lattice site $x$ when putting the sources $\psi\overline{\psi}$ back
on the lattice, whereas the factor $2$ compensates for the proposition
probability to shift in forward or backward direction when the fermion
string is still open. Note that there are no ratios of $Q$-weights
involved, since no monomer term is added or removed by the hybrid
shift/remove update step.
To complete our discussion of the fermionic bond update, we note that
the algorithm provides improved estimators for the fermionic two-point
function $g^{f}(x)$ and the partition functions $Z_{F}$. Because the
algorithm samples directly the configuration space $\mathcal{G}^{f}$, every open
fermion string configuration contributes unity to the stochastic Monte
Carlo estimator for $g^{f}(x)$. To be precise we have
$$g^{f}(x|\mathcal{C}\in\mathcal{G}^{f})=\delta_{x,x_{1}-x_{2}}\,,$$
(18)
where $x_{1}$ and $x_{2}$ are the end and starting point of the open
fermion string, i.e., the positions of the sink $\psi$ and the source
$\overline{\psi}$, respectively. Similarly, every bond configuration in
$\mathcal{Z}_{F}$ is generated with its proper weight and hence contributes unity
to the stochastic estimator for $Z_{F}$, i.e., the Monte Carlo
estimator for $Z_{F}$ is simply
$$Z_{F}(\mathcal{C}\in\mathcal{Z}_{F})=1\,.$$
(19)
Finally we note that the factors of $L_{t}$ appearing in the acceptance
ratios above may become inconvenient in practice, especially towards
the continuum limit when $L_{t}\rightarrow\infty$. The factors only
occur when contact between $\mathcal{Z}_{F}$ and $\mathcal{G}^{f}$ is made, i.e., they are
responsible for getting the relative normalisation between $Z_{F}$ and
$g^{f}$ right. However, since we make use of translational invariance in
eq.(18) the factors of $L_{t}$ are in fact cancelled
and can hence be omitted.
2.2 Updating the bosonic bond configuration
In this section we now discuss the update steps which relate the bond
configuration spaces $\mathcal{Z}_{F}$ and $\mathcal{G}^{b}_{F}$ for a fixed fermionic bond
configuration with fermion number $F=0,1$.
We point out that for an arbitrary superpotential there are in general
no restrictions on the bosonic bond configurations. This is for
example the case for the superpotential $P_{b}$ which we consider in our
series of papers, cf. eq.(53). In contrast,
the superpotential $P_{u}$ in eq.(54) yields
the local constraint $N(x)=0\mod 2$ on the site occupation number,
due to the parity symmetry $\phi\rightarrow-\phi$. In the following
discussion, we always present the generic case first, and then specify
the modifications or simplifications due to the constraint. In
analogy to the fermionic bond update, the ‘put/remove’ and the ‘shift’
updates are the main steps for updating the bosonic bond configurations.
The ‘put/remove’ step introduces or removes one or two sources $\phi$,
while the ‘shift’ step shifts the sources by one lattice spacing. If
there are no restrictions on the bond configuration, we are free to
decide for each Monte Carlo step whether to proceed by a ‘remove’ update
or a ‘shift’ update. With probability $p_{\mathrm{rm}}$, we propose to
remove the sources from the lattice, while the proposition to continue
the worm update with a ‘shift’ step is chosen with probability $1-p_{\mathrm{rm}}$.
The step from $\mathcal{G}_{F}^{b}$ to $\mathcal{Z}_{F}$ (and vica versa) is induced by
removing (or introducing) a bosonic source $\phi$ at sites $x_{1}$ and
$x_{2}$, with $x_{1}=x_{2}$ not excluded. The step does not alter the bond
configuration, but only the site occupation numbers at sites $x_{1}$ and
$x_{2}$. Thus, only the ratios of the site weights $Q_{F}$ are involved in
the acceptance probability for e.g. the ‘remove’ step,
$$\displaystyle P_{\mathrm{rm}}(x_{1},x_{2}\rightarrow\varnothing)$$
$$\displaystyle=$$
$$\displaystyle\!\!\!\left\{\begin{array}[]{ll}\displaystyle\frac{1}{p_{\mathrm{%
rm}}L_{t}^{2}}\frac{Q_{F}(N(x_{1})-2)}{Q_{F}(N(x_{1}))}&\mathrm{if}\ x_{1}=x_{%
2},\\
\displaystyle\frac{1}{p_{\mathrm{rm}}L_{t}^{2}}\frac{Q_{F}(N(x_{1})-1)}{Q_{F}(%
N(x_{1}))}\frac{Q_{F}(N(x_{2})-1)}{Q_{F}(N(x_{2}))}&\mathrm{if}\ x_{1}\neq x_{%
2}.\end{array}\right.$$
(20)
The prefactor $1/(p_{\mathrm{rm}}L_{t}^{2})$ is motivated as follows. The
factor $1/L_{t}^{2}$ balances the probability for the proposition of putting
the bosonic sources at the sites $x_{1}$ and $x_{2}$ when re-entering the
configuration space $\mathcal{G}^{b}_{F}$, while the factor $1/p_{\mathrm{rm}}$
balances the proposition probability for the choice of proceeding by
the shift update instead of the remove update, as discussed above.
The acceptance ratios for re-entering the configuration space $\mathcal{G}^{b}_{F}$
from $\mathcal{Z}_{F}$ are given by
$$\displaystyle P_{\mathrm{put}}(\varnothing\rightarrow x_{1},x_{2})$$
$$\displaystyle=$$
$$\displaystyle\left\{\begin{array}[]{ll}\displaystyle p_{\mathrm{rm}}L_{t}^{2}%
\frac{Q_{F}(N(x_{1})+2)}{Q_{F}(N(x_{1}))}&\mathrm{if}\ x_{1}=x_{2},\\
\displaystyle p_{\mathrm{rm}}L_{t}^{2}\frac{Q_{F}(N(x_{1})+1)}{Q_{F}(N(x_{1}))%
}\frac{Q_{F}(N(x_{2})+1)}{Q_{F}(N(x_{2}))}&\mathrm{if}\ x_{1}\neq x_{2}.\end{%
array}\right.$$
(21)
Two remarks are in order. Firstly, if there are no constraints on the
bond configuration, one can in principle introduce just a single
source $\phi$ which subsequently is shifted around. In effect, the
algorithm then samples the one-point function which in this situation
is indeed nonvanishing. Secondly, we note that if the constraint $N=0\mod 2$ is in place, the two sources can only be placed or removed
when $x_{1}=x_{2}$. As a consequence, only the first of the two
acceptance ratios in eq.(20) and
eq.(21) are relevant, while the second
ones are zero by definition.
Next, we discuss the bosonic ‘shift’ update. With this step we now
change the bosonic bond configuration. Shifting the source from site
$x$ to a next neighbouring site $y$ is always associated with an
increase or a decrease of the bosonic bond occupation number between
the sites $x$ and $y$ by one. Whether or not the occupation number is
increased or decreased is decided with probability $1/2$. Similarly,
the source can move forward or backward, and we propose both
directions with equal probability $1/2$. In addition, when there are
several types of bosonic bonds $b_{i}$ with $i\in\{j\rightarrow k|j,k\in\mathbb{N}\}$, we need to decide in each step which
bond is updated. We do so by choosing the proposition probabilities
$p_{j\rightarrow k}$ with $\sum_{j,k}p_{j\rightarrow k}=1$. However, because the proposals are completely symmetric, these
probabilities do not affect the acceptance ratios. In the following,
we will use the shorthand notation
$$\displaystyle n^{j\rightarrow k}_{xy}$$
$$\displaystyle=$$
$$\displaystyle\left\{\begin{array}[]{ll}n^{b}_{j\rightarrow k}(x)&\mathrm{if}\ %
y=x+1,\\
n^{b}_{j\rightarrow k}(y)&\mathrm{if}\ y=x-1,\end{array}\right.$$
(22)
for the occupation number of the bosonic bonds $b_{j\rightarrow k}$
between the sites $x$ and $y$. The shifts $x\rightarrow y$ and
$n^{j\rightarrow k}_{xy}\rightarrow n^{j\rightarrow k}_{xy}+1$ are
balanced with shifts $y\rightarrow x$ and $n^{j\rightarrow k}_{xy}\rightarrow n^{j\rightarrow k}_{xy}-1$, which gives the acceptance
ratios
$$\displaystyle P_{\mathrm{sh}}(x\rightarrow y,n^{j\rightarrow k}_{xy}%
\rightarrow n^{j\rightarrow k}_{xy}+1)\\
\displaystyle=\!\left\{\!\!\!\begin{array}[]{ll}\displaystyle\frac{w_{j%
\rightarrow k}}{n^{j\rightarrow k}_{xy}+1}\frac{Q_{F}(N(x)+j-1)}{Q_{F}(N(x))}%
\cdot\frac{Q_{F}(N(y)+k+1)}{Q_{F}(N(y))}&\mathrm{if}\ y=x+1,\\
\displaystyle\frac{w_{j\rightarrow k}}{n^{j\rightarrow k}_{xy}+1}\frac{Q_{F}(N%
(x)+k-1)}{Q_{F}(N(x))}\cdot\frac{Q_{F}(N(y)+j+1)}{Q_{F}(N(y))}&\mathrm{if}\ y=%
x-1,\\
\end{array}\right.$$
(23)
$$\displaystyle P_{\mathrm{sh}}(x\rightarrow y,n^{j\rightarrow k}_{xy}%
\rightarrow n^{j\rightarrow k}_{xy}-1)\\
\displaystyle=\!\left\{\!\!\!\begin{array}[]{ll}\displaystyle\frac{n^{j%
\rightarrow k}_{xy}}{w_{j\rightarrow k}}\frac{Q_{F}(N(x)-j-1)}{Q_{F}(N(x))}%
\cdot\frac{Q_{F}(N(y)-k+1)}{Q_{F}(N(y))}&\mathrm{if}\ y=x+1,\\
\displaystyle\frac{n^{j\rightarrow k}_{xy}}{w_{j\rightarrow k}}\frac{Q_{F}(N(x%
)-k-1)}{Q_{F}(N(x))}\cdot\frac{Q_{F}(N(y)-j+1)}{Q_{F}(N(y))}&\mathrm{if}\ y=x-%
1.\end{array}\right.$$
(24)
Of course, these generic ratios simplify considerably for the specific
bonds $b_{i},i\in\{1\rightarrow 1,1\rightarrow 2,1\rightarrow 3\}$ relevant for the superpotentials considered in our series of
papers. For example, the acceptance ratios for updating the bond
$b_{1\rightarrow 1}$ read
$$\displaystyle P_{\mathrm{sh}}(x\rightarrow y,n^{1\rightarrow 1}_{xy}%
\rightarrow n^{1\rightarrow 1}_{xy}+1)$$
$$\displaystyle=$$
$$\displaystyle\frac{w_{1\rightarrow 1}}{n^{1\rightarrow 1}_{xy}+1}\cdot\frac{Q_%
{F}(N(y)+2)}{Q_{F}(N(y))},$$
(25)
$$\displaystyle P_{\mathrm{sh}}(x\rightarrow y,n^{1\rightarrow 1}_{xy}%
\rightarrow n^{1\rightarrow 1}_{xy}-1)$$
$$\displaystyle=$$
$$\displaystyle\frac{n^{1\rightarrow 1}_{xy}}{w_{1\rightarrow 1}}\cdot\frac{Q_{F%
}(N(x)-2)}{Q_{F}(N(x))}\,.$$
(26)
Because the bond is symmetric, there is no need to distinguish whether
$y=x+1$ or $y=x-1$.
To complete the discussion of the bosonic bond update, we point out
that the algorithm again provides improved estimators for the bosonic
two-point function $g^{b}_{F}(x)$ and the partition functions $Z_{F}$. As in
the fermionic case, the algorithm samples directly the configuration
space $\mathcal{G}^{b}_{F}$ with the correct weighting when the sources are
present. Therefore, every configuration contributes unity to the
stochastic Monte Carlo estimator for $g^{b}_{F}(x)$, and we have
$$g_{F}^{b}(x|\mathcal{G}^{b}_{F})=\delta_{x_{1}-x_{2},x}\,,$$
(27)
where $x_{1}$ and $x_{2}$ are the positions of the two sources. Whenever
the bosonic update decides to remove the sources, we have a
configuration in $\mathcal{Z}_{F}$ and hence a contribution of unity to the
stochastic estimator for $Z_{F}$, that is, we have
$$Z_{F}(\mathcal{C}\in\mathcal{Z}_{F})=1\,.$$
(28)
In complete analogy to the fermionic update we note that the factors
of $L_{t}$ appearing in the acceptance ratios of the ‘put/remove’ step
can be compensated by adjusting the overall normalisation of the
two-point function, e.g. by making use of translational invariance.
3 Calculation of the site weight ratios
In order to calculate the weight of a bond configuration, it is
necessary to know the site weights
$$Q_{F}(n)=\int_{-\infty}^{\infty}d\phi\ \phi^{n}\mathrm{e}^{-V(\phi)}M(\phi)^{1%
-F},$$
(29)
where $V(\phi)$ and $M(\phi)$ depend on the superpotential and the
discretisation employed, and $F=0,1$ is the fermion number, for
arbitrary values of the site occupation number $n$. The values of $n$
required in practice are usually limited to $\mathcal{O}(10^{3})$. However, it
turns out that even for moderate values of $n$ of order $\mathcal{O}(100)$ the
site weights $Q_{F}(n)$ can quickly grow larger than $10^{100}$ or
more. As a consequence, the calculation of the site weights quickly
becomes numerically unstable for growing $n$. In fact, even for simple
potentials when the weights can be calculated analytically in terms of
confluent hypergeometric functions, the numerical evaluation of these
functions is difficult for large $n$, and even specialised libraries
such as the ones available in Wolfram’s Mathematica
[22] appear not to be accurate enough.
Fortunately, for the Monte Carlo simulations we only need ratios of
the site weights, such as $Q_{F}(n+1)/Q_{F}(n),Q_{F}(n+2)/Q_{F}(n)$ and
$Q_{1}(n)/Q_{0}(n)$, and these ratios usually do not become larger than
$\mathcal{O}(10)$ even for large $n$. In addition, also the transfer matrix
elements can be rewritten in terms of these ratios as discussed in th
appendix of our second paper of the series [5].
Therefore, we now present a numerically stable computational strategy
to calculate the site weight ratios reliably for arbitrary values of
the site occupation numbers.
We start by defining an arbitrary polynomial superpotential
$$P(\phi)=\sum_{i=0}^{p}p_{i}\phi^{i},$$
(30)
and the corresponding bosonic self-interaction potential $V(\phi)$ as
well as the monomer weight $M(\phi)$,
$$V(\phi)=\sum_{i=0}^{2(p-1)}k_{i}\phi^{i},\quad\quad M(\phi)=\sum_{i=0}^{p-2}m_%
{i}\phi^{i}.$$
(31)
Explicitly, the weights in each sector are then given by
$$Q_{1}(n)=\int_{-\infty}^{\infty}d\phi\ \phi^{n}\mathrm{e}^{-V(\phi)}$$
(32)
and
$$Q_{0}(n)=\sum_{i=0}^{p-2}m_{i}Q_{1}(n+i).$$
(33)
For convenience we also define the ratios of the site weights
$Q_{F}(n)$,
$$\displaystyle R^{\prime}_{F}(n)$$
$$\displaystyle=$$
$$\displaystyle\frac{Q_{F}(n+1)}{Q_{F}(n)},$$
(34)
$$\displaystyle R_{F}(n)$$
$$\displaystyle=$$
$$\displaystyle\frac{Q_{F}(n+2)}{Q_{F}(n)},$$
(35)
$$\displaystyle R_{m}(n)$$
$$\displaystyle=$$
$$\displaystyle\frac{Q_{0}(n)}{Q_{1}(n)}$$
(36)
which are used for the acceptance ratios in the Monte Carlo
simulations. In principle, only the ratios $R_{1}^{\prime}(n)$ need to be
calculated since all other ratios can be derived from those. For
example, $R_{1}(n)$ can be expressed in terms of $R_{1}^{\prime}(n)$ as
$$R_{1}(n)=R_{1}^{\prime}(n+1)R_{1}^{\prime}(n)\,,$$
(37)
but since in some cases $Q_{1}(n\,\text{odd})=0$ the introduction of
$R_{1}(2n)$ is nevertheless necessary. $R_{m}(n)$ can be expressed via the
ratios $R_{1}(n)$ and $R_{1}^{\prime}(n)$ and appropriate products thereof,
$$R_{m}(n)=m_{0}+R_{1}^{\prime}(n)\left(m_{1}+R_{1}(n+2)\left(m_{3}+\ldots\right%
)\right)+R_{1}(n)\left(m_{2}+R_{1}(n+2)\left(m_{4}+\ldots\right)\right),$$
(38)
and the ratios $R_{0}^{\prime}(n)$ and $R_{0}(n)$ via $R_{m}(n)$, $R_{1}(n)$ and $R_{1}^{\prime}(n)$ by
$$\displaystyle R_{0}(n)$$
$$\displaystyle=$$
$$\displaystyle\frac{R_{m}(n+2)}{R_{m}(n)}R_{1}(n),$$
(39)
$$\displaystyle R_{0}^{\prime}(n)$$
$$\displaystyle=$$
$$\displaystyle\frac{R_{m}(n+1)}{R_{m}(n)}R_{1}^{\prime}(n).$$
(40)
First, we now discuss how to gain numerical stability for the special
case of an even superpotential $P(\phi)$. In a second step we will
then adapt the idea to treat the somewhat more subtle case of an
arbitrary superpotential.
3.1 Even superpotential
Unbroken supersymmetric quantum mechanics requires a superpotential
$P(\phi)$ with $\deg(P(\phi))$ = 0 mod 2. In particular, in our series
of papers we investigate the superpotential
$$P(\phi)=p_{2}\phi^{2}+p_{4}\phi^{4}$$
(41)
which is symmetric w.r.t. the parity transformation $\phi\rightarrow-\phi$. As a consequence of the symmetry, $Q_{F}(n\,\text{odd})=0$
for both $F=0,1$ and the ratios $R_{F}^{\prime}(n)$ need not be considered –
instead, it is sufficient to determine $R_{1}(2n)$ with $n\in\mathbb{N}_{0}$ only.
For the the potential $V(\phi)$ we then have the form
$$V(\phi)=k_{2}\phi^{2}+k_{4}\phi^{4}+k_{6}\phi^{6}\,,$$
(42)
consistent with both the standard discretisation and the $Q$-exact
one. To keep the integrals numerically under control, for fixed $n$
we apply a variable transformation $\phi\rightarrow\phi/\widetilde{\phi}$ to obtain rescaled weights $\widetilde{Q}_{1}(2n)$ as
$$Q_{1}(2n)=\widetilde{\phi}\,^{2n+1}\widetilde{Q}_{1}(2n).$$
(43)
Since we have $Q_{1}(2n)\geq 0$, we can choose the rescaling factor to
be $\widetilde{\phi}=Q_{1}(2n)^{1/(2n+1)}$ and the rescaled weight
becomes $\widetilde{Q}_{1}(2n)=1$. Calculating the ratio of rescaled
weights as
$$\widetilde{R}_{1}(2n)=\frac{\widetilde{Q}_{1}(2n+2)}{\widetilde{Q}_{1}(2n)}=%
\widetilde{Q}_{1}(2n+2),$$
(44)
where both integrals $\widetilde{Q}_{1}(2n+2)$ and
$\widetilde{Q}_{1}(2n)$ are rescaled with the same factor
$\widetilde{\phi}=Q_{1}(2n)^{1/(2n+1)}$, we find that
$$R_{1}(2n)=\widetilde{\phi}\,^{2}\,\widetilde{R}_{1}(2n)\,.$$
(45)
In addition, the rescaled weight $\widetilde{Q}_{1}(2n+2)$ is now of
$\mathcal{O}(1)$ and can be evaluated reliably via numerical integration. So
if we start by integrating directly the numerically stable site
weights $Q_{1}(0)$ and $Q_{1}(2)$, we can recursively generate ratios
$R_{1}(2n)$ with higher and higher $n$. Note that after each calculation
of a ratio $R_{1}(2n)$, one needs to update the rescaling factor
$\widetilde{\phi}\rightarrow\widetilde{\phi}\,^{\prime}$. This can be
achieved most easily via
$$\widetilde{\phi}\,^{\prime}=\widetilde{\phi}\,^{\frac{2n+1}{2n+3}}\,R_{1}(2n)^%
{\frac{1}{2n+3}}.$$
(46)
Our procedure guarantees that all involved quantities are of
$\mathcal{O}(1)$. Once all ratios $R_{1}(2n)$ are known, one can calculate the
ratios $R_{m}(2n)$, noting that for the specific superpotential we
consider, eq.(38) simplifies to
$$R_{m}(2n)=m_{0}+m_{2}R_{1}(2n).$$
(47)
The calculation of the ratios $R_{0}(2n)$ as given in eq.(39) is
then straightforward.
3.2 Arbitrary Superpotential
In the context of broken supersymmetric quantum mechanics, one
encounters superpotentials with $\deg(P(\phi))$ = 1 mod 2. Therefore,
we now adapt the procedure from above to superpotentials of this
form. For simplicity, we restrict ourselves to the odd superpotential
we consider as the example in our series of papers,
$$P(\phi)=\sum_{i=1}^{3}p_{i}\phi^{i}.$$
(48)
If at least one of the coefficients $p_{1}$ and $p_{2}$ is nonzero, which
is always the case for the superpotentials we use, $V(\phi)$ reads
$$V(\phi)=k_{1}\phi+k_{2}\phi^{2}+k_{3}\phi^{3}+k_{4}\phi^{4},$$
(49)
and at least one of the coefficients $k_{1}$ and $k_{3}$ is nonzero
either. This has a two important consequences. Firstly, the moments
defined in eq.(32) are nonzero for $n$ odd, from which it
follows that the ratios $R_{F}^{\prime}(n)$ defined in eq.(36) have to
be calculated as well. Secondly, the weights $Q_{1}(n)$ are no longer
necessarily positive. It turns out, however, that for all pratical
purposes it does not affect the simulations. We will discuss this
further in Section 4.
For the evaluation of the integrals, we apply the same variable
transformation $\phi\rightarrow\phi/\widetilde{\phi}$ as before,
such that we have rescaled weights $\widetilde{Q}_{1}(n)$ given by
$$Q_{1}(n)=\widetilde{\phi}\,^{n+1}\,\widetilde{Q}_{1}(n).$$
(50)
We now choose $\widetilde{\phi}=|Q_{1}(n)|^{1/(n+1)}\cdot\mathop{\rm sgn}(Q_{1}(n))$. Then, the integral becomes $\widetilde{Q}_{1}(n)=1$
again as before. Furthermore, defining the rescaled ratios
$\widetilde{R}_{1}^{\prime}(n)$ to be
$$\widetilde{R}_{1}^{\prime}(n)=\frac{\widetilde{Q}_{1}(n+1)}{\widetilde{Q}_{1}(%
n)}=\widetilde{Q}_{1}(n+1),$$
(51)
where both integrals $\widetilde{Q}_{1}(n+1)$ and $\widetilde{Q}_{1}(n)$
are rescaled with the same factor $\widetilde{\phi}=|Q_{1}(n)|^{1/(n+1)}\cdot\mathop{\rm sgn}(Q_{1}(n))$, we find $R_{1}^{\prime}(n)=\widetilde{\phi}\,\widetilde{R}_{1}^{\prime}(n)$. We proceed analogously to the case of the even
superpotential by recursive iteration, with the only exception that we
generate the ratios $R_{1}^{\prime}(n)$ instead of the ratios $R_{1}(n)$. The
update for the rescaling factor $\widetilde{\phi}\rightarrow\widetilde{\phi}\,^{\prime}$ is done via
$$\widetilde{\phi}\,^{\prime}=|\widetilde{\phi}|^{\frac{n+1}{n+2}}\,|R_{1}^{%
\prime}(n)|^{\frac{1}{n+2}}\cdot\mathop{\rm sgn}(R_{1}^{\prime}(n)).$$
(52)
Once all the ratios $R_{1}^{\prime}(n)$ are known, one can calculate the ratios
$R_{1}(n)$ via eq.(37), the ratios $R_{m}(n)$ via eq.(38),
and the ratios $R_{0}(n)$ and $R_{0}^{\prime}(n)$ via eq.(39) and
(40), respectively.
4 Results of the Monte Carlo simulations
The results in this section are merely thought of as a proof of the
feasibility of the algorithm and as a test of its efficiency.
Comparing the Monte Carlo results with the exact solution of the
system at finite lattice spacing provided in our second paper
[5] of course also serves as a validation for
the algorithm. We refer to that paper for a thorough discussion and
physical interpretation of the results.
For the following Monte Carlo simulations, we consider the same
superpotentials and discretisations as in the previous two papers. In
particular, we simulate the system using the action with counterterm
for both unbroken and broken supersymmetry as well as the $Q$-exact
action for unbroken supersymmetry. Details for the various actions can
be found in the first paper of our series. Here we only give the
details of the superpotentials for broken and unbroken supersymmetry,
respectively,
$$\displaystyle P_{b}(\phi)$$
$$\displaystyle=-\frac{\mu^{2}}{4\lambda}\phi+\frac{1}{3}\lambda\phi^{3}\,,$$
(53)
$$\displaystyle P_{u}(\phi)$$
$$\displaystyle=\frac{1}{2}\mu\phi^{2}+\frac{1}{4}g\phi^{4}\,,$$
(54)
and we recall that the continuum limit is taken by fixing the
dimensionful parameters $\mu,g,\lambda$ and $L$ while taking the
lattice spacing $a\rightarrow 0$. In practice, the dimensionless
ratios $f_{u}=g/\mu^{2}$ and $f_{b}=\lambda/\mu^{3/2}$ fix the couplings
and $\mu L$ the extent of the system in units of $\mu$, while $a\mu$
and $a/L$ are subsequently sent to zero. In analogy to the number of
sweeps for a standard Monte Carlo simulation, we count the number of
times the algorithm is in either one of the two configuration spaces
$\mathcal{Z}_{F},F=0,1$. The statistics for a simulation is therefore given by
$Z_{0}+Z_{1}=Z_{a}$.
First, we consider the standard discretisation with the superpotential
$P_{u}$ such that supersymmetry is unbroken. As a first observable, we
show the results for the bosonic and fermionic correlation functions
for $\mu L=10$, $L/a=60$ and $f_{u}=1$ for $Z_{a}=10^{7}$ in
figure 5.
This is essentially the same plot as figure 10(b) in our second paper
[5], but now with the additional data from the
Monte Carlo simulation and plotted on a logarithmic scale. Note that
we use the notation $x=t$ in accordance with
[5]. The simulation indeed reproduces the exact
result within very small statistical errors which demonstrate the
efficiency of the algorithm. The exponential error reduction is due to
the use of the improved estimators for the two-point function which
are available in the context of the worm algorithms. The improvement
is particularly impressive for the fermionic correlator where the
error reduction allows to follow the correlator over more than seven
orders of magnitude without loss of statistical significance. In fact
the relative error for the lowest value of the fermionic correlator is
still only $4\%$.
As a second example, we show the mass gaps for different $\mu L$ at a
coupling $f_{u}=1$ with statistics of $Z_{a}=10^{6}$ in figure
6.
The $\mu L$ considered are in the region where thermal effects are
negligible and essentially only $Z_{0}$ contributes to the total
partition function, such that $Z_{a}\simeq Z_{0}$. We extract the
masses from the asymptotic behaviour of the correlation functon at
large $t$, i.e., we extract the lowest energy gap. Because of the
extremely good signal-to-noise ratio the asymptotic behaviour can be
truly reached and, in doing so, systematic errors from contributions
of excited states are essentially excluded. Of course, we know from
our exact results that the overlap of the simple operators we use to
construct the two-point function is close to maximal. This is clearly
visible in figure 5 where we observe an almost
purely exponential decay for all $t/L$. Because the energy gaps are
independent of $\mu L$, they are expected to fall on top of each other
for all values of $\mu L$ at fixed lattice spacing $a\mu$. This is
indeed the case within our numerical accuracy, and the extracted
masses, when expressed in units of $\mu$, indeed extrapolate to the
correct zero-temperature continuum limit. The inset of figure
6 shows a detailed comparison of the
simulation results with our exact solution from
[5] represented by the dashed line and we
observe a beautiful agreement even very close to the continuum.
Next, we consider the action with counterterm and the superpotential
$P_{b}$ for which the supersymmetry is broken. In this case we encounter
an issue concerning the potential non-positivity of the weights which
we already mentioned in Section 3.2. This potentially dangerous sign problem is not of
fermionic origin, but is instead related to the bond formulation of
the bosonic degrees of freedom. As a matter of fact it occurs already
in the purely bosonic system, independent of the dimensionality of the
system. However, negative weights only occur in a region of parameter
space which becomes irrelevant towards the continuum limit. In that
sense, the sign problem is a lattice artefact and can be avoided
straightforwardly. Nevertheless, in order to eliminate any systematic
error we deal with this bosonic sign problem by incorporating the sign
of the configuration into the observables, even though it has no
practical consequences.
As a first observable in the broken case, we show the bosonic and
fermionic two-point functions, $\langle\phi_{t}\phi_{0}\rangle$ and
$\langle\psi_{t}\overline{\psi}_{0}\rangle$, for periodic and antiperiodic
b.c. for $\mu L=10$ at fixed coupling $f_{b}=1$ in figure
7 for a statistics of $Z_{a}=10^{8}$.
The exact results from [5] are shown as dashed
lines. The simulation yields results which agree with the exact
results within the very small statistical errors on the level of
1$\permil$. Note that the correlators for periodic and antiperiodic
b.c. are constructed a posteriori from the simulation results in the
bosonic and fermionic sectors $Z_{0}$ and $Z_{1}$, respectively, and it is
crucial to sample the relative weight between the two sectors
correctly in order to get the final values right. The relative
sampling is solely in the responsibility of the fermion simulation
algorithm. Our results in figure 7 show that the
open fermion string algorithm indeed transits sufficiently well
between the two sectors.
This statement can be made more quantitative by looking at the ratio
$Z_{p}/Z_{a}$ which represents the Witten index in our field theoretic
setup. From our exact results in [5] we expect
a nonzero Witten index at finite lattice spacing which however
extrapolates to zero in the continuum limit. So the behaviour of the
algorithm towards the continuum limit is particularly interesting,
because for vanishing lattice spacing the would-be Goldstino at finite
lattice spacing turns into a true, massless Goldstino. In such a
situation one usually encounters critical slowing down of the
simulation algorithms, such that the errors on the results grow large
and the results become unreliable. The massless Goldstino is directly
related to the tunnelling between the bosonic and the fermionic sector,
and the reproduction of a Witten index $W=0$ in the continuum with
small errors is hence a true demonstration of the efficiency of the
open fermion string algorithm to transit between the bosonic and
fermionic sector. In addition, we know from [5]
that the lattice artefacts are exponentially enhanced towards zero
temperature and it is interesting to see how the simulation algorithm
handles this situation at coarse lattice spacing.
In figure 8 we show the ratio $Z_{p}/Z_{a}$
as a function of the lattice spacing $a\mu$ for different values of
$\mu L$ at fixed coupling $f_{b}=1$.
For this quantity, too, the simulation yields results which agree with
the exact results within the small statistical errors. Moreover, the
efficiency of the algorithm does not appear to deteriorate towards the
continuum limit or for small values of $\mu L$ where the Witten index
is very close to zero. This can for example be seen from the fact that
the errors obtained with fixed statistics essentially remain constant
towards the continuum limit and are also independent of the system
size. This nicely demonstrates the efficiency of the algorithm also
for a system with broken supersymmetry.
The last system we investigate with the worm algorithm is unbroken
supersymmetry formulated with the $Q$-exact action111For Monte
Carlo simulations using the $Q$-exact action for broken
supersymmetry, we encounter the very same problems we ran into in
the transfer matrix approach. The bond occupation number grows
extremely large even on small lattices and for coarse lattice
spacings such that the generation of reliable results turns out to
be impossible.. We first consider the ratio of partition functions
$Z_{p}/Z_{a}$ which in the limit of $\mu L\rightarrow\infty$ yields
the Witten index. From a simulational point of view, the ratio
essentially calculates the fraction of configurations in sector $\mathcal{Z}_{0}$
versus the ones in $\mathcal{Z}_{1}$. For unbroken supersymmetry the system is
almost exclusively in the bosonic sector, and hence the ratio is very
close to one except when the size of the system becomes very small,
i.e., in the high temperature limit. Moreover, from our exact results
in [5] we know that the lattice artefacts in
this quantity are very small and the continuum limit is not very
interesting. For these reasons, we consider in figure
9 the dependence of the ratio $Z_{p}/Z_{a}$ on
$\mu L$ for different values of the lattice spacing $a/L$ with a
statistics of $Z_{a}=10^{8}$.
Also for this quantity, we find that the results agree with the exact
result within the very small statistical errors. Again, the open
fermion string algorithm proves to be very efficient even close to
$\mu L\simeq 0$ where the tunnelling from the bosonic to the fermionic
sector and vice versa becomes important and dominates the behaviour of
the system. Thus, even in this somewhat extreme situation of very high
temperature, the algorithm does not show any signs of critical slowing
down despite the fact that there is a quasi-zero mode in the system.
Note that the algorithm is capable of handling negative bare masses
independent of the discretisation used and
fig.9 is simply also an illustration of this
fact.
The last quantity we calculate are the lowest bosonic and fermionic
mass gaps for different $\mu L$ at fixed coupling of $f_{u}=1$ from a
statistics of $Z_{a}=10^{6}$. The mass gaps are extracted from the
two-point correlation functions exactly in the same way as before for
the standard action, and in figure 10 we show
the results of this analysis.
As expected, the masses for the boson and the fermion are indeed
indistinguishable within statistical errors. The degeneracy of the
masses at finite lattice spacing due to the $Q$-exactness of the
action emerges also for the results from Monte Carlo simulations. Note
that the chosen values for $\mu L$ lie well within the region where
thermal effects are negligible and the masses extrapolate nicely to
the correct zero-temperature continuum limit. The inset in figure
10 shows a detailed comparison with our exact
results from [5] and we again observe beautiful
agreement.
5 Conclusions
In this paper we present an algorithm for simulating ${\cal N}=2$
supersymmetric quantum mechanics on the lattice. The algorithm is
based on the reformulation of the system in terms of bosonic and
fermionic bonds, and in essence represents an efficient Monte Carlo
scheme for updating fermionic and bosonic bond configurations. The
updating of the fermionic degrees of freedom is of specific interest,
because this in in general the most challenging part of a
simulation. This is particularly true for systems with broken
supersymmetry, where standard simulation algorithms suffer from
critical slowing down due to the massless Goldstino mode. In addition,
these systems inevitably also suffer from a sign problem related to
the Goldstino and the vanishing Witten index.
In contrast, the fermion simulation algorithm proposed in
[2] eliminates critical slowing down by directly
sampling the fermionic two-point correlation function. It is based on
introducing a fluctuating open fermion string which efficiently
updates the bond configurations on all length scales up to the
correlation length associated with the fermionic correlation
function. As a consequence, the fermion string induces frequent
tunnellings between the bosonic and fermionic vacuum when that
correlation length becomes large. Since the two vacua contribute to
the partition function with opposite signs, the frequent tunnelling
guarantees sufficiently small statistical fluctuations for the average
sign, and hence a solution to the fermion sign problem. In fact, the
more severe the sign problem gets towards the continuum limit, the
more efficiently the algorithm tunnels between the bosonic and
fermionic sectors. This is of course due to the growing correlation
length associated with the vanishing Goldstino mass. The bosonic
degrees of freedom can be expressed in terms of bonds as
well. Therefore, we also give the details of an updating algorithm for
the bosonic bond configurations. Since we consider $Q$-exact
discretisations in addition to the standard one, the algorithm
involves updating generic types of bonds.
The simulation algorithm requires the calculation of the site weights
$Q_{F}(N)$. Their numerical evaluation, however, turns out to be
numerically unstable for large site occupation numbers $N$. Hence, in
Section 3, we devise a computational strategy which
allows to reliably evaluate the ratios of weights for arbitrarily
large occupation numbers. Since this is a generic problem occuring in
the bond formulation of field theories with real scalar fields, such
an computational scheme is useful also in other situations.
Finally, we present a selection of results obtained using the open
fermion string algorithm. We concentrate on two specific realisations
of supersymmetric quantum mechanics, one with broken and one with
unbroken supersymmetry. In addition, we consider both the standard and
the $Q$-exact discretisation. Since exact results are available at
finite lattice spacing from our investigation in
[5], we can benchmark our stochastic results
and directly validate them. The calculation of the bosonic and
fermionic correlation functions shows that they can be determined very
accurately over several orders of magnitude. This allows for a very
precise computation of the boson and fermion masses, the latter in
many cases with a smaller error than the former. In general, a
precision of 1‰ can be reached with a very modest
computational effort. In systems with broken supersymmetry it is
crucial that the simulation algorithm efficiently samples the relative
weights between the bosonic and fermionic sectors. Our results for the
partition function ratio $Z_{p}/Z_{a}$, i.e., the Witten index, show that
this is indeed the case. For fixed statistics, the errors do not grow
towards the continuum limit. In that limit the index gets very close
to zero and the sign problem would therefore be most
severe. Similarly, the error is essentially independent of the system
size, which shows that the sign problem is truly solved.
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Wolfram Research Inc., Mathematica Version 9.0, 2012. |
Solitons and geometrical structures
in a perfect fluid spacetime
Adara M. Blaga
()
Abstract
Geometrical aspects of a perfect fluid spacetime are described in terms of different curvature tensors and $\eta$-Ricci and $\eta$-Einstein solitons in a perfect fluid spacetime are determined. Conditions for the Ricci soliton to be steady, expanding or shrinking are also given. In a particular case when the potential vector field $\xi$ of the soliton is of gradient type, $\xi:=grad(f)$, we derive from the soliton equation a nonlinear second order PDE satisfied by $f$.
†† 2010 Mathematics Subject Classification.
53B50, 53C44, 53C50, 83C02.
†† Key words and phrases.
Ricci soliton, Einstein soliton, perfect fluid, Lorentz space.
1 Introduction
Lorentzian manifolds form a special subclass of pseudo-Riemannian manifolds of great importance in general relativity, where spacetime can be modeled as a $4$-dimensional Lorentzian manifold of signature (3,1) or, equivalently, (1,3).
Relativistic fluid models are of great interest in different branches of astrophysics, plasma physics, nuclear physics etc. Perfect fluids are often used in general relativity to model idealized distributions of matter, such as the interior of a star or an isotropic universe. Einstein’s gravitational equation can describe the behavior of a perfect fluid inside of a spherical object and the Friedmann-Lemaître-Robertson-Walker equations are used to describe the evolution of the universe. In general relativity, the source for the gravitational field is the energy-momentum tensor.
A perfect fluid can be completely characterized by its rest frame mass density and isotropic pressure. It has no shear stresses, viscosity, nor heat conduction and is characterized by an energy-momentum tensor of the form:
(1)
$$T(X,Y)=pg(X,Y)+(\sigma+p)\eta(X)\eta(Y),$$
for any $X$, $Y\in\chi(M)$, where $p$ is the isotropic pressure, $\sigma$ is the energy-density, $g$ is the metric tensor of Minkowski spacetime, $\xi:=\sharp(\eta)$ is the velocity vector of the fluid and $g(\xi,\xi)=-1$. If $\sigma=-p$,
the energy-momentum tensor is Lorentz-invariant ($T=-\sigma g$) and in this case we talk about the vacuum. If $\sigma=3p$, the medium is a radiation fluid.
The field equations governing the perfect fluid motion are Einstein’s gravitational equations:
(2)
$$kT(X,Y)=S(X,Y)+(\lambda-\frac{scal}{2})g(X,Y),$$
for any $X$, $Y\in\chi(M)$, where $\lambda$ is the cosmological constant,
$k$ is the gravitational constant (which can be taken $8\pi G$, with $G$ the universal gravitational constant), $S$ is the Ricci tensor and $scal$ is the scalar curvature of $g$. They are obtained from Einstein’s equations by adding a cosmological constant in order to get a static universe, according to Einstein’s idea. In modern cosmology, it is considered as a candidate for dark energy, the cause of the acceleration of the expansion of the universe.
Replacing $T$ from (1) we obtain:
(3)
$$S(X,Y)=-(\lambda-\frac{scal}{2}-kp)g(X,Y)+k(\sigma+p)\eta(X)\eta(Y),$$
for any $X$, $Y\in\chi(M)$. Recall that a manifold having the property that the Ricci tensor $S$ is a functional combination of $g$ and $\eta\otimes\eta$, for $\eta$ a $1$-form $g$ dual to a unitary vector field, is called quasi-Einstein [4]. Quasi-Einstein manifolds arose during the study of exact solutions of Einstein field equations. For example, the Robertson-Walker spacetime are quasi-Einstein manifolds [11]. They also can be taken as a
model of the perfect fluid spacetime in general relativity [9], [10].
Ricci flow and Einstein flow are intrinsec geometric flows on a pseudo-Riemannian manifold, whose fixed points are solitons. In our paper, we are interested in a generalized version of the following two types of solitons:
1.
Ricci solitons [12], which generates self-similar solutions to the Ricci flow:
(4)
$$\frac{\partial}{\partial t}g=-2S,$$
2.
Einstein solitons [3], which generate self-similar solutions to the Einstein flow:
(5)
$$\frac{\partial}{\partial t}g=-2(S-\frac{scal}{2}g).$$
Perturbing the equations that define these types of solitons by a multiple of a certain $(0,2)$-tensor field $\eta\otimes\eta$, we obtain two slightly more general notions, namely, $\eta$-Ricci solitons and $\eta$-Einstein solitons, which we shall consider in a perfect fluid spacetime, i.e. in a $4$-dimensional pseudo-Riemannian manifold $M$ with a Lorentzian metric $g$ whose content is a perfect fluid.
2 Basic properties of a perfect fluid spacetime
Let $(M,g)$ be a general relativistic perfect fluid spacetime satisfying (3).
Consider $\{E_{i}\}_{1\leq i\leq 4}$ an orthonormal frame field i.e. $g(E_{i},E_{j})=\varepsilon_{ij}\delta_{ij}$, $i$, $j\in\{1,2,3,4\}$ with $\varepsilon_{11}=-1$, $\varepsilon_{ii}=1$, $i\in\{2,3,4\}$, $\varepsilon_{ij}=0$, $i,j\in\{1,2,3,4\}$, $i\neq j$. Let $\xi=\sum_{i=1}^{4}\xi^{i}E_{i}$. Then
$$-1=g(\xi,\xi)=\sum_{1\leq i,j\leq 4}\xi^{i}\xi^{j}g(E_{i},E_{j})=\sum_{i=1}^{4%
}\varepsilon_{ii}(\xi^{i})^{2}$$
and
$$\eta(E_{i})=g(E_{i},\xi)=\sum_{j=1}^{4}\xi^{j}g(E_{i},E_{j})=\varepsilon_{ii}%
\xi^{i}.$$
Contracting (3) and taking into account that $g(\xi,\xi)=-1$, we get:
(6)
$$scal=4\lambda+k(\sigma-3p).$$
Therefore:
(7)
$$S(X,Y)=(\lambda+\frac{k(\sigma-p)}{2})g(X,Y)+k(\sigma+p)\eta(X)\eta(Y),$$
for any $X$, $Y\in\chi(M)$.
Example 2.1.
A radiation fluid has constant scalar curvature equal to $4\lambda$.
If we denote by $\nabla$ the Levi-Civita connection associated to $g$, then for any $X$, $Y$, $Z\in\chi(M)$:
$$(\nabla_{X}S)(Y,Z):=X(S(Y,Z))-S(\nabla_{X}Y,Z)-S(Y,\nabla_{X}Z)=$$
$$=k(\sigma+p)\{\eta(Y)g(\nabla_{X}\xi,Z)+\eta(Z)g(\nabla_{X}\xi,Y)\}=$$
(8)
$$=k(\sigma+p)\{\eta(Y)(\nabla_{X}\eta)Z+\eta(Z)(\nabla_{X}\eta)Y\}.$$
Imposing different conditions on the covariant differential of $S$, we have that:
i) $M$ is Ricci symmetric if $\nabla S=0$;
ii) $S$ is Codazzi tensor if $(\nabla_{X}S)(Y,Z)=(\nabla_{Y}S)(X,Z)$, for any $X$, $Y$, $Z\in\chi(M)$;
iii) $S$ is $\alpha$-recurrent if $(\nabla_{X}S)(Y,Z)=\alpha(X)S(Y,Z)$, for any $X$, $Y$, $Z\in\chi(M)$, for $\alpha$ a nonzero $1$-form;
iv) $S$ is weakly pseudo Ricci symmetric if $S$ is not identically zero and $(\nabla_{X}S)(Y,Z)=\alpha(X)S(Y,Z)+\alpha(Y)S(Z,X)+\alpha(Z)S(X,Y)$, for any $X$, $Y$, $Z\in\chi(M)$, for $\alpha$ a nonzero $1$-form;
v) $S$ is pseudo Ricci symmetric if $S$ is not identically zero and $(\nabla_{X}S)(Y,Z)=2\alpha(X)S(Y,Z)+\alpha(Y)S(Z,X)+\alpha(Z)S(X,Y)$, for any $X$, $Y$, $Z\in\chi(M)$, for $\alpha$ a nonzero $1$-form.
Proposition 2.2.
Let $(M,g)$ be a general relativistic perfect fluid spacetime satisfying (7).
1.
If $M$ is Ricci symmetric or $S$ is Codazzi tensor, then $p=-\sigma$ or $\nabla\xi=0$.
2.
If $S$ is $\alpha$-recurrent, then $p=-\sigma=\frac{\lambda}{k}$ or $\nabla\xi=0$.
3.
If $S$ is (weakly) pseudo Ricci symmetric, then $p=\frac{2}{3}(\frac{\lambda}{k})-\frac{\sigma}{3}$. In this cases, $\xi$ is torse-forming (in particular, irrotational and geodesic) vector field and $\eta$ is closed (and Codazzi) $1$-form.
Proof..
1.
If $\nabla S=0$, writing (2) for $Y=Z$ we get $(\sigma+p)\eta(Y)g(\nabla_{X}\xi,Y)=0$, for any $X$, $Y\in\chi(M)$. It follows $\nabla\xi=0$ or $p=-\sigma$.
If $(\nabla_{X}S)(Y,Z)=(\nabla_{Y}S)(X,Z)$, for any $X$, $Y$, $Z\in\chi(M)$, using (2) for $Y=Z:=\xi$ we get $(\sigma+p)g(\nabla_{\xi}\xi,X)=0$, for any $X\in\chi(M)$. It follows $\nabla_{\xi}\xi=0$ or $p=-\sigma$. If $\nabla_{\xi}\xi=0$, using (2) for $X:=\xi$ we get $(\sigma+p)g(\nabla_{Y}\xi,Z)=0$, for any $Y$, $Z\in\chi(M)$. It follows $\nabla\xi=0$ or $p=-\sigma$.
2.
If $(\nabla_{X}S)(Y,Z)=\alpha(X)S(Y,Z)$, for any $X$, $Y$, $Z\in\chi(M)$, using (7) and (2) and writing the obtained relation for $Y=Z:=\xi$ we get
$$(\lambda-\frac{k(\sigma+3p)}{2})\alpha(X)=0,$$
for any $X\in\chi(M)$. It follows $p=\frac{2\lambda-k\sigma}{3k}$. Writing the same relation for $Z:=\xi$ we get
$$-k(\sigma+p)g(\nabla_{X}\xi,Y)=(\lambda-\frac{k(\sigma+3p)}{2})\alpha(X)\eta(Y%
)=0,$$
for any $X$, $Y\in\chi(M)$. It follows $\nabla\xi=0$ or $p=-\sigma$ (so $p=-\sigma=\frac{\lambda}{k}$).
3.
If $(\nabla_{X}S)(Y,Z)=\alpha(X)S(Y,Z)+\alpha(Y)S(Z,X)+\alpha(Z)S(X,Y)$, for any $X$, $Y$, $Z\in\chi(M)$, using (7) and (2) and writing the obtained relation for $Y=Z:=\xi$ we get
$$(\lambda-\frac{k(\sigma+3p)}{2})[\alpha(X)-2\alpha(\xi)\eta(X)]=0,$$
for any $X\in\chi(M)$. Writing the same relation for $X=Y=Z:=\xi$ we get
$$(\lambda-\frac{k(\sigma+3p)}{2})\alpha(\xi)=0.$$
If $\lambda-\frac{k(\sigma+3p)}{2}\neq 0$ follows $\alpha(\xi)=0$ and $\alpha(X)=\alpha(\xi)\eta(X)=0$, for any $X\in\chi(M)$, which contradicts the fact that $\alpha$ is nonzero. Therefore, $\lambda-\frac{k(\sigma+3p)}{2}=0$, so $p=\frac{2\lambda-k\sigma}{3k}$.
Writing now the obtained relation for $Z:=\xi$ we get
$$k(\sigma+p)\{g(\nabla_{X}\xi,Y)+\alpha(\xi)[g(X,Y)+\eta(X)\eta(Y)]\}=0,$$
for any $X$, $Y\in\chi(M)$.
If $\sigma+p=0$ follows $p=-\sigma=\frac{\lambda}{k}$ and from (7), $S(X,Y)=0$, for any $X$, $Y\in\chi(M)$, which contradicts the fact that $S$ is nonzero. Therefore:
$$g(\nabla_{X}\xi,Y)+\alpha(\xi)[g(X,Y)+\eta(X)\eta(Y)]=0,$$
for any $X$, $Y\in\chi(M)$ which implies
$$\nabla_{X}\xi=-\alpha(\xi)[X+\eta(X)\xi],$$
for any $X\in\chi(M)$ i.e. $\xi$ is a torse-forming vector field. Then
$$(curl(\xi))(X,Y):=g(\nabla_{X}\xi,Y)-g(\nabla_{Y}\xi,X)=0,$$
for any $X$, $Y\in\chi(M)$ i.e. $\xi$ is irrotational. Also
$$g(\nabla_{\xi}\xi,X)=g(\nabla_{X}\xi,\xi)=\frac{1}{2}X(g(\xi,\xi))=0,$$
for any $X\in\chi(M)$ i.e. $\xi$ is geodesic.
Concerning $\eta$, notice that
$$(\nabla_{X}\eta)Y-(\nabla_{Y}\eta)X:=X(\eta(Y))-\eta(\nabla_{X}Y)-Y(\eta(X))+%
\eta(\nabla_{Y}X):=(d\eta)(X,Y).$$
Also
$$X(\eta(Y))-\eta(\nabla_{X}Y)-Y(\eta(X))+\eta(\nabla_{Y}X)=g(\nabla_{X}\xi,Y)-g%
(\nabla_{Y}\xi,X)=0$$
which implies $d\eta=0$.
If $(\nabla_{X}S)(Y,Z)=2\alpha(X)S(Y,Z)+\alpha(Y)S(Z,X)+\alpha(Z)S(X,Y)$, for any $X$, $Y$, $Z\in\chi(M)$, following the steps of computations above, we get the same conclusions.
∎
Remark the following facts:
1) If in the general relativistic perfect fluid spacetime $\xi$ is not $\nabla$-parallel and it has $\alpha$-recurrent Ricci tensor field, we have the vacuum case. The energy-momentum tensor is $T_{vac}=\frac{\lambda}{k}g$.
2) $S(X,\xi)=-S(\xi,\xi)\eta(X)=[\lambda-\frac{k(\sigma+3p)}{2}]g(X,\xi)$, for any $X\in\chi(M)$ which shows that $\lambda-\frac{k(\sigma+3p)}{2}$ is the eigenvalue of $Q$ corresponding to the eigenvector $\xi$, where $Q$ is defined by $g(QX,Y):=S(X,Y)$, $X$, $Y\in\chi(M)$. From Proposition 2.2 we deduce that if $\xi$ is not $\nabla$-parallel and $M$ is Ricci symmetric or $S$ is Codazzi tensor, then $(M,g)$ is Einstein and $S(X,\xi)=(\lambda+k\sigma)\eta(X)$, for any $X\in\chi(M)$, and if $S$ is (weakly) pseudo Ricci symmetric, then $S(X,\xi)=0$, for any $X\in\chi(M)$, hence $\xi\in\ker Q$; also, $div(\xi):=\sum_{i=1}^{4}\varepsilon_{ii}g(\nabla_{E_{i}}\xi,E_{i})=-3\alpha(\xi)$. Notice that in all these cases, if $\sigma>-\frac{\lambda}{k}$, the scalar curvature $scal:=\sum_{i=1}^{4}\varepsilon_{ii}S(E_{i},E_{i})=4\lambda+k(\sigma-3p)>3(%
\lambda-kp)$ is positive.
3) Considering Plebanski energy conditions $\sigma\geq 0$ and $-\sigma\leq p\leq\sigma$ for perfect fluids, when $S$ is (weakly) pseudo Ricci symmetric, the energy-density is lower bounded by $\max\{-\frac{\lambda}{k},\frac{\lambda}{2k}\}$. It was observed that a positive cosmological constant $\lambda$ acts as repulsive gravity, explaining the accelerating universe. The observations of Edwin Hubble confirmed that the universe is expanding, therefore, this seems to be the real case, so $\sigma\geq\frac{\lambda}{2k}$.
4) Let $hX:=X+\eta(X)\xi$ be the projection tensor, $X\in\chi(M)$. If $S$ is (weakly) pseudo Ricci symmetric and $\alpha(\xi)\neq 0$, then $(h,\xi,\eta,g)$ is a Lorentzian concircular structure [17]. In the particular case $\alpha(\xi)=-1$ it becomes LP-Sasakian structure [13]. In the other case, if $\alpha(\xi)=0$, then $\xi$ and $\eta$ are $\nabla$-parallel, $\xi$ is divergence free, hence harmonic vector field.
Applying the covariant derivative to (2) we obtain:
(9)
$$k(\nabla_{X}T)(Y,Z)=(\nabla_{X}S)(Y,Z)-\frac{1}{2}X(scal)g(Y,Z),$$
for any $X$, $Y$, $Z\in\chi(M)$.
Proposition 2.3.
Let $(M,g)$ be a general relativistic perfect fluid spacetime satisfying (7).
1.
If $T$ is covariantly constant, then the scalar curvature is constant.
2.
If $T$ is Codazzi tensor, then the scalar curvature satisfies $grad(scal)+(dscal)(\xi)\xi+2k(\sigma+p)\nabla_{\xi}\xi=0$.
3.
If $T$ is $\alpha$-recurrent, then the scalar curvature satisfies $dscal+[4\lambda-scal-k(\sigma+3p)]\alpha=0$.
Proof..
1.
If $T$ is covariantly constant i.e. $\nabla T=0$ follows $(\nabla_{X}S)(Y,Z)=\frac{1}{2}X(scal)g(Y,Z)$, for any $X$, $Y$, $Z\in\chi(M)$. For $Y=Z:=\xi$ we have:
(10)
$$2S(\nabla_{X}\xi,\xi)=\frac{1}{2}X(scal),$$
for any $X\in\chi(M)$.
Replacing $S$ from (7) we obtain $X(scal)=0$, for any $X\in\chi(M)$, which implies that the scalar curvature is constant. Therefore, $\nabla S=0$ (i.e. $M$ is Ricci symmetric) and from Proposition 2.2 follows $p=-\sigma$ (the vacuum case) or $\nabla\xi=0$. If $\xi$ is not $\nabla$-parallel:
(11)
$$S=-(\lambda-\frac{scal}{2}-kp)g$$
and from (6) we get:
(12)
$$p=\frac{4\lambda-scal}{4k},$$
so the pressure and the energy-density are constant.
2.
The condition $(\nabla_{X}T)(Y,Z)=(\nabla_{Y}T)(X,Z)$, for any $X$, $Y$, $Z\in\chi(M)$ is equivalent to:
$$(\nabla_{X}S)(Y,Z)-(\nabla_{Y}S)(X,Z)=\frac{1}{2}[X(scal)g(Y,Z)-Y(scal)g(Z,X)],$$
for any $X$, $Y$, $Z\in\chi(M)$. Replacing $\nabla S$ from (2) we get:
(13)
$$2k(\sigma+p)[\eta(Y)g(\nabla_{X}\xi,Z)+\eta(Z)g(\nabla_{X}\xi,Y)-\eta(X)g(%
\nabla_{Y}\xi,Z)-\eta(Z)g(\nabla_{Y}\xi,X)]=$$
$$=X(scal)g(Y,Z)-Y(scal)g(Z,X),$$
for any $X$, $Y$, $Z\in\chi(M)$. For $Y=Z:=\xi$ and taking into account that $g(\nabla_{X}\xi,\xi)=0$, for any $X\in\chi(M)$, the above relation becomes:
$$2k(\sigma+p)g(\nabla_{\xi}\xi,X)+X(scal)+\xi(scal)\eta(X)=0,$$
for any $X\in\chi(M)$. Also $X(scal)=(dscal)(X)=g(grad(scal),X)$ and we obtain:
$$2k(\sigma+p)\nabla_{\xi}\xi+grad(scal)+(dscal)(\xi)\xi=0.$$
If the scalar curvature is constant, then $\nabla_{\xi}\xi=0$ or $p=-\sigma$. If $\nabla_{\xi}\xi=0$, taking $X:=\xi$ in (13) we get
$(\sigma+p)g(\nabla_{Y}\xi,Z)=0$ which implies $p=-\sigma$ or $\nabla\xi=0$.
If $\xi$ is geodesic vector field, then $grad(\sigma)+(d\sigma)(\xi)\xi=3[grad(p)+(dp)(\xi)\xi]$ and the gradient of the scalar curvature is collinear with $\xi$.
3.
The condition $(\nabla_{X}T)(Y,Z)=\alpha(X)T(Y,Z)$, for any $X$, $Y$, $Z\in\chi(M)$, with $\alpha$ a nonzero $1$-form, is equivalent to:
$$(\nabla_{X}S)(Y,Z)-\frac{X(scal)}{2}g(Y,Z)=\alpha(X)S(Y,Z)+\alpha(X)(\lambda-%
\frac{scal}{2})g(Y,Z),$$
for any $X$, $Y$, $Z\in\chi(M)$. Replacing $S$ from (7) and $\nabla S$ from (2) we get:
(14)
$$k(\sigma+p)[\eta(Y)g(\nabla_{X}\xi,Z)+\eta(Z)g(\nabla_{X}\xi,Y)-\alpha(X)\eta(%
Y)\eta(Z)]=$$
$$=\frac{1}{2}\{X(scal)+\alpha(X)[4\lambda-scal+k(\sigma-p)]\}g(Y,Z),$$
for any $X$, $Y$, $Z\in\chi(M)$. For $Y=Z:=\xi$ and taking into account that $g(\nabla_{X}\xi,\xi)=0$, for any $X\in\chi(M)$, the above relation becomes:
$$X(scal)=\alpha(X)[-4\lambda+scal+k(\sigma+3p)],$$
for any $X\in\chi(M)$.
If the scalar curvature is constant, then $\sigma=0$ and the pressure is constant $p=\frac{4\lambda-scal}{3k}$.
∎
Remark the following facts:
1) If the energy-momentum tensor $T$ of a general relativistic perfect fluid spacetime is covariantly constant, then we have the vacuum case or $\xi$ is $\nabla$-parallel (in particular, geodesic).
2) If $T$ is Codazzi and the scalar curvature is not constant but $\xi$ is a geodesic vector field, then the gradient of the scalar curvature is collinear with $\xi$. It was proved that for a Codazzi energy-momentum tensor, the Ricci tensor $S$ is conserved [2].
3) For an $\alpha$-recurrent energy-momentum tensor, if the scalar curvature is constant, then the pressure is constant ($p=\frac{4\lambda-scal}{3k}$), but the energy-density vanishes.
3 Perfect fluid spacetime with torse-forming vector field $\xi$
We shall treat the special case when $\xi$ is a torse-forming vector field [20] of the form:
(15)
$$\nabla\xi=I_{\chi(M)}+\eta\otimes\xi.$$
Then $\nabla_{\xi}\xi=\xi+\eta(\xi)\xi=0$, for any $X\in\chi(M)$ i.e. $\xi$ is geodesic, $g(\nabla_{X}\xi,\xi)=0$ and $(d\eta)(X,Y):=X(\eta(Y))-Y(\eta(X))-\eta([X,Y])=X(g(Y,\xi))-Y(g(X,\xi))-g(%
\nabla_{X}Y,\xi)+g(\nabla_{Y}X,\xi)=g(\nabla_{X}\xi,Y)-g(\nabla_{Y}\xi,X)=0$. We also have:
(16)
$$R(X,Y)\xi=\eta(Y)X-\eta(X)Y,$$
(17)
$$\eta(R(X,Y)Z)=-\eta(Y)g(X,Z)+\eta(X)g(Y,Z),$$
for any $X$, $Y$, $Z\in\chi(M)$.
In this case, we shall see which are the consequences of certain conditions imposed to different types of curvatures of this space, namely, when the curvature satisfies conditions of the type $\mathcal{T}(\xi,\cdot)\cdot S=0$ and $S(\xi,\cdot)\cdot\mathcal{T}=0$, where
$\mathcal{T}$ is the Riemann curvature tensor $R$, the projective curvature tensor $\mathcal{W}$ [14], the concircular curvature tensor $\mathcal{P}$ [19], the conformal curvature tensor $\mathcal{C}$ [1] and the conharmonic curvature tensor $\mathcal{H}$ [16] defined as follows:
(18)
$$\mathcal{W}(X,Y)Z:=R(X,Y)Z+\frac{1}{\dim(M)-1}[g(Z,X)QY-g(Y,Z)QX]$$
(19)
$$\mathcal{P}(X,Y)Z:=R(X,Y)Z+\frac{scal}{\dim(M)(\dim(M)-1)}[g(Z,X)Y-g(Y,Z)X]$$
$$\mathcal{C}(X,Y)Z:=R(X,Y)Z+\frac{1}{\dim(M)-2}\{-\frac{scal}{\dim(M)-1}[g(Z,X)%
Y-g(Y,Z)X]+$$
(20)
$$+g(Z,X)QY-g(Y,Z)QX+S(Z,X)Y-S(Y,Z)X\}$$
(21)
$$\mathcal{H}(X,Y)Z:=R(X,Y)Z+\frac{1}{\dim(M)-2}[g(Z,X)QY-g(Y,Z)QX+S(Z,X)Y-S(Y,Z%
)X]$$
for any $X$, $Y$, $Z\in\chi(M)$.
Remark that for an empty gravitational field characterized by vanishing Ricci tensor, the curvature tensors $R$, $\mathcal{W}$ and $\mathcal{H}$ coincide.
Let us denote by $\mathcal{T}$ a curvature tensor of type $(1,3)$ and ask for certain Ricci-semisymmetry curvature conditions, namely, $\mathcal{T}(\xi,\cdot)\cdot S=0$ and $S(\xi,\cdot)\cdot\mathcal{T}=0$, where by $\cdot$ we denote the derivation of the tensor algebra at each point of the tangent space:
•
$(\mathcal{T}(\xi,X)\cdot S)(Y,Z):=((\xi\wedge_{\mathcal{T}}X)\cdot S)(Y,Z):=S(%
(\xi\wedge_{\mathcal{T}}X)Y,Z)+S(Y,(\xi\wedge_{\mathcal{T}}X)Z)$, for $(X\wedge_{\mathcal{T}}Y)Z:=\mathcal{T}(X,Y)Z$;
•
$(S(\xi,X)\cdot\mathcal{T})(Y,Z)W:=((\xi\wedge_{S}X)\cdot\mathcal{T})(Y,Z)W:=%
\mathcal{T}((\xi\wedge_{S}X)Y,Z)W+\linebreak\mathcal{T}(Y,(\xi\wedge_{S}X)Z)W+%
\mathcal{T}(Y,Z)(\xi\wedge_{S}X)W$, for $(X\wedge_{S}Y)Z:=S(Y,Z)X-S(X,Z)Y$.
3.1 Perfect fluid spacetime satisfying $\mathcal{T}(\xi,\cdot)\cdot S=0$
The condition $\mathcal{T}(\xi,\cdot)\cdot S=0$ is equivalent to
(22)
$$S(\mathcal{T}(\xi,X)Y,Z)+S(Y,\mathcal{T}(\xi,X)Z)=0,$$
for any $X$, $Y$, $Z\in\chi(M)$ and from (7) with:
$$(\lambda+\frac{k(\sigma-p)}{2})[g(\mathcal{T}(\xi,X)Y,Z)+g(Y,\mathcal{T}(\xi,X%
)Z)]+$$
$$+k(\sigma+p)[\eta(\mathcal{T}(\xi,X)Y)\eta(Z)+\eta(Y)\eta(\mathcal{T}(\xi,X)Z)%
]=0,$$
for any $X$, $Y$, $Z\in\chi(M)$.
Theorem 3.1.
Let $(M,g)$ be a general relativistic perfect fluid spacetime satisfying (3) with torse-forming vector field $\xi$.
1.
If $R(\xi,\cdot)\cdot S=0$, then $p=-\sigma$.
2.
If $\mathcal{W}(\xi,\cdot)\cdot S=0$, then $p=-\sigma$ or $p=\frac{2\lambda-3}{2k}$.
3.
If $\mathcal{P}(\xi,\cdot)\cdot S=0$, then $p=-\sigma$ or $p=\frac{4\lambda+k\sigma-12}{3k}$.
4.
If $\mathcal{C}(\xi,\cdot)\cdot S=0$, then $p=-\sigma$ or $p=\frac{2\lambda-k\sigma-6}{3k}$.
5.
If $\mathcal{H}(\xi,\cdot)\cdot S=0$, then $p=-\sigma$ or $p=\frac{\lambda-1}{k}$.
Proof..
1.
From the symmetries of $R$ and (16) and (17) we obtain:
(23)
$$k(\sigma+p)[\eta(Z)g(X,Y)+\eta(Y)g(Z,X)+2\eta(X)\eta(Y)\eta(Z)]=0,$$
for any $X$, $Y$, $Z\in\chi(M)$. Take $Z:=\xi$ and (23) becomes:
(24)
$$k(\sigma+p)[g(X,Y)+\eta(X)\eta(Y)]=0,$$
for any $X$, $Y\in\chi(M)$ and we obtain $p=-\sigma$.
2.
From (18), from the symmetries of $R$ and (16) and (17) we obtain:
(25)
$$k(\sigma+p)(2\lambda-2kp-3)[\eta(Z)g(X,Y)+\eta(Y)g(Z,X)+2\eta(X)\eta(Y)\eta(Z)%
]=0,$$
for any $X$, $Y$, $Z\in\chi(M)$. Take $Z:=\xi$ and (25) becomes:
(26)
$$k(\sigma+p)(2\lambda-2kp-3)[g(X,Y)+\eta(X)\eta(Y)]=0,$$
for any $X$, $Y\in\chi(M)$ and we obtain $p=-\sigma$ or the pressure is constant $p=\frac{2\lambda-3}{2k}$.
3.
From (19), from the symmetries of $R$ and (16) and (17) we obtain:
(27)
$$k(\sigma+p)[4\lambda+k(\sigma-3p)-12][\eta(Z)g(X,Y)+\eta(Y)g(Z,X)+2\eta(X)\eta%
(Y)\eta(Z)]=0,$$
for any $X$, $Y$, $Z\in\chi(M)$. Take $Z:=\xi$ and (27) becomes:
(28)
$$k(\sigma+p)[4\lambda+k(\sigma-3p)-12][g(X,Y)+\eta(X)\eta(Y)]=0,$$
for any $X$, $Y\in\chi(M)$ and we obtain $p=-\sigma$ or $p=\frac{4\lambda+k\sigma-12}{3k}$.
4.
From (3), from the symmetries of $R$ and (16) and (17) we obtain:
(29)
$$k(\sigma+p)[2\lambda-k(\sigma+3p)-6][\eta(Z)g(X,Y)+\eta(Y)g(Z,X)+2\eta(X)\eta(%
Y)\eta(Z)]=0,$$
for any $X$, $Y$, $Z\in\chi(M)$. Take $Z:=\xi$ and (32) becomes:
(30)
$$k(\sigma+p)[2\lambda-k(\sigma+3p)-6][g(X,Y)+\eta(X)\eta(Y)]=0,$$
for any $X$, $Y\in\chi(M)$ and we obtain $p=-\sigma$ or $p=\frac{2\lambda-k\sigma-6}{3k}$.
5.
From (21), from the symmetries of $R$ and (16) and (17) we obtain:
(31)
$$k(\sigma+p)(\lambda-kp-1)[\eta(Z)g(X,Y)+\eta(Y)g(Z,X)+2\eta(X)\eta(Y)\eta(Z)]=0,$$
for any $X$, $Y$, $Z\in\chi(M)$. Take $Z:=\xi$ and (32) becomes:
(32)
$$k(\sigma+p)(\lambda-kp-1)[g(X,Y)+\eta(X)\eta(Y)]=0,$$
for any $X$, $Y\in\chi(M)$ and we obtain $p=-\sigma$ or $p=\frac{\lambda-1}{k}$.
∎
3.2 Perfect fluid spacetime satisfying $S(\xi,\cdot)\cdot\mathcal{T}=0$
Denote by
$$A=\lambda+\frac{k(\sigma-p)}{2},\ \ B=k(\sigma+p),$$
$$a=\frac{1}{3},\ \ b=\frac{scal}{12}(=\frac{4\lambda+k(\sigma-3p)}{12}),\ \ c=-%
\frac{scal}{6}(=-\frac{4\lambda+k(\sigma-3p)}{6}),\ \ d=\frac{1}{2}$$
and the curvature tensors can be written:
(33)
$$S(X,Y)=Ag(X,Y)+B\eta(X)\eta(Y)$$
(34)
$$\mathcal{W}(X,Y)Z=R(X,Y)Z+a[g(Z,X)QY-g(Y,Z)QX]$$
(35)
$$\mathcal{P}(X,Y)Z=R(X,Y)Z+b[g(Z,X)Y-g(Y,Z)X]$$
$$\mathcal{C}(X,Y)Z=R(X,Y)Z+c[g(Z,X)Y-g(Y,Z)X]+$$
(36)
$$+d[g(Z,X)QY-g(Y,Z)QX]+d[S(Z,X)Y-S(Y,Z)X]$$
(37)
$$\mathcal{H}(X,Y)Z=R(X,Y)Z+d[g(Z,X)QY-g(Y,Z)QX]+d[S(Z,X)Y-S(Y,Z)X]$$
for any $X$, $Y$, $Z\in\chi(M)$.
The condition $S(\xi,\cdot)\cdot\mathcal{T}=0$ is equivalent to
$$S(X,\mathcal{T}(Y,Z)W)\xi-S(\xi,\mathcal{T}(Y,Z)W)X+S(X,Y)\mathcal{T}(\xi,Z)W-$$
$$-S(\xi,Y)\mathcal{T}(X,Z)W+S(X,Z)\mathcal{T}(Y,\xi)W-S(\xi,Z)\mathcal{T}(Y,X)W+$$
(38)
$$+S(X,W)\mathcal{T}(Y,Z)\xi-S(\xi,W)\mathcal{T}(Y,Z)X=0,$$
for any $X$, $Y$, $Z$, $W\in\chi(M)$.
Taking the inner product with $\xi$, the relation (3.2) becomes:
$$-S(X,\mathcal{T}(Y,Z)W)-S(\xi,\mathcal{T}(Y,Z)W)\eta(X)+$$
$$+S(X,Y)\eta(\mathcal{T}(\xi,Z)W)-S(\xi,Y)\eta(\mathcal{T}(X,Z)W)+S(X,Z)\eta(%
\mathcal{T}(Y,\xi)W)-$$
(39)
$$-S(\xi,Z)\eta(\mathcal{T}(Y,X)W)+S(X,W)\eta(\mathcal{T}(Y,Z)\xi)-S(\xi,W)\eta(%
\mathcal{T}(Y,Z)X)=0,$$
for any $X$, $Y$, $Z$, $W\in\chi(M)$.
Theorem 3.2.
Let $(M,g)$ be a general relativistic perfect fluid spacetime satisfying (3) with torse-forming vector field $\xi$.
1.
If $S(\xi,\cdot)\cdot R=0$, then $p=\frac{\lambda}{k}$.
2.
If $S(\xi,\cdot)\cdot\mathcal{W}=0$, then $p_{1,2}=\frac{\lambda-6+3(\lambda+k\sigma)\pm 2\sqrt{3[(\lambda+k\sigma)^{2}-3%
(\lambda+k\sigma)+3]}}{k}$.
3.
If $S(\xi,\cdot)\cdot\mathcal{P}=0$, then $p=\frac{\lambda}{k}$ or $p=\frac{4\lambda+k\sigma-12}{3k}$.
4.
If $S(\xi,\cdot)\cdot\mathcal{C}=0$, then $p=\frac{\lambda}{k}$ or $p=\frac{2\lambda-k\sigma-6}{3k}$.
5.
If $S(\xi,\cdot)\cdot\mathcal{H}=0$, then $p=\frac{\lambda}{k}$ or $p=\frac{\lambda-1}{k}$.
Proof..
1.
From (3.2) and (33) we obtain:
$$-Ag(X,R(Y,Z)W)+A[g(X,Z)g(Y,W)-g(X,Y)g(Z,W)]+$$
$$+2A[\eta(X)\eta(Z)g(Y,W)-\eta(X)\eta(Y)g(Z,W)]+$$
(40)
$$+B[\eta(Y)\eta(W)g(X,Z)-\eta(Z)\eta(W)g(X,Y)]=0,$$
for any $X$, $Y$, $Z$, $W\in\chi(M)$. Taking $Z=W:=\xi$ we get:
(41)
$$(2A-B)[\eta(X)\eta(Y)+g(X,Y)]=0,$$
for any $X$, $Y\in\chi(M)$. It follows $p=\frac{\lambda}{k}$.
2.
From (3.2), (33) and (34) we obtain:
$$-Ag(X,R(Y,Z)W)+A(1+aB-2aA)[g(X,Z)g(Y,W)-g(X,Y)g(Z,W)]+$$
$$+A(2+aB-2aA)[\eta(X)\eta(Z)g(Y,W)-\eta(X)\eta(Y)g(Z,W)]+$$
(42)
$$+B(1+aB-aA)[\eta(Y)\eta(W)g(X,Z)-\eta(Z)\eta(W)g(X,Y)]=0,$$
for any $X$, $Y$, $Z$, $W\in\chi(M)$. Taking $Z=W:=\xi$ we get:
(43)
$$[2A-B+a(2AB-2A^{2}-B^{2})][\eta(X)\eta(Y)+g(X,Y)]=0,$$
for any $X$, $Y\in\chi(M)$. It follows $p_{1,2}=\frac{\lambda-6+3(\lambda+k\sigma)\pm 2\sqrt{3[(\lambda+k\sigma)^{2}-3%
(\lambda+k\sigma)+3]}}{k}$.
3.
From (3.2), (33) and (35) we obtain:
$$-Ag(X,R(Y,Z)W)+A(1-2b)[g(X,Z)g(Y,W)-g(X,Y)g(Z,W)]+$$
$$+2A(1-b)[\eta(X)\eta(Z)g(Y,W)-\eta(X)\eta(Y)g(Z,W)]+$$
(44)
$$+B(1-b)[\eta(Y)\eta(W)g(X,Z)-\eta(Z)\eta(W)g(X,Y)]=0,$$
for any $X$, $Y$, $Z$, $W\in\chi(M)$. Taking $Z=W:=\xi$ we get:
(45)
$$(2A-B)(1-b)[\eta(X)\eta(Y)+g(X,Y)]=0,$$
for any $X$, $Y\in\chi(M)$. It follows $p=\frac{\lambda}{k}$ or $p=\frac{4\lambda+k\sigma-12}{3k}$.
4.
From (3.2), (33) and (3.2) we obtain:
$$-Ag(X,R(Y,Z)W)+A(1-2c+dB-4dA)[g(X,Z)g(Y,W)-g(X,Y)g(Z,W)]+$$
$$+A(2-2c+dB-4dA)[\eta(X)\eta(Z)g(Y,W)-\eta(X)\eta(Y)g(Z,W)]+$$
(46)
$$+B(1-c+dB-3dA)[\eta(Y)\eta(W)g(X,Z)-\eta(Z)\eta(W)g(X,Y)]=0,$$
for any $X$, $Y$, $Z$, $W\in\chi(M)$. Taking $Z=W:=\xi$ we get:
(47)
$$(2A-B)[1-c-d(2A-B)][\eta(X)\eta(Y)+g(X,Y)]=0,$$
for any $X$, $Y\in\chi(M)$. It follows $p=\frac{\lambda}{k}$ or $p=\frac{2\lambda-k\sigma-6}{3k}$.
5.
From (3.2), (33) and (37) we obtain:
$$-Ag(X,R(Y,Z)W)+A(1+dB-4dA)[g(X,Z)g(Y,W)-g(X,Y)g(Z,W)]+$$
$$+A(2+dB-4dA)[\eta(X)\eta(Z)g(Y,W)-\eta(X)\eta(Y)g(Z,W)]+$$
(48)
$$+B(1+dB-3dA)[\eta(Y)\eta(W)g(X,Z)-\eta(Z)\eta(W)g(X,Y)]=0,$$
for any $X$, $Y$, $Z$, $W\in\chi(M)$. Taking $Z=W:=\xi$ we get:
(49)
$$(2A-B)[1-d(2A-B)][\eta(X)\eta(Y)+g(X,Y)]=0,$$
for any $X$, $Y\in\chi(M)$. It follows $p=\frac{\lambda}{k}$ or $p=\frac{\lambda-1}{k}$.
∎
Remark the following facts:
1) If for a general relativistic perfect fluid spacetime $(M,g)$ satisfying (3) with torse-forming vector field $\xi$ the projective curvature tensor $\mathcal{W}$ or the conharmonic curvature tensor $\mathcal{H}$ satisfy $\mathcal{W}(\xi,\cdot)\cdot S=0$, respectively $\mathcal{H}(\xi,\cdot)\cdot S=0$, then we have the vacuum case or the pressure is constant, but $R(\xi,\cdot)\cdot S=0$ leads only to the vacuum case.
2) Under the same assumptions, the condition $S(\xi,\cdot)\cdot R=0$ or $S(\xi,\cdot)\cdot\mathcal{H}=0$ implies a constant pressure of the fluid.
4 Solitons in a perfect fluid spacetime
4.1 $\eta$-Ricci solitons
Consider the equation:
(50)
$$\mathcal{L}_{\xi}g+2S+2ag+2b\eta\otimes\eta=0,$$
where $g$ is a pseudo-Riemannian metric, $S$ is the Ricci curvature, $\xi$ is a vector field, $\eta$ is a $1$-form and $a$ and $b$ are real constants. The data $(g,\xi,a,b)$ which satisfy
the equation (50) is said to be an $\eta$-Ricci soliton in $M$ [6]; in particular, if $b=0$, $(g,\xi,a)$ is a Ricci soliton [12] and it is called shrinking, steady or expanding according as $a$ is negative, zero or positive, respectively [7].
Writing explicitly the Lie derivative $\mathcal{L}_{\xi}g$ we get $(\mathcal{L}_{\xi}g)(X,Y)=g(\nabla_{X}\xi,Y)+g(X,\nabla_{Y}\xi)$ and from (50) we obtain:
(51)
$$S(X,Y)=-ag(X,Y)-b\eta(X)\eta(Y)-\frac{1}{2}[g(\nabla_{X}\xi,Y)+g(X,\nabla_{Y}%
\xi)],$$
for any $X$, $Y\in\chi(M)$.
Contracting (51) we get:
(52)
$$scal=-a\dim(M)+b-div(\xi).$$
Let $(M,g)$ be a general relativistic perfect fluid spacetime and $(g,\xi,a,b)$ be an $\eta$-Ricci soliton in $M$. From (7) and (51) we obtain:
(53)
$$[\lambda+\frac{k(\sigma-p)}{2}+a]g(X,Y)+[k(\sigma+p)+b]\eta(X)\eta(Y)+\frac{1}%
{2}[g(\nabla_{X}\xi,Y)+g(X,\nabla_{Y}\xi)]=0,$$
for any $X$, $Y\in\chi(M)$.
Consider $\{E_{i}\}_{1\leq i\leq 4}$ an orthonormal frame field and let $\xi=\sum_{i=1}^{4}\xi^{i}E_{i}$. We have shown in the previous section that $\sum_{i=1}^{4}\varepsilon_{ii}(\xi^{i})^{2}=-1$ and $\eta(E_{i})=\varepsilon_{ii}\xi^{i}$.
Multiplying (53) by $\varepsilon_{ii}$ and summing over $i$ for $X=Y:=E_{i}$, we get:
(54)
$$4a-b=-4\lambda-k(\sigma-3p)-div(\xi).$$
Writing (53) for $X=Y:=\xi$, we obtain:
(55)
$$a-b=-\lambda+\frac{k(\sigma+3p)}{2}.$$
Therefore:
(56)
$$\left\{\begin{array}[]{ll}a=-\lambda-\frac{k(\sigma-p)}{2}-\frac{div(\xi)}{3}%
\\
b=-k(\sigma+p)-\frac{div(\xi)}{3}\end{array}\right..$$
Theorem 4.1.
Let $(M,g)$ be a $4$-dimensional pseudo-Riemannian manifold and
$\eta$ be the $g$-dual $1$-form of the gradient vector field $\xi:=grad(f)$ with $g(\xi,\xi)=-1$. If (50) defines an $\eta$-Ricci soliton in $M$, then the Laplacian equation satisfied by $f$ becomes:
(57)
$$\Delta(f)=-3[b+k(\sigma+p)].$$
Remark 4.2.
If $b=0$ in (50), we obtain the Ricci soliton with $a=-\lambda+\frac{k(\sigma+3p)}{2}$ which is steady if $p=\frac{2}{3}(\frac{\lambda}{k})-\frac{\sigma}{3}$, expanding if $p>\frac{2}{3}(\frac{\lambda}{k})-\frac{\sigma}{3}$ and shrinking if $p<\frac{2}{3}(\frac{\lambda}{k})-\frac{\sigma}{3}$. In these cases, $div(\xi)=-3k(\sigma+p)$. From Plebanski energy conditions for perfect fluids we deduce that $\sigma\geq\max\{-\frac{\lambda}{k},\frac{\lambda}{2k}\}$ for the steady case, $\sigma>\frac{\lambda}{2k}$ and $\sigma>-\frac{\lambda}{k}$ for the expanding and shrinking case, respectively.
Example 4.3.
An $\eta$-Ricci soliton $(g,\xi,a,b)$ in a radiation fluid is given by
$$\left\{\begin{array}[]{ll}a=-\lambda-kp-\frac{div(\xi)}{3}\\
b=-4kp-\frac{div(\xi)}{3}\end{array}\right..$$
4.2 $\eta$-Einstein solitons
Consider the equation:
(58)
$$\mathcal{L}_{\xi}g+2S+(2a-scal)g+2b\eta\otimes\eta=0,$$
where $g$ is a pseudo-Riemannian metric, $S$ is the Ricci curvature, $scal$ is the scalar curvature, $\xi$ is a vector field, $\eta$ is a $1$-form and $a$ and $b$ are real constants. The data $(g,\xi,a,b)$ which satisfy
the equation (58) is said to be an $\eta$-Einstein soliton in $M$; in particular, if $b=0$, $(g,\xi,a)$ is an Einstein soliton [3].
Writing explicitly the Lie derivative $\mathcal{L}_{\xi}g$ we get $(\mathcal{L}_{\xi}g)(X,Y)=g(\nabla_{X}\xi,Y)+g(X,\nabla_{Y}\xi)$ and from (58) we obtain:
(59)
$$S(X,Y)=-(a-\frac{scal}{2})g(X,Y)-b\eta(X)\eta(Y)-\frac{1}{2}[g(\nabla_{X}\xi,Y%
)+g(X,\nabla_{Y}\xi)],$$
for any $X$, $Y\in\chi(M)$.
Contracting (59) we get:
(60)
$$\frac{2-\dim(M)}{2}scal=-a\dim(M)+b-div(\xi).$$
Let $(M,g)$ be a general relativistic perfect fluid spacetime and $(g,\xi,a,b)$ be an $\eta$-Einstein soliton in $M$. From (6), (7) and (59) we obtain:
(61)
$$(\lambda-kp-a)g(X,Y)-[k(\sigma+p)+b]\eta(X)\eta(Y)-\frac{1}{2}[g(\nabla_{X}\xi%
,Y)+g(X,\nabla_{Y}\xi)]=0,$$
for any $X$, $Y\in\chi(M)$.
Consider $\{E_{i}\}_{1\leq i\leq 4}$ an orthonormal frame field and let $\xi=\sum_{i=1}^{4}\xi^{i}E_{i}$. We have shown in the previous section that $\sum_{i=1}^{4}\varepsilon_{ii}(\xi^{i})^{2}=-1$ and $\eta(E_{i})=\varepsilon_{ii}\xi^{i}$.
Multiplying (61) by $\varepsilon_{ii}$ and summing over $i$ for $X=Y:=E_{i}$, we get:
(62)
$$4a-b=4\lambda+k(\sigma-3p)-div(\xi).$$
Writing (61) for $X=Y:=\xi$, we obtain:
(63)
$$a-b=\lambda+k\sigma.$$
Therefore:
(64)
$$\left\{\begin{array}[]{ll}a=\lambda-kp-\frac{div(\xi)}{3}\\
b=-k(\sigma+p)-\frac{div(\xi)}{3}\end{array}\right..$$
Theorem 4.4.
Let $(M,g)$ be a $4$-dimensional pseudo-Riemannian manifold and
$\eta$ be the $g$-dual $1$-form of the gradient vector field $\xi:=grad(f)$ with $g(\xi,\xi)=-1$. If (58) defines an $\eta$-Einstein soliton in $M$, then the Laplacian equation satisfied by $f$ becomes:
(65)
$$\Delta(f)=-3[b+k(\sigma+p)].$$
Remark 4.5.
If $b=0$ in (58), we obtain the Einstein soliton with $a=\lambda+k\sigma$ which is steady if $p=\frac{2}{3}(\frac{\lambda}{k})-\frac{\sigma}{3}$, expanding if $p>\frac{2}{3}(\frac{\lambda}{k})-\frac{\sigma}{3}$ and shrinking if $p<\frac{2}{3}(\frac{\lambda}{k})-\frac{\sigma}{3}$. In these cases, $div(\xi)=-3k(\sigma+p)$. From Plebanski energy conditions for perfect fluids we deduce that $\sigma\geq\max\{-\frac{\lambda}{k},\frac{\lambda}{2k}\}$ for the steady case, $\sigma>\frac{\lambda}{2k}$ and $\sigma>-\frac{\lambda}{k}$ for the expanding and shrinking case, respectively.
Example 4.6.
An $\eta$-Einstein soliton $(g,\xi,a,b)$ in a radiation fluid is given by
$$\left\{\begin{array}[]{ll}a=\lambda-kp-\frac{div(\xi)}{3}\\
b=-4kp-\frac{div(\xi)}{3}\end{array}\right..$$
Remark the following facts:
1) From Examples 4.3 and 4.6 we deduce that the Ricci soliton in a radiation fluid is steady if $p=\frac{\lambda}{3k}$,
expanding if $p>\frac{\lambda}{3k}$ and shrinking if $p<\frac{\lambda}{3k}$.
2) In a general relativistic perfect fluid spacetime, if the vector field $\xi$ is torse-forming i.e. $\nabla_{X}\xi=\alpha[X+\eta(X)\xi]$, for any $X\in\chi(M)$ with $\alpha$ a nonzero real number, then $div(\xi)=3\alpha$. In this case, the existence of a Ricci soliton given by (50) for $b=0$, from Plebanski energy conditions implies $-2k\sigma\leq\alpha<0$ (precisely, $\alpha=-k(\sigma+p)$).
3) If the vector field $\xi$ is conformally Killing i.e. $L_{\xi}g=rg$ with $r$ a nonzero real number, then the existence of a Ricci soliton given by (50) for $b=0$, implies the vacuum case. Moreover, the soliton is steady if $p=\frac{\lambda}{k}+\frac{r}{2k}$, expanding if $p>\frac{\lambda}{k}+\frac{r}{2k}$ and shrinking if $p<\frac{\lambda}{k}+\frac{r}{2k}$.
4) The existence of a steady Ricci soliton or a (weakly) pseudo Ricci symmetric Ricci tensor field imply the same condition on the pressure of the fluid, a surprisingly fact being that the $1$-form $\alpha$, arbitrary chosen, does not appear effectively.
References
[1]
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Department of Mathematics
West University of Timişoara
Bld. V. Pârvan nr. 4, 300223, Timişoara, România
[email protected] |
Can We Assess Mental Health through Social Media and Smart Devices? Addressing Bias in Methodology and Evaluation
Adam Tsakalidis
1University of Warwick, UK
1{a.tsakalidis, m.liakata, t.damoulas, a.i.cristea}@warwick.ac.uk
2The Alan Turing Institute, UK
2
Maria Liakata
1University of Warwick, UK
1{a.tsakalidis, m.liakata, t.damoulas, a.i.cristea}@warwick.ac.uk
2The Alan Turing Institute, UK
2
Theo Damoulas
1University of Warwick, UK
1{a.tsakalidis, m.liakata, t.damoulas, a.i.cristea}@warwick.ac.uk
2The Alan Turing Institute, UK
2
Alexandra I. Cristea
1University of Warwick, UK
1{a.tsakalidis, m.liakata, t.damoulas, a.i.cristea}@warwick.ac.uk
3Durham University, UK3
Abstract
Predicting mental health from smartphone and social media data on a longitudinal basis has recently attracted great interest, with very promising results being reported across many studies [3, 9, 13, 26]. Such approaches have the potential to revolutionise mental health assessment, if their development and evaluation follows a real world deployment setting. In this work we take a closer look at state-of-the-art approaches, using different mental health datasets and indicators, different feature sources and multiple simulations, in order to assess their ability to generalise. We demonstrate that under a pragmatic evaluation framework, none of the approaches deliver or even approach the reported performances. In fact, we show that current state-of-the-art approaches can barely outperform the most naïve baselines in the real-world setting, posing serious questions not only about their deployment ability, but also about the contribution of the derived features for the mental health assessment task and how to make better use of such data in the future.
Keywords:mental health bias evaluation wellbeing natural language processing smartphones sensors social media
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capbtabboxtable[][\FBwidth]
1 Introduction
Establishing the right indicators of mental well-being is a grand challenge posed by the World Health Organisation [7]. Poor mental health is highly correlated with low motivation, lack of satisfaction, low productivity and a negative economic impact [20]. The current approach is to combine census data at the population level [19], thus failing to capture well-being on an individual basis. The latter is only possible via self-reporting on the basis of established psychological scales, which are hard to acquire consistently on a longitudinal basis, and they capture long-term aggregates instead of the current state of the individual.
The widespread use of smart-phones and social media offers new ways of assessing mental well-being, and recent research [1, 2, 3, 5, 9, 10, 13, 14, 22, 23, 26] has started exploring the effectiveness of these modalities for automatically assessing the mental health of a subject, reporting very high accuracy. What is typically done in these studies is to use features based on the subjects’ smart phone logs and social media, to predict some self-reported mental health index (e.g., “wellbeing”, “depression” and others), which is provided either on a Likert scale or on the basis of a psychological questionnaire (e.g., PHQ-8 [12], PANAS [29], WEMWBS [25] and others).
Most of these studies are longitudinal, where data about individuals is collected over a period of time and predictions of mental health are made over a sliding time window. Having such longitudinal studies is highly desirable, as it can allow fine-grained monitoring of mental health. However, a crucial question is what constitutes an appropriate evaluation framework, in order for such approaches to be employable in a real world setting. Generalisation to previously unobserved users can only be assessed via leave-N-users-out cross-validation setups, where typically, N is equal to one (LOUOCV, see Table 1). However, due to the small number of subjects that are available, such generalisation is hard to achieve by any approach [13]. Alternatively, personalised models [3, 13] for every individual can be evaluated via a within-subject, leave-N-instances-out cross-validation (for N=1, LOIOCV), where an instance for a user $u$ at time $i$ is defined as a {$X_{ui}$, $y_{ui}$} tuple of {features(u, i), mental-health-score(u, i)}. In a real world setting, a LOIOCV model is trained on some user-specific instances, aiming to predict her mental health state at some future time points. Again however, the limited number of instances for every user make such models unable to generalize well. In order to overcome these issues, previous work [2, 5, 9, 10, 22, 26] has combined the instances {$X_{u_{j}i}$, $y_{u_{j}i}$} from different individuals $u_{j}$ and performed evaluation using randomised cross validation (MIXED). While such approaches can attain optimistic performance, the corresponding models fail to generalise to the general population and also fail to ensure effective personalised assessment of the mental health state of a single individual.
In this paper we demonstrate the challenges that current state-of-the-art models face, when tested in a real-world setting. We work on two longitudinal datasets with four mental health targets, using different features derived from a wide range of heterogeneous sources. Following the state-of-the-art experimental methods and evaluation settings, we achieve very promising results, regardless of the features we employ and the mental health target we aim to predict. However, when tested under a pragmatic setting, the performance of these models drops heavily, failing to outperform the most naïve – from a modelling perspective – baselines: majority voting, random classifiers, models trained on the identity of the user, etc. This poses serious questions about the contribution of the features derived from social media, smartphones and sensors for the task of automatically assessing well-being on a longitudinal basis. Our goal is to flesh out, study and discuss such limitations through extensive experimentation across multiple settings, and to propose a pragmatic evaluation and model-building framework for future research in this domain.
2 Related Work
Research in assessing mental health on a longitudinal basis aims to make use of relevant features extracted from various modalities, in order to train models for automatically predicting a user’s mental state (target), either in a classification or a regression manner [1, 2, 3, 9, 10, 13, 26]. Examples of state-of-the-art work in this domain are listed in Table 2, along with the number of subjects that was used and the method upon which evaluation took place. Most approaches have used the “MIXED” approach to evaluate models [1, 2, 5, 9, 10, 22, 26], which, as we will show, is vulnerable to bias, due to the danger of recognising the user in the test set and thus simply inferring her average mood score. LOIOCV approaches that have not ensured that their train/test sets are independent are also vulnerable to bias in a realistic setting [3, 13]. From the works listed in Table 2, only Suhara et al. [23] achieves unbiased results with respect to model generalisability; however, the features employed for their prediction task are derived from self-reported questionnaires of the subjects and not by automatic means.
3 Problem Statement
We first describe three major problems stemming from unrealistic construction and evaluation of mental health assessment models and then we briefly present the state-of-the-art in each case, which we followed in our experiments.
P1
Training on past values of the target variable: This issue arises when the past $N$ mood scores of a user are required to predict his/her next mood score in an autoregressive manner. Since such an approach would require the previous N scores of past mood forms, it would limit its ability to generalise without the need of manual user input in a continuous basis. This makes it impractical for a real-world scenario. Most importantly, it is difficult to measure the contribution of the features towards the prediction task, unless the model is evaluated using target feature ablation. For demonstration purposes, we have followed the experimental setup by LiKamWa et al. [13], which is one of the leading works in this field.
P2
Inferring test set labels: When training personalised models (LOIOCV) in a longitudinal study, it is important to make sure that there are no overlapping instances across consecutive time windows. Some past works have extracted features {$f(t-N)$, …, $f(t)$} over $N$ days, in order to predict the $score_{t}$ on day $N+1$ [3, 13]. Such approaches are biased if there are overlapping days of train/test data. To illustrate this problem we have followed the approach by Canzian and Musolesi [3], as one of the pioneering works on predicting depression with GPS traces, on a longitudinal basis.
P3
Predicting users instead of mood scores: Most approaches merge all the instances from different subjects, in an attempt to build user-agnostic models in a randomised cross-validation framework [2, 9, 10, 26]. This is problematic, especially when dealing with a small number of subjects, whose behaviour (as captured through their data) and mental health scores differ on an individual basis. Such approaches are in danger of “predicting” the user in the test set, since her (test set) features might be highly correlated with her features in the training set, and thus infer her average well-being score, based on the corresponding observations of the training set. Such approaches cannot guarantee that they will generalise on either a population-wide (LOUOCV) or a personalised (LOIOCV) level. In order to examine this effect in both a regression and a classification setting, we have followed the experimental framework by Tsakalidis et al. [26] and Jaques et al. [9].
3.1 P1: Training on past values of the target (LOIOCV, LOUOCV)
LiKamWa et al. [13] collected smartphone data from $32$ subjects over a period of two months. The subjects were asked to self-report their “pleasure” and “activeness” scores at least four times a day, following a Likert scale (1 to 5), and the average daily scores served as the two targets. The authors aggregated various features on social interactions (e.g., number of emails sent to frequently interacting contacts) and routine activities (e.g., browsing and location history) derived from the smartphones of the participants. These features were extracted over a period of three days, along with the two most recent scores on activeness and pleasure. The issue that naturally arises is that such a method cannot generalise to new subjects in the LOUOCV setup, as it requires their last two days of self-assessed scores. Moreover, in the LOIOCV setup, the approach is limited in a real world setting, since it requires the previous mental health scores by the subject to provide an estimate of her current state. Even in this case though, the feature extraction should be based on past information only – under LOIOCV in [13], the current mood score we aim at predicting is also used as a feature in the (time-wise) subsequent two instances of the training data.
Experiments in [13] are conducted under LOIOCV and LOUOCV, using Multiple Linear Regression (LR) with Sequential Feature Selection (in LOUOCV, the past two pairs of target labels of the test user are still used as features). In order to better examine the effectiveness of the features for the task, the same model can be tested without any ground-truth data as input. Nevertheless, a simplistic model predicting the per-subject average outperforms their LR in the LOUOCV approach, which poses the question of whether the smartphone-derived features can be used effectively to create a generalisable model that can assess the mental health of unobserved users. Finally, the same model tested in the LOIOCV setup achieves the lowest error; however, this is trained not only on target scores overlapping with the test set, but also on features derived over a period of three days, introducing further potential bias, as discussed in the following.
3.2 P2: Inferring Test Labels (LOIOCV)
Canzian and Musolesi [3] extracted mobility metrics from $28$ subjects to predict their depressive state, as derived from their daily self-reported PHQ-8 questionnaires. A 14-day moving average filter is first applied to the PHQ-8 scores and the mean value of the same day (e.g. Monday) is subtracted from the normalised scores, to avoid cyclic trends. This normalisation results into making the target score $s_{t}$ on day $t$ dependent on the past {$s_{t-14},...,s_{t-1}$} scores. The normalised PHQ-8 scores are then converted into two classes, with the instances deviating more than one standard deviation above the mean score of a subject being assigned to the class “1” (“0”, otherwise). The features are extracted over various time windows (looking at $T_{HIST}=\{0,...,14\}$ days before the completion of a mood form) and personalised model learning and evaluation are performed for every $T_{HIST}$ separately, using a LOIOCV framework.
What is notable is that the results improve significantly when features are extracted from a wider $T_{HIST}$ window. This could imply that the depressive state of an individual can be detected with a high accuracy if we look back at her history. However, by training and testing a model on instances whose features are derived from the same days, there is a high risk of over-fitting the model to the timestamp of the day in which the mood form was completed. In the worst-case scenario, there will be an instance in the train set whose features (e.g. total covered distance) are derived from the 14 days, 13 of which will also be used for the instance in the test set. Additionally, the target values of these two instances will also be highly correlated due to the moving average filter, making the task artificially easy for large $T_{HIST}$ and not applicable in a real-world setting.
While we focus on the approach in [3], a similar approach with respect to feature extraction was also followed in LiKamWa et al. [13] and Bogomolov et al. [2], extracting features from the past 2 and 2 to
5 days, respectively.
3.3 P3: Predicting Users (LOUOCV)
Tsakalidis et al. [26] monitored the behaviour of $19$ individuals over four months. The subjects were asked to complete two psychological scales [25, 29] on a daily basis, leading to three target scores (positive, negative, mental well-being); various features from smartphones (e.g., time spent on the preferred locations) and textual features (e.g., ngrams) were extracted passively over the 24 hours preceding a mood form timestamp. Model training and evaluation was performed in a randomised (MIXED) cross-validation setup, leading to high accuracy ($R^{2}=0.76$). However, a case demonstrating the potential user bias is when the models are trained on the textual sources: initially the highest $R^{2}$ ($0.22$) is achieved when a model is applied to the mental-wellbeing target; by normalising the textual features on a per-user basis, the $R^{2}$ increases to $0.65$. While this is likely to happen because the vocabulary used by different users is normalised, there is also the danger of over-fitting the trained model to the identity of the user. To examine this potential, the LOIOCV/LOUOCV setups need to be studied alongside the MIXED validation approach, with and without the per-user feature normalisation step.
A similar issue is encountered in Jaques et al. [9] who monitored $68$ subjects over a period of a month. Four types of features were extracted from survey and smart devices carried by subjects. Self-reported scores on a daily basis served as the ground truth. The authors labelled the instances with the top $30\%$ of all the scores as “happy” and the lowest $30\%$ as “sad” and randomly separated them into training, validation and test sets, leading to the same user bias issue. Since different users exhibit different mood scores on average [26], by selecting instances from the top and bottom scores, one might end up separating users and convert the mood prediction task into a user identification one. A more suitable task could have been to try to predict the highest and lowest scores of every individual separately, either in a LOIOCV or in a LOUOCV setup.
While we focus on the works of Tsakalidis et al. [26] and Jaques et al. [9], similar experimental setups were also followed in [10], using the median of scores to separate the instances and performing five-fold cross-validation, and by Bogomolov et al. in [2], working on a user-agnostic validation setting on 117 subjects to predict their happiness levels, and in [1], for the stress level classification task.
4 Experiments
4.1 Datasets
By definition, the aforementioned issues are feature-, dataset- and target-independent (albeit the magnitude of the effects may vary). To illustrate this, we run a series of experiments employing two datasets, with different feature sources and four different mental health targets.
Dataset 1: We employed the dataset obtained by Tsakalidis et al. [26], a pioneering dataset which contains a mix of longitudinal textual and mobile phone usage data for 30 subjects. From a textual perspective, this dataset consists of social media posts (1,854/5,167 facebook/twitter posts) and private messages (64,221/132/47,043 facebook/twitter/ SMS messages) sent by the subjects. For our ground truth, we use the {positive, negative, mental well-being} mood scores (in the ranges of [10-50], [10-50], [14-70], respectively) derived from self-assessed psychological scales during the study period.
Dataset 2: We employed the StudentLife dataset [28], which contains a wealth of information derived from the smartphones of 48 students during a 10-week period. Such information includes samples of the detected activity of the subject, timestamps of detected conversations, audio mode of the smartphone, status of the smartphone (e.g., charging, locked), etc. For our target, we used the self-reported stress levels of the students (range [0-4]), which were provided several times a day. For the approach in LiKamWa et al. [13], we considered the average daily stress level of a student as our ground-truth, as in the original paper; for the rest, we used all of the stress scores and extracted features based on some time interval preceding their completion, as described next, in 4.3111For P3, this creates the P2 cross-correlation issue in the MIXED/LOIOCV settings. For this reason, we ran the experiments by considering only the last entered score in a given day as our target. We did not witness any major differences that would alter our conclusions..
4.2 Task Description
We studied the major issues in the following experimental settings (see Table 3):
P1: Using Past Labels:
We followed the experimental setting in [13] (see section 3.1): we treated our task as a regression problem and used Mean Squared Error (MSE) and classification accuracy222Accuracy is defined in [13] as follows: 5 classes are assumed (e.g., [0, …, 4]) and the squared error $e$ between the centre of a class halfway towards the next class is calculated (e.g., 0.25). If the squared error of a test instance is smaller than $e$, then it is considered as having been classified correctly. for evaluation. We trained a Linear Regression (LR) model and performed feature selection using Sequential Feature Selection under the LOIOCV and LOUOCV setups; feature extraction is performed over the previous 3 days preceding the completion of a mood form. For comparison, we use the same baselines as in [13]: Model A always predicts the average mood score for a certain user (AVG); Model B predicts the last entered scores (LAST); Model C makes a prediction using the LR model trained on the ground-truth features only (-feat). We also include Model D, trained on non-target features only (-mood) in an unbiased LOUOCV setting.
P2: Inferring Test Labels: We followed the experimental setting presented in [3]. We process our ground-truth in the same way as the original paper (see section 3.2) and thus treat our task as a binary classification problem. We use an SVM${}_{RBF}$ classifier, using grid search for parameter optimisation, and perform evaluation using specificity and sensitivity. We run experiments in the LOIOCV and LOUOCV settings, performing feature extraction at different time windows ($T_{HIST}=\{1,...,14\}$). In order to better demonstrate the problem that arises here, we use the previous label classifier (LAST) and the SVM classifier to which we feed only the mood timestamp as a feature (DATE) for comparison. Finally, we replace our features with completely random data and train the same SVM with $T_{HIST}=14$ by keeping the same ground truth, performing 100 experiments and reporting averages of sensitivity and specificity (RAND).
P3: Predicting Users: We followed the evaluation settings of two past works (see section 3.3), with the only difference being the use of 5-fold CV instead of a train/dev/test split that was used in [9]. The features of every instance are extracted from the past day before the completion of a mood form. In Experiment 1 we follow the setup in [26]: we perform 5-fold CV (MIXED) using SVM (SVR${}_{RBF}$) and evaluate performance based on $R^{2}$ and $RMSE$. We compare the performance when tested under the LOIOCV/LOUOCV setups, with and without the per-user feature normalisation step. We also compare the performance of the MIXED setting, when our model is trained on the one-hot-encoded user id only. In Experiment 2 we follow the setup in [9]: we label the instances as “high” (“low”), if they belong to the top-30% (bottom-30%) of mood score values (“UNIQ” – for “unique” – setup). We train an SVM classifier in 5-fold CV using accuracy for evaluation and compare performance in the LOIOCV and LOUOCV settings. In order to further examine user bias, we perform the same experiments, this time by labelling the instances on a per-user basis (‘‘PERS’’ -- for ‘‘personalised’’ -- setup), aiming to predict the per-user high/low mood days333In cases where the lowest of the top-30% scores ($s$) was equal to the highest of the bottom-30% scores, we excluded the instances with score $s$..
4.3 Features
For Dataset 1, we first defined a ‘‘user snippet’’ as the concatenation of all texts generated by a user within a set time interval, such that the maximum time difference between two consecutive document timestamps is less than 20 minutes. We performed some standard noise reduction steps (converted text to lowercase, replaced URLs/user mentions and performed language identification444https://pypi.python.org/pypi/langid and tokenisation [6]). Given a mood form and a set of snippets produced by a user before the completion of a mood form, we extracted some commonly used feature sets for every snippet written in English [26], which were used in all experiments. To ensure sufficient data density, we excluded users for whom we had overall fewer than $25$ snippets on the days before the completion of the mood form or fewer than $40$ mood forms overall, leading to $27$ users and $2,368$ mood forms. For Dataset 2, we extracted the features presented in Table 4. We only kept the users that had at least 10 self-reported stress questionnaires, leading to $44$ users and $2,146$ instances. For our random experiments used in P2, in Dataset 1 we replaced the text representation of every snippet with random noise ($\mu=0,\sigma=1$) of the same feature dimensionality; in Dataset 2, we replaced the actual inferred value of every activity/audio sample with a random inference class; we also replaced each of the detected conversation samples and samples detected in a dark environment/locked/charging, with a random number ($<$100, uniformly distributed) indicating the number of pseudo-detected samples.
5 Results
5.1 P1: Using Past Labels
Table 5 presents the results on the basis of the methodology by LiKamWa et al. [13], along with the average scores reported in [13] – note that the range of the mood scores varies on a per-target basis; hence, the reported results of different models should be compared among each other when tested on the same target.
As in [13], always predicting the average score (AVG) for an unseen user performs better than applying a LR model trained on other users in a LOUOCV setting. If the same LR model used in LOUOCV is trained without using the previously self-reported ground-truth scores (Model D, -mood), its performance drops further. This showcases that personalised models are needed for more accurate mental health assessment (note that the AVG baseline is, in fact, a personalised baseline) and that there is no evidence that we can employ effective models in real-world applications to predict the mental health of previously unseen individuals, based on this setting.
The accuracy of LR under LOIOCV is higher, except for the “stress” target, where the performance is comparable to LOUOCV and lower than the AVG baseline. However, the problem in LOIOCV is the fact that the features are extracted based on the past three days, thus creating a temporal cross-correlation in our input space. If a similar correlation exists in the output space (target), then we end up in danger of overfitting our model to the training examples that are temporally close to the test instance. This type of bias is essentially present if we force a temporal correlation in the output space, as studied next.
5.2 P2: Inferring Test Labels
The charts in Fig. 1 (top) show the results by following the LOIOCV approach from Canzian and Musolesi [3]. The pattern that these metrics take is consistent and quite similar to the original paper: specificity remains at high values, while sensitivity increases as we increase the time window from which we extract our features. The charts on the bottom in Fig. 1 show the corresponding results in the LOUOCV setting. Here, such a generalisation is not feasible, since the increases in sensitivity are accompanied by sharp drops in the specificity scores.
The arising issue though lies in the LOIOCV setting. By training and testing on the same days (for $T_{HIST}>1$), the kernel matrix takes high values for cells which are highly correlated with respect to time, making the evaluation of the contribution of the features difficult. To support this statement, we train the same model under LOIOCV, using only on the mood form completion date (Unix epoch) as a feature. The results are very similar to those achieved by training on $T_{HIST}=14$ (see Table 6). We also include the results of another naïve classifier (LAST), predicting always the last observed score in the training set, which again achieves similar results. The clearest demonstration of the problem though is by comparing the results of the RAND against the FEAT classifier, which shows that under the proposed evaluation setup we can achieve similar performance if we replace our inputs with random data, clearly demonstrating the temporal bias that can lead to over-optimistic results, even in the LOIOCV setting.
5.3 P3: Predicting Users
Experiment 1:
Table 3 shows the results based on the evaluation setup of Tsakalidis et al. [26]. In the MIXED cases, the pattern is consistent with [26], indicating that normalising the features on a per-user basis yields better results, when dealing with sparse textual features (positive, negative, wellbeing targets).
The explanation of this effect lies within the danger of predicting the user’s identity instead of her mood scores. This is why the per-user normalisation does not have any effect for the stress target, since for that we are using dense features derived from smartphones: the vocabulary used by the subjects for the other targets is more indicative of their identity. In order to further support this statement, we trained the SVR model using only the one-hot encoded user id as a feature, without any textual features. Our results yielded $R^{2}$={$0.64$, $0.50$, $0.66$} and $RMSE$={$5.50$, $5.32$, $6.50$} for the {positive, negative, wellbeing} targets, clearly demonstrating the user bias in the MIXED setting.
The RMSEs in LOIOCV are the lowest, since different individuals exhibit different ranges of mental health scores. Nevertheless, $R^{2}$ is slightly negative, implying again that the average predictor for a single user provides a better estimate for her mental health score. Note that while the predictions across all individuals seem to be very accurate (see Fig. 3), by separating them on a per-user basis, we end up with a negative $R^{2}$.
In the unbiased LOUOCV setting the results are, again, very poor. The reason for the high differences observed between the three settings is provided by the $R^{2}$ formula itself ($1-(\sum_{i}(pred_{i}-y_{i})^{2})/(\sum_{i}(y_{i}-\bar{y})^{2}))$. In the MIXED case, we train and test on the same users, while $\bar{y}$ is calculated as the mean of the mood scores across all users, whereas in the LOIOCV/LOUOCV cases, $\bar{y}$ is calculated for every user separately. In MIXED, by identifying who the user is, we have a rough estimate of her mood score, which is by itself a good predictor, if it is compared with the average predictor across all mood scores of all users. Thus, the effect of the features in this setting cannot be assessed with certainty.
Experiment 2:
Table 7 displays our results based on Jaques et al. [9] (see section 3.3). The average accuracy on the “UNIQ” setup is higher by $14\%$ compared to the majority classifier in MIXED. The LOIOCV setting also yields very promising results (mean accuracy: $81.17\%$). As in all previous cases, in LOUOCV our models fail to outperform the majority classifier. A closer look at the LOIOCV and MIXED results though reveals the user bias issue that is responsible for the high accuracy. For example, $33\%$ of the users had all of their “positive” scores binned into one class, as these subjects were exhibiting higher (or lower) mental health scores throughout the experiment, whereas another $33\%$ of the subjects had $85\%$ of their instances classified into one class. By recognising the user, we can achieve high accuracy in the MIXED setting; in the LOIOCV, the majority classifier can also achieve at least $85\%$ accuracy for $18$/$27$ users.
In the “PERS” setup, we removed the user bias, by separating the two classes on a per-user basis. The results now drop heavily even in the two previously well-performing settings and can barely outperform the majority classifier. Note that the task in Experiment 2 is relatively easier, since we are trying to classify instances into two classes which are well-distinguished from each other from a psychological point of view. However, by removing the user bias, the contribution of the user-generated features to this task becomes once again unclear.
6 Proposal for Future Directions
Our results emphasize the difficulty of automatically predicting individuals’ mental health scores in a real-world setting and demonstrate the dangers due to flaws in the experimental setup. Our findings do not imply that the presented issues will manifest themselves to the same degree in different datasets – e.g., the danger of predicting the user in the MIXED setting is higher when using the texts of 27 users rather than sensor-based features of more users [1, 2, 9, 22]. Nevertheless, it is crucial to establish appropriate evaluation settings to avoid providing false alarms to users, if our aim is to build systems that can be deployed in practice. To this end, we propose model building and evaluation under the following:
•
LOUOCV: By definition, training should be performed strictly on features and target data derived from a sample of users and tested on a completely new user, since using target data from the unseen user as features violates the independence hypothesis. A model trained in this setting should achieve consistently better results on the unseen user compared to the naïve (from a modelling perspective) model that always predicts his/her average score.
•
LOIOCV: By definition, the models trained under this setting should not violate the iid hypothesis. We have demonstrated that the temporal dependence between instances in the train and test set can provide over-optimistic results. A model trained on this setting should consistently outperform naïve, yet competitive, baseline methods, such as the last-entered mood score predictor, the user’s average mood predictor and the auto-regressive model.
Models that can be effectively applied in any of the above settings could revolutionise the mental health assessment process while providing us in an unbiased setting with great insights on the types of behaviour that affect our mental well-being. On the other hand, positive results in the MIXED setting cannot guarantee model performance in a real-world setting in either LOUOCV or LOIOCV, even if they are compared against the user average baseline [4].
Transfer learning approaches can provide significant help in the LOUOCV setting. However, these assume that single-domain models have been effectively learned beforehand – but all of our single-user (LOIOCV) experiments provided negative results. Better feature engineering through latent feature representations may prove to be beneficial. While different users exhibit different behaviours, these behaviours may follow similar patterns in a latent space. Such representations have seen great success in recent years in the field of natural language processing [15], where the aim is to capture latent similarities between seemingly diverse concepts and represent every feature based on its context. Finally, working with larger datasets can help in providing more data to train on, but also in assessing the model’s ability to generalise in a more realistic setting.
7 Conclusion
Assessing mental health with digital media is a task which could have great impact on monitoring of mental well-being and personalised health. In the current paper, we have followed past experimental settings to evaluate the contribution of various features to the task of automatically predicting different mental health indices of an individual. We find that under an unbiased, real-world setting, the performance of state-of-the-art models drops significantly, making the contribution of the features impossible to assess. Crucially, this holds for both cases of creating a model that can be applied in previously unobserved users (LOUOCV) and a personalised model that is learned for every user individually (LOIOCV).
Our major goal for the future is to achieve positive results in the LOUOCV setting. To overcome the problem of having only few instances from a diversely behaving small group of subjects, transfer learning techniques on latent feature representations could be beneficial. A successful model in this setting would not only provide us with insights on what types of behaviour affect mental state, but could also be employed in a real-world system without the danger of providing false alarms to its users.
Acknowledgements
The current work was supported by the EPSRC through the University of Warwick’s CDT in Urban Science and Progress (grant EP/L016400/1) and through The Alan Turing Institute (grant EP/N510129/1). We would like to thank the anonymous reviewers for their detailed feedback and the authors of the works that were analysed in our paper (N. Jaques, R. LiKamWa, M. Musolesi) for the fruitful discussions over several aspects of the presented challenges.
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Quillen–Suslin Theory for Classical
Groups: Revisited over Graded Rings
Rabeya Basu
and
Manish Kumar Singh
(Date:: )
Corresponding Author: Rabeya Basu – [email protected]
2010 Mathematics Subject Classification:
11E57, 13A02, 13B25, 13B99, 13C10, 13C99, 15B99.
Key words: Bilinear forms, Symplectic and
Orthogonal forms, Graded rings.
Abstract: In this paper we deduce a graded version of Quillen–Suslin’s Local-Global Principle for the traditional classical groups, viz. general linear,
symplectic and orthogonal groups and establish its equivalence of the normality property of the respective elementary subgroups. This
generalizes previous result of Basu–Rao–Khanna; cf. [3]. Then, as an application,
we establish an analogue Local-Global Principle for the commutator subgroups of the special linear and symplectic groups.
Finally, by using Swan-Weibel’s homotopy trick, we establish
graded analogue of the Local-Global Principle
for the transvection subgroups of the full automorphism groups of the linear, symplectic and orthogonal modules. This generalizes
the previous result of Bak–Basu–Rao; cf. [5].
1. Introduction
In [3], the first author with Ravi A. Rao and Reema Khanna revisited Quillen-Suslin’s Local-Global Principle
for the linear, symplectic and orthogonal groups to show the equivalence of the
Local-Global Principle and Normality property of their Elementary subgroups. A “relative version”
is also established recently by the same authors; cf. [6]. In [3], the authors have also deduced a Local-Global Principle for
the commutator subgroups of the special linear groups, and symplectic groups. In this article we aim to revisit those results over commutative $\mathbb{N}$-graded
rings with identity. We follow the line of proof described in [3].
For the linear case, the graded version of the Local-Global Principle was studied by Chouinard in [7], and by J. Gubeladze in [8], [9].
Though the analogue results are expected for the symplectic and orthogonal groups, according to our best knowledge,
that is not written explicitly in any existing literatures. By deducing the equivalence, we establish the graded version of the
Local-Global Principle for the above two types of classical groups.
To generalize the existing results, from the polynomial rings to the graded rings, one has to use the line of proof of Quillen–Suslin’s patching argument
(cf. [12], [13]),
and Swan–Weibel’s homotopy trick.
For a nice exposition we refer to [8] by J. Gubeladze.
Though, we are writing this article for commutative graded rings, one may also
consider standard graded algebras which are finite over its center. For a commutative $\mathbb{N}$-graded ring with 1,
we establish:
$(1)$ (Theorem 3.8) Normality of the elementary subgroups is equivalent to the graded
Local-Global Principle for the linear, symplectic and orthogonal groups.
$(2)$ (Theorem 4.8) Analogue of Quillen–Suslin’s Local-Global
Principle for the commutator subgroups of the special linear and symplectic groups over graded rings.
In [5], the following was established by the first author with
A. Bak and R.A. Rao:
(Local-Global Principle for the Transvection Subgroups)
An analogue
of Quillen–Suslin’s Local-Global Principle for the transvection subgroup of
the automorphism group of projective,
symplectic and orthogonal modules of global rank at least 1 and
local rank at least 3, under the assumption that the projective module has
constant local rank and that the symplectic and orthogonal modules are locally
an orthogonal sum of a constant number of hyperbolic planes.
In this article, we observe that by using Swan-Weibel’s homotopy trick,
one gets an analogue statement for the graded case. More precisely, we deduce the following fact:
$(3)$ (Theorem 5.4) Let $A=\bigoplus_{i=0}^{\infty}A_{i}$ be a graded ring and $Q\simeq P\oplus A_{0}$ be a
projective $A_{0}$-module ($Q\simeq P\oplus\mathbb{H}(A_{0})$ for symplectic and orthogonal modules). If an automorphism of $Q\otimes_{A_{0}}A$
is locally in the transvection subgroup, then it is globally in the transvection subgroup.
2. Definitions and Notations
Let us start by recalling the following well-known
fact: Given a ring $R$ and a subring $R_{0}$, one can express $R$ as a
direct limit of subrings which are finitely generated over $R_{0}$ as
rings. Considering $R_{0}$ to be the minimal subring (i.e. the
image of $\mathbb{Z}$), it follows that every ring is a direct limit of
Noetherian rings. Hence we may consider $R$ to be Noetherian (cf. Pg 271 [11]).
Throughout this paper we assume $A$ to be a Noetherian, commutative graded ring with identity $1$. We shall write
$A=A_{0}\oplus A_{1}\oplus A_{2}\oplus\cdots$. As we know the multiplication in a graded ring satisfies the following property:
For all $i,j$, $A_{i}A_{j}\subset A_{i+j}$. An element $a\in A$ will be denoted by $a=a_{0}+a_{1}+a_{2}+\cdots$, where $a_{i}\in A_{i}$ for each $i$, and
and all but finitely many $a_{i}$ are zero. Let $A_{+}=A_{1}\oplus A_{2}\oplus\cdots$.
Graded structure of $A$ induces a graded structure on ${\rm M}_{n}(A)$ (ring of $n\times n$ matrices).
Let $S$ be a multiplicatively closed subset of $A_{0}$.
Then for a non-zero divisor $s\in S$ we shall denote the
localization of a matrix $\alpha\in{\rm M}_{n}(A)$ to ${\rm M}_{n}(A_{s})$ as $\alpha_{s}$.
Otherwise, $\alpha_{i},\;i\in\mathbb{Z}$ will represent the $i$-th component of $\alpha$.
We shall use standard notations ${\rm GL}_{n}(A)$ and ${\rm SL}_{n}(A)$ to denote the group of invertible matrices and its subgroup of all
invertible matrices with determinant $1$ respectively.
We recall the well-known “Swan-Weibel’s homotopy trick”, which is the main ingredient to handle the graded case.
Definition 2.1.
Let $a\in A_{0}$ be a fixed element. We fix an element $b=b_{0}+b_{1}+\cdots$ in $A$ and define a ring homomorphism
$\epsilon:A\rightarrow A[X]$ given by
$$\epsilon(b)=\epsilon(b_{0}+b_{1}+\cdots)\;=\;b_{0}+b_{1}X+b_{2}X^{2}+\cdots+b_%
{i}X^{i}+\cdots.$$
Then we evaluate the polynomial $\epsilon(b)(X)$ at $X=a$ and denote the image by $b^{+}(a)$ i.e. $b^{+}(a)=\epsilon(b)(a)$.
Note that $\big{(}b^{+}(x)\big{)}^{+}(y)=b^{+}(xy)$. Observe, $b_{0}=b^{+}(0)$.
We shall use this fact frequently.
The above ring homomorphism $\epsilon$ induces a group homomorphism at the ${\rm GL}_{n}(A)$ level for every $n\geq 1$, i.e. for
$\alpha\in{\rm GL}_{n}(A)$ we get a map
$$\epsilon:{\rm GL}_{n}(A)\rightarrow{\rm GL}_{n}(A[X])\text{ defined by}$$
$$\alpha=\alpha_{0}\oplus\alpha_{1}\oplus\alpha_{2}\oplus\cdots\mapsto\alpha_{0}%
\oplus\alpha_{1}X\oplus\alpha_{2}X^{2}\cdots,$$
where $\alpha_{i}\in{\rm GL}_{n}(A_{i})$.
As above for $a\in A_{0}$, we define $\alpha^{+}(a)$ as
$$\alpha^{+}(a)=\epsilon(\alpha)(a).$$
Now we are going to recall the definitions of the traditional classical groups, viz. the general linear groups, the symplectic and orthogonal groups
(of even size) and their
type subgroups, viz. elementary (symplectic and orthogonal elementary resp.) subgroups.
Let $e_{ij}$ be the matrix with $1$ in the $ij$-position and $0$’s elsewhere.
The matrices of the form $\{{\rm E}_{ij}(\lambda):\lambda\in A\mid i\neq j\},$ where
$${\rm E}_{ij}(\lambda)={\rm I}_{n}+\lambda e_{ij}$$
are called the elementary generators.
Definition 2.2.
The subgroup generated by the set $\{{\rm E}_{ij}(\lambda):\lambda\in A\mid i\neq j\}$, is called the
elementary subgroup of the general linear group and is denoted by ${\rm E}_{n}(A)$.
Observe that ${\rm E}_{n}(A)\subseteq{\rm SL}_{n}(A)\subseteq{\rm GL}_{n}(A)$.
Let $\sigma$ be a permutation defined by: For $i\in\{1,\ldots,2m\}$
$$\sigma(2i)=2i-1\text{ and }\sigma(2i-1)=2i.$$
With respect to this permutation, we define two $2m\times 2m$ forms (viz.) $\psi_{m}$ and $\widetilde{\psi}_{m}$ as follows: For $m>1$, let
$$\psi_{1}=\begin{bmatrix}0&1\\
-1&0\end{bmatrix}\text{ and for $m>1$},\,\,\psi_{m}=\begin{bmatrix}\psi_{m-1}&%
0\\
0&{\rm I}_{2}\end{bmatrix},$$
$$\widetilde{\psi}_{1}=\begin{bmatrix}0&1\\
1&0\end{bmatrix}\text{ and for $m>1$},\,\,\widetilde{\psi}_{m}=\begin{bmatrix}%
\widetilde{\psi}_{m-1}&0\\
0&{\rm I}_{2}\end{bmatrix}.$$
Using above two forms, we define the following traditional classical groups:
Definition 2.3.
A matrix $\alpha\in{\rm GL}_{2m}(A)$ is called symplectic if it fixes $\psi_{m}$ under the action of conjugation, i.e.
$$\alpha^{t}\psi_{m}\alpha=\psi_{m}.$$
The group generated by all symplectic matrices is called the symplectic group and is denoted by ${\rm Sp}_{2m}(A)={\rm Sp}_{n}(A)$, where $n=2m$.
Definition 2.4.
A matrix $\alpha\in{\rm GL}_{2m}(A)$ is called orthogonal if it fixes $\widetilde{\psi}_{m}$ under the action of conjugation, i.e.
$$\alpha^{t}\widetilde{\psi}_{m}\alpha=\widetilde{\psi}_{m}.$$
The group generated by all orthogonal matrices is called orthogonal group and is denoted by ${\rm O}_{2m}(A)={\rm O}_{n}(A)$, where $n=2m$.
Definition 2.5.
The matrices of the form
$$\{{se}_{ij}(z)\in{\rm GL}_{2m}(A):z\in A\mid i\neq j\},$$
where
$$se_{ij}(z)={\rm I}_{2m}+ze_{ij}\text{ if }i=\sigma(j)$$
or
$$se_{ij}(z)={\rm I}_{2m}+ze_{ij}-(-1)^{i+j}ze_{\sigma(j)\sigma(i)}\text{ if }i%
\neq\sigma(j)\text{ and }i<j.$$
are called symplectic elementary generators.
The subgroup generated by symplectic elementary generators is called symplectic elementary group.
Definition 2.6.
The matrices of the form $\{{oe}_{ij}(z):z\in A\mid i\neq j\},$ where
$$oe_{ij}(z)={\rm I}_{2m}+ze_{ij}-ze_{\sigma(j)\sigma(i)},\text{ if }i\neq\sigma%
(j)\text{ and }i<j.$$
are called orthogonal elementary generators.
The subgroup generated by orthogonal elementary generators is called orthogonal elementary group.
Remark 2.7.
It is a well known fact that the elementary subgroups are normal subgroups of the respective classical groups; for $n\geq 3$ in the
linear case, $n=2m\geq 4$ in the symplectic case, and $n=2m\geq 6$ in the orthogonal case.
The linear case was due to A. Suslin (cf. [15]), symplectic was proved by V. Kopeiko (cf. [15]).
Finally the orthogonal case was proved by Suslin–Kopeiko (cf. [14]).
We use the following notations
to treat above three groups uniformly:
$${\rm G}(n,A)\text{ will denote }{\rm GL}_{n}(A)\text{ or }{\rm Sp}_{2m}(A)%
\text{ or }{\rm O}_{2m}(A).$$
$${\rm E}(n,A)\text{ will denote }{\rm E}_{n}(A)\text{ or }{\rm EO}_{2m}(A)\text%
{ or }{\rm ESp}_{2m}(A).$$
$${\rm S}(n,A)\text{ will denote }{\rm SL}_{n}(A)\text{ or }{\rm SO}_{2m}(A),%
\text{ and }$$
$$ge_{ij}\text{ will denote the elementary generators of }{\rm E}_{n}(A)\text{ %
or }{\rm ESp}_{n}(A)\text{or }{\rm EO}_{n}(A).$$
Here ${\rm SO}_{n}(A)$ is the subgroup of ${\rm O}_{n}(A)$ with determinant $1$. Throughout the paper we will assume $n=2m$
to treat the non-linear cases. We shall assume $n\geq 3$ while treating the linear case, and
$n\geq 6$; i.e. $m\geq 3$ while treating the symplectic and orthogonal cases.
Let
$$\widetilde{v}=\begin{cases}v\psi_{m}\text{ in the symplectic case, and }\\
v\widetilde{\psi}_{m}\text{ in the orthogonal case. }\end{cases}$$
Definition 2.8.
Let $v,w\in A^{n}$ be vectors of length $n$, then we define the inner product $\langle v,w\rangle$ as follows:
(1)
$\langle v,w\rangle$ = $v^{t}w$ in the linear case,
(2)
$\langle v,w\rangle$ = $\widetilde{v}w$ in the symplectic and orthogonal cases.
Definition 2.9.
Let $v,w\in A$, we define the map $M:A^{n}\times A^{n}\rightarrow M_{n}(A)$ as follows:
(1)
$M(v,w)=vw^{t}$ in the linear case,
(2)
$M(v,w)=v\widetilde{w}+w\widetilde{v}$ in the symplectic case,
(3)
$M(v,w)=v\widetilde{w}-w\widetilde{v}$ in the orthogonal case.
By $[G,G]$, we denote the commutator subgroup of $G$ i.e. group generated by elements of the form $ghg^{-1}h^{-1}$, for $g,h\in G$.
Definition 2.10.
A row $(v_{1},\ldots,v_{n})\in A^{n}$ is called unimodular row of length $n$
if there exist $b_{1},\ldots,b_{n}\in A$ such that
$\Pi_{i=1}^{n}v_{i}b_{i}=1$; i.e. If $I$ is the ideal generated by
$\langle v_{1},\ldots,v_{n}\rangle$, then $(v_{1},\ldots,v_{n})$ is unimodular
if and only if $I=A$. In the above case we say $A$ has $n$-cover.
3. Local-Global Principle for classical groups
Before discussing the main theorem we recall following standard facts.
Lemma 3.1 (Splitting Lemma).
The elementary generators of $n\times n$
matrices satisfy the following property:
$$ge_{ij}(x+y)=ge_{ij}(x)\;ge_{ij}(y)$$
for all $x,y\in A$
and for all $i,j=1,\ldots,n$ with $i\neq j$.
Proof.
Standard.
(cf. [3], $3, Lemma $3.2$).
∎
Lemma 3.2.
Let $G$ be a group, and $a_{i},b_{i}\in G$ for $i=1,\ldots,r$.
Let $J_{k}=\underset{j=1}{\overset{k}{\Pi}}a_{j}$.
Then
$$\underset{i=1}{\overset{r}{\huge\Pi}}a_{i}b_{i}=\underset{i=1}{\overset{r}{\Pi%
}}J_{i}b_{i}J_{i}^{-1}\underset{i=1}{\overset{r}{\Pi}}a_{i}.$$
Proof.
cf. [3], $3, Lemma $3.4.$
∎
Following structural lemma pays a key role in the proof of main theorem.
It is well-known for the polynomial rings; cf. [3] ($3, Lemma $3.6$).
Here, we deduce the analogue for the graded rings.
Lemma 3.3.
Let ${\rm G}(n,A,A_{+})$ denote the subgroup of ${\rm G}(n,A)$ which
is equal to ${\rm I}_{n}$ modulo $A_{+}$.
Then the group ${\rm G}(n,A,A_{+})\cap{\rm E}(n,A)$ is generated by the elements of the type
$\epsilon\,ge_{ij}(A_{+})\,\epsilon^{-1}$ for some $\epsilon\in{\rm E}(n,A_{0})$.
Proof.
Let $\alpha\in{\rm E}(n,A)\cap{\rm G}(n,A,A_{+})$. Then we can write
$$\alpha=\underset{k=1}{\overset{r}{\Pi}}ge_{i_{k}j_{k}}(a_{k})$$
for some elements $a_{k}\in A$, $k=1,\ldots,r$.
As $a_{k}=({a_{0}})_{k}+({a_{+}})_{k}$ for some $(a_{0})_{k}\in A_{0}$ and $(a_{+})_{k}\in A_{+}$. Using the splitting lemma (Lemma 3.1) we can rewrite the expression as:
$$\alpha=\underset{k=1}{\overset{r}{\Pi}}\big{(}ge_{i_{k}j_{k}}{(a_{0})}_{k}\big%
{)}\,\big{(}ge_{i_{k}j_{k}}{(a_{+})}_{k}\big{)}.$$
We write $\epsilon_{t}=\Pi_{k=1}^{t}ge_{i_{k}j_{k}}\big{(}{(a_{0})}_{k}\big{)}$ for $t\in\{1,2,\ldots,r\}$. Observe that $\epsilon_{r}={\rm I}_{n}$, as $\alpha\in{\rm G}(n,A,A_{+})$.
Then
$$\alpha=\Big{(}\underset{k=1}{\overset{r}{\Pi}}\epsilon_{k}ge_{i_{k}j_{k}}\big{%
(}{(a_{+})}_{k}\big{)}\epsilon_{k}^{-1}\Big{)}\Big{(}\underset{k=1}{\overset{r%
}{\Pi}}ge_{i_{k}j_{k}}\big{(}{(a_{0})}_{k}\big{)}\Big{)}=\mathcal{A}\mathcal{B%
}\text{ (say) },$$
where $\mathcal{A}=\Big{(}\underset{k=1}{\overset{r}{\Pi}}\epsilon_{k}ge_{i_{k},j_{k}%
}\big{(}{(a_{+})}_{k}\big{)}\epsilon_{k}^{-1}\Big{)}$
and $\mathcal{B}=\Big{(}\underset{k=1}{\overset{r}{\Pi}}ge_{i_{k}j_{k}}\big{(}{(a_{%
0})}_{k}\big{)}\Big{)}.$
Now go modulo $A_{+}$. Let bar (–) denote the quotient ring modulo $A_{+}$. Then,
$$\overline{\alpha}=\overline{{\rm I}}_{n}=\overline{\mathcal{A}}\overline{%
\mathcal{B}}=\overline{{\rm I}}_{n}\overline{\mathcal{B}}=\bar{{\rm I}}_{n}%
\implies\overline{\mathcal{B}}=\overline{{\rm I}}_{n},$$
as $\alpha\in{\rm G}(n,A,A_{+})$. Since entries of $\mathcal{B}$
are in $A_{0}$, it follows that $\mathcal{B}={\rm I}_{n}$.
$$\alpha=\underset{k=1}{\overset{r}{\Pi}}\epsilon_{k}\,\big{(}ge_{i_{k}j_{k}}%
\big{(}{(a_{+})}_{k}\big{)}\big{)}\,\epsilon_{k}^{-1};$$
as desired.
∎
Now we prove a variant of “Dilation Lemma” mentioned in the statement of $(3)$ of Theorem 3.8.
Lemma 3.4.
Assume the “Dilation Lemma” $($Theorem 3.8 - $($4$)$$)$
to be true. Let $\alpha_{s}\in{\rm E}(n,A_{s}),\;$ with $\alpha^{+}(0)={{\rm I}_{n}}$. Then one gets
$$\alpha^{+}(b+d)\alpha^{+}(d)^{-1}\in{\rm E}(n,A)$$
for some $s,d\in A_{0}$ and $b=s^{l},l\gg 0$.
Proof.
We have $\alpha_{s}\in{\rm E}(n,A_{s})$.
Hence $\alpha_{s}^{+}(X)\in{\rm E}(n,A_{s}[X])$. Let $\beta^{+}(X)=\alpha^{+}(X+d){\alpha^{+}(d)}^{-1}$, where $d\in A_{0}$.
Then $\beta^{+}_{s}(X)\in{\rm E}(n,A_{s}[X])$ and $\beta^{+}(0)={{\rm I}_{n}}$.
Hence by Theorem 3.8 - (4) their exists
$\widetilde{\beta}(X)\in{\rm E}(n,A[X])$ such that $\widetilde{\beta}_{s}(X)=\beta_{s}^{+}(bX)$. Putting $X=1$, we get the required result.
∎
Following is a very crucial result we need for our method of proof. There are many places we use this fact in a very subtle way.
In particular, we mainly use this lemma for the step $(4)\Rightarrow(3)$ of 3.8.
Lemma 3.5.
(cf.[10], Lemma 5.1) Let $R$ be a Noetherian ring and $s\in R$.
Then there exists a natural number $k$ such that the homomorphism
${\rm G}(n,s^{k}R)\rightarrow{\rm G}(n,R_{s})$ $($induced by localization
homomorphism $R\rightarrow R_{s})$ is injective. Moreover, it follows that
the induced map ${\rm E}(n,R,s^{k}R)\rightarrow{\rm E}(n,R_{s})$ is
injective.
The next two lemma’s will be used in intermediaries to prove the equivalent conditions mentioned in Theorem 3.8. We state it without proof.
Lemma 3.6.
Let $v=(v_{1},v_{2},\ldots,v_{n})$ be a unimodular row over over a
commutative semilocal ring $R$. Then the row $(v_{1},v_{2},\ldots,v_{n})$ is completable; i.e. $(v_{1},v_{2},\ldots,v_{n})$
elementarily equivalent to the row $(1,0,\ldots,0)$;
i.e. their exists $\epsilon\in{\rm E}_{n}(R)$ such that $(v_{1},v_{2},\ldots,v_{n})\epsilon=(1,0,\ldots,0)$.
Proof.
cf. [4], Lemma $1.2.21$.
∎
Lemma 3.7.
Let $A$ be a ring and $v\in{\rm E}(n,A)e_{1}$. Let $w\in A^{n}$ be a
column vector such that $\langle v,w\rangle=0$. Then ${\rm I}_{n}+M(v,w)\in E(n,A)$.
Proof.
cf. [4], Lemma $2.2.4$.
∎
Theorem 3.8.
The followings are equivalent for any graded ring $A=\oplus_{i=0}^{\infty}A_{i}$ for
$n\geq 3$ in the linear case and $n\geq 6$ otherwise.
(1)
(Normality):
${\rm E}(n,A)$ is a normal subgroup of ${\rm G}(n,A)$.
(2)
If $v\in{\rm Um}_{n}(A)$ and $\langle v,w\rangle=0$, then ${\rm I}_{n}+M(v,w)\in{\rm E}(n,A)$.
(3)
(Local-Global Principle):
Let $\alpha\in{\rm G}(n,A)$ with $\alpha^{+}(0)={\rm I}_{n}$. If
for every maximal ideal $\mathfrak{m}$ of $A_{0}$
$\alpha_{\mathfrak{m}}\in{\rm E}(n,A_{\mathfrak{m}})$,
then $\alpha\in{\rm E}(n,A)$.
(4)
(Dilation Lemma):
Let $\alpha\in{\rm G}(n,A)$ with $\alpha^{+}(0)={\rm I}_{n}$ and $\alpha_{s}\in{\rm E}(n,A_{s})$ for some non-zero divisor $s\in A_{0}$.
Then there exists $\beta\in{\rm E}(n,A)$ such that
$$\beta_{s}^{+}(b)=\alpha_{s}^{+}(b)$$
for some $b=s^{l}$; $l\gg 0$. i.e.
$\alpha^{+}_{s}(s^{l})$ will be defined over $A$ for $l\gg 0$.
(5)
If $\alpha\in{\rm E}(n,A)$, then $\alpha^{+}(a)\in{\rm E}(n,A)$, for every $a\in A_{0}$.
(6)
If $v\in{\rm E}(n,A)e_{1},$ and $\langle v,w\rangle=0$, then ${\rm I}_{n}+M(v,w)\in{\rm E}(n,A)$.
Proof.
(6) $\Rightarrow$ (5):
Let $\alpha=\underset{k=1}{\overset{t}{\Pi}}\big{(}{\rm I}_{n}+a\;M(e_{i_{k}},e_{j_%
{k}})\big{)}\text{ where }a\in A$, and $t\geq 1$, a positive integer.
Then
$$\alpha^{+}(b)=\underset{k=1}{\overset{t}{\Pi}}\big{(}{\rm I}_{n}+a^{+}(b)\;M(e%
_{i_{k}},e_{j_{k}})\big{)},$$
where
$b\in A_{0}$. Take $v=e_{i}$ and $w=a^{+}(b)e_{j}$, Then $\alpha={\rm I}_{n}+M(v,w)$ and $\langle v,w\rangle=0$. Indeed,
Linear case:
$$\displaystyle{\rm M}(v,w)=$$
$$\displaystyle vw^{t}=e_{i}(a^{+}(b)e_{j})^{t}=a^{+}(b)e_{i}e_{j}^{t}=a^{+}(b)%
\;{\rm M}(e_{i},e_{j}),$$
$$\displaystyle\langle v,w\rangle=$$
$$\displaystyle v^{t}w=e_{i}^{t}(a^{+}(b)e_{j})^{t}=a^{+}(b)\;e_{i}^{t}e_{j}=a^{%
+}(b)\langle e_{i},e_{j}\rangle=0.$$
Symplectic case:
$$\displaystyle{\rm M}(v,w)=$$
$$\displaystyle vw^{t}\psi_{m}+wv^{t}\psi_{m}=e_{i}(a^{+}(b)e_{j})^{t}\psi_{m}+a%
^{+}(b)e_{j}e_{i}^{t}\psi_{m}$$
$$\displaystyle=$$
$$\displaystyle a^{+}(b)e_{i}e_{j}^{t}\psi_{m}+a^{+}(b)e_{j}e_{i}^{t}\psi_{m}=a^%
{+}(b)\;{\rm M}(e_{i},e_{j}),$$
$$\displaystyle\langle e_{i},e_{j}\rangle=e_{i}^{t}\psi_{m}e_{j}=(e_{i}^{t}\psi_%
{m}e_{i})(e_{i}^{t}e_{j})=\psi_{m}(e_{i}^{t}e_{j})=0\text{ hence, }$$
$$\displaystyle\langle v,w\rangle=\;v^{t}\psi_{m}w=e_{i}^{t}\psi_{m}a^{+}(b)e_{j%
}=a^{+}(b)e_{i}^{t}e_{j}=a^{+}(b)\langle e_{i},e_{j}\rangle=0.$$
Orthogonal case:
$$\displaystyle{\rm M}(v,w)=$$
$$\displaystyle v.w^{t}\widetilde{\psi}_{m}-w.v^{t}\widetilde{\psi}_{m}=e_{i}.(a%
^{+}(b)e_{j})^{t}\widetilde{\psi}_{m}-a^{+}(b)e_{j}.e_{i}^{t}\widetilde{\psi}_%
{m}$$
$$\displaystyle=$$
$$\displaystyle a^{+}(b)e_{i}.e_{j}^{t}\widetilde{\psi}_{m}-a^{+}(b)e_{j}.e_{i}^%
{t}\widetilde{\psi}_{m}=a^{+}(b)\;{\rm M}(e_{i},e_{j}),$$
$$\displaystyle\langle e_{i},e_{j}\rangle=e_{i}^{t}\widetilde{\psi}_{m}e_{j}=(e_%
{i}^{t}\widetilde{\psi}_{m}e_{i})(e_{i}^{t}e_{j})=\widetilde{\psi}_{m}(e_{i}^{%
t}e_{j})=0\text{ hence, }$$
$$\displaystyle\langle v,w\rangle=\;v^{t}\widetilde{\psi}_{m}w=e_{i}^{t}%
\widetilde{\psi}_{m}a^{+}(b)e_{j}=a^{+}(b)e_{i}^{t}e_{j}=a^{+}(b)\langle e_{i}%
,e_{j}\rangle=0.$$
Hence applying (6) over ring $A$, we have $\alpha^{+}(b)\in{\rm E}(n,A)$. Therefore $\alpha^{+}(b)\in{\rm E}(n,A)$ for $b\in A_{0}$.
(5) $\Rightarrow$ (4):
Since $\alpha_{s}\in{\rm E}(n,A_{s})$ with $({\alpha_{0}}_{s})={\rm I}_{n}$, the diagonal entries of $\alpha$
are of the form $1+g_{ii}$, where $g_{ii}\in({A_{+}})_{s}$ and off-diagonal entries are of the
form $g_{ij}$, where $i\neq j$ and $g_{ij}\in({A_{+})}_{s}$.
We choose $l$ to be large enough such that $s^{l}$ is greater than the common denominator of all
$g_{ii}$ and $g_{ij}$. Then using (5), we get
$$\alpha_{s}^{+}(s^{l})\in{\rm E}(n,A_{s}).$$
Since that $\alpha^{+}(s^{l})$ permits a natural pullback (as denominators are cleared), we have $\alpha^{+}(s^{l})\in{\rm E}(n,A)$.
(4) $\Rightarrow$ (3):
Since $\alpha_{\mathfrak{m}}\in{\rm E}(n,A_{\mathfrak{m}})$, we have an element $s\in A_{0}-\mathfrak{m}$ such that
$\alpha_{s}\in{\rm E}(n,A_{s})$. Let $s_{1},s_{2},\ldots,s_{r}\in A_{0}$ be non-zero divisors with $s_{i}\in A_{0}-\mathfrak{m}_{i}$
such that $\langle s_{1},s_{2},\ldots,s_{r}\rangle=A$. From (4) we have $\alpha^{+}(b_{i})\in{\rm E}(n,A),$
for some $b_{i}={{s_{i}}^{l_{i}}}$ with $b_{1}+\cdots+b_{r}=1$. Now consider $\alpha_{s_{1}s_{2}\ldots s_{r}}$,
which is the image of $\alpha$ in $A_{s_{1}\cdots s_{r}}$. Due to Lemma 3.5, $\alpha\mapsto\alpha_{s_{1}s_{2}\cdots s_{r}}$
is injective and hence we can perform our calculation in $A_{s_{1}\cdots s_{r}}$ and then pull it back to $A$.
$$\displaystyle\alpha_{s_{1}s_{2}\ldots s_{r}}=$$
$$\displaystyle\alpha_{s_{1}s_{2}\ldots s_{r}}^{+}(b_{1}+b_{2}\cdots+b_{r})$$
$$\displaystyle=$$
$$\displaystyle{{\big{(}(\alpha_{s_{1}})}_{s_{2}s_{3}\ldots}}\big{)}^{+}(b_{1}+%
\cdots+b_{r}){{\big{(}(\alpha_{s_{1}})}_{s_{2}s_{3}\ldots}}\big{)}^{+}(b_{2}+%
\cdots+b_{r})^{-1}\cdots$$
$$\displaystyle{\big{(}(\alpha_{s_{i}})}_{s_{1}\ldots\hat{s_{i}}\ldots s_{r}}%
\big{)}^{+}(b_{i}+\cdots+b_{r}){\big{(}(\alpha_{s_{i}})}_{s_{1}\ldots\hat{s_{i%
}}\ldots s_{r}}\big{)}^{+}(b_{i+1}+\cdots+b_{r})^{-1}$$
$$\displaystyle\cdots{{\big{(}(\alpha_{s_{r}})}_{s_{1}s_{2}\ldots s_{r-1}}}\big{%
)}^{+}(b_{r}){{\big{(}(\alpha_{s_{r}})}_{s_{1}s_{2}\ldots s_{r-1}}}\big{)}^{+}%
(0)^{-1}$$
Observing that each
$${\big{(}(\alpha_{s_{i}})}_{s_{1}s_{2}\ldots\hat{s_{i}}\ldots s_{r}}\big{)}^{+}%
(b_{i}+\cdots+b_{r}){\big{(}(\alpha_{s_{i}})}_{s_{1}\ldots\hat{s_{i}}\ldots s_%
{r}}\big{)}^{+}(b_{i+1}+\cdots+b_{r})^{-1}\in{\rm E}(n,A)$$
due to Lemma 3.4 (here $\hat{s}_{i}$ means we omit $s_{i}$ in the product $s_{1}\cdots\hat{s_{i}}\cdots s_{r}$),
we have $\alpha_{s_{1}\cdots s_{r}}\in{\rm E}(n,A_{s_{1}\cdots s_{r}})$ and hence $\alpha\in{\rm E}(n,A)$.
Remark 3.9.
Following is a commutative diagram (here we are assuming $\langle s_{i},s_{j}\rangle=A$):
\UseComputerModernTips
$$\xymatrix{A\ar[r]\ar[d]&A_{s_{i}}\ar[d]\\
A_{s_{j}}\ar[r]&A_{s_{i}s_{j}}}$$
Let $\theta_{i}=\alpha_{s_{i}}^{+}(b_{i}+\cdots+b_{r})\alpha_{s_{i}}^{+}(b_{i-1}+%
\cdots+b_{r})\in A_{s_{i}}$ and $\theta_{ij}$
be the image of $\theta_{i}$ in $A_{s_{i}s_{j}}$ and similarly $\pi_{j}=\alpha_{s_{j}}^{+}(b_{j}+\cdots+b_{r})\alpha_{s_{j}}^{+}(b_{j-1}+%
\cdots+b_{r})\in A_{s_{j}}$
and $\pi_{s_{i},s_{j}}$ be its image in $A_{s_{i}s_{j}}$. Then due to Lemma 3.5 the product $\big{(}\theta_{s_{i}s_{j}}\big{)}\big{(}\pi_{s_{i}s_{j}}\big{)}$
can be identified with the product $\theta_{i}\pi_{j}$.
(3) $\Rightarrow$ (2):
Since polynomial rings are special case of graded rings, the result follows by using $3\Rightarrow 2$ in [3], $\lx@sectionsign 3$.
(2) $\Rightarrow$ (1) $\Rightarrow$ (6):
The proof goes as in [3]. ($$3$, $2\Rightarrow 1\Rightarrow 7\Rightarrow 6$).
∎
4. Local-Global Principle for commutator subgroup
In this section we deduce the analogue of Local-Global Principle for the commutator subgroup of the linear and the symplectic group over graded rings.
Unless mentioned otherwise, we assume $n\geq 3$ for the linear case and $n\geq 6$ for the symplectic case.
Let us begin with the following well-known fact for semilocal rings.
Lemma 4.1.
Let $A$ be a semilocal commutative ring with identity. Then for $n\geq 2$ in the linear case and
$n\geq 4$ in the symplectic case, one gets
$${\rm S}(n,A)={\rm E}(n,A).$$
Proof.
cf. Lemma $1.2.25$ in [4].
∎
Remark 4.2.
If $\alpha=(\alpha_{ij}/1)\in{\rm G}(n,A_{s})$, then $\alpha$ has a natural pullback $\beta=(\alpha_{ij})\in{\rm G}(n,A)$ such that $\beta_{s}=\alpha$.
If $\alpha_{s}\in{\rm S}(n,A_{s})$ such that it admits a natural pullback $\beta\in{\rm G}(n,A)$,
then $\beta\in{\rm S}(n,A)$.
The next lemma deduces an analogue result of “Dilation Lemma” (Theorem 3.8-($3$)) for ${\rm S}(n,A)$; the special linear (resp. symplectic) group.
Lemma 4.3.
Let $\alpha=(\alpha)_{ij}\in{\rm S}(n,A)$. Then $\alpha^{+}(a)\in{\rm S}(n,A),$
where $a\in A_{0}$. Hence if $\alpha\in G(n,A)$ with $\alpha^{+}(0)={{\rm I}_{n}}$ and $\alpha_{s}\in{\rm S}(n,A_{s})\text{ then }\alpha_{s}^{+}(b)\in{\rm S}(n,A)$ $($after identifying $\alpha_{s}^{+}(b)$ with its pullback by using Lemma 3.5$)$,
where $s\in A_{0}$ is a non-zero divisor and $b=s^{l}$ with $l\gg 0$, for $n\geq 1$ in linear case and $n\geq 2$ in symplectic case.
Proof.
Since $\text{ det }:A\rightarrow A^{*}\text{ (units of }A)\subset A_{0},$ and $A_{i}A_{j}\subset A_{i+j}$,
the non-zero component of $\alpha$ doesn’t contribute for the value of the determinant and hence
$$\text{det }\alpha=\text{det }\alpha^{+}(0).$$
Therefore, if $\beta=\alpha^{+}(a)$, then
$$\text{det }\beta=\text{det }\beta^{+}(0)=\text{det }\big{(}\alpha^{+}(a)\big{)%
}^{+}(0)=\text{det }\alpha^{+}(a\cdot 0)=\text{det }\alpha^{+}(0)=1.$$
Hence $\alpha^{+}(a)\in{\rm S}(n,A)$.
Since $\alpha_{s}\in{\rm S}(n,A_{s})$ with $\alpha^{+}(0)={{\rm I}_{n}}$, the diagonal entries are of the form $1+g_{ii}$, and
off diagonal entries are $g_{ij}$, where $g_{ij}\in{(A_{+}})_{s}$ for all $i,j$.
We choose $b\in(s)$ such that $b=s^{l}$ with $l\gg 0$, so that $b$ can dilute
the denominator of each entries. Then $\alpha_{s}^{+}(b)\in{\rm S}(n,A)$.
∎
The next lemma gives some structural information about commutators.
Lemma 4.4.
Let $\alpha,\beta\in{\rm S}(n,A)$ and $A_{0}$ be a commutative semilocal ring.
Then the commutator subgroup
$$[\alpha,\beta]\in[\alpha\alpha_{0}^{-1},\beta\beta_{0}^{-1}]\;{\rm E}(n,A).$$
Proof.
Since $A_{0}$ is semilocal, we have ${\rm S}(n,A_{0})={\rm E}(n,A_{0})$ by Lemma 4.1. Hence $\alpha^{+}(0),\beta^{+}(0)\in{\rm E}(n,A_{0})$. Let $a=\alpha{\alpha^{+}(0)}^{-1}$ and $b=\beta{\beta^{+}(0)}^{-1}$. Then
$$\displaystyle[\alpha,\beta]$$
$$\displaystyle=[\alpha{\alpha^{+}(0)}^{-1}\alpha^{+}(0),\beta{\beta^{+}(0)}^{-1%
}\beta^{+}(0)]$$
$$\displaystyle=a\alpha^{+}(0)b\beta^{+}(0){\alpha^{+}(0)}^{-1}a^{-1}{\beta^{+}(%
0)}^{-1}b^{-1}$$
$$\displaystyle=\big{(}aba\!^{-1}b\!^{-1}\big{)}\big{(}bab\!^{-1}\alpha^{+}(0)ba%
\!^{-1}b\!^{-1}\big{)}\big{(}ba\beta^{+}(0){\alpha^{+}(0)}\!^{-1}a\!^{-1}b^{-1%
}\big{)}\big{(}b{\beta^{+}(0)}^{-1}b\!^{-1}\big{)}.$$
Since ${\rm E}(n,A)$ is a normal subgroup of ${\rm S}(n,A)$, the elements $bab^{-1}\alpha^{+}(0)ba^{-1}b^{-1}$,
$ba\beta^{+}(0){\alpha^{+}(0)}^{-1}a^{-1}b^{-1}$ and $b{\beta^{+}(0)}^{-1}b^{-1}$ are in ${\rm E}(n,A).$
∎
Corollary 4.5.
Let $\alpha\in[{\rm S}(n,A),{\rm S}(n,A)]$ with $\alpha^{+}(0)={\rm I}_{n},$ and
let $A_{0}$ be a semilocal commutative ring. Then using the normality property of the elementary $($resp. elementary symplectic$)$ subgroup
$\alpha$ can be written as
$$\underset{k=1}{\overset{t}{\Pi}}\;[\beta_{k},\gamma_{k}]\;\epsilon,$$
for some $t\geq 1$, and $\beta_{k},\gamma_{k}\in{\rm S}(n,A)$, with $\beta_{k}^{+}(0)=\gamma_{k}^{+}(0)={\rm I}_{n}$, and $\epsilon\in{\rm E}(n,A)$ with $\epsilon^{+}(0)={\rm I}_{n}$.
Proof.
Since $\alpha\in[{\rm S}(n,A),{\rm S}(n,A)]$, $\alpha=\underset{k=1}{\overset{t}{\Pi}}[a_{k},b_{k}]$ for some $t\geq 1$.
Using Lemma 4.4 we identify $\beta_{k}$ with $a_{k}{{a_{k}}^{+}(0)}^{-1}$ and
$\gamma_{k}$ with $b_{k}{{b_{k}}^{+}(0)}^{-1}$. This gives
$$\alpha=\underset{k=1}{\overset{t}{\Pi}}\;[\beta_{k},\gamma_{k}]\;\epsilon.$$
Then it follows that $\epsilon^{+}(0)={\rm I}_{n}$, as $\alpha^{+}(0)={\rm I}_{n}$.
∎
The next lemma gives a variant of “Dilation Lemma” (Lemma 4.3)
for ${\rm S}(n,A)$; the special linear (resp. symplectic) group.
Lemma 4.6.
If $\alpha\in{\rm S}(n,A_{s})$ with $\alpha^{+}(0)={{\rm I}_{n}}$, then
$$\alpha^{+}(b+d){\alpha^{+}(d)}^{-1}\in{\rm S}(n,A),$$
where $s$ is a non-zero divisor and $b,d\in A_{0}$ with $b=s^{i}$ for some $i\gg 0$.
Proof.
Let $\alpha^{+}(X)\in{\rm G}(n,A[X])$, then $\alpha_{s}^{+}(X)\in{\rm E}(n,A_{s}[X])$.
Let $\beta^{+}(X)=\alpha^{+}(X+d){\alpha^{+}(d)}^{-1}$. Then $\beta_{s}^{+}(X)\in{\rm S}(n,A_{s}[X])$.
Hence by Lemma 4.3, $\beta^{+}(bX)\in{\rm S}(n,A)$. Putting $X=1$,
we get result.
∎
The following lemma makes use of Lemma 4.3 to deduce “Dilation Lemma” for the commutator subgroup $[{\rm S}(n,A),{\rm S}(n,A)]$.
Lemma 4.7.
Let $\alpha_{s}=\underset{i=1}{\overset{t}{\Pi}}[{\beta_{i}}_{s},{\gamma_{i}}_{s}]%
\epsilon_{s}$ for some $t\geq 1$,
such that ${\beta_{i}}_{s}$ and ${\gamma_{i}}_{s}\in{\rm S}(n,A_{s})$ and $\epsilon_{s}\in{\rm E}(n,A_{s})$
with $\gamma_{i_{s}}^{+}(0)=\beta_{i_{s}}^{+}(0)={\epsilon_{s}}^{+}(0)={\rm I}_{n}$.
Then $($by identifying with its pullback by using Lemma 3.5$)$
$$\alpha_{s}^{+}(b+d)\alpha_{s}^{+}(d)^{-1}\in[{\rm S}(n,A),{\rm S}(n,A)],$$
where $b,d\in A_{0}$ with $b=s^{l}$ for some $l\gg 0$.
Proof.
Without loss of generality (as we would conclude by the end of the proof),
we can assume
$$\alpha_{s}=[\beta_{s},\gamma_{s}]\epsilon_{s}=\beta_{s}\gamma_{s}{\beta_{s}}^{%
-1}{\gamma_{s}}^{-1}\epsilon_{s}.$$
Since $\beta_{s}^{+}(0)={\rm I}_{n}$, by Lemma 4.3 ${\beta_{s}}^{+}(b),{\gamma_{s}}^{+}(b)\in{\rm S}(n,A)$ for some $b=s^{l}$ with $l\gg 0$. Also,
${\epsilon_{s}}^{+}(b)\in{\rm E}(n,A)$ by (5) of Theorem 3.8.
Hence
$$\displaystyle\alpha_{s}^{+}(b+d){\alpha_{s}^{+}(d)}^{-1}=$$
$$\displaystyle\;\beta_{s}^{+}(b+d)\gamma_{s}^{+}(b+d){\beta_{s}^{+}(b+d)}^{-1}{%
\gamma_{s}^{+}(b+d)}^{-1}$$
$$\displaystyle\epsilon_{s}^{+}(b+d){\epsilon_{s}^{+}(d)}^{-1}\gamma_{s}^{+}(d)%
\beta_{s}^{+}(d){\gamma_{s}^{+}(d)}^{-1}{\beta_{s}^{+}(d)}^{-1}.$$
Since ${\rm E}(n,A_{s})$ and ${\rm S}(n,A_{s})$ are normal subgroups in ${\rm G}(n,A_{s})$, by rearranging (using $\epsilon_{s}^{+}(b+d){\epsilon_{s}^{+}(d)}^{-1}$
as intermediary) we can consider
$\gamma_{s}^{+}(b+d){\gamma_{s}^{+}(d)}^{-1}$ and $\beta_{s}^{+}(b+d){\beta_{s}^{+}(d)}^{-1}$ together. Now by using Lemma 4.6
we get $\gamma_{s}^{+}(b+d){\gamma_{s}^{+}(d)}^{-1}$ and $\beta_{s}^{+}(b+d){\beta_{s}^{+}(d)}^{-1}\in{\rm S}(n,A)$.
Hence $($by identifying with its pullback by using Lemma 3.5$)$ it follows that
$$\alpha_{s}^{+}(b+d){\alpha_{s}^{+}(d)}^{-1}\in[{\rm S}(n,A),{\rm S}(n,A)].$$
∎
Now we deduce the graded version of the Local-Global Principle for the commutators subgroups.
Theorem 4.8.
Let $\alpha\in{\rm S}(n,A)$ with $\alpha^{+}(0)={\rm I}_{n}$.
If
$$\alpha_{\mathfrak{p}}\in[{\rm S}(n,A_{\mathfrak{p}}),{\rm S}(n,A_{\mathfrak{p}%
})]\text{ for all }\mathfrak{p}\in{\rm Spec(A_{0})},$$
then
$$\alpha\in[{\rm S}(n,A),{\rm S}(n,A)].$$
Proof.
Since $\alpha_{\mathfrak{p}}\in[{\rm S}(n,A_{\mathfrak{p}}),{\rm S}(n,A_{\mathfrak{p}%
})]\subset[{\rm S}(n,A_{\mathfrak{m}}),{\rm S}(n,A_{\mathfrak{m}})]$,
we have for a maximal ideal $\mathfrak{m}\supseteq\mathfrak{p}$, $s\in A_{0}-\mathfrak{m}$
such that
$$\alpha_{s}\in[{\rm S}(n,A_{s}),{\rm S}(n,A_{s})],$$
hence $\alpha_{s}$ can be decomposed as
$$\alpha_{s}=\underset{i=1}{\overset{t}{\Pi}}[{\beta_{i_{s}}},{\gamma_{i_{s}}}]%
\epsilon_{s}$$
by Corollary
4.5, where $\beta_{i_{s}},\gamma_{i_{s}}\in{\rm S}(n,A_{s})$ and $\epsilon_{s}\in{\rm E}(n,A_{s})$.
Now using the “Dilation Lemma” (Lemma 4.3 for ${\rm S}(n,A_{s})$ and theorem 3.8 $(4)$ for
${\rm E}(n,A_{s})$) on each of these elements, we have for $b=s^{i},i\gg 0$
$$\alpha_{s}^{+}(b)=\underset{i=1}{\overset{t}{\Pi}}[\beta_{i_{s}}^{+}(b),\gamma%
_{i_{s}}^{+}(b)]\epsilon_{s}^{+}(b)\in[{\rm S}(n,A),{\rm S}(n,A)]E(n,A).$$
Since ${\rm E}(n,A_{s})\subseteq[{\rm S}(n,A_{s}),{\rm S}(n,A_{s})]$, we have $\alpha_{s}\in[{\rm S}(n,A_{s}),{\rm S}(n,A_{s})]$.
Let $s_{1},\ldots,s_{r}\in A_{0}$ non-zero divisors such that $s_{i}\in A_{0}-\mathfrak{m}_{i}$ and $\langle s_{1},\ldots,s_{r}\rangle=A_{0}$. Then
it follows that $b_{1}+\cdots+b_{r}=1$ for suitable (as before) $b_{i}\in(s_{i})$; and $i=1,\ldots r$. Now
$$\displaystyle\alpha_{s_{1}s_{2}\ldots s_{r}}=$$
$$\displaystyle\alpha_{s_{1}s_{2}\ldots s_{r}}^{+}(1)$$
$$\displaystyle=$$
$$\displaystyle\alpha_{s_{1}s_{2}\ldots s_{r}}^{+}(b_{1}+b_{2}+\cdots+b_{r})$$
$$\displaystyle=$$
$$\displaystyle{{\big{(}(\alpha_{s_{1}})}_{s_{2}s_{3}s_{4}\ldots}}\big{)}^{+}(b_%
{1}+\cdots+b_{r}){{\big{(}(\alpha_{s_{1}})}_{s_{2}s_{3}s_{4}\ldots}}\big{)}^{+%
}(b_{2}+\cdots+b_{r})^{-1}\!\!\cdots$$
$$\displaystyle{\big{(}(\alpha_{s_{i}})}_{s_{1}\ldots\hat{s_{i}}\ldots s_{r}}%
\big{)}^{+}\!\!(b_{i}+\cdots+b_{r}){\big{(}(\alpha_{s_{i}})}_{s_{1}\ldots\hat{%
s_{i}}\ldots s_{r}}\big{)}^{+}\!\!(b_{i+1}+\cdots+b_{r})^{-1}$$
$$\displaystyle\cdots{{\big{(}(\alpha_{s_{r}})}_{s_{1}s_{2}\ldots s_{r-1}}}\big{%
)}^{+}\!\!(b_{r}){{\big{(}(\alpha_{s_{r}})}_{s_{1}s_{2}\ldots s_{r-1}}}\big{)}%
^{+}\!\!(0)^{-1}.$$
Using Lemma 3.5 the product is well defined (see Remark 3.9) and using Lemma 4.7, we conclude that $\alpha\in[{\rm S}(n,A),{\rm S}(n,A)].$
∎
5. Auxiliary Result for Transvection subgroup
Let $R$ be a commutative ring with $1$. We recall that a finitely generated projective $R$-module $Q$ has a unimodular element
if their exists $q\in Q$ such that $qR\simeq R$ and $Q\simeq qR\oplus P$ for some projective $R$-module $P$.
In this section, we shall consider three types of (finitely generated) classical modules; viz. projective,
symplectic and orthogonal modules over graded rings. Let $A=\bigoplus_{i=0}^{\infty}{A_{i}}$ be a commutative graded Noetherian ring with identity.
For definitions and related facts we refer
[5]; $1, $2. In that paper, the first author with A. Bak and R.A. Rao has established an
analogous Local-Global Principle for the elementary transvection subgroup of the
automorphism group of projective, symplectic and orthogonal modules of global
rank at least 1 and local rank at least 3. In this article we deduce an analogous statement for the above classical groups over graded rings.
We shall assume for every maximal ideal $\mathfrak{m}$ of $A_{0}$, the symplectic and orthogonal module $Q_{\mathfrak{m}}\simeq(A_{0}^{2m})_{\mathfrak{m}}$
with the standard form (for suitable integer $m$).
Remark 5.1.
By definition the global rank or simply rank of a finitely generated projective $R$-module
(resp. symplectic or orthogonal $R$-module) is the largest integer $k$ such that $\oplus^{k}R$
(resp. $\perp^{k}$ $\mathbb{H}(R)$) is a direct summand (resp. orthogonal summand) of the module. Here
$\mathbb{H}(R)$ denotes the hyperbolic plane.
Let $Q$ denote a projective, symplectic or orthogonal $A_{0}$-module of global rank
$\geq 1$, and total (or local)
rank $r+1$ in the linear case and $2r+2$ otherwise and let $Q_{1}=Q\otimes_{A_{0}}A$.
We use the following notations to deal with the above three classical modules uniformly.
(1)
${\rm G}(Q_{1})$ := the full automorphism group of $Q_{1}$.
(2)
${\rm T}(Q_{1})$ := the subgroup generated by transvections of $Q_{1}$.
(3)
${\rm ET}(Q_{1})$ := the subgroup generated by elementary-transvections of $Q_{1}$.
(4)
$\widetilde{\rm E}(r,A)$ := ${\rm E}(r+1,A)$ for linear case and ${\rm E}(2r+2,A)$ otherwise.
In [5], the first author with Ravi A. Rao and A. Bak established the “Dilation Lemma” and the “Local-Global Principle” for
the transvection subgroups of the automorphism
group over polynomial rings. In this section, we generalize their results over graded rings. For the statements over polynomial rings
we request the reader to look at Proposition 3.1 and Theorem 3.6 in [5].
Following is the graded version of the “Dilation Lemma”.
Proposition 5.2 (Dilation Lemma).
Let $Q$ be a projective $A_{0}$-module, and $Q_{1}=Q\otimes_{A_{0}}A$.
Let $s$ be a non-nilpotent element in $A_{0}$. Let $\alpha\in G(Q_{1})$ with $\alpha^{+}(0)={\rm I}_{n}$. Suppose
$$\alpha_{s}\in\begin{cases}{\rm E}\big{(}r+1,(A_{0})_{s}\big{)}&\text{ in the %
linear case,}\\
{\rm E}\big{(}2r+2,(A_{0})_{s}\big{)}&\text{ otherwise. }\end{cases}$$
Then there exists $\widetilde{\alpha}\in{\rm ET}(Q_{1})$ and $l\gg 0$ such that $\widetilde{\alpha}$
localizes at $\alpha^{+}(b)$ for some $b=(s^{l})$ and $\widetilde{\alpha}^{+}(0)={\rm I}_{n}$.
Proof.
. Let $\alpha\in{\rm G}(Q_{1})$ and $\alpha(X):=\alpha^{+}(X)\in{\rm G}(Q_{1}[X])$.
Since $\alpha_{s}\in\widetilde{{\rm E}}(r,A_{s})$, $\alpha_{s}(X)\in\widetilde{{\rm E}}(r,A_{s}[X])$. Since $\alpha(0):=\alpha^{+}(0)={\rm I}_{n}$, we can apply “Dilation Lemma” for the ring $A[X]$ (cf.[5], Proposition 3.1) and hence there exists
$\widetilde{\alpha}(X)\in{\rm ET}(Q_{1}[X])$ such that $\widetilde{\alpha}_{s}(X)=\alpha_{s}^{+}(bX)$ for some
$b\in(s^{l})$, $l\gg 0$. Substituting $X=1$, we get $\widetilde{\alpha}\in{\rm ET}(Q_{1})$ and
$\widetilde{\alpha}_{s}:=\widetilde{\alpha}_{s}(1)=\alpha_{s}(b)=\alpha_{s}^{+}%
(b)$. This proves the proposition.
∎
Lemma 5.3.
If $\alpha\in{\rm G}(Q_{1})$ with $\alpha^{+}(0)={\rm I}_{n}$ such that $\alpha_{s}\in{\rm E}(n,A_{s})$,
then $\alpha^{+}(b+d)\alpha^{+}(d)^{-1}\in{\rm ET}(Q_{1})$ for some $b\in(s^{l})$ with $l\gg 0$, and $d\in A_{0}$.
Proof.
Let $\beta=\alpha_{s}^{+}(b+d){\alpha_{s}^{+}(d)}^{-1}$. Since $\alpha_{s}\in{\rm E}(n,A_{s})$
implies $\alpha^{+}(a)\in{\rm E}(n,A_{s})$ for $a\in A_{s}$ and hence $\beta\in{\rm E}(n,A_{s})$.
Hence by Theorem 5.2, their exists a $\widetilde{\beta}\in{\rm ET}(Q_{1})$ such that $\widetilde{\beta}_{s}^{+}(b)=\beta$. Hence the lemma follows.
∎
Theorem 5.4 (Local-Global Principle).
Let $Q$ be a projective $A_{0}$-module, and $Q_{1}=Q\otimes_{A_{0}}A$.
Suppose $\alpha\in G(Q_{1})$ with $\alpha^{+}(0)={\rm I}_{n}$. If
$$\alpha_{\mathfrak{m}}\in\begin{cases}{\rm E}\big{(}r+1,(A_{0})_{\mathfrak{m}}%
\big{)}&\text{ in the linear case, }\\
{\rm E}\big{(}2r+2,(A_{0})_{\mathfrak{m}}\big{)}&\text{ otherwise. }\end{cases}$$
for all $\mathfrak{m}\in$ ${\rm max}\;(A_{0})$, then $\alpha\in{\rm ET\big{(}Q_{1}\big{)}}\subseteq{\rm T\big{(}Q_{1}\big{)}}$.
Proof.
Let $\alpha\in{\rm G}(Q_{1})$ with $\alpha^{+}(0)={\rm I}_{n}$. Since $\alpha_{\mathfrak{m}}\in\widetilde{{\rm E}}(r,A_{\mathfrak{m}})$.
Hence their exists a non-nilpotent $s\in A_{0}-\mathfrak{m}$, such that $\alpha_{s}\in\widetilde{{\rm E}}(r,A_{s})$. Hence by the above “Dilation Lemma”
there exists $\widetilde{\alpha}\in{\rm ET}(Q_{1})$ such that $\widetilde{\alpha}_{s}=\alpha_{s}^{+}(b)$, for some $b=s^{l}$ with $l\gg 0$.
For each maximal ideal $\mathfrak{m}_{i}$ we can find a suitable $b_{i}$. Since $A$ is Noetherian, it follows that $b_{1}+\cdots+b_{r}=1$ for some positive integer $r$.
Now we observe that $\alpha^{+}(b_{i}+\cdots+b_{r}){\alpha^{+}(b_{i+1}+\cdots+b_{r})}^{-1}\in{\rm ET%
}(Q_{1})$ and hence
calculating in the similar manner as we did it in Remark 3.9, we get
$$\displaystyle\alpha=\alpha^{+}(1)=$$
$$\displaystyle\alpha^{+}(b_{1}+\cdots+b_{r}){\alpha^{+}(b_{2}+\cdots+b_{r})}^{-%
1}\cdots\alpha^{+}(b_{i}+\cdots+b_{r})$$
$$\displaystyle{\alpha^{+}(b_{i+1}+\cdots+b_{r})}^{-1}\cdots\alpha^{+}(0)^{-1}%
\in{\rm ET}(Q_{1})\subseteq{\rm T}(Q_{1});$$
as desired.
∎
Acknowledgement
The second author expresses his sincere thanks to IISER Pune for allowing him to work on this project and
providing him their infrastructural facilities.
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Indian Institute of Science Education and Research $($IISER Pune$)$, India,
email: [email protected], [email protected]
Indian Institute of Science Education and Research $($IISER Pune$)$, India,
email: [email protected] |
Ab-initio coupled-cluster effective interactions for the shell model:
Application to neutron-rich oxygen and carbon isotopes
G. R. Jansen
Physics Division, Oak Ridge National
Laboratory, Oak Ridge, TN 37831, USA
Department of Physics and Astronomy, University of Tennessee,
Knoxville, TN 37996, USA
J. Engel
Deptartment of Physics and Astronomy, University
of North Carolina, Chapel Hill, NC, 27516-3255, USA
G. Hagen
Physics Division, Oak Ridge National Laboratory,
Oak Ridge, TN 37831, USA
Department of Physics and Astronomy, University of Tennessee,
Knoxville, TN 37996, USA
P. Navratil
TRIUMF, 4004 Wesbrook Mall, Vancouver,
British Columbia, V6T 2A3 Canada
A. Signoracci
Physics Division, Oak Ridge National
Laboratory, Oak Ridge, TN 37831, USA
Department of Physics and Astronomy, University of Tennessee,
Knoxville, TN 37996, USA
Abstract
We derive and compute effective valence-space shell-model
interactions from ab-initio coupled-cluster theory and apply them to
open-shell and neutron-rich oxygen and carbon isotopes. Our
shell-model interactions are based on nucleon-nucleon and
three-nucleon forces from chiral effective-field theory. We compute
the energies of ground and low-lying states, and find good agreement
with experiment. In particular our calculations are consistent with
the $N=14,16$ shell closures in ${}^{22,24}$O, while for ${}^{20}$C the
corresponding $N=14$ closure is weaker. We find good agreement
between our coupled-cluster effective-interaction results with those
obtained from standard single-reference coupled-cluster calculations
for up to eight valence neutrons.
pacs: 21.30.Fe, 21.60.Cs, 21.60.De, 21.10.-k
Introduction. – The nuclear shell model is the
foundation on which our understanding of nuclei is built.
One of the most important problems in nuclear structure
today is to understand how shell structure changes with
neutron-to-proton ratio throughout the nuclear chart. Shell structure
influences the locations of the neutron and proton drip lines and the
stability of matter. Examples of changes in shell structure are the
appearance of new magic numbers $N=14$ and $N=16$ in the neutron-rich
oxygen isotopes Stanoiu et al. (2004); Kanungo et al. (2009), and the emergence of
an $N=34$ sub-shell closure in ${}^{54}$Ca
Holt et al. (2012); Hagen et al. (2012a); Steppenbeck
et al. (2013a, b).
Phenomenological shell-model Hamiltonians such as the $sd$ Hamiltonian of Brown
and Wildenthal Wildenthal (1984); Brown and Wildenthal (1988) (abbreviated USD) and
the $p$-$sd$ Hamiltonian of Warturburton and Brown Warburton and Brown (1992) (abbreviated
WBP), have successfully described properties of nuclei with proton number $Z$
and neutron number $N$ less than about 20. To understand the origin of shell
structure, however, researchers are now trying to derive the shell model from
realistic nucleon-nucleon (NN) and three-nucleon forces (3NFs), without
further phenomenology Hjorth-Jensen et al. (1995); Otsuka et al. (2010); Holt et al. (2012). Within the
last few years, for example, Otsuka et al. (2010) showed that 3NFs play a
pivotal role in placing the drip line (correctly) in the oxygen isotopes at
${}^{24}$O, and Holt et al. (2012) showed that the inclusion of 3NFs are
necessary to explain the high $2^{+}$ state in ${}^{48}$Ca .
Until recently, all work to compute effective shell-model interactions was
perturbative. Lately, however, nonperturbative calculations have become
possible. Some have been based on the ab-initio no-core shell model
Navrátil et al. (2009); Barrett et al. (2013), via a valence-cluster expansion
Navrátil et al. (1997); Lisetskiy et al. (2008); Shukla et al. (2011), and others on the in-medium
similarity renormalization group Tsukiyama et al. (2012). In this Letter we use a
third approach, the ab-initio coupled-cluster method
Coester (1958); Coester and Kümmel (1960); Čížek (1966, 2007); Kümmel et al. (1978); Hagen et al. (2013), to construct
effective shell-model interactions for use in open-shell and neutron-rich
nuclei. Starting from NN interactions and 3NFs generated by chiral
effective-field-theory, we compute the ground- and excited-state energies of
neutron-rich carbon and oxygen isotopes with up to eight neutrons in the valence
space. Intense theoretical and experimental interest surround the structure of
both these isotope chains, and particularly the neutron-rich carbon isotopes.
Separation energies, spin assignments for low-lying states, the energies of
$2^{+}$ states, and transition rates in these isotopes all depend on the
locations of shell gaps.
Kondo et al. (2009); Strongman et al. (2009); Ozawa et al. (2011); Kobayashi et al. (2012); Petri et al. (2012); Voss et al. (2012); Petri et al. (2011); Forssén et al. (2013).
At present there is no evidence for a shell closure at the $N=14$ nucleus
${}^{20}$C Stanoiu et al. (2008), despite the $N=14$ shell closure at ${}^{22}$O.
Furthermore, Efimov physics may be at play in ${}^{22}$C
Tanaka et al. (2010); Ershov et al. (2012). This Letter takes the first steps towards an
ab-initio shell-model description of the neutron-rich carbon isotopes, and
addresses the role of 3NFs in these isotopes.
Hamiltonian and model space. – Our coupled-cluster calculations start
from the intrinsic $A$-nucleon Hamiltonian,
$$\hat{H}=\sum_{i<j}\left({({\bf p}_{i}-{\bf p}_{j})^{2}\over 2mA}+\hat{V}_{NN}^%
{(i,j)}\right)+\sum_{i<j<k}\hat{V}_{\rm 3N}^{(i,j,k)}.$$
(1)
Here the intrinsic kinetic energy (the first term) depends on the mass number
$A\equiv Z+N$. The potential $\hat{V}_{NN}$ denotes the chiral NN interaction
at next-to-next-to-next-to leading order Entem and Machleidt (2003); Machleidt and Entem (2011) (with
cutoff $\Lambda=500$ MeV), and $\hat{V}_{\rm 3N}$ is the 3NF that enters at
next-to-next-to leading order with a local regulator Navrátil (2007) (with
cutoff $\Lambda_{\rm 3N}=400$ MeV). The low-energy constants of the 3NF are
given by $c_{E}=0.098$ and $c_{D}=-0.2$. These were initially determined from a
fit to the triton half-life and binding energy with a cutoff $\Lambda_{\rm 3N}=500$ MeV Gazit et al. (2009), and then, with $\Lambda_{\rm 3N}=400$ MeV, $c_{E}$
was re-adjusted to reproduce the ${}^{4}$He binding energy while $c_{D}$ was kept
fixed Roth et al. (2012). To achieve faster convergence as model space size is
increased, we use the similarity renormalization group (SRG) to evolve
$\hat{V}_{\rm NN}$ and $\hat{V}_{\rm 3N}$ to the lower momentum scale
$\lambda_{\rm SRG}=2.0\ \mathrm{fm}^{-1}$ Jurgenson et al. (2009). For the
coupled-cluster calculations themselves, we use a Hartree-Fock basis, with each
orbital expanded in another basis containing up to $N_{\rm max}+1=13$ major
shells from a harmonic-oscillator potential with frequency $\hbar\omega=20$MeV. We limit the number of 3NF matrix elements through the additional cut
$N_{1}+N_{2}+N_{3}\leq 14$. The resulting model-space is sufficient to obtain
well converged results for energies of the states reported in this Letter. We
use the normal-ordered two-body approximation for the 3NF
Hagen et al. (2007); Roth et al. (2012), which has been shown to work well in light-
and medium mass nuclei Roth et al. (2012). By using a Hartree-Fock basis we
minimize the oscillator-frequency dependence arising from the neglect of
residual 3NFs.
Formalism. – To derive an effective shell-model Hamiltonian in a valence
space from ab-initio coupled-cluster theory, we use the valence-cluster
expansion first applied in the no-core shell model
Navrátil and Barrett (1996); Lisetskiy et al. (2008). We expand the Hamiltonian in
Eq. (1) in a form suitable for the shell model:
$$H^{A}_{\rm CCEI}=H_{0}^{A,A_{c}}+H_{1}^{A,A_{c}+1}+H_{2}^{A,A_{c}+2}+\ldots,$$
(2)
where CCEI stands for coupled-cluster effective interaction, $A$ is
the mass of the nucleus we wish to treat, and $A_{c}$ is the mass of the
nucleus with a closed core below the valence space. In
Eq. (2), $H_{0}^{A,A_{c}}$ is the Hamiltonian for the core,
$H_{1}^{A,A_{c}+1}$ is the valence one-body Hamiltonian, and
$H_{2}^{A,A_{c}+2}$ is the additional two-body piece. In this work we
limit ourselves to one- and two-body terms in the valence-space
shell-model Hamiltonian. To solve for the ground-state of the core
nucleus $A_{c}$ we use the coupled-cluster method in the
singles-and-doubles approximation (CCSD) with the $\Lambda$-triples
correction treated perturbatively ($\Lambda$-CCSD(T))
Kucharski and Bartlett (1998); Taube and Bartlett (2008). To obtain the ground and excited
states for the $A_{c}+1$ and $A_{c}+2$ nuclei we use the one- and
two-particle-attached equation-of-motion (EOM) coupled-cluster methods
Gour et al. (2006); Jansen et al. (2011); Jansen (2013). For the particle-attached
EOM we truncate at one-particle and two-particle-one-hole excitations,
and for the two-particle-attached EOM we truncate at two-particle and
three-particle-one-hole excitations. In coupled-cluster theory the
basic ingredient is the similarity-transformed Hamiltonian
$\overline{H}=e^{-T}He^{T}$, which is inherently non-Hermitian
Bartlett and Musiał (2007), thus we need to solve for the left- and the right
eigenstates to obtain a complete bi-orthogonal set of
states. From the left and right eigen-states we can write
Eq. (2) in a spectral representation.
The valence space representation of Eq. (2) consists of a core
energy term (which we compute from $H_{0}^{A,A_{c}}$ using
$\Lambda$-CCSD(T)), a one-body term (built from the particle-attached
eigenvalues of $H_{1}^{A,A_{c}+1}$), and a two-body term. The two-body
term is computed using the Okubo-Lee-Suzuki similarity transformation
Okubo (1954); Suzuki and Lee (1980); Suzuki (1982); Kvaal (2008) by projecting the
two-particle attached EOM eigen states onto two-body valence-space
states. The Okubo-Lee-Suzuki
projection of $H_{2}^{A,A_{c}+2}$ onto the model-space is
Navrátil and Barrett (1996); Kvaal (2008),
$$\langle\alpha_{P}|\overline{H_{\rm eff}^{A}}|\alpha_{P^{\prime}}\rangle=\sum_{%
k=1}^{d}\langle\alpha_{P}|R_{k}^{A,A_{c}+2}\rangle e_{k}\overline{\langle%
\alpha_{P^{\prime}}|R_{k}^{A,A_{c}+2}\rangle}.$$
(3)
Here the $|R_{k}^{A,A_{c}+2}\rangle$ are the two-particle attached
EOM eigenstates with eigenvalue $e_{k}$ for $A_{c}+2$ (with mass $A$ of
the target nucleus in the kinetic energy), $|\alpha_{P}\rangle$
are the model-space states, the sum is over the $k$ two-particle
attached eigenstates that have the largest overlap with the model
space. The $\langle\alpha_{P}|R_{k}^{A,A_{c}+2}\rangle$ are the
matrix elements of the unitary operator ${\bf U}$. $\overline{\langle\alpha_{P^{\prime}}|R_{k}^{A,A_{c}+2}\rangle}$denote the matrix elements
of the inverse of ${\bf U}$. To obtain the effective two-body
shell-model interaction, we subtract the one-body part from
Eq. (3) to avoid double counting.
Note, that we could also construct an effective Hamiltonian using the
corresponding left eigenvectors. This introduces an ambiguity in the
construction of $\overline{H_{\rm eff}^{A}}$. We have verified,
however, that in this work the matrix elements of the effective
operator are almost identical for either choice. The effective
Hamiltonian in Eq. (3) is not Hermitian. Current
shell-model codes require Hermitian matrices. To obtain a Hermitian
representation of the effective shell-model Hamiltonian we construct
the metric operator $S^{\dagger}S$ where $S$ is a matrix that
diagonalizes ${H}^{A}_{\rm CCEI}$; the Hermitian shell-model
Hamiltonian is then $\left[S^{\dagger}S\right]{H}^{A}_{\rm CCEI}\left[S^{\dagger}S\right]^{-1/2}$ Scholtz et al. (1992); Navrátil and Barrett (1996).
Results. – Here we report our CCEI results for ground and low-lying
states in oxygen and carbon isotopes. We choose ${}^{16}$O in oxygen and ${}^{14}$C
in carbon as the closed-shell cores. We then project the one- and
two-particle-attached coupled-cluster wave functions onto the one- and
two-particle model space states in the valence space — the
$d_{5/2},s_{1/2},d_{3/2}$ shell — and proceed to use the resulting
shell-model Hamiltonians in heavier isotopes.
We would like to gauge the accuracy of our CCEI approach by comparing
with full $\Lambda$-CCSD(T) calculations, the results of which we
refer to as reference values. Figure 1 shows the
ground-state energies of all oxygen isotopes ${}^{16-24}$O computed with
the CCEI (red squares), experimental ground-state energies (black
circles), and the $\Lambda$-CCSD(T) ground-state energies in
${}^{21,22,23,24}$O. Our $\Lambda$-CCSD(T) calculations use the model
space mentioned earlier, while the calculations that determine our
CCEI use $N_{\rm max}=12$ and $N_{1}+N_{2}+N_{3}=12$. We believe that
our CCEI results are converged to within $\sim 100$ keV. Both our
$\Lambda$-CCSD(T) and CCEI results are in good agreement with
experimental binding energies. Our CCEI and $\Lambda$-CCSD(T)
calculations also agree well with a variety of recent calculations in
the oxygen isotopes that start with the same Hamiltonian
Hergert et al. (2013); Cipollone et al. (2013).
If we look more closely, we see that the reference $\Lambda$-CCSD(T) results in
${}^{21,22}$O are in excellent agreement with our CCEI results. In ${}^{23,24}$O
the CCEI results start to deviate from the $\Lambda$-CCSD(T) reference values.
In ${}^{24}$O the CCEI ground-state is less bound by about $3.5$ MeV than
obtained with $\Lambda$-CCSD(T). The difference indicates that effective
three-body interactions induced by the Okubo-Lee-Suzuki transformation (which
we neglect) start to play a role in the CCEI approach when the number of
valence nucleons gets too large. The problem can be remedied by including these
interactions or by increasing the valence space size.
Next, we compare low-lying CCEI excited-state energies in ${}^{22}$O
with an EOM coupled-cluster calculation that includes singles and
doubles excitations Stanton and Bartlett (1993). EOM-CCSD can accurately
describe low-lying states that are dominated by one-particle-one-hole
excitations Bartlett and Musiał (2007), and we therefore choose those states
for comparison. In ${}^{22}$O we obtain low-lying $2^{+}$ and $3^{+}$
states with $2.5$ MeV and $3.5$ MeV of excitation energy. The CCEI
result for the same states is $2.7$ MeV and $4.0$ MeV, though the CCEI
result for the $3^{+}$ state in ${}^{22}$O is not yet converged; it moves
down by $\sim 150$ keV when we increase the model space size from
$N=10$ to $N=12$ oscillator shells. The $2^{+}$ state changes only by
$\sim 5$ keV indicating that it, by contrast, is well
converged. Standard EOM-CCSD works well for states that are dominated
by one-particle-one-hole excitations. In our CCEI calculations,
correlations between all particles in the valence space are treated
exactly. Therefore, we expect to see some differences in the computed
spectra. For example, in CCEI we are able to compute the second $0^{+}$
state in ${}^{22}$O, which is dominated by two-particle-two-hole
excitations from the ground-state.
We turn now to carbon. The $\Lambda$-CCSD(T) ground-state energies of
${}^{14,15,16}$C are $-104.0$ MeV, $-104.2$ MeV, and $-106.6$ MeV,
respectively. In ${}^{14}$C the result agrees well with the
experimental ground-state energy of $-105.3$ MeV, but for ${}^{15,16}$C
our particle-attached and two-particle-attached EOM results are
$2.3$ MeV and $4.2$ MeV under-bound with respect to experimental data.
The under-binding persists throughout the chain of carbon isotopes in
our CCEI calculations.
Figures 2 and 3 summarize our CCEI results for
the excited states in the neutron-rich oxygen and carbon isotopes. Figure
2 shows the excitation spectra for ${}^{19-24}$O. Our results are
overall in very good agreement with the experimental excited-state energies.
And without any adjustment of parameters we obtain spectra that are
qualitatively similar to that produced by the phenomenological USD
Hamiltonian. Our $2^{+}$ states in the even oxygen isotopes are on
target and consistent with $N=14$ sub-shell closure in ${}^{22}$O and the $N=16$
shell closure in ${}^{24}$O.
Figure 3 shows excitation spectra in ${}^{17-22}$C. Here
the right column contains results produced by the phenomenological
$p$-$sd$ WBP shell-model Hamiltonian Warburton and Brown (1992) with
$2\hbar\omega$ excitations. As in the oxygen isotopes we agree very
well overall with experiment, and our $2^{+}$ states are remarkably
close to the data. Our $2^{+}$ energy in ${}^{20}$C is at $1.72$ MeV while
the corresponding $2^{+}$ state in ${}^{22}$O lies at $2.78$ MeV; thus our
results are consistent with a weaker $N=14$ sub-shell closure in the
carbon isotopes.
In the odd isotopes ${}^{17,19,21}$C we get the $1/2^{+}$ state in the wrong
position. Our calculations, however, rely on an underlying harmonic-oscillator
basis and therefore do not account for the particle continuum. The $1/2^{+}$
state is dominated by $s$-waves and is located close to the particle emission
threshold, where continuum effects are obviously important
Hagen et al. (2012b, a). The $3/2^{+}$ and the $5/2^{+}$ states are dominated
by $d$-waves, which couple somewhat less to the continuum because of the $l=2$
centrifugal barrier Hagen et al. (2012b). Overall, we expect continuum effects to
be quite significant for the ${}^{17,19,21}$C isotopes. Preliminary calculations
within the no-core shell model with continuum Baroni et al. (2013) for ${}^{17}$C,
with the same chiral NN+3N interaction used here, show that the $1/2^{+}$ state
(unbound in our calculation) gains about 2 MeV in energy and becomes bound. At
the same time, the $3/2^{+}$ and $5/2^{+}$ states are lowered in energy by more
than 1 MeV. We anticipate similar or even stronger continuum effects ${}^{19}$C
and ${}^{21}$C; these would most likely make the $1/2^{+}$ states the ground
states, as they are in reality.
Summary. – We have used coupled cluster theory to derive
shell-model Hamiltonians that depend on no parameters other than those
in the initial NN and $3N$ chiral interactions. We have reproduced
ground and excitation energies with good accuracy in carbon and oxygen
isotopes. The results demonstrate both the predictive power of
Hamiltonians from chiral EFT and the accuracy of the coupled cluster
framework. Finally, our shell model calculations can and will be
systematically improved (e.g. by including induced three-body
interactions), extended to include effective operators other than the
Hamiltonian, and applied to heavier nuclei, where accurate
phenomenological Hamiltonians are harder to obtain.
Acknowledgements.
We thank Scott Bogner, Morten Hjorth-Jensen and Thomas Papenbrock
for useful discussions. This work was supported by the Office of
Nuclear Physics, U.S. Department of Energy (Oak Ridge National
Laboratory), under DE-FG02-96ER40963 (University of Tennessee),
DE-FG02-97ER41019 (University of North Carolina), DE-SC0008499
(NUCLEI SciDAC collaboration), NERRSC Grant No. 491045-2011, and
the Field Work Proposal ERKBP57 at Oak Ridge National Laboratory.
Computer time was provided by the Innovative and Novel Computational
Impact on Theory and Experiment (INCITE) program. TRIUMF receives
funding via a contribution through the National Research Council
Canada. This research used resources of the Oak Ridge Leadership
Computing Facility located in the Oak Ridge National Laboratory,
which is supported by the Office of Science of the Department of
Energy under Contract No. DE-AC05-00OR22725, and used computational
resources of the National Center for Computational Sciences, the
National Institute for Computational Sciences, and the Notur project
in Norway.
Note added. – Very recently Bogner et al
Bogner et al. (2014) applied the in-medium
similarity-renormalization-group method to construct non-perturbative
shell-model interactions and applied it to neutron-rich oxygen
isotopes.
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Abstract
It follows from the famous Fagin’s theorem that all problems in NP are expressible in
existential second-order logic ($\exists$SO), and vice versa. Indeed, there are well-known
$\exists$SO characterizations of NP-complete problems such as $3$-colorability, Hamiltonicity and clique.
Furthermore, the $\exists$SO sentences that characterize those problems are simple and elegant.
However, there are also NP problems that do not seem to possess equally simple and elegant $\exists$SO characterizations.
In this work, we are mainly interested in this latter class of problems.
In particular, we characterize in second-order logic the class of hypercube graphs and the classes SATQBF${}_{k}$ of satisfiable quantified Boolean formulae with $k$ alternations of quantifiers. We also provide detailed descriptions of the strategies followed to obtain the corresponding nontrivial second-order sentences. Finally, we sketch a third-order logic sentence that defines the
class SATQBF = $\bigcup_{k\geq 1}$SATQBF${}_{k}$. The sub-formulae
used in the construction of these complex second- and third-order logic sentences, are good candidates to form part of a library of formulae. Same as libraries
of frequently used functions simplify the writing of complex
computer programs, a library of formulae could
potentially simplify the writing of complex second- and third-order queries,
minimizing the probability of error.
Expressing Properties in Second and Third Order Logic: Hypercube Graphs and SATQBF 111Pre-print of article submitted to an special issue of the Logic Journal of the IGPL with selected papers from the 16th Brazilian Logic Conference.
\@afterheading
School of Information Management, Victoria University of Wellington,
PO Box 600, Wellington 6140, New Zealand. E-mail: [email protected]
School of Engineering and Advanced Technology, Massey University,
Private Box 756, Wellington 6140, New Zealand. E-mail: [email protected]
ICTIC, Universidad de la Cuenca del Plata, Corrientes, Argentina and
Department of Informatics, Universidad Nacional de San Luis,
Ejercito de Los Andes 950, D5700HHW, San Luis, Argentina.
E-mail: [email protected]
Keywords: second-order logic, third-order logic, quantified Boolean formulae, queries, finite
model theory, hypercube graphs
1 Introduction
Examples of second-order formulae expressing different properties of
graphs are fairly common in the literature. Classical
examples are $3$-colorability,
Hamiltonicity, and clique (see [8, 10]
among others). These properties can be expressed by simple and
elegant second-order formulae. Likewise, there are graph properties
that can be expressed by simple and elegant third-order
formulae. One of those properties is that of being a hypercube graph
(see [5]).
An $n$-hypercube graph ${\bf Q}_{n}$, also called
an $n$-cube, is
an undirected graph whose vertices are binary $n$-tuples.
Two vertices of ${\bf Q}_{n}$ are adjacent iff they differ in exactly one bit.
The expressive power of third-order
logic is not actually required to characterize hypercube graphs, since
they can be recognized in nondeterministic polynomial
time. Recall that by Fagin’s theorem [4], $\exists$SO captures NP.
Thus there are formulae in existential second-order logic
($\exists$SO) which can express this property. Nevertheless,
to define the class of hypercube graphs in
second-order logic is certainly more challenging than to define it
in third-order logic.
From an applied perspective, this indicates that it makes sense to investigate higher-order quantifiers in the context of database query languages. Despite the fact that most of the queries commonly used in the industry are in P, the use of higher-order quantifiers can potentially simplify the way in which many of those queries are expressed.
Let SATQBF${}_{k}$ denote the class of satisfiable quantified Boolean formulae with $k$ alternating blocks of quantifiers. From Fagin-Stockmeyer characterization of the polynomial-time hierarchy [13] and the fact that SATQBF${}_{k}$ is complete for the level $\Sigma^{p}_{k}$ of that hierarchy [14], it follows that for every $k\geq 1$, SATQBF${}_{k}$ can be defined by a formula in the prenex fragment $\Sigma^{1}_{k}$ of second-order logic with $k$ alternating blocks of quantifiers. SATQBF${}_{k}$ provides a prime example of a property (or query) whose expression in the language of second-order logic is possible but challenging. Indeed, it is not a trivial task to write a second-order logic sentence that evaluates to true precisely on those word models that represent sentences in SATQBF${}_{k}$. As usual in finite model theory [3], the term word model refers here to a finite relational structure formed by a binary relation and a finite number of unary relations. By contrast, if we restrict our attention to quantified Boolean formulae in which the quantified free part is in conjunctive normal form and has exactly three Boolean variables in each conjunct, then the problem is expressible in monadic second-order logic provided that the formulae are encoded using a different kind of finite relational structures which include ternary relations (see [10]).
Thus, on the one hand there are well-known NP-complete problems such as
$3$-colorability, Hamiltonicity and clique, that have corresponding well-known characterizations
in $\exists$SO which are simple and elegant. Those characterizations
have in common that the existential second-order quantifiers
can be identified with the guessing stage of the NP algorithm, and that the
remaining first-order formula corresponds to the polynomial time
deterministic verification stage. On the other hand, there are well-known problems such as
hypercube graph (which can also be characterized in $\exists$SO) and SATQBF${}_{k}$ (which can
be characterized in $\Sigma^{1}_{k}$) that do not appear to have a straightforward characterization in second-order logic,
even if we consider the full second-order language.
This observation prompted us to write second-order characterizations of hypercube graph and SATQBF${}_{k}$.
The resulting second-order sentence for hypercube graph can be found in [11]. The corresponding sentence for
SATQBF${}_{k}$ was included in [12]. Both sentences are complex and several pages long.
In this article we present a detailed description of the strategies followed to write these
sentences. The sub-formulae used for the
implementation of these strategies could be part of a future library of
second-order formulae.
Same as libraries
of frequently used functions simplify the writing of complex
computer programs, a library of formulae could
potentially simplify the writing of complex second-order queries,
minimizing the probability of error.
The minimization of the probability of error constitutes an important objective in the context of this work,
since given a query $q$ and a second-order formula $\varphi$, it is not possible to formally prove whether $\varphi$ expresses $q$.
For this reason, we make use of full second-order logic to present the characterizations of hypercube graph and SATQBF${}_{k}$, even though its $\exists$SO and $\Sigma^{1}_{k}$ fragments, respectively, already have the expressive power required for these tasks.
This has permitted us to write relatively clear and intuitive formulae as well as to follow a top-down strategy, similar to that commonly used in the development of computer programs, to further reduce the chance of error.
If we consider the whole class SATQBF = $\bigcup_{k\geq 1}$SATQBF${}_{k}$ of satisfiable quantified Boolean formulae, then the problem becomes PSPACE-complete. Since PSPACE can be captured by second-order logic extended with a transitive closure operator, and furthermore this logic is widely conjectured to be strictly more expressive than the standard second-order logic, the existence of a second-order logic characterization of this problem is unlikely. Thus, we decided to look for a characterization in third-order logic. Note that it is a well-known fact that third-order logic is powerful enough as to characterize every problem in PSPACE.
We conclude the paper presenting a sketch of a third-order logic sentence that defines the
class SATQBF. That is, we present a strategy to write a third-order sentence
that evaluates to true precisely on those word models that represent sentences in SATQBF.
We strongly believe that in many respects the descriptive approach
to Complexity is more convenient than the classical one. That is,
using formulae of some logic to study upper bounds in the time or
space complexity of a given problem, instead of Turing machines.
There are many different measures which can be taken on the formulae
that express a given problem such as quantifier rank, quantifier
blocks alternation, number of variables, number of binary
connectives, and arity of quantified relation variables. It has been proved
that bounds on those measures impact on the
expressive power of logics over finite models (see
[10], [3], [8]). Furthermore, it is
rather obvious that all those measures are decidable, in contrast to
the use of Turing machines, where the usual measures relevant to
computation power such as time, space, treesize, and number of
alternations, are clearly undecidable. Regarding lower bounds there
are also several well studied and powerful techniques in Descriptive
Complexity which proved to be extremely useful in the last decades,
such as Ehrenfeucht-Fraisse games and their variations (see
[9] in particular) and 0-1 Laws (again see [10],
[3], [8]).
Hence, it is important to learn how to build
formulae which are large, but still intuitive and clearly
understandable in a top down approach, in the same way that this is
important in the construction of algorithms in the classical
approach to Complexity, which are also clear and intuitive no matter
their size. The work reported in this article is to the authors’
knowledge one of the first steps in that direction.
In the next section, we introduce the necessary notation and
formally describe by means of a third-order logic sentence, the
class of hypercube graphs. In Section 3, we define in second-order
logic the basic arithmetic operations that we need for this work. We
describe the strategy used to characterize the class of hypercube
graphs in the language of second-order logic in Section 4. In
Section 5 we formally describe the problems SATQBF${}_{k}$ and SATQBF,
and we consider their complexity. In Section 6, we explain in full
detail how to build for each $k\geq 1$, a second-order sentence that
expresses $SATQBF_{k}$. In Section 7 we explain how to build a third-order
logic sentence which expresses SATQBF, and we give a sketch of
such formula. Finally in Section 8, we present some final considerations.
2 Background
We assume that the reader is acquainted with the basic concepts
and the framework of Finite Model Theory [3, 10]. We use the notation from [10].
We work on the vocabulary $\sigma=\{E\}$ of
graphs. An undirected graph ${\bf G}$ is a finite relational structure of vocabulary
$\sigma$ satisfying $\varphi_{1}\equiv\forall xy(E(x,y)\rightarrow E(y,x))$ and $\varphi_{2}\equiv\forall x(\neg E(x,x))$. If we do not require
${\bf G}$ to satisfy neither $\varphi_{1}$ nor $\varphi_{2}$, then we speak of a directed graph (or
digraph). We denote as $V$ the domain of the structure ${\bf G}$, i.e., the set of vertices
of the graph ${\bf G}$. The edge relation of ${\bf G}$ is denoted as $E^{\bf G}$.
By second-order logic we refer to the logic
that is obtained when first-order
logic is extended with second-order variables which range over subsets and
relations defined over the domain, and quantification over such variables. As usual, we use uppercase
letters $X,Y,Z,\ldots$ to denote second-order variables and lower case
letters $x,y,z,\ldots$ to denote first-order variables. The arity of the
second-order variables that we use in our formulae is always clear from the context.
See [10] or [3] for a formal definition of second-order logic in the
context of finite model theory. We include an example of a second-order
formula that defines a simple graph property instead.
Example 1
An undirected graph ${\bf G}$ is regular if
all its vertices have the same degree. It is well known that the class of
regular graphs is not definable in first-order logic [3, 8]. In
second-order logic, this class can be defined as follows:
$\exists A\big{(}\forall x\big{(}\exists B(\mathrm{A1}\wedge\mathrm{A2})\big{)}%
\big{)}$ where
•
A1 expresses “$B$ is the set of vertices which are adjacent to $x$”.
$\mathrm{A1}\equiv\forall z\big{(}B(z)\leftrightarrow E(x,z)\big{)}$
•
A2 expresses “the sets $A$ and $B$ have the same cardinality”
with a formula stating that there is a bijection $F$ from $A$ to $B$.
$\mathrm{A2}\equiv\exists F\forall xyz\big{(}\mathrm{A2.1}\wedge\mathrm{A2.2}%
\wedge\mathrm{A2.3}\wedge\mathrm{A2.4}\wedge\mathrm{A2.5}\big{)}$ where
–
A2.1 means “$F$ is a subset of $A\times B$”.
$\mathrm{A2.1}\equiv\big{(}F(x,y)\rightarrow A(x)\wedge B(y)\big{)}$
–
A2.2 means “$F$ is a function”.
$\mathrm{A2.2}\equiv\big{(}F(x,y)\wedge F(x,z)\rightarrow y=z\big{)}$
–
A2.3 means “$F$ is total”.
$\mathrm{A2.3}\equiv\big{(}A(x)\rightarrow\exists y(F(x,y))\big{)}$
–
A2.4 means “$F$ is injective”.
$\mathrm{A2.4}\equiv\big{(}F(x,z)\wedge F(y,z)\rightarrow x=y\big{)}$
–
A2.5 means “$F$ is surjective”.
$\mathrm{A2.5}\equiv\big{(}B(y)\rightarrow\exists x(F(x,y))\big{)}$
We say that a sentence $\varphi$ expresses a Boolean
query $q$ (or property) over finite relational structures of vocabulary
$\sigma$, if for every finite relational structure ${\bf G}$ of vocabulary
$\sigma$, $q({\bf G})=\mathrm{true}$ iff
${\bf G}\models\varphi$. For instance the sentence in Example 1 expresses
the Boolean query: Is ${\bf G}$ a regular graph?
We denote by
$Mod(\varphi)$ the class of finite $\sigma$-structures ${\bf G}$ such
that ${\bf G}\models\varphi$. A class of finite $\sigma$-structures
${\cal C}$ is definable in a logic $\cal L$, if ${\cal C}=Mod(\varphi)$ for some ${\cal L}$-sentence $\varphi$ of vocabulary $\sigma$. For instance
the class of regular graphs is definable in second-order logic, as shown by the formula given in Example 1.
Next, we define the class of hypercube graphs using a relatively simple and
elegant formula in third-order logic. This logic extends second-order
logic with third-order variables which range over subsets and relations
defined over the powerset of the domain, and quantification over such variables. We use uppercase
calligraphic letters ${\cal X},{\cal Y},{\cal Z},\ldots$ to denote
third-order variables. A formal definition of higher-order logics in the
context of finite model theory can be found in [7] among others.
Example 2
An $n$-hypercube (or $n$-cube for short) ${\bf Q}_{n}$ can be
defined as an undirected graph whose vertices are all the binary $n$-tuples.
Two vertices of ${\bf Q}_{n}$ are adjacent iff they differ in exactly one bit.
A $1$-cube ${\bf Q}_{1}$, a $2$-cube ${\bf Q}_{2}$ and a $3$-cube ${\bf Q}_{3}$ are displayed in Figure 2.1.
We can build an $(n+1)$-cube ${\bf Q}_{n+1}$ starting with two
isomorphic copies of an $n$-cube ${\bf Q}_{n}$ and adding edges between
corresponding vertices. Using this fact, we can define in
third-order logic the so called class of hypercube graphs, as
follows:
$\exists{\cal C}\exists{\cal O}\big{(}\mathrm{A1}\wedge\mathrm{A2}\wedge\forall
G%
_{1}\forall G_{2}\big{(}({\cal C}(G_{1})\wedge{\cal C}(G_{2})\wedge\mathrm{A3}%
)\rightarrow\mathrm{A4}\big{)}\wedge\mathrm{A5}\wedge\mathrm{A6}\big{)}$ where
•
A1 expresses “${\cal C}$ is a class of undirected graphs”.
•
A2 expresses “${\cal O}$ is a total order on ${\cal C}$”.
•
A3 expresses “$G_{1}$ is the immediate predecessor of $G_{2}$ in the order ${\cal O}$”.
•
A4 expresses “$G_{2}$ can be built from two isomorphic copies of $G_{1}$ by adding edges between the corresponding vertices”.
•
A5 expresses “the first graph in the order ${\cal O}$ is a $Q_{1}$”.
•
A6 expresses “the last graph in the order ${\cal O}$ is the input graph”.
In turn, we can express A4 as follows:
$\exists F_{1}\exists F_{2}\big{(}\mathrm{A4.1}\wedge\mathrm{A4.2}\wedge\mathrm%
{A4.3}\wedge\forall x(x\in dom(G_{1})\rightarrow\mathrm{A4.4})\wedge$
$\neg\exists xy(x,y\in dom(G_{1})\wedge x\neq y\wedge\mathrm{A4.5})\big{)}$ where
•
A4.1 expresses “$F_{1}$ and $F_{2}$ are injective and total functions from $dom(G_{1})$ to $dom(G_{2})$”.
•
A4.2 expresses “the ranges of $F_{1}$ and $F_{2}$ form a partition of $dom(G_{2})$”.
•
A4.3 expresses “$F_{1}$ and $F_{2}$ are isomorphisms from $G_{1}$ to the sub-graphs of $G_{2}$ induced by the ranges of $F_{1}$ and $F_{2}$, respectively”.
•
A4.4 expresses “there is an edge in $G_{2}$ which connects $F_{1}(x)$ and $F_{2}(x)$”.
•
A4.5 expresses “there is an edge in $G_{2}$ which connects $F_{1}(x)$ and $F_{2}(y)$”.
Note that, if there is an edge $(a,b)$ in $G_{2}$ such that $a$ belongs to the
range of $F_{1}$ and $b$ belongs to the range of $F_{2}$, or vice versa, then either $F^{-1}_{1}(a)=F^{-1}_{2}(b)$ or $F^{-1}_{1}(b)=F^{-1}_{2}(a)$.
The missing logic formulae in this example are left as an exercise for the reader.
The property of a graph being an $n$-cube for some $n$, is known to be in NP.
A nondeterministic Turing machine can decide in polynomial time whether an
input structure ${\bf G}$ of the vocabulary $\sigma$ of graphs is an
hypercube, by simply computing the following steps:
i.
Compute the logarithm in base $2$ of the size $n$ of the domain of the input
structure ${\bf G}$ which must be a positive integer;
ii.
Guess a sequence $s_{1},\ldots,s_{n}$ of $n$ binary strings, each of length $\log_{2}n$;
iii.
Check in polynomial time that all binary strings are unique, that the sequence contains all binary strings of length $\log_{2}n$ and that,
for some ordering $a_{s_{1}},\ldots,a_{s_{n}}$ of the nodes in $V$, a string $s_{i}$ differs
from a string $s_{j}$ in exactly $1$ bit iff there is an edge
$(a_{s_{i}},a_{s_{j}})\in E^{\bf G}$.
Thus, as we mentioned in the introduction, the full expressive power of third-order
logic is not actually needed to characterize the class of hypercube graphs. In fact, there is a
formula in $\exists$SO which can express this property. Recall that by Fagin’s
theorem [4], $\exists$SO captures $NP$. However, it is very
unlikely that there is a formula in second-order logic, not to mention in
$\exists$SO, that expresses the property
in a way which is as intuitive and simple as in the example above.
3 Arithmetic in Second-Order Logic
We define in this section the basic arithmetic operations of addition,
multiplication and exponentiation in second-order logic over finite structures. We encode initial
segments of natural number as finite relational structures by using linear
digraphs. Let $\bf G$ be a linear digraph. The first (root) element of the domain in the order determined by
the edge relation $E^{\bf G}$ represents the $0$, the second element in this
order represents the $1$, the third element represents the $2$ and so on.
Since in a linear digraph, $E^{\bf G}$ is the successor relation, for clarity we use $\mathrm{succ}(x,y)$ to denote $E(x,y)$. We also use $x=n$ where $n>0$ to denote the formula of the form
$$\exists y(\mathrm{succ}(y,x)\wedge\exists x(\mathrm{succ}(x,y)\wedge\exists y(%
\mathrm{succ}(y,x)\wedge\cdots\wedge\varphi)\cdots))$$
with $n$ nested quantifiers and $\varphi\equiv\neg\exists x(\mathrm{succ}(x,y))$ if $n$ is odd or $\varphi\equiv\neg\exists y(\mathrm{succ}(y,x))$ if $n$ is even. Likewise, $x=0$ denotes $\neg\exists y(\mathrm{succ}(y,x))$.
We assume a total order
$\leq$ of the nodes in $V$ such that $x\leq y$ iff there is a path from
$x$ to $y$ in $\bf G$ or $x=y$. This total order is easily definable in
second-order logic.
Let us start by defining the operation of addition. The strategy is depicted in
Figure 3.1 in which we show the result $z$ of
adding $x$ and $y$ along a linear graph.
The predicate $\mathrm{sum}(x,y,z)$, which is true iff $z=x+y$, can be defined in second-order logic as follows.
$\big{(}x=0\wedge z=y\big{)}\vee\big{(}y=0\wedge z=x\big{)}\vee$
$\big{(}x\neq 0\wedge y\neq 0\wedge\exists F\big{(}\mathrm{A1}\wedge F(z,y)%
\wedge\exists x^{\prime}y^{\prime}(\hskip 2.845276pt\mathrm{succ}(x,x^{\prime}%
)\wedge F(x^{\prime},y^{\prime})\wedge y^{\prime}=1)\wedge$
$\forall x^{\prime}y^{\prime}x^{\prime\prime}y^{\prime\prime}((\hskip 2.845276%
pt\mathrm{succ}(x^{\prime},y^{\prime})\wedge F(x^{\prime},x^{\prime\prime})%
\wedge F(y^{\prime},y^{\prime\prime}))\rightarrow\hskip 2.845276pt\mathrm{succ%
}(x^{\prime\prime},y^{\prime\prime}))\big{)}\big{)}$
where A1 expresses “$F$ is an injective function with domain $\{n\in V\hskip 2.845276pt|\hskip 2.845276pt\mathrm{succ}(x)\leq n\leq z\}$”.
It is an easy and supplementary task to write the actual formula corresponding to A1. For the sake of clarity, we avoid this kind of supplementary details from now on.
The next
arithmetic operation that we define is multiplication. The strategy is
depicted in Figure 3.2 in which we show the result $z$ of $x$ times
$y$. Each of the nodes in the subset $S=\{2,\ldots,x\}$ can be considered as a root of a different ordered tree in a forest. Each
root node in the forest has $y$ children and the result $z$ is the last child
of node $x$.
The predicate $\mathrm{times}(x,y,z)$, which is true if $z=x\times y$, can be defined in second-order logic as follows.
$(x=1\wedge y\neq 0\wedge z=y)\vee(y=1\wedge x\neq 0\wedge z=x)\vee((x=0\vee y=%
0)\wedge z=0)\vee$
$\big{(}x\neq 0\wedge y\neq 0\wedge x\neq 1\wedge y\neq 1\wedge$
$\exists S\big{(}\forall u\big{(}\big{(}(2\leq u\wedge u\leq x)\rightarrow[%
\exists y^{\prime}(S(u,y^{\prime}))\wedge$
$\forall x^{\prime}y^{\prime}((x^{\prime}\leq y^{\prime}\wedge x^{\prime}\neq y%
^{\prime}\wedge S(u,x^{\prime})\wedge S(u,y^{\prime}))\rightarrow$
$\neg\exists z^{\prime}(x^{\prime}\leq z^{\prime}\wedge z^{\prime}\leq y^{%
\prime}\wedge\neg S(u,z^{\prime})))\wedge$
$\exists F(\mathrm{A1})\wedge\mathrm{A2}\wedge\mathrm{A3}\wedge\mathrm{A4}]\big%
{)}\wedge$
$\mathrm{A5}\wedge\forall uv(S(u,v)\rightarrow(2\leq u\wedge u\leq x))\big{)}%
\big{)}$ where
•
A1 expresses “$F$ is a bijection from $\{n\in V\hskip 2.845276pt|\hskip 2.845276ptS(u,n)\}$ to $\{n\in V\hskip 2.845276pt|\hskip 2.845276pt1\leq n\leq y\}$, which means that the output degree of $u$ is $y$”.
•
A2 expresses “if $u=2$ then the first child of $u$ is $\mathrm{succ}(y)$”.
•
A3 expresses “if $u=x$ then the last child of $u$ is $z$”.
•
A4 expresses “if $u\neq 2$ then $\mathrm{succ}(c_{u-1},c_{u})$ for $c_{u-1}$ the last child of $u-1$ and $c_{u}$ the first child of $u$”.
•
A5 expresses “the input degree of every node in $S$ is $\leq 1$”.
Finally, we need to define the arithmetic operation of exponentiation in second-order logic. In this case, the strategy is depicted in
Figure 3.3. Note that, the first node in the linear digraph is
$x^{1}$, the second node is $x^{2}$, and so on till node $y$-th (the final
node) which is $x^{y}$.
The predicate $\mathrm{exp}(x,y,z)$, which is true if $z=x^{y}$, can be
defined in second-order logic as follows.
$(x\neq 0\wedge y=0\wedge z=1)\vee(y=1\wedge z=x)\vee(x=1\wedge z=1)\vee$
$\big{(}x\geq 2\wedge y\geq 2\wedge\exists V^{\prime}E^{\prime}\big{(}\mathrm{A%
1}\wedge\exists F(\mathrm{A2})\wedge$
$\forall u(\neg V^{\prime}(u)\vee(u=x\vee\exists x^{\prime}(E^{\prime}(x^{%
\prime},u)\wedge\mathrm{times}(x,x^{\prime},u))))\big{)}\big{)}$ where
•
A1 expresses “$(V^{\prime},E^{\prime})$ is a linear digraph whose first (root) node is $x$ and whose last (leaf) node is $z$”.
•
A2 expresses “$F$ is a bijection from $V^{\prime}$ to $\{1,\ldots,y\}$, i.e., $|V^{\prime}|=y$”.
4 Hypercube Graph in Second-Order Logic
We describe in this section two different strategies to define in second-order logic the
class of hypercube graphs. The first strategy is based in the usual definition of Hypercube graph which identifies the nodes of the graph with binary strings. This definition was explained and expressed by means of a third-order logic formula in Example 2. The second strategy is based in the following definition: An $n$-hypercube graph is a graph with $2^{n}$ nodes, which correspond to the subsets of a set with $n$ elements. Two nodes labelled by subsets $S_{i}$ and $S_{j}$ are joined by an edge if and only if $S_{i}$ can be obtained from $S_{j}$ by adding or removing a single element. The first strategy resulted in a more cumbersome formula than the formula produced by the second strategy. However, the descriptive complexity of the formula produced by this latter strategy is higher.
4.1 First Strategy
The idea is to use binary encodings to represent
each node in the graph, and then to compare the binary encodings of two
connected nodes to identify whether they differ exactly in $1$ bit.
Following a top down approach to the problem, we start with a very general
schema of the formula and then we explore the main sub-formulae involved
in the solution. We aim for a good balance between
level of detail and clarity of presentation. Consequently,
we leave out of the presentation some trivial sub-formulae which are not central to the general strategy.
Let $\bf G$ be an undirected graph with $|V|=n$. The following second-order formula is satisfied by $\bf G$ iff $\bf G$ is an $m$-hypercube graph for some $m$.
$$\displaystyle\varphi_{1}\equiv\exists\leq\big{(}\mathrm{A1}\wedge\exists F\,%
\exists m\big{(}\mathrm{A2}\wedge\forall xy(E(x,y)\leftrightarrow\mathrm{A3})%
\wedge\mathrm{A4}\big{)}\big{)}\text{ where}$$
•
A1 expresses “$\leq$ is a total order of the domain $V$ of $\bf G$”.
•
A2 expresses “$F$ is a bijection on $V$”.
•
A3 expresses “The binary encodings of $F(x)$ and $F(y)$ have both length $m$ and differ exactly in one bit”.
•
A4 expresses “There is a node whose binary encoding contains no zeros”.
The total order $\leq$ is used to identify each individual node of $V$. Thus,
we can assume that $V=\{0,\ldots,n-1\}$. This is needed for the
binary encoding of the nodes in $V$, as it will become clear latter on.
It should be clear how to express A1 and A2 in the language of second-order
logic. Thus we concentrate our effort in explaining the strategy to express A3.
Finally, note that A4 means that all binary encodings (of length $m$) correspond to some node in $V$, which implies that the number of nodes of ${\bf G}$ is a power of $2$, and also that $m=\log_{2}n$. A sub-formula that expresses A4 can be easily built by using the same ideas that we use for A3 below. That is, we can existentially quantify for some node $z$, a linear digraph $(V_{z},E_{z})$ and a Boolean assignment $B_{z}$ which assigns $1$ to each node, and such that the binary string represented by $(V_{z},E_{z},B_{z})$ is the binary encoding of $F(z)$.
The following formula expresses A3.
$$\displaystyle\exists V_{x}E_{x}V_{y}E_{y}B_{x}B_{y}\big{(}$$
$$\displaystyle\mathrm{A3.1}$$
$$\displaystyle\wedge\mathrm{A3.2}\wedge\mathrm{A3.3}\wedge\mathrm{A3.4}\wedge%
\mathrm{A3.5}\wedge$$
$$\displaystyle\exists G\big{(}$$
$$\displaystyle\mathrm{A3.6}\wedge$$
$$\displaystyle\forall uv((E_{x}(u,v)\rightarrow\exists u^{\prime}v^{\prime}(G(u%
,u^{\prime})\wedge G(v,v^{\prime})\wedge E_{y}(u^{\prime},v^{\prime})))\wedge$$
$$\displaystyle\quad\quad(E_{y}(u,v)\rightarrow\exists u^{\prime}v^{\prime}(G(u^%
{\prime},u)\wedge G(v^{\prime},v)\wedge E_{x}(u^{\prime},v^{\prime}))))\wedge$$
$$\displaystyle\exists v\forall v^{\prime}((\mathrm{A3.7}\rightarrow v^{\prime}%
\neq v)\wedge$$
$$\displaystyle\quad\quad\quad(\mathrm{A3.8}\rightarrow v^{\prime}=v))\big{)}%
\big{)}\text{ where}$$
•
A3.1 expresses “$(V_{x},E_{x})$ and $(V_{y},E_{y})$ are linear digraphs”.
•
A3.2 expresses “$B_{x}$ is a function from $V_{x}$ to $\{0,1\}$”.
•
A3.3 expresses “$B_{y}$ is a function from $V_{y}$ to $\{0,1\}$”.
•
A3.4 expresses “$(V_{x},E_{x},B_{x})$ is the binary encoding of $F(x)$”.
•
A3.5 expresses “$(V_{y},E_{y},B_{y})$ is the binary encoding of $F(y)$”.
•
A3.6 expresses “$G$ is a bijection from $V_{x}$ to $V_{y}$”.
•
A3.7 expresses “$B_{x}(v^{\prime})=B_{y}(G(v^{\prime}))$”.
•
A3.8 expresses “$B_{x}(v^{\prime})\neq B_{y}(G(v^{\prime}))$”.
To complete the picture, we need to explain how to write A3.4 and A3.5 in the language of second-order logic.
Since both can be expressed in second-order logic in a similar way, we only show the formula for A3.4. Let $x_{i}$ be the $i$-th node in the
linear graph $(V_{x},E_{x})$ defined in the previous formula. We say that $(V_{x},E_{x},B_{x})$ is the binary
encoding of $F(x)$ if
$$F(x)=b_{1}\times 2^{m-1}+b_{2}\times 2^{m-2}+\cdots+b_{m}\times 2^{0}\text{, %
where $b_{i}=B_{x}(x_{i})$.}$$
In second-order logic, we use a
function $W_{x}$ which
assigns to each node $x_{i}$ in $V_{x}$ its corresponding value $b_{i}\times 2^{m-i}$ in the encoding. This function is depicted in
Figure 4.1. The following formula defines the encoding.
$\exists W_{x}\,I_{x}\,n_{x}\,v\,w\,\forall x^{\prime}\big{(}\mathrm{A3.4.1}\wedge$
$\forall s\,s^{\prime}\,q\,q^{\prime}((E_{x}(s,q)\wedge I_{x}(s,s^{\prime})%
\wedge I_{x}(q,q^{\prime}))\rightarrow\mathrm{succ}(s^{\prime},q^{\prime}))\wedge$
$\mathrm{A3.4.2}\wedge\mathrm{A3.4.3}\wedge\,I_{x}(w,m)\wedge$
$\exists V^{\prime}\,E^{\prime}\,v_{1}\,v_{2}\,w[\mathrm{A3.4.4}\wedge\mathrm{A%
3.4.5}\wedge$
$\forall u\big{(}\neg V^{\prime}(u)\vee((u=v_{1}\rightarrow u=0)\wedge(u=v_{2}%
\rightarrow u=1)\wedge(u=w\rightarrow\mathrm{A3.4.6})\wedge$
$((u\neq v_{1}\wedge u\neq v_{2})\rightarrow\exists y^{\prime}(E^{\prime}(y^{%
\prime},u)\wedge\mathrm{A3.4.7})))\big{)}\wedge$
$V_{x}(x^{\prime})\rightarrow[(\mathrm{A3.4.8})\vee(\mathrm{A3.4.9}\,\wedge%
\exists t(W_{x}(x^{\prime},t)\,\wedge\,\mathrm{A3.4.10}))]\wedge$
$\mathrm{A3.4.11}\wedge\mathrm{A3.4.12}]\big{)}$ where
•
A3.4.1 expresses “$I_{x}$ is a bijection from $V_{x}$ to $\{1,\ldots,m\}$”.
•
A3.4.2 expresses “$v$ and $w$ are the first and last nodes of $(V_{x},E_{x})$, respectively”.
•
A3.4.3 expresses “$I_{x}(v,1)$”.
•
A3.4.4 expresses “$(V^{\prime},E^{\prime})$ is a linear graph”.
•
A3.4.5 expresses “$v_{1},v_{2}$ and $w$ are the $1$-st, $2$-nd and last nodes in $(V^{\prime},E^{\prime})$, respectively”.
•
A3.4.6 expresses “$\mathrm{exp}(2,m-1,u)$”.
•
A3.4.7 expresses “$\mathrm{times}(2,y^{\prime},u)$”.
•
A3.4.8 expresses “$B_{x}(x^{\prime})=0$” $\wedge$ “$W_{x}(x^{\prime})=0$”.
•
A3.4.9 expresses “$B_{x}(x^{\prime})=1$” $\wedge$ “$\mathrm{sum}(n_{x},I_{x}(x^{\prime}),m)$”.
•
A3.4.10 expresses “$\mathrm{exp}(2,n_{x},t)$”
•
A3.4.11 expresses “$F(x)=W_{x}(x_{1})+W_{x}(x_{2})+\cdots+W_{x}(x_{m})$ for $x_{i}$ the $i$-th node in $(V_{x},E_{x})$”.
•
A3.4.12 expresses “$W_{x}$ is a function from $V_{x}$ to $V^{\prime}$”.
Finally, we note that A3.4.11 can be expressed as follows.
$\exists U_{x}\big{(}\mathrm{A3.4.11.1}\wedge\forall x^{\prime}\big{(}\neg V_{x%
}(x^{\prime})\vee(\mathrm{A3.4.11.2}\wedge\mathrm{A3.4.11.3}\wedge$
$(\mathrm{A3.4.11.4}\rightarrow\exists x^{\prime\prime}(E_{x}(x^{\prime\prime},%
x^{\prime})\wedge\mathrm{A3.4.11.5})))\big{)}\big{)}$ where
•
A3.4.11.1 expresses “$U_{x}$ is a function from $V_{x}$ to $V$”.
•
A3.4.11.2 expresses “if $x^{\prime}$ is the first node in $(V_{x},E_{x})$ then $U_{x}(x^{\prime})=W_{x}(x^{\prime})$”.
•
A3.4.11.3 expresses “if $x^{\prime}$ is the last node in $(V_{x},E_{x})$ then $U_{x}(x^{\prime})=F(x)$”.
•
A3.4.11.4 expresses “$x^{\prime}$ is not the first node in $(V_{x},E_{x})$”.
•
A3.4.11.5 expresses “$\mathrm{sum}(U_{x}(x^{\prime\prime}),W_{x}(x^{\prime}),U_{x}(x^{\prime}))$”.
4.2 Second Strategy
The second strategy to define the class of hypercube graphs can be described in two steps.
i.
To identify every node $x$ in the input graph $\bf G$ with a different subset $S_{x}$ of a set $V^{\prime}\subset V$ of cardinality $\log_{2}|V|$, making sure that every subset of $V^{\prime}$ is assigned to some node of $\bf G$.
ii.
To check that for every pair of nodes $x$ and $y$ in $\bf G$, there is an edge between $x$ and $y$ iff $S_{x}$ can be obtained from $S_{y}$ by adding or removing a single element.
In second-order logic we can express this strategy as follows.
$$\displaystyle\varphi_{2}\equiv\exists R\big{(}$$
$$\displaystyle\exists V^{\prime}\big{(}\mathrm{A1}\wedge\forall S(\mathrm{A2}%
\rightarrow(\exists x(\mathrm{A3}\wedge\mathrm{A4})))\wedge\exists z(\mathrm{A%
5})\big{)}\wedge$$
$$\displaystyle\forall xy((E(x,y)\wedge E(y,x))\leftrightarrow\mathrm{A6})\big{)}$$
where
•
A1 expresses “$V^{\prime}\subset V\wedge V^{\prime}\neq\emptyset$”.
•
A2 expresses “$S\subseteq V^{\prime}\wedge S\neq\emptyset$”.
•
A3 expresses “$x$ is identified with $S$ via $R$”.
$\mathrm{A3}\equiv\forall v(R(x,v)\leftrightarrow S(v))$
•
A4 expresses “no other node $y\neq x$ can be identified with $S$ via $R$”.
$\mathrm{A4}\equiv\neg\exists y(x\neq y\wedge\forall v(R(y,v)\leftrightarrow S(%
v)))$
•
A5 expresses “all nodes, with the only exception of node $z$, are identified with some nonempty subset of $V^{\prime}$ via R”.
$\mathrm{A5}\equiv\neg\exists v\big{(}R(z,v)\big{)}\wedge\forall z^{\prime}\big%
{(}z\neq z^{\prime}\rightarrow\exists S(\mathrm{A5.1}\wedge\forall v(R(z^{%
\prime},v)\leftrightarrow S(v)))\big{)}$ where
–
A5.1 expresses “$S\neq\emptyset\wedge S\subseteq V^{\prime}$”.
•
A6 expresses “the set $S_{x}$ identified with $x$ can be obtained from the set $S_{y}$ identified with $y$ by adding or removing a single element”.
$\mathrm{A6}\equiv\exists v\big{(}\big{(}(R(x,v)\wedge\neg R(y,v))\vee(R(y,v)%
\wedge\neg R(x,v))\big{)}\wedge$
$\forall v^{\prime}\big{(}v^{\prime}\neq v\rightarrow(R(x,v^{\prime})%
\leftrightarrow R(y,v^{\prime}))\big{)}$
Remark 1
The formula $\varphi_{2}$ that expresses the second strategy has a prefix of second-order quantifiers of the form $\exists R\exists V^{\prime}\forall S$. Thus, it is in the class $\Sigma^{1}_{2}$. The existence of a formula in $\Sigma^{1}_{1}$ that expresses this second strategy is unlikely, since we must express that every subset $S$ is identified with some node in the graph. On the other hand, the formula $\varphi_{1}$ that expresses the first strategy, while considerably more cumbersome than $\varphi_{2}$, only uses existential second-order quantifiers and can be translated in a rather straightforward way into an equivalent $\Sigma^{1}_{1}$ formula. That is, we could transform the current quantification schema of the form
$$\forall xy\big{(}\exists V^{1}_{x}E^{2}_{x}B^{2}_{x}V^{1}_{y}E^{2}_{y}B^{2}_{y%
}\ldots\exists W^{2}_{x}U^{2}_{x}W^{2}_{y}U^{2}_{y}\ldots\big{)},$$
where the superindices added to the relation variables denote their arity, into an schema of the form
$$\big{(}\exists V^{2}_{x}E^{3}_{x}B^{3}_{x}V^{2}_{y}E^{3}_{y}B^{3}_{y}\ldots%
\exists W^{3}_{x}U^{3}_{x}W^{3}_{y}U^{3}_{y}\ldots\big{)},$$
where the prefix “$\forall xy$” is eliminated and the arity of every relation variable is increased in $1$, so that we can incorporate all nodes. Thus, for instance, every set $V^{1}_{x}$ corresponding to some node $x$ in a graph ${\bf G}$ is now encoded in the binary relation $V^{2}_{x}$ in such a way that $V^{1}_{x}=\{y|(x,y)\in V^{2}_{x}\}$. Then, we can simply express that the set $\{x|(x,y)\in V^{2}_{x}\}$ contains every node in the graph $\bf G$. Moreover, we can now omit $V_{y},E_{y},B_{y},W_{y}$ and $U_{y}$, since for every pair of nodes $x$ and $y$, their corresponding sets $V^{1}_{x}$ and $V^{1}_{y}$ will be both encoded into the binary relation $V^{2}_{x}$, and something similar will happen for the relations $E$, $B$, $W$ and $U$.
This is an important consideration since by Fagin-Stockmeyer characterization of the polynomial-time hierarchy [13] $\Sigma^{1}_{1}$ captures NP while $\Sigma^{1}_{2}$ captures $\mathrm{NP}^{\mathrm{NP}}$.
5 Quantified Boolean Formulae
A Boolean variable is any symbol to which we
can associate the truth values 0 and 1. Let $V$ be a countable set of Boolean
variables. The class of Boolean formulae over $V$ is the smallest class which
is defined by:
•
The Boolean constants 0 and 1 are Boolean formulae.
•
Every Boolean variable $x$ in $V$ is a Boolean formula.
•
If $\varphi$ and $\psi$ are Boolean formulae then $(\varphi\wedge\psi)$, $(\varphi\vee\psi)$ and $\neg(\varphi)$ are Boolean formulae.
The semantics of the Boolean formulae is given by the well-known semantics of the propositional logic.
A quantified Boolean formula over $V$, as defined by the influential Garey and
Johnson book on the theory of NP-Completeness [6], is a formula of the form
$$Q_{1}x_{1}Q_{2}x_{2}\ldots Q_{n}x_{n}(\varphi),$$
where $\varphi$ is a Boolean formula over $V$, $n\geq 0$, $x_{1},\ldots,x_{n}\in V$ and, for $1\leq i\leq n$, $Q_{i}$ is either “$\exists$” or “$\forall$”.
A variable that occurs in the Boolean formula but does not occur in the
prefix of quantifiers is called a free variable. We call QBF the set of
quantified Boolean formulae without free variables. As usual, for $k\geq 1$, $\mathrm{QBF}_{k}$ denotes
the fragment of QBF which consists of those formulae which start with an
existential block and have $k$ alternating blocks of quantifiers. Let $X\subset V$ be a
finite set of Boolean variables, we assume w.l.o.g. that a formula in
$\mathrm{QBF}_{k}$ over $X$ is of the form
$$\exists\bar{x}_{1}\forall\bar{x}_{2}\ldots Q\bar{x}_{k}(\varphi),$$
where for $1\leq i\leq k$, $\bar{x}_{i}=(x_{i1},\ldots,x_{il_{i}})$ is a
vector of $l_{i}$ different variables from $X$, $\exists\bar{x}_{i}$ denotes a
block of $l_{i}$ quantifiers of the form $\exists x_{i1},\ldots,\exists x_{il_{i}}$, $\forall\bar{x}_{i}$ denotes a
block of $l_{i}$ quantifiers of the form $\forall x_{i1},\ldots,\forall x_{il_{i}}$, $\varphi$ is a (quantifier free) Boolean formula
over $X$, $Q$ is “$\exists$” if $k$ is odd and “$\forall$” if $k$ is even, and the sets $X_{1},\ldots,X_{k}$ of variables
in $\bar{x}_{1},\ldots,\bar{x}_{k}$, respectively, form a partition of $X$.
We define next the notion of satisfiability of quantified Boolean
formulae. But first we introduce the concept of alternating valuations which
uses rooted binary trees to represent all possible valuations for a given formula, and paths
from the root to the leaves of such trees to represent individual valuations.
This unusual way of representing valuations is motivated by the way in
which we express in second-order logic the satisfiability problem for the
classes $\mathrm{QBF}_{k}$.
Let $\mathbf{T}_{v}$ be a rooted binary tree of
vocabulary $\sigma_{\mathbf{T}_{v}}=\{E,B,0,1\}$. That is, $\mathbf{T}_{v}$
is a maximally connected acyclic digraph in which every vertex has at most two child
vertices and, except for the root, has a unique parent. Here, $0$ and $1$ are
constant symbols which are interpreted as truth values and $B^{\mathbf{T}_{v}}$
is a total function which assigns a truth value $0^{\mathbf{T}_{v}}$ or
$1^{\mathbf{T}_{v}}$ to each vertex in $V$.
We say that $\mathbf{T}_{v}$ is an alternating valuation if the
following holds:
•
Every leaf of $\mathbf{T}_{v}$ is at the same depth $d$.
•
All vertices at a given depth, i.e., in the same level, have the same
out-degree.
•
If two vertices $a,b\in V$ are siblings, then $B^{\mathbf{T}_{v}}(a)\neq B^{\mathbf{T}_{v}}(b)$.
Let $\varphi\equiv\exists\bar{x}_{1}\forall\bar{x}_{2}\ldots Q\bar{x}_{k}(\psi)$ be a formula in $\mathrm{QBF}_{k}$, where $Q$ is “$\exists$” if $k$ is odd and
“$\forall$” if $k$ is even, and let $l_{j}$ for $1\leq j\leq k$ be the
length of the $j$-th alternating block of quantifiers. We say that an
alternating valuation $\mathrm{T}_{v}$ is applicable to $\varphi$, if the
depth of $\mathrm{T}_{v}$ is $l_{1}+\cdots+l_{k}-1$ and for every $1\leq i\leq l_{1}+\cdots+l_{k}$, it holds that:
•
All vertices at depth $i-1$ have no siblings if $1\leq i\leq l_{1}$ or
$l_{1}+l_{2}+1\leq i\leq l_{1}+l_{2}+l_{3}$ or $\cdots$ or $l_{1}+l_{2}+\cdots+l_{k^{\prime}-1}+1\leq i\leq l_{1}+l_{2}+\cdots+l_{k^{%
\prime}}$, where $k^{\prime}=k$ if the $k$-th
block of quantifiers is existential and $k^{\prime}=k-1$ otherwise.
•
All vertices at depth $i-1$ have exactly one sibling if $l_{1}+1\leq i\leq l_{1}+l_{2}$ or $l_{1}+l_{2}+l_{3}+1\leq i\leq l_{1}+l_{2}+l_{3}+l_{4}$ or
$\cdots$ or $l_{1}+l_{2}+\cdots+l_{k^{\prime\prime}-1}+1\leq i\leq l_{1}+l_{2}+\cdots+l_{k^%
{\prime\prime}}$, where $k^{\prime\prime}=k$ if the $k$-th
block of quantifiers is universal and $k^{\prime\prime}=k-1$ otherwise.
Let $\gamma=\exists\bar{x}_{1}\forall\bar{x}_{2}\ldots Q\bar{x}_{k}(\varphi)$ be a formula in $\mathrm{QBF}_{k}$ over $X$, and let $\mathbf{T}_{v}$ be an alternating
valuation applicable to $\gamma$. A leaf valuation $\mathbf{L}_{v}$ is a linear
subgraph of $\mathbf{T}_{v}$ of vocabulary $\sigma_{\mathbf{T}_{v}}$ which corresponds
to a path from the root to a leaf in $\mathbf{T}_{v}$. Let $v$ be a mapping from
the set of variables $X$ to $\{0,1\}$, i.e., a Boolean assignment, such that
for $x_{i}\in X$ the $i$-th variable in the prefix of quantifiers
of $\gamma$, it holds that $v(x_{i})=1$ iff $B^{\mathbf{L}_{v}}(n_{i})=1^{\mathbf{L}_{v}}$ for $n_{i}$ the $i$-th node in the linear
order induced by $E^{\mathbf{L}_{v}}$. We say that $\mathbf{L}_{v}$ satisfies
$\gamma$,
written $\mathbf{L}_{v}\models\gamma$, if the Boolean assignment $v$
satisfies $\varphi$. That is, if $\varphi$ is a Boolean variable $x_{i}$ in $X$, then $\mathbf{L}_{v}\models\varphi$ if $v(x_{i})=1$; if $\varphi=\neg(\psi)$, then $\mathbf{L}_{v}\models\varphi$ if $\mathbf{L}_{v}\not\models\psi$ (i.e., if it is not the
case that $\mathbf{L}_{v}\models\psi$); if $\varphi=(\psi\vee\alpha)$, then
$\mathbf{L}_{v}\models\varphi$ if either $\mathbf{L}_{v}\models\psi$ or
$\mathbf{L}_{v}\models\alpha$; and if $\varphi=(\psi\wedge\alpha)$, then
$\mathbf{L}_{v}\models\varphi$ if both $\mathbf{L}_{v}\models\psi$ and
$\mathbf{L}_{v}\models\alpha$.
Finally, we say that the alternating valuation $\mathbf{T}_{v}$ satisfies
$\gamma$ if every leaf valuation $\mathbf{L}_{v}$ of $\mathbf{T}_{v}$ satisfies
$\gamma$.
A Boolean formulae $\varphi$ in $\mathrm{QBF}_{k}$ is satisfiable if and only if
there is an alternating valuation $\mathbf{T}_{v}$ which satisfies $\varphi$;
otherwise $\varphi$ is unsatisfiable. $\mathrm{SATQBF}_{k}$ is the set of
$\mathrm{QBF}_{k}$ formulae that are satisfiable. $\mathrm{SATQBF}=\bigcup_{k\geq 1}\mathrm{SATQBF}_{k}$.
It is well known that $\mathrm{SATQBF}_{k}$ is complete for the level
$\Sigma^{p}_{k}$ of the polynomial-time hierarchy (see [6, 1] among
others sources). It is also well known that second-order logic captures the
polynomial-time hierarchy. In fact, there is an exact correspondence between
the prenex fragments of second-order logic with up to $k$ alternations of
quantifiers $\Sigma^{1}_{k}$ and the levels $\Sigma^{p}_{k}$ of the polynomial time
hierarchy [13]. Thus, for every $k$, $\mathrm{SATQBF}_{k}$ can be
defined in second-order logic, in fact, it can even be defined in
$\Sigma^{1}_{k}$. Regarding $\mathrm{SATQBF}$, we note that it is
$\mathrm{PSPACE}$-complete [13]. Since existential third-order logic
captures $\mathrm{NTIME}(2^{n^{{\cal O}(1)}})$ (see [7]) and
$\mathrm{PSPACE}\subseteq\mathrm{DTIME}(2^{n^{{\cal O}(1)}})\subseteq\mathrm{%
NTIME}(2^{n^{{\cal O}(1)}})$, we know that $\mathrm{SATQBF}$ can be
defined in existential third-order logic. In the following sections we present
a second-order formula that defines $\mathrm{SATQBF_{k}}$ and a
third-order formula that defines $\mathrm{SATQBF}$, respectively.
6 $\mathrm{SATQBF_{k}}$ in Second-Order Logic
Following a top-down approach, we present a detailed construction of a second-order formula
that defines $\mathrm{SATQBF}_{k}$. But first, we need to fix an encoding of
quantified Boolean formulae as relational structures.
There is a well-known correspondence between words and finite
structures. Let $A$ be a finite alphabet and let $\pi(A)$ be
the vocabulary $\{\leq\}\cup\{R_{a}:a\in A\}$, where $\leq$ is a binary
relation symbol and the $R_{a}$ are unary relation symbols. We can identify any word $v=a_{1}\ldots a_{n}$ in $A^{*}$ with a $\pi(A)$-structure
${\bf B}$, where the cardinality of $B$ equals the length of $v$,
$\leq^{\bf B}$ is a total order on ${\bf B}$, and, for each $R_{a}\in\pi(A)$,
$R_{a}^{\bf B}$ contains the positions in $v$ carrying an $a$,
$R_{a}^{\bf B}=\{b\in B:\textrm{for some $j$ $(1\leq j\leq n)$,}$
$\textrm{$b$ is the $j$-th element in the order $\leq^{\bf B}$ and $a_{j}=a$}\}$
Such structures are usually known as word models for $v$ ([3]).
As any two word models for $v$ are isomorphic, we can speak of the
word model for $v$.
Note that we can represent Boolean variables of the form $x_{n}$ by using a
symbol “$X$” followed
by a sequence of $n$ symbols “$|$”. For instance, we can write $X|||$ for
$x_{3}$. Thus using word models,
every quantified Boolean formula $\varphi$ can be viewed as a finite
relational structure $G_{\varphi}$ of the
following vocabulary.
$$\pi=\{\leq,P_{\neg},P_{\vee},P_{\wedge},P_{\exists},P_{\forall},P_{(},P_{)},P_%
{X},P_{|}\}$$
Example 3
If $\varphi$ is the quantified Boolean formula $\exists x_{1}\forall x_{2}((\neg x_{1})\vee x_{2})$, which using our notation for the variables
corresponds to $\exists X|\forall X||((\neg X|)\vee X||)$,
then the following $\pi$-structure ${\bf G}_{\varphi}$ (note that ${\bf G}_{\varphi}$ is a linear graph)
where $G_{\varphi}=\{1,\ldots,18\}$, $\leq^{{\bf G}_{\varphi}}$ is a total order on ${\bf G}_{\varphi}$,
$P^{{\bf G}_{\varphi}}_{\neg}=\{10\}$,
$P^{{\bf G}_{\varphi}}_{\vee}=\{14\}$, $P^{{\bf G}_{\varphi}}_{\wedge}=\emptyset$,
$P^{{\bf G}_{\varphi}}_{\exists}=\{1\}$, $P^{{\bf G}_{\varphi}}_{\forall}=\{4\}$, $P^{{\bf G}_{\varphi}}_{(}=\{8,9\}$, $P^{{\bf G}_{\varphi}}_{)}=\{13,18\}$, $P^{{\bf G}_{\varphi}}_{X}=\{2,5,11,15\}$, $P^{{\bf G}_{\varphi}}_{|}=\{3,6,7,12,16,17\}$, encodes $\varphi$.
We show next how to build a second-order logic formula $\varphi_{\mathrm{SATQBF}_{k}}$
such that, given a relational structure ${\bf G}_{\varphi}$ of vocabulary $\pi$, it
holds that ${\bf G}_{\varphi}\models\varphi_{\mathrm{SATQBF}_{k}}$ iff the
quantified Boolean formula $\varphi$ represented by ${\bf G}_{\varphi}$, is
satisfiable. That is, we show next how to build a second-order formula
$\varphi_{\mathrm{SATQBF}_{k}}$ of vocabulary $\pi$ that defines
$\mathrm{SATQBF}_{k}$. As mentioned earlier, we follow a top-down approach for the
construction of this formula. At the highest level of abstraction,
we can think of $\varphi_{\mathrm{SATQBF}_{k}}$ as a second-order formula that
expresses the following:
“There is an alternating valuation $${\bf T}_{v}$$ applicable to $$\varphi$$ that satisfies
$$\varphi$$”.
(A)
Recall that an alternating valuation ${\bf T}_{v}$ satisfies $\varphi$ iff every
leaf valuation ${\bf L}_{v}$ of ${\bf T}_{v}$ satisfies the quantifier-free part
$\varphi^{\prime}$ of $\varphi$. Also recall that every leaf
valuation ${\bf T}_{v}$ corresponds to a Boolean assignment $v$. Thus, if
$\varphi=\exists\bar{x}_{1}\forall\bar{x}_{2}\ldots Q\bar{x}_{k}(\varphi^{%
\prime})$, where for $1\leq i\leq k$, $\bar{x}_{i}=(x_{i1},\ldots,x_{il_{i}})$, $Q$ is “$\exists$” if $k$ is odd and “$\forall$” if $k$ is
even, $X_{1},\ldots,X_{k}$ are the set of variables
in $\bar{x}_{1},\ldots,\bar{x}_{k}$, respectively, and $\varphi^{\prime}$ is a
(quantifier free) Boolean formulae
over $X=X_{1}\cup\cdots\cup X_{k}$, then the expression in (A) can be divided in two parts:
AVS1
(Alternating Valuation that Satisfies $\varphi$, Part 1) which expresses
“There is a partial Boolean assignment $v_{1}$ on $X_{1}$,
such that for all partial Boolean assignments $v_{2}$ on $X_{2}$,
…,
there is (or “for all” if k is even) a partial Boolean assignment $v_{k}$ on $X_{k}$”.
AVS2
(Alternating Valuation that Satisfies $\varphi$, Part 2) which expresses
“The Boolean assignment $v=v_{1}\cup v_{2}\cup\cdots\cup v_{k}$ satisfies the (quantifier free) Boolean formula $\varphi^{\prime}$”.
For each partial Boolean assignment $v_{i}$ ($1\leq i\leq k$), we use
a second-order variable $V_{i}$ of arity one and two second-order variables
$E_{i}$ and $B_{i}$ of arity two, to store the encoding of each $v_{i}$ as a linear graph
$G_{i}=(V_{i},E_{i})$ with an associated function $B_{i}:V_{i}\rightarrow\{0,1\}$ (see Figure 6.1).
Correspondingly, we use a second-order variable $V_{t}$ of arity one and two
second-order variables $E_{t}$ and $B_{t}$ of arity two, to store the encoding of
each Boolean assignment $v$ (leaf valuation ${\bf T}_{v}$) as a linear graph
$G_{t}=(V_{t},E_{t})$ with an associated function $B_{t}:V_{t}\rightarrow\{0,1\}$. Figure 6.2 illustrates an alternating
valuation applicable to $\varphi$ and its corresponding encoding.
In the next subsection we describe the process followed to build
a second-order formula to express Statement AVS1. Then we describe in
Subsection 6.2, the corresponding process for Statement AVS2.
6.1 Expressing Statement AVS1
Let $k_{\exists}$ and $k_{\forall}$ be the index of the last existential
quantifier block and
the last universal quantifier block, respectively, in the prefix of $k$ blocks
of quantifiers of $\varphi$. We can express Statement AVS1 as follows:
$\exists V_{1}E_{1}B_{1}\forall V_{2}E_{2}B_{2}\cdots Q_{k}V_{k}E_{k}B_{k}%
\exists V_{t}E_{t}B_{t}U_{1},U_{2},\ldots,U_{k}\Big{(}\mathrm{A1}\wedge\mathrm%
{A2}\wedge\mathrm{A3}\wedge\mathrm{A4}\wedge\mathrm{A5}\wedge$
$\big{(}\big{(}\mathrm{A6}\wedge\mathrm{A7}\wedge\mathrm{A8}\wedge\mathrm{A9}%
\wedge\mathrm{A10}\wedge\mathrm{A11}\big{)}\rightarrow\mathrm{AVS2}\big{)}\Big%
{)}$ where
•
A1 expresses “$G_{t}=(V_{t},E_{t})$ is a linear graph”.
•
A2 expresses “The length of $G_{t}$ equals the number of variables that appear in the prefix of quantifiers of $\varphi$”.
•
A3 expresses “$G_{1}=(V_{1},E_{1}),G_{3}=(V_{2},E_{2}),\ldots,G_{k_{\exists}}=(V_{k_{\exists}%
},L_{k_{\exists}})$ are linear graphs”.
•
A4 expresses “$B_{1}:V_{1}\rightarrow\{0,1\}$, $B_{3}:V_{3}\rightarrow\{0,1\}$, …, $B_{k_{\exists}}:V_{k_{\exists}}\rightarrow\{0,1\}$ are total functions”.
•
A5 expresses “The lengths of the linear graphs $G_{1},G_{3},\ldots,G_{k_{\exists}}$ equal the lengths of their corresponding blocks of quantifiers in $\varphi$”.
•
A6 expresses “$V_{1},V_{2},\ldots,V_{k}$ are pairwise disjoint sets”.
•
A7 expresses “$G_{2}=(V_{2},E_{2}),G_{4}=(V_{4},E_{4}),\ldots,G_{k_{\forall}}=(V_{k_{\forall}%
},L_{k_{\forall}})$ are linear graphs”
•
A8 expresses “$B_{2}:V_{2}\rightarrow\{0,1\}$, $B_{4}:V_{4}\rightarrow\{0,1\}$, …, $B_{k_{\forall}}:V_{k_{\forall}}\rightarrow\{0,1\}$ are total functions”
•
A9 expresses “The lengths of the linear graphs $G_{2},G_{4},\ldots,G_{k_{\forall}}$ equal the lengths of their corresponding blocks of quantifiers in $\varphi$”.
•
A10 expresses “$U_{1}$ is a total injection from $G_{1}$ to the first part of $G_{t}$ and $U_{2}$ is a total injection from $G_{2}$ to the second part of $G_{t}$ …and $U_{k}$ is a total injection from $G_{k}$ to the $k$-th part of $G_{t}$”.
•
A11 expresses “$B_{t}:V_{t}\rightarrow\{0,1\}$ is a total function that coincides with $B_{1}$, $B_{2}$, $\ldots$, $B_{k}$”.
•
AVS2 expresses Statement AVS2 as described in Subsection 6.2.
Next, we discuss how to write the sub-formulae A1–A11 in second-order logic.
A1.
This is expressed by the auxiliary formula $\mathrm{LINEAR}(V_{t},E_{t})$, which is defined
in Subsection 6.3 below.
A2.
This is implied by the following statement which is expressed in further
detail in Subsection 6.2.1 (A).
“There is a partial surjective injection $V_{p}$ from the quantifier prefix of
$\varphi$ to $G_{t}$, which maps every $X$ in the prefix to its
corresponding node in $G_{t}$, and which preserves $\leq^{{\bf G}_{\varphi}}$ and
$E_{t}$”.
A3.
$\mathrm{LINEAR}(V_{1},E_{1})\wedge\mathrm{LINEAR}(V_{3},E_{3})\wedge\cdots%
\wedge\mathrm{LINEAR}(V_{k_{\exists}},E_{k_{\exists}})$, where the sub-formulae $\mathrm{LINEAR}(V_{i},E_{i})$
are as defined in Subsection 6.3.
A4.
$\forall t,p,p^{\prime}\Big{(}\bigwedge_{i=1,3,\ldots,k_{\exists}}\big{(}%
\mathrm{A4.1}\wedge\mathrm{A4.2}\wedge\mathrm{A4.3}\big{)}\Big{)}$
•
A4.1 expresses “$B_{i}$ is a function”.
$\mathrm{A4.1}\equiv((B_{i}(t,p)\wedge B_{i}(t,p^{\prime}))\rightarrow p=p^{%
\prime})$
•
A4.2 expresses “$B_{i}$ is total”.
$\mathrm{A4.2}\equiv(V_{i}(t)\rightarrow\exists p(B_{i}(t,p)))$
•
A4.3 expresses “the range of $B_{i}$ is $\{0,1\}$”.
$\mathrm{A4.3}\equiv(B_{i}(t,p)\rightarrow(\text{$p=1$ $\vee$ $p=0$}))$
where $p=0$ and $p=1$ have the obvious meaning and are defined in
Subsection 6.3.
A5.
If $k_{\exists}\neq k$, then
$\bigwedge_{1,3,\ldots,k_{\exists}}\big{(}\exists L^{\prime}v_{1}v_{2}\ldots v_%
{k_{\exists}}v_{k_{\exists}+1}(\alpha_{k_{\exists}}\wedge\zeta_{i})\big{)}$
where $\alpha_{k_{\exists}}$ is the formula template $\alpha_{i}$ instantiated
with $i=k_{\exists}$.
If $k_{\exists}=k$, then
$\big{(}\bigwedge_{1,3,\ldots,k_{\exists}-2}\big{(}\exists L^{\prime}v_{1}v_{2}%
\ldots v_{k_{\exists}-1}(\alpha_{k_{\exists-2}}\wedge\zeta_{i})\big{)}\big{)}%
\wedge\exists L^{\prime}v_{1}v_{2}\ldots v_{k}v_{e}(\beta_{1}\wedge\beta_{2}%
\wedge\beta_{3})$
where $\alpha_{k_{\exists}-2}$ is the formula template $\alpha_{i}$ instantiated
with $i=k_{\exists}-2$ (Note that $k_{\exists}-2$ is the previous to the last
existential block, and the subformulae $\beta_{1},\beta_{2}$ and $\beta_{3}$ take
care of the last block of quantifiers).
Next, we define the subformulae $\alpha_{i}$, $\beta_{1}$ $\zeta_{i}$, $\beta_{2}$ and $\beta_{3}$ in the listed order. For their definitions we use an auxiliary formula $\mathrm{PATH}_{\leq}(x,y)$ which is in turn defined in Subsection 6.3 below, and which expresses “the pair $(x,y)$ is in the transitive closure of the relation $\leq$”.
The subformula $\alpha_{i}$ is satisfied if, for $1\leq j\leq i$, $v_{j}$ is the position of the first quantifier of
the $j$-th block (when $i$ is not the last block of quantifiers).
$\alpha_{i}\equiv\big{(}P_{\exists}(v_{1})\wedge P_{\forall}(v_{2})\wedge\ldots%
\wedge P_{Q}(v_{i+1})\wedge\neg\exists x(x\neq v_{1}\wedge x\leq v_{1})\wedge$
$\mathrm{PATH}_{\leq}(v_{1},v_{2})\wedge\mathrm{PATH}_{\leq}(v_{2},v_{3})\wedge%
\cdots\wedge\mathrm{PATH}_{\leq}(v_{i},v_{i+1})\wedge$
$\neg\exists x(\mathrm{PATH}_{\leq}(v_{1},x)\wedge\mathrm{PATH}_{\leq}(x,v_{2})%
\wedge x\neq v_{1}\wedge x\neq v_{2}\wedge P_{\forall}(x))\wedge$
$\neg\exists x(\mathrm{PATH}_{\leq}(v_{2},x)\wedge\mathrm{PATH}_{\leq}(x,v_{3})%
\wedge x\neq v_{2}\wedge x\neq v_{3}\wedge P_{\exists}(x))\wedge$
$\ldots\wedge$
$\neg\exists x(\mathrm{PATH}_{\leq}(v_{i},x)\wedge\mathrm{PATH}_{\leq}(x,v_{i+1%
})\wedge x\neq v_{i}\wedge x\neq v_{i+1}\wedge P_{Q}(x))\big{)}$
where $P_{Q}$ is $P_{\forall}$ if $i$ is odd or $P_{\exists}$ if $i$ is even.
The subformula $\beta_{1}$ is satisfied if, for $1\leq j\leq i$, $v_{j}$ is the position of the first quantifier of
the $j$-th block.
$\beta_{1}\equiv\big{(}P_{\exists}(v_{1})\wedge P_{\forall}(v_{2})\wedge\ldots%
\wedge\mathrm{P1}(v_{k})\wedge P_{|}(v_{e})\wedge\neg\exists x(x\leq v_{1})\wedge$
$\mathrm{PATH}_{\leq}(v_{1},v_{2})\wedge\mathrm{PATH}_{\leq}(v_{2},v_{3})\wedge%
\cdots\wedge\mathrm{PATH}_{\leq}(v_{k},v_{e})\wedge$
$\neg\exists x(\mathrm{PATH}_{\leq}(v_{1},x)\wedge\mathrm{PATH}_{\leq}(x,v_{2})%
\wedge x\neq v_{1}\wedge x\neq v_{2}\wedge P_{\forall}(x))\wedge$
$\neg\exists x(\mathrm{PATH}_{\leq}(v_{2},x)\wedge\mathrm{PATH}_{\leq}(x,v_{3})%
\wedge x\neq v_{2}\wedge x\neq v_{3}\wedge P_{\exists}(x))\wedge$
$\ldots\wedge$
$\neg\exists x(\mathrm{PATH}_{\leq}(v_{k},x)\wedge\mathrm{PATH}_{\leq}(x,v_{e})%
\wedge x\neq v_{k}\wedge x\neq v_{e}\wedge\mathrm{P2}(x))\big{)}$
where $\mathrm{P1}$ is $P_{\exists}$ if $k$ is odd or $P_{\forall}$ if $k$ is even, and $\mathrm{P2}$ is $P_{\forall}$ if $k$ is
odd or $P_{\exists}$ if $k$ is even.
When $i$ is not the index of the last block of quantifiers, the subformula $\zeta_{i}$ is satisfied if $L^{\prime}$ is a bijection from the
indices of the symbols $X$ in the $i$-th
alternating block of quantifiers to $V_{i}$, which preserves $E_{i}$ and
$\mathrm{Next}_{X}=\{(a,b)\in\leq^{{\bf G}_{\varphi}}\,|\,\text{$a$ and $b$
are indices of symbols in the $i$-th block}\wedge P_{X}(a)\wedge P_{X}(b)%
\wedge\forall c((a\leq c\wedge c\leq b)\rightarrow\neg P_{X}(c))\}$ (i.e., the
order of appearance of the $X$’s in the $i$-th block of quantifiers in the prefix of
$\varphi$). This is illustrated in Figure 6.3. Recall that we encode in $G_{i}=(V_{i},E_{i},B_{i})$ a partial truth
assignment for the variables in the $i$-th alternating block of
quantifiers.
$\zeta_{i}\equiv\big{(}\mathrm{A5.1}\wedge\mathrm{A5.2}\wedge\mathrm{A5.3}%
\wedge\mathrm{A5.4}\wedge\mathrm{A5.5}\big{)}$ where
•
A5.1 defines the “domain of $L^{\prime}$”.
$\mathrm{A5.1}\equiv\forall x\big{(}(\mathrm{PATH}_{\leq}(v_{i},x)\wedge\mathrm%
{PATH}_{\leq}(x,v_{i+1})\wedge x\neq v_{i+1}\wedge P_{X}(x))\leftrightarrow%
\exists y(L^{\prime}(x,y))\big{)}$
•
A5.2 expresses “$L^{\prime}$ is surjective”.
$\mathrm{A5.2}\equiv\forall y\big{(}V_{i}(y)\rightarrow\exists z(L^{\prime}(z,y%
))\big{)}$
•
A5.3 expresses “$L^{\prime}$ preserves $\mathrm{Next}_{X}$ and $E_{i}$” which implies injectivity.
$\mathrm{A5.3}\equiv\forall sts^{\prime}t^{\prime}\Big{(}\big{(}L^{\prime}(s,t)%
\wedge L^{\prime}(s^{\prime},t^{\prime})\wedge s\neq s^{\prime}\wedge\mathrm{%
PATH}_{\leq}(v_{i},s)\wedge\mathrm{PATH}_{\leq}(s^{\prime},v_{i+1})$
$\wedge\mathrm{PATH}_{\leq}(s,s^{\prime})\wedge\neg\exists z(\mathrm{PATH}_{%
\leq}(s,z)\wedge\mathrm{PATH}_{\leq}(z,s^{\prime})\wedge$
$z\neq s\wedge z\neq s^{\prime}\wedge P_{X}(z))\big{)}\rightarrow E_{i}(t,t^{%
\prime})\Big{)}$
•
A5.4 defines the “range of $L^{\prime}$”.
$\mathrm{A5.4}\equiv\forall xy(L^{\prime}(x,y)\rightarrow V_{i}(y))$
•
A5.5 expresses “$L^{\prime}$ is a function”.
$\mathrm{A5.5}\equiv\forall xyz\big{(}(L^{\prime}(x,y)\wedge L^{\prime}(x,z))%
\rightarrow y=z\big{)}$
The subformula $\beta_{2}$ is satisfied if $L^{\prime}$ is a bijection from the
indices of the symbols $X$ in the $k$-th
alternating block of quantifiers to $V_{k}$, which preserves $E_{k}$ and
$\mathrm{Next}_{X}$ (i.e., the
order of appearance of the $X$’s in the $k$-th block of quantifiers in the prefix of
$\varphi$).
$\beta_{2}\equiv\big{(}\mathrm{A5.1^{\prime}}\wedge\mathrm{A5.2^{\prime}}\wedge%
\mathrm{A5.3^{\prime}}\wedge\mathrm{A5.4^{\prime}}\wedge\mathrm{A5.5^{\prime}}%
\big{)}$ where
•
$\mathrm{A5.1^{\prime}}$ defines the “domain of $L^{\prime}$”.
$\mathrm{A5.1^{\prime}}\equiv\forall x\big{(}(\mathrm{PATH}_{\leq}(v_{k},x)%
\wedge\mathrm{PATH}_{\leq}(x,v_{e})\wedge P_{X}(x))\leftrightarrow\exists y(L^%
{\prime}(x,y))\big{)}$
•
$\mathrm{A5.2^{\prime}}$ expresses “$L^{\prime}$ is surjective”.
$\mathrm{A5.2^{\prime}}\equiv\forall y\big{(}V_{k}(y)\rightarrow\exists z(L^{%
\prime}(z,y))\big{)}$
•
$\mathrm{A5.3^{\prime}}$ expresses “$L^{\prime}$ preserves $\mathrm{Next}_{X}$ and $E_{k}$” which implies injectivity.
$\mathrm{A5.3^{\prime}}\equiv\forall sts^{\prime}t^{\prime}\Big{(}\big{(}L^{%
\prime}(s,t)\wedge L^{\prime}(s^{\prime},t^{\prime})\wedge s\neq s^{\prime}%
\wedge\mathrm{PATH}_{\leq}(v_{k},s)\wedge\mathrm{PATH}_{\leq}(s^{\prime},v_{e})$
$\wedge\mathrm{PATH}_{\leq}(s,s^{\prime})\wedge\neg\exists z(\mathrm{PATH}_{%
\leq}(s,z)\wedge\mathrm{PATH}_{\leq}(z,s^{\prime})\wedge$
$z\neq s\wedge z\neq s^{\prime}\wedge P_{X}(z))\big{)}\rightarrow E_{k}(t,t^{%
\prime})\Big{)}$
•
$\mathrm{A5.4^{\prime}}$ defines the “range of $L^{\prime}$”.
$\mathrm{A5.4^{\prime}}\equiv\forall xy(L^{\prime}(x,y)\rightarrow V_{k}(y))$
•
$\mathrm{A5.5^{\prime}}$ expresses “$L^{\prime}$ is a function”.
$\mathrm{A5.5^{\prime}}\equiv\forall xyz\big{(}(L^{\prime}(x,y)\wedge L^{\prime%
}(x,z))\rightarrow y=z\big{)}$
The last subformula $\beta_{3}$ is satisfied if $v_{e}$ is the last symbol “$|$”
in the prefix of quantifiers of $\varphi$. We use $\mathrm{SUC}_{\leq}(x,y)$ to
denote that $x$ is the immediate successor of $y$ in the total order $\leq^{{\bf G}_{\varphi}}$. The formula that expresses $\mathrm{SUC}_{\leq}(x,y)$ is defined in Subsection 6.3.
$\beta_{3}\equiv\Big{(}\forall v^{\prime}\big{(}\mathrm{SUC}_{\leq}(v_{e},v^{%
\prime})\rightarrow\neg P_{|}(v^{\prime})\big{)}\wedge P_{|}(v_{e})\wedge$
$\forall v^{\prime}\big{(}\mathrm{PATH}_{\leq}(v_{e},v^{\prime})\rightarrow(%
\neg P_{\exists}(v^{\prime})\wedge\neg P_{\forall}(v^{\prime}))\big{)}\wedge$
$\exists xyw\forall v^{\prime}\big{(}P_{X}(x)\wedge P_{Q}(w)\wedge\mathrm{SUC}_%
{\leq}(x,y)\wedge\mathrm{SUC}_{\leq}(w,x)\wedge\mathrm{PATH}_{\leq}(y,v_{e})\wedge$
$((\mathrm{PATH}_{\leq}(v^{\prime},v_{e})\wedge\mathrm{PATH}_{\leq}(y,v^{\prime%
}))\rightarrow P_{|}(v^{\prime}))\big{)}\Big{)}$
where $P_{Q}$ is $P_{\exists}$ if $k$ is odd, or $P_{\forall}$ if $k$ is even.
A6.
Let $V_{i}\cap V_{j}=\emptyset$ denote $\forall x\big{(}(V_{i}(x)\rightarrow\neg V_{j}(x))\wedge(V_{j}(x)\rightarrow%
\neg V_{i}(x))\big{)}$, we can express that $V_{1},V_{2},\ldots,V_{k}$ are pairwise disjoint sets as follows.
$(V_{1}\cap V_{2}=\emptyset)\wedge(V_{1}\cap V_{3}=\emptyset)\wedge(V_{1}\cap V%
_{4}=\emptyset)\wedge\cdots\wedge(V_{1}\cap V_{k}=\emptyset)\wedge$
$(V_{2}\cap V_{3}=\emptyset)\wedge(V_{2}\cap V_{4}=\emptyset)\wedge\cdots\wedge%
(V_{2}\cap V_{k}=\emptyset)\wedge$
$\ldots\wedge(V_{k-1}\cap V_{k}=\emptyset)$
A7.
$\mathrm{LINEAR}(V_{2},E_{2})\wedge\mathrm{LINEAR}(V_{4},E_{4})\wedge\cdots%
\wedge\mathrm{LINEAR}(V_{k_{\forall}},E_{k_{\forall}})$,
where $\mathrm{LINEAR}(V_{i},E_{i})$
is as defined in Subsection 6.3.
A8.
$\forall t,p,p^{\prime}\Big{(}\bigwedge_{i=2,4,\ldots,k_{\forall}}\big{(}%
\mathrm{A8.1}\wedge\mathrm{A8.2}\wedge\mathrm{A8.3}\big{)}\Big{)}$ where
•
A8.1 expresses “$B_{i}$ is a function”.
$\mathrm{A8.1}\equiv((B_{i}(t,p)\wedge B_{i}(t,p^{\prime}))\rightarrow p=p^{%
\prime})$
•
A8.2 expresses “$B_{i}$ is total”.
$\mathrm{A8.2}\equiv(V_{i}(t)\rightarrow\exists p(B_{i}(t,p)))$
•
A8.3 expresses “the range of $B_{i}$ is $\{0,1\}$”.
$\mathrm{A8.3}\equiv(B_{i}(t,p)\rightarrow(p=1\vee p=0))$
where $p=0$ and $p=1$ have the obvious meaning and are defined in
Subsection 6.3.
A9.
If $k_{\forall}\neq k$, then
$\bigwedge_{2,4,\ldots,k_{\forall}}\big{(}\exists L^{\prime}v_{1}v_{2}\ldots v_%
{k_{\forall}}v_{k_{\forall}+1}(\alpha_{k_{\forall}}\wedge\zeta_{i})\big{)}$
where $\alpha_{k_{\forall}}$ is the formula template $\alpha_{i}$ instantiated
with $i=k_{\forall}$.
If $k_{\forall}=k$, then
$\big{(}\bigwedge_{2,4,\ldots,k_{\forall}-2}\big{(}\exists L^{\prime}v_{1}v_{2}%
\ldots v_{k_{\forall}-1}(\alpha_{k_{\forall-2}}\wedge\zeta_{i})\big{)}\big{)}%
\wedge\exists L^{\prime}v_{1}v_{2}\ldots v_{k}v_{e}(\beta_{1}\wedge\beta_{2}%
\wedge\beta_{3})$
where $\alpha_{k_{\forall}-2}$ is the formula template $\alpha_{i}$ instantiated
with $i=k_{\forall}-2$ (Note that $k_{\forall}-2$ is the previous to the last
universal block, and the subformulae $\beta_{1},\beta_{2}$ and $\beta_{3}$ take
care of the last block of quantifiers).
The subformulae $\alpha_{i}$, $\zeta_{i}$, $\beta_{1}$, $\beta_{2}$ and $\beta_{3}$ are
the same as in (A5).
A10.
$\Big{(}\mathrm{A10.1}\wedge\bigwedge_{2\leq i\leq k-1}\big{(}\mathrm{A10.2.i}%
\big{)}\wedge\mathrm{A10.3}\Big{)}$ where
•
A10.1 expresses “$U_{1}$ is a total injection from $V_{1}$ to $V_{t}$ such that: (a) preserves $E_{1}$ and $E_{t}$ and (b) $U_{1}($“first node in
the order $E_{1}$”$)=$“first node in the order $E_{t}$””.
•
A10.2.i expresses “$U_{i}$ is a total injection from $V_{i}$ to $V_{t}$ such that: (a) preserves $E_{i}$ and $E_{t}$ and (b) $U_{i}($“first node in the order $E_{i}$”$)=\mathrm{SUC}_{E_{t}}(U_{i-1}($“last node in the order $E_{i-1}$”$))$”.
•
A10.3 expresses “$U_{k}$ is a total injection from $V_{k}$ to $V_{t}$ such that: (a) preserves $E_{k}$ and $E_{t}$ and (b) $U_{k}($“first node in order $E_{k}$”$)=\mathrm{SUC}_{E_{t}}(U_{k-1}($“last node in order $E_{k-1}$”$))$”.
We describe next the second-order formula for A10.3 which is in turn illustrated in
Figure 6.4. Note that the node labeled $x$ in Figure 6.4 corresponds to the last node in the linear graph $G_{k-1}$ and that $x$ is mapped by the function $U_{k-1}$ to the node labelled $y$ in the linear graph $G_{t}$. Accordingly, $U_{k}$ maps the first node in the linear graph $G_{k}$ (i.e. the node labeled $u$), to the successor of node $y$ in $G_{t}$ (i.e. to the node labelled $t$).
$\mathrm{A10.3}\equiv\forall xytu\big{(}\mathrm{A10.3.1}\wedge\mathrm{A10.3.2}%
\wedge\mathrm{A10.3.3}\wedge\mathrm{A10.3.4}\big{)}$ where
•
A10.3.1 expresses “$U_{k}$ is a total injection from $V_{k}$ to $V_{t}$”.
$\mathrm{A10.3.1}\equiv((U_{k}(x,y)\wedge U_{k}(x,t))\rightarrow y=t)\wedge$
$((U_{k}(x,y)\wedge U_{k}(u,y))\rightarrow x=u)\wedge$
$(V_{k}(x)\rightarrow\exists y(U_{k}(x,y)))\wedge$
$(U_{k}(x,y)\rightarrow(V_{k}(x)\wedge V_{t}(y)))$
•
A10.3.2 expresses “preserves $E_{t}$”.
$\mathrm{A10.3.2}\equiv((U_{k}(x,y)\wedge U_{k}(u,t)\wedge E_{t}(y,t))%
\rightarrow E_{k}(x,u))$
•
A10.3.3 expresses “preserves $E_{k}$”.
$\mathrm{A10.3.3}\equiv((U_{k}(x,y)\wedge U_{k}(u,t)\wedge E_{k}(x,u))%
\rightarrow E_{t}(y,t))$
•
A10.3.4 expresses “$U_{k}($“first node in order $E_{k}$”$)=\mathrm{SUC}_{E_{t}}(U_{k-1}($“last node in order $E_{k-1}$”$))$”.
$\mathrm{A10.3.4}\equiv\big{(}\big{(}U_{k-1}(x,y)\wedge\neg\exists v(E_{k-1}(x,%
v))\wedge E_{t}(y,t)\wedge\neg\exists v(E_{k}(v,u)\wedge V_{k}(u))\big{)}$
$\rightarrow U_{k}(u,t)\big{)}$
A11.
$\forall xytpp^{\prime}\big{(}(B_{1}(t,p)\wedge U_{1}(t,y)\wedge B_{t}(y,p^{%
\prime}))\rightarrow p=p^{\prime}\big{)}\wedge$
$\forall xytpp^{\prime}\big{(}(B_{2}(t,p)\wedge U_{2}(t,y)\wedge B_{t}(y,p^{%
\prime}))\rightarrow p=p^{\prime}\big{)}\wedge$
$\ldots\wedge$
$\forall xytpp^{\prime}\big{(}(B_{k}(t,p)\wedge U_{k}(t,y)\wedge B_{t}(y,p^{%
\prime}))\rightarrow p=p^{\prime}\big{)}$
6.2 Expressing Statement AVS2
Statement AVS2 can be rephrased as follows:
$\exists\,V_{p}\,C\,E_{C}\,\mathit{ST}\,E_{\mathit{ST}}\,M\,C_{\wedge}\,C_{\vee%
}\,C_{\neg}\,C_{(}\,C_{)}\,C_{1}\,C_{0}\,H_{\phi}\,\big{(}\mathrm{AVS2.1}%
\wedge\mathrm{AVS2.2}\big{)}$
where
•
AVS2.1 expresses “There is a Boolean expression $\phi$ which is obtained from the quantifier-free part of $\varphi$ by replacing each occurrence of a variable by the corresponding truth value in $\{0,1\}$ assigned by the leaf valuation represented by $(G_{t},B_{t})$”.
•
AVS2.2 expresses “The Boolean expression $\phi$ evaluates to true”.
We describe next how to express AVS2.1 and AVS2.2 in
second-order logic.
6.2.1 Expressing AVS2.1
The idea is to define mappings to represent the relationships among the input
graph ${\bf G}_{\varphi}$, the graph $G_{t}$ and the quantifier-free part of the
input formulae. This is illustrated in Figure 6.5.
We can express AVS2.1 as follows:
$\mathrm{AVS2.1}\equiv A\wedge B\wedge C$ where
•
A expresses “$V_{p}$ is a partial bijection from the prefix of quantifiers of
$\varphi$ (restricted to the $X$’s that appear in the prefix) to $V_{t}$, which
maps every $X$ to its corresponding node in $G_{t}$, and which preserves $\leq^{{\bf G}_{\varphi}}$ and $E_{t}$”.
$\mathrm{A}\equiv\forall xyz\big{(}\mathrm{A1}\wedge\mathrm{A2}\wedge\mathrm{A3%
}\big{)}\wedge\forall sts^{\prime}t^{\prime}\big{(}\mathrm{A4}\big{)}$ where
–
A1 expresses “$V_{p}$ is a function”.
$\mathrm{A1}\equiv((V_{p}(x,y)\wedge V_{p}(x,z))\rightarrow y=z)$
–
A2 expresses “$V_{p}$ is injective”
$\mathrm{A2}\equiv((V_{p}(x,y)\wedge V_{p}(z,y))\rightarrow x=z)$
–
A3 defines the “domain and range of $V_{p}$”
$\mathrm{A3}\equiv\big{(}(P_{X}(x)\wedge\mathrm{PRED}_{\leq}(x,z)\wedge(P_{%
\exists}(z)\vee P_{\forall}(z)))\leftrightarrow\exists y(V_{t}(y)\wedge V_{p}(%
x,y))\big{)}$
where $\mathrm{PRED}_{\leq}(x,z)$ denotes the subformula that expresses that $z$ is the strict predecessor of
$x$ in the order $\leq^{{\bf G}_{\varphi}}$ (see Subsection 6.3).
–
A4 expresses “$V_{p}$ preserves $\leq^{{\bf G}_{\varphi}}$ and $E_{t}$”.
$(V_{p}(s,s^{\prime})\wedge V_{p}(t,t^{\prime})\wedge E_{t}(s^{\prime},t^{%
\prime}))\rightarrow$
$\big{(}\mathrm{PATH}_{\leq}(s,t)\wedge\forall z^{\prime}\big{(}(z^{\prime}\neq
s%
\wedge z^{\prime}\neq t\wedge\mathrm{PATH}_{\leq}(s,z^{\prime})\wedge\mathrm{%
PATH}_{\leq}(z^{\prime},t))\rightarrow$
$\neg P_{X}(z^{\prime})\big{)}\big{)}$
•
B expresses “$H_{\phi}$ is a partial surjective injection from the quantifier
free part of $\varphi$ to the formula $\phi$, encoded as the
first formula in $(C,E_{C})$ (see Figures 6.7 and 6.8),
which maps every $X$ in the quantifier-free part of $\varphi$ to the
corresponding position in the first formula in $(C,E_{C})$ (i.e. $\phi$),
which preserves $\wedge$, $\vee$, $\neg$, $($, $)$, $\leq^{{\bf G}_{\varphi}}$ and
$E_{C}$, and which ignores $|$”.
$\mathrm{B}\equiv\forall xy_{1}y_{2}z_{1}z_{2}\big{(}\mathrm{B1}\wedge\mathrm{B%
2}\wedge\mathrm{B3}\wedge\mathrm{B4}\big{)}\wedge\forall xx^{\prime}zy_{1}y_{2%
}z_{1}z_{2}\big{(}\mathrm{B5}\big{)}$ where
–
B1 expresses “$H_{\phi}$ is a function”.
$\mathrm{B1}\equiv\big{(}(H_{\phi}(x,y_{1},y_{2})\wedge H_{\phi}(x,z_{1},z_{2}))\rightarrow$
$(y_{1}=z_{1}\wedge y_{2}=z_{2}\wedge\exists x^{\prime}(P_{(}(x^{\prime})\wedge%
\mathrm{PATH}_{\leq}(x^{\prime},x))\wedge C(y_{1},y_{2}))\big{)}$
–
B2 expresses “$H_{\phi}$ is injective”.
$\mathrm{B2}\equiv\big{(}H_{\phi}(x,y_{1},y_{2})\wedge H_{\phi}(z,y_{1},y_{2})%
\rightarrow x=z\big{)}$
–
B3 expresses “the range of $H_{\phi}$ is the first formula in $(C,E_{C})$”.
$\mathrm{B3}\equiv\forall y_{1}^{\prime}y_{2}^{\prime}z_{1}^{\prime}z_{2}^{%
\prime}t_{1}^{\prime}t_{2}^{\prime}v^{\prime}v_{2}\big{(}\big{(}\mathit{ST}(v^%
{\prime})\wedge\neg\exists y(E_{\mathit{ST}}(y,v^{\prime}))\wedge$
$E_{\mathit{ST}}(v^{\prime},v_{2})\wedge M(v^{\prime},y_{1}^{\prime},y_{2}^{%
\prime})\wedge M(v_{2},z_{1}^{\prime},z_{2}^{\prime})\wedge$
$E_{C}(t_{1}^{\prime},t_{2}^{\prime},z_{1}^{\prime},z_{2}^{\prime})\wedge%
\mathrm{PATH}_{E_{C}}(y_{1}^{\prime},y_{2}^{\prime},y_{1},y_{2})\wedge$
$\mathrm{PATH}_{E_{C}}(y_{1},y_{2},t_{1}^{\prime},t_{2}^{\prime})\big{)}%
\rightarrow\exists x^{\prime}(H_{\phi}(x^{\prime},y_{1},y_{2}))\big{)}\wedge$
–
B4 expresses “the domain of $H_{\phi}$ corresponds to the quantifier free part of $\varphi$”.
$\mathrm{B4}\equiv\big{(}\exists x^{\prime}(P_{(}(x^{\prime})\wedge\mathrm{PATH%
}_{\leq}(x^{\prime},x))\rightarrow\exists y_{1}^{\prime}y_{2}^{\prime}(H_{\phi%
}(x,y_{1}^{\prime},y_{2}^{\prime}))\big{)}$
–
B5 expresses “$H_{\phi}$ preserves $\leq^{{\bf G}_{\varphi}}$ (ignoring “$|$”), $E_{C}$, $\wedge,\vee$, $(,)$ and $\neg$, and maps X to
$0/1$”.
$\mathrm{B5}\equiv\big{(}(H_{\phi}(x,y_{1},y_{2})\wedge H_{\phi}(z,z_{1},z_{2})%
\wedge E_{C}(y_{1},y_{2},z_{1},z_{2}))$
$\rightarrow(\mathrm{SUC_{\leq}}(x,z)\vee(\mathrm{PATH}_{\leq}(x,z)\wedge%
\forall x^{\prime}(\mathrm{PATH}_{\leq}(x,x^{\prime})\wedge$
$\mathrm{PATH}_{\leq}(x^{\prime},z)\wedge x^{\prime}\neq x\wedge x^{\prime}\neq
z%
)\rightarrow P_{|}(x^{\prime})))\big{)}\wedge$
$\big{(}H_{\phi}(x,y_{1},y_{2})\rightarrow((P_{(}(x)\wedge C_{(}(y_{1},y_{2}))\vee$
$(P_{)}(x)\wedge C_{)}(y_{1},y_{2}))\vee$
$(P_{\wedge}(x)\wedge C_{\wedge}(y_{1},y_{2}))\vee$
$(P_{\vee}(x)\wedge C_{\vee}(y_{1},y_{2}))\vee$
$(P_{\neg}(x)\wedge C_{\neg}(y_{1},y_{2}))\vee$
$(P_{X}(x)\wedge(C_{0}(y_{1},y_{2})\vee C_{1}(y_{1},y_{2}))))\big{)}$
•
C expresses “for every bijection $V_{0}$ from “$|\cdots|$” in “$QX|\cdots|$” (where Q is “$\exists$” or “$\forall$”)
to “$|\cdots|$” in “$(\ldots X|\cdots|\ldots)$”
that links a variable in the quantifier prefix of $\varphi$ with an occurrence of that variable in the quantifier-free part of it,
the variable in the quantifier free part of $\varphi$ which corresponds to the function $V_{0}$ is replaced in $\phi$ by the
value assigned to that variable by the leaf valuation $(G_{t},B_{t})$ (see Figures 6.5)”.
Note that in the formula below, $z_{0}$ represents the root in
$\mathrm{dom}(V_{0})$, $z_{f}$ represents the leaf in $\mathrm{dom}(V_{0})$, $y_{0}$ represents the root in
$\mathrm{ran}(V_{0})$, and $y_{f}$ represents the leaf in $\mathrm{ran}(V_{0})$ (see Figure 6.6).
Also note that $\phi$ is encoded in $(C,E_{C})$ starting in the node $M$(“first node in $(\mathit{ST},E_{\mathit{ST}})$”) and ending in the node $E^{-1}_{C}(M(\text{``second node in $(\mathit{ST},E_{\mathit{ST}})$''}))$, and that it is equivalent to the quantifier-free part of $\varphi$
with the variables replaced by $0$ or $1$ according to the leaf valuation
$(G_{t},B_{t})$ (this is further clarified in Subsection 6.2.2, also note Figures 6.7 and 6.8).
$\mathrm{C}\equiv\forall\,V_{0}\,\exists\,z_{0}\,y_{0}\,z_{f}\,y_{f}\,z_{0}^{%
\prime}\,y_{0}^{\prime}\,z_{f}^{\prime}\,y_{f}^{\prime}\,\big{(}(\mathrm{C1}%
\wedge\mathrm{C2}\wedge\mathrm{C3}\wedge\mathrm{C4}\wedge\mathrm{C5}\wedge%
\mathrm{C6})\rightarrow\mathrm{C7}\big{)}$
where
–
C1 expresses “$z_{0}$ is the root in $\mathrm{dom}(V_{0})$, $z_{f}$ is the leaf in $\mathrm{dom}(V_{0})$, $y_{0}$ is the root in $\mathrm{ran}(V_{0})$ and $y_{f}$ is the leaf in $\mathrm{ran}(V_{0})$”.
$\mathrm{C1}\equiv V_{0}(z_{0},y_{0})\wedge\neg\exists z^{\prime}y^{\prime}(%
\mathrm{PRED}_{\leq}(z_{0},z^{\prime})\wedge V_{0}(z^{\prime},y^{\prime}))\wedge$
$V_{0}(z_{f},y_{f})\wedge\neg\exists z^{\prime}y^{\prime}(\mathrm{SUC}_{\leq}(z%
_{f},z^{\prime})\wedge V_{0}(z^{\prime},y^{\prime}))\wedge$
$\forall z^{\prime}\big{(}(\mathrm{PATH}_{\leq}(z_{0},z^{\prime})\wedge\mathrm{%
PATH}_{\leq}(z^{\prime},z_{f}))\rightarrow\exists y^{\prime}(V_{0}(z^{\prime},%
y^{\prime}))\big{)}\wedge$
$\forall y^{\prime}\big{(}(\mathrm{PATH}_{\leq}(y_{0},y^{\prime})\wedge\mathrm{%
PATH}_{\leq}(y^{\prime},y_{f}))\rightarrow\exists z^{\prime}(V_{0}(z^{\prime},%
y^{\prime}))\big{)}$
–
C2 expresses “$V_{0}$ is a bijection from “$|\cdots|$” in “$QX|\cdots|$”
to “$|\cdots|$” in
“$(\ldots X|\cdots|\ldots)$” which preserves $\leq^{{\bf G}_{\varphi}}$”.
$\mathrm{C2}\equiv\forall xyvw\big{(}(V_{0}(x,y)\rightarrow(P_{|}(x)\wedge P_{|%
}(y)))\wedge$
$((V_{0}(x,y)\wedge V_{0}(x,v))\rightarrow y=v)\wedge$
$((V_{0}(x,y)\wedge V_{0}(w,y))\rightarrow x=w)\wedge$
$((V_{0}(x,y)\wedge V_{0}(v,w)\wedge\mathrm{SUC}_{\leq}(x,v))\rightarrow\mathrm%
{SUC}_{\leq}(y,w))\big{)}$
–
C3 expresses “$z_{0}^{\prime}$ is the predecessor of the root in $\mathrm{dom}(V_{0})$, i.e., it is the $X$ in the prefix of quantifiers”.
$\mathrm{C3}\equiv\mathrm{PRED}_{\leq}(z_{0},z_{0}^{\prime})\wedge P_{X}(z_{0}^%
{\prime})$
–
C4 expresses “$y_{0}^{\prime}$ is the predecessor of the root in $\mathrm{ran}(V_{0})$, i.e., it is the $X$ in the quantifier-free part”.
$\mathrm{C4}\equiv\mathrm{PRED}_{\leq}(y_{0},y_{0}^{\prime})\wedge P_{X}(y_{0}^%
{\prime})$
–
C5 expresses “$z_{f}^{\prime}$ is the successor of the leaf in $\mathrm{dom}(V_{0})$”.
$\mathrm{C5}\equiv\mathrm{SUC}_{\leq}(z_{f},z_{f}^{\prime})\wedge\neg P_{|}(z_{%
f}^{\prime})$
–
C6 expresses “$y_{f}^{\prime}$ is the successor of the leaf in $\mathrm{ran}(V_{0})$”.
$\mathrm{C6}\equiv\mathrm{SUC}_{\leq}(y_{f},y_{f}^{\prime})\wedge\neg P_{|}(y_{%
f}^{\prime})$
–
C7 expresses “$B_{t}(V_{p}(z_{0}^{\prime}))=H_{\phi}(y_{0}^{\prime})$”.
$\mathrm{C7}\equiv\forall\,x\,x^{\prime}\,\big{(}(V_{p}(z_{0}^{\prime},x)\wedge
B%
_{t}(x,x^{\prime}))\rightarrow$
$\exists z_{1}z_{2}(H_{\phi}(y_{0}^{\prime},z_{1},z_{2})\wedge$
$((\text{``$x^{\prime}=0$''}\wedge C_{0}(z_{1},z_{2}))\vee(\text{``$x^{\prime}=%
1$''}\wedge C_{1}(z_{1},z_{2}))))\big{)}$
6.2.2 Expressing AVS2.2
Now we need to check whether the formula $\phi$ built in the previous step,
evaluates to true. The idea is to evaluate one connective at a time, and one pair of
matching parenthesis at a time, until the final result becomes $1$.
Let us look at the example in Figure 6.7. Note that there are ten
evaluation steps, which correspond to ten “operators” (i.e., either
connectives or pairs of parenthesis). If there are at most $n$ symbols in
$\phi$, that means that the whole evaluation process needs at most $n$
evaluation steps. This is the reason for using pairs of elements to represent
the nodes of the graph $(C,E_{C})$, and quadruples to represent the edges. This
allows the whole evaluation process to take up to $n$ steps (where $n$ is the
length of the input formula). In each step, we have a Boolean sentence on
$\{0,1\}$ with up to $n$ symbols. Each node in the graph $(\mathit{ST},E_{\mathit{ST}})$ represents one such formula, and the function $M$ (for Marker)
is a pointer which tells us in which node in $(C,E_{C})$ that formula begins.
Note that in each evaluation step, either one or
two symbols are removed from the formula at the previous step.
Figure 6.8 further illustrates the graphs (A) and (B) of
Figure 6.7 with a horizontal orientation. Each evaluation step is called a stage. And the first
symbol in each stage is given by the marker function $M$.
Based on this description, we can express AVS2.2 in Section 6.2 as follows:
$\mathrm{A1}\wedge\mathrm{A2}\wedge\mathrm{A3}\wedge\mathrm{A4}\wedge\mathrm{A5}$ where
•
A1 expresses “$(C,E_{C})$ is a linear graph”.
•
A2 expresses “$(\mathit{ST},E_{\mathit{ST}})$ is a linear graph”.
•
A3 expresses “$M:\mathit{ST}\rightarrow C$ is an injective and total function that preserves $\mathrm{PATH}$ in $E_{\mathit{ST}}$ and $E_{C}$”.
•
A4 expresses “$C_{\wedge},C_{\vee},C_{\neg},C_{(},C_{)},C_{0},C_{1}$ are pairwise disjoint, and $C_{\wedge}\cup C_{\vee}\cup C_{\neg}\cup C_{(}\cup C_{)}\cup C_{0}\cup C_{1}=C$”.
•
A5 expresses “For every stage $x$, from stage $x$ to stage $x+1$, we need to follow the rules of evaluation (see Figure 6.7 part A). The formula in $(C,E_{C})$ at stage $x+1$ is the same as the formula at stage $x$, except for one of three possible sorts of changes, which correspond to the cases (a), (b) and (c) of Figure 6.9”.
We describe next how to express A1–A5 above in second-order
logic. See Section 6.3 for the auxiliary formulae used below.
$\mathrm{A1}\equiv\mathrm{LINEAR}(C,E_{C})$
$\mathrm{A2}\equiv\mathrm{LINEAR}_{2}(\mathit{ST},E_{\mathit{ST}})$
$\mathrm{A3}\equiv\forall s\,s^{\prime}\,t_{1}\,t_{2}\,k_{1}\,k_{2}\big{(}%
\mathrm{A3.1}\wedge\mathrm{A3.3}\wedge\mathrm{A3.3}\wedge\mathrm{A3.4}\big{)}$ where
•
A3.1 expresses “$M$ is a function, $M:\mathit{ST}\rightarrow C$”.
$\mathrm{A3.1}\equiv((M(s,t_{1},t_{2})\wedge M(s,k_{1},k_{2}))\rightarrow((t_{1%
}=k_{1}\wedge t_{2}=k_{2})\wedge\mathit{ST}(s)\wedge C(t_{1},t_{2})))$
•
A3.2 expresses “$M$ is injective”.
$\mathrm{A3.2}\equiv((M(s,k_{1},k_{2})\wedge M(t_{1},k_{1},k_{2}))\rightarrow s%
=t_{1})$
•
A3.3 expresses “$M$ is total”.
$\mathrm{A3.3}\equiv(\mathit{ST}(s)\rightarrow\exists t_{1}^{\prime}t_{2}^{%
\prime}(M(s,t_{1}^{\prime},t_{2}^{\prime})))$
•
A3.4 expresses “$M$ preserves PATH in $E_{\mathit{ST}}$ and $E_{C}$”.
$\mathrm{A3.4}\equiv((M(s,t_{1},t_{2})\wedge M(s^{\prime},k_{1},k_{2})\wedge%
\mathrm{PATH}_{\mathit{ST}}(s,s^{\prime}))\rightarrow\mathrm{PATH}_{E_{C}}(t_{%
1},t_{2},k_{1},k_{2}))$
$\mathrm{A4}\equiv\forall s_{1}s_{2}\big{(}(C_{\wedge}(s_{1},s_{2})\rightarrow%
\neg C_{\vee}(s_{1},s_{2}))\wedge(C_{\wedge}(s_{1},s_{2})\rightarrow\neg C_{%
\neg}(s_{1},s_{2}))\wedge$
$(C_{\wedge}(s_{1},s_{2})\rightarrow\neg C_{(}(s_{1},s_{2}))\wedge(C_{\wedge}(s%
_{1},s_{2})\rightarrow\neg C_{)}(s_{1},s_{2}))\wedge$
$(C_{\wedge}(s_{1},s_{2})\rightarrow\neg C_{0}(s_{1},s_{2}))\wedge(C_{\wedge}(s%
_{1},s_{2})\rightarrow\neg C_{1}(s_{1},s_{2}))\wedge\cdots\big{)}\wedge$
$\forall s_{1}s_{2}\big{(}C(s_{1},s_{2})\rightarrow(C_{\wedge}(s_{1},s_{2})\vee
C%
_{\vee}(s_{1},s_{2})\vee C_{\neg}(s_{1},s_{2})\vee C_{(}(s_{1},s_{2})\vee C_{)%
}(s_{1},s_{2})\vee$
$C_{0}(s_{1},s_{2})\vee C_{1}(s_{1},s_{2}))\big{)}\wedge$
$\forall s_{1}s_{2}\big{(}(C_{\wedge}(s_{1},s_{2})\rightarrow C(s_{1},s_{2}))%
\wedge(C_{\vee}(s_{1},s_{2})\rightarrow C(s_{1},s_{2}))\wedge$
$(C_{\neg}(s_{1},s_{2})\rightarrow C(s_{1},s_{2}))\wedge(C_{(}(s_{1},s_{2})%
\rightarrow C(s_{1},s_{2}))\wedge$
$(C_{)}(s_{1},s_{2})\rightarrow C(s_{1},s_{2}))\wedge(C_{0}(s_{1},s_{2})%
\rightarrow C(s_{1},s_{2}))\wedge$
$(C_{1}(s_{1},s_{2})\rightarrow C(s_{1},s_{2}))\big{)}$
$\mathrm{A5}\equiv\forall x\big{(}\mathit{ST}(x)\rightarrow\exists E_{v}\,f_{1}%
\,f_{2}\,l_{1}\,l_{2}\,f_{1}^{\prime}\,f_{2}^{\prime}\,l_{1}^{\prime}\,l_{2}^{%
\prime}\big{(}\alpha_{d}\vee\alpha_{e}\vee(\alpha_{0}\wedge(\alpha_{a}\vee%
\alpha_{b}\vee\alpha_{c}))\big{)}\big{)}$ where
The function $E_{v}$ maps the formula at stage $x$ to the formula at stage
$x+1$. The subformula $\alpha_{d}$ corresponds to the last transition, i.e., the
transition to the last formula in $(C,E_{C})$ (“$0$” or “$1$”). The
subformula $\alpha_{e}$ corresponds to the last formula in $(C,E_{C})$. The
subformulae $\alpha_{a}$, $\alpha_{b}$ and $\alpha_{c}$ correspond to the three
possible cases ($a$), ($b$) and ($c$) as in Figure 6.9, according
to which sort of operation is the one involved in the transition from the
formula in stage $x$ to the next formula in $(C,E_{C})$. Note that the
transition to the last formula $\alpha_{d}$ is necessarily an instance of case
($c$) in Figure 6.9. For case ($c$) in Figure 6.9, $E_{v}$
is not total in its domain, since $(v_{1},v_{2})\big{(}(\big{)}$ and $(w_{1},w_{2})\big{(})\big{)}$ are not mapped. For the last formula, $E_{v}$ is not injective,
since $(f_{1}^{\prime},f_{2}^{\prime})=(l_{1}^{\prime},l_{2}^{\prime})$ (i.e., $f_{1}^{\prime}=l_{1}^{\prime}$ and $f_{2}^{\prime}=l_{2}^{\prime}$) (see Figure 6.11).
$\alpha_{0}\equiv\mathrm{A5.1}\wedge\mathrm{A5.2}\wedge\mathrm{A5.3}\wedge%
\mathrm{A5.4}$ where
•
A5.1 expresses “$x$ is not the leaf in $E_{\mathit{ST}}$, and it is not the predecessor of the leaf”.
$\mathrm{A5.1}\equiv\exists yy_{1}(E_{\mathit{ST}}(x,y)\wedge E_{\mathit{ST}}(y%
,y_{1}))$
•
A5.2 expresses “$E_{v}:C\rightarrow C$ is a partial injection mapping the formula in $(C,E_{C})$ in stage $x$ to the formula in $(C,E_{C})$ in stage $E_{\mathit{ST}}(x)$”
$\mathrm{A5.2}\equiv\forall s_{1}s_{2}t_{1}t_{2}k_{1}k_{2}\big{(}((E_{v}(s_{1},%
s_{2},t_{1},t_{2})\wedge E_{v}(s_{1},s_{2},k_{1},k_{2}))\rightarrow$
$((t_{1}=k_{1}\wedge t_{2}=k_{2})\wedge C(s_{1},s_{2})\wedge C(t_{1},t_{2})))\wedge$
$((E_{v}(s_{1},s_{2},k_{1},k_{2})\wedge E_{v}(t_{1},t_{2},k_{1},k_{2}))\rightarrow$
$(s_{1}=t_{1}\wedge s_{2}=t_{2}))\big{)}$
•
A5.3 expresses “$((f_{1},f_{2}),(l_{1},l_{2}))$ and $((f_{1}^{\prime},f_{2}^{\prime}),(l_{1}^{\prime},l_{2}^{\prime}))$ are the delimiters of the two formulae as in Figure 6.10”.
$\mathrm{A5.3}\equiv M(x,f_{1},f_{2})\wedge\mathrm{A5.3.1}\wedge\mathrm{A5.3.2}%
\wedge\mathrm{5.3.3}$ where
–
A5.3.1 expresses “$M(E_{\mathit{ST}}(x),E_{C}(l_{1},l_{2}))$”.
–
A5.3.2 expresses “$E_{C}(l_{1},l_{2})=(f^{\prime}_{1},f^{\prime}_{2})$”.
–
A5.3.3 expresses “$E^{-1}_{C}(M(E_{\mathit{ST}}(E_{\mathit{ST}}(x))),l_{1}^{\prime},l_{2}^{\prime})$”.
•
A5.4 expresses “$E_{v}$ maps nodes from the subgraph induced by $((f_{1},f_{2}),(l_{1},l_{2}))$ to the subgraph induced by $((f_{1}^{\prime},f_{2}^{\prime}),(l_{1}^{\prime},l_{2}^{\prime}))$”.
$\mathrm{A5.4}\equiv\forall y_{1}y_{2}z_{1}z_{2}\big{(}E_{v}(y_{1},y_{2},z_{1},%
z_{2})\rightarrow$
$\big{(}\mathrm{PATH}_{E_{C}}(f_{1},f_{2},y_{1},y_{2})\wedge\mathrm{PATH}_{E_{C%
}}(y_{1},y_{2},l_{1},l_{2})\wedge$
$\mathrm{PATH}_{E_{C}}(f_{1}^{\prime},f_{2}^{\prime},z_{1},z_{2})\wedge\mathrm{%
PATH}_{E_{C}}(z_{1},z_{2},l_{1}^{\prime},l_{2}^{\prime})\big{)}\big{)}\wedge$
$E_{v}(f_{1},f_{2},f_{1}^{\prime},f_{2}^{\prime})\wedge E_{v}(l_{1},l_{2},l_{1}%
^{\prime},l_{2}^{\prime})$
$\alpha_{a}\equiv\exists v_{1}v_{2}w_{1}w_{2}v_{1}^{\prime}v_{2}^{\prime}w_{1}^%
{\prime}w_{2}^{\prime}p_{11}p_{12}p_{21}p_{22}p_{31}p_{32}p_{11}^{\prime}p_{12%
}^{\prime}\big{(}$
$\mathrm{A5.5}\wedge\mathrm{A5.6}\wedge\mathrm{A5.7}\wedge\mathrm{A5.8}\wedge%
\mathrm{A5.9}\big{)}$ where
•
A5.5 expresses “$((v_{1},v_{2}),(w_{1},w_{2}))$ and $((v_{1}^{\prime},v_{2}^{\prime}),(w_{1}^{\prime},w_{2}^{\prime}))$
define the window of change, that is the segment of the formula that is affected (changed) in the transition from stage $x$ to stage $x+1$ of the evaluation (see Cases (a) and (b) in Figure 6.10)”.
$\mathrm{A5.5}\equiv\mathrm{PATH}_{E_{C}}(f_{1},f_{2},v_{1},v_{2})\wedge\mathrm%
{PATH}_{E_{C}}(w_{1},w_{2},l_{1},l_{2})\wedge E_{C}(p_{11},p_{12},p_{21},p_{22%
})\wedge$
$E_{C}(p_{21},p_{22},p_{31},p_{32})\wedge E_{C}(v_{1},v_{2},p_{11},p_{12})%
\wedge E_{C}(p_{31},p_{32},w_{1},w_{2})$
$C_{(}(v_{1},v_{2})\wedge C_{)}(w_{1},w_{2})\wedge\mathrm{PATH}_{E_{C}}(f_{1}^{%
\prime},f_{2}^{\prime},v_{1}^{\prime},v_{2}^{\prime})\wedge\mathrm{PATH}_{E_{C%
}}(w_{1}^{\prime},w_{2}^{\prime},l_{1}^{\prime},l_{2}^{\prime})\wedge$
$E_{C}(v_{1}^{\prime},v_{2}^{\prime},p_{11}^{\prime},p_{12}^{\prime})\wedge E_{%
C}(p_{11}^{\prime},p_{12}^{\prime},w_{1}^{\prime},w_{2}^{\prime})\wedge E_{v}(%
p_{11},p_{12},p_{11}^{\prime},p_{12}^{\prime})\wedge$
$E_{v}(v_{1},v_{2},v_{1}^{\prime},v_{2}^{\prime})\wedge E_{v}(w_{1},w_{2},w_{1}%
^{\prime},w_{2}^{\prime})\wedge C_{(}(v_{1}^{\prime},v_{2}^{\prime})\wedge C_{%
)}(w_{1}^{\prime},w_{2}^{\prime})$
•
A5.6 expresses “$E_{v}$ preserves $E_{C}$ outside of the window of change, and preserves
left and right side of the window of change (see Figure 6.10)”.
$\mathrm{A5.6}\equiv\forall z_{11}z_{12}z_{21}z_{22}z_{11}^{\prime}z_{12}^{%
\prime}z_{21}^{\prime}z_{22}^{\prime}\big{(}$
$\big{(}(\mathrm{PATH}_{E_{C}}(f_{1},f_{2},z_{11},z_{12})\wedge\mathrm{PATH}_{E%
_{C}}(z_{21},z_{22},v_{1},v_{2})\wedge$
$E_{C}(z_{11},z_{12},z_{21},z_{22})\wedge E_{v}(z_{11},z_{12},z_{11}^{\prime},z%
_{12}^{\prime})\wedge E_{v}(z_{21},z_{22},z_{21}^{\prime},z_{22}^{\prime}))\rightarrow$
$(\mathrm{PATH}_{E_{C}}(f_{1}^{\prime},f_{2}^{\prime},z_{11}^{\prime},z_{12}^{%
\prime})\wedge\mathrm{PATH}_{E_{C}}(z_{21}^{\prime},z_{22}^{\prime},v_{1}^{%
\prime},v_{2}^{\prime})\wedge$
$E_{C}(z_{11}^{\prime},z_{12}^{\prime},z_{21}^{\prime},z_{22}^{\prime}))\big{)}\wedge$
$\big{(}(\mathrm{PATH}_{E_{C}}(w_{1},w_{2},z_{11},z_{12})\wedge\mathrm{PATH}_{E%
_{C}}(z_{21},z_{22},l_{1},l_{2})\wedge$
$E_{C}(z_{11},z_{12},z_{21},z_{22})\wedge E_{v}(z_{11},z_{12},z_{11}^{\prime},z%
_{12}^{\prime})\wedge E_{v}(z_{21},z_{22},z_{21}^{\prime},z_{22}^{\prime}))\rightarrow$
$(\mathrm{PATH}_{E_{C}}(w_{1}^{\prime},w_{2}^{\prime},z_{11}^{\prime},z_{12}^{%
\prime})\wedge\mathrm{PATH}_{E_{C}}(z_{21}^{\prime},z_{22}^{\prime},l_{1}^{%
\prime},l_{2}^{\prime})\wedge$
$E_{C}(z_{11}^{\prime},z_{12}^{\prime},z_{21}^{\prime},z_{22}^{\prime}))\big{)}%
\big{)}$
•
A5.7 expresses “$E_{v}$ preserves symbols in left side of the window of change”.
$\mathrm{A5.7}\equiv\forall z_{11}z_{12}z_{11}^{\prime}z_{12}^{\prime}\big{(}$
$\big{(}\mathrm{PATH}_{E_{C}}(f_{1},f_{2},z_{11},z_{12})\wedge\mathrm{PATH}_{E_%
{C}}(z_{11},z_{12},v_{1},v_{2})\wedge E_{v}(z_{11},z_{12},z_{11}^{\prime},z_{1%
2}^{\prime})\big{)}$
$\rightarrow\big{(}\mathrm{PATH}_{E_{C}}(f_{1}^{\prime},f_{2}^{\prime},z_{11}^{%
\prime},z_{12}^{\prime})\wedge\mathrm{PATH}_{E_{C}}(z_{11}^{\prime},z_{12}^{%
\prime},v_{1}^{\prime},v_{2}^{\prime})\wedge$
$\big{(}(C_{(}(z_{11},z_{12})\wedge C_{(}(z_{11}^{\prime},z_{12}^{\prime}))\vee%
(C_{)}(z_{11},z_{12})\wedge C_{)}(z_{11}^{\prime},z_{12}^{\prime}))\vee$
$(C_{\wedge}(z_{11},z_{12})\wedge C_{\wedge}(z_{11}^{\prime},z_{12}^{\prime}))%
\vee(C_{\vee}(z_{11},z_{12})\wedge C_{\vee}(z_{11}^{\prime},z_{12}^{\prime}))\vee$
$(C_{0}(z_{11},z_{12})\wedge C_{0}(z_{11}^{\prime},z_{12}^{\prime}))\vee(C_{1}(%
z_{11},z_{12})\wedge C_{1}(z_{11}^{\prime},z_{12}^{\prime}))\vee$
$(C_{\neg}(z_{11},z_{12})\wedge C_{\neg}(z_{11}^{\prime},z_{12}^{\prime}))\big{%
)}\big{)}\big{)}$
•
A5.8 expresses “$E_{v}$ preserves symbols in right side of the window of change”.
$\mathrm{A5.8}\equiv\forall z_{11}z_{12}z_{11}^{\prime}z_{12}^{\prime}\big{(}$
$\big{(}\mathrm{PATH}_{E_{C}}(w_{1},w_{2},z_{11},z_{12})\wedge\mathrm{PATH}_{E_%
{C}}(z_{11},z_{12},l_{1},l_{2})\wedge E_{v}(z_{11},z_{12},z_{11}^{\prime},z_{1%
2}^{\prime})\big{)}$
$\rightarrow\big{(}\mathrm{PATH}_{E_{C}}(w_{1}^{\prime},w_{2}^{\prime},z_{11}^{%
\prime},z_{12}^{\prime})\wedge\mathrm{PATH}_{E_{C}}(z_{11}^{\prime},z_{12}^{%
\prime},l_{1}^{\prime},l_{2}^{\prime})\wedge$
$\big{(}(C_{(}(z_{11},z_{12})\wedge C_{(}(z_{11}^{\prime},z_{12}^{\prime}))\vee%
(C_{)}(z_{11},z_{12})\wedge C_{)}(z_{11}^{\prime},z_{12}^{\prime}))\vee$
$(C_{\wedge}(z_{11},z_{12})\wedge C_{\wedge}(z_{11}^{\prime},z_{12}^{\prime}))%
\vee(C_{\vee}(z_{11},z_{12})\wedge C_{\vee}(z_{11}^{\prime},z_{12}^{\prime}))\vee$
$(C_{0}(z_{11},z_{12})\wedge C_{0}(z_{11}^{\prime},z_{12}^{\prime}))\vee(C_{1}(%
z_{11},z_{12})\wedge C_{1}(z_{11}^{\prime},z_{12}^{\prime}))\vee$
$(C_{\neg}(z_{11},z_{12})\wedge C_{\neg}(z_{11}^{\prime},z_{12}^{\prime}))\big{%
)}\big{)}\big{)}$
•
A5.9 expresses “In $(p_{11}^{\prime},p_{12}^{\prime})$ we get the result of applying the operator $\theta$ in $(p_{21},p_{22})$ to the Boolean values $b_{1}$, in $(p_{11},p_{12})$, and $b_{2}$ in $(p_{31},p_{32})$ (see (a) in Figure 6.9)”.
$\mathrm{A5.9}\equiv\big{(}(C_{0}(p_{11},p_{12})\wedge C_{0}(p_{31},p_{32})%
\wedge C_{\wedge}(p_{21},p_{22})\wedge C_{0}(p_{11}^{\prime},p_{12}^{\prime}))\vee$
$(C_{0}(p_{11},p_{12})\wedge C_{0}(p_{31},p_{32})\wedge C_{\vee}(p_{21},p_{22})%
\wedge C_{0}(p_{11}^{\prime},p_{12}^{\prime}))\vee$
$(C_{0}(p_{11},p_{12})\wedge C_{1}(p_{31},p_{32})\wedge C_{\wedge}(p_{21},p_{22%
})\wedge C_{0}(p_{11}^{\prime},p_{12}^{\prime}))\vee$
$(C_{0}(p_{11},p_{12})\wedge C_{1}(p_{31},p_{32})\wedge C_{\vee}(p_{21},p_{22})%
\wedge C_{1}(p_{11}^{\prime},p_{12}^{\prime}))\vee$
$(C_{1}(p_{11},p_{12})\wedge C_{0}(p_{31},p_{32})\wedge C_{\wedge}(p_{21},p_{22%
})\wedge C_{0}(p_{11}^{\prime},p_{12}^{\prime}))\vee$
$(C_{1}(p_{11},p_{12})\wedge C_{0}(p_{31},p_{32})\wedge C_{\vee}(p_{21},p_{22})%
\wedge C_{1}(p_{11}^{\prime},p_{12}^{\prime}))\vee$
$(C_{1}(p_{11},p_{12})\wedge C_{1}(p_{31},p_{32})\wedge C_{\wedge}(p_{21},p_{22%
})\wedge C_{1}(p_{11}^{\prime},p_{12}^{\prime}))\vee$
$(C_{1}(p_{11},p_{12})\wedge C_{1}(p_{31},p_{32})\wedge C_{\vee}(p_{21},p_{22})%
\wedge C_{1}(p_{11}^{\prime},p_{12}^{\prime}))\big{)}$
The subformulae $\alpha_{b}$ and $\alpha_{c}$ that correspond to the cases ($b$)
and ($c$) in Figure 6.9, are similar to $\alpha_{a}$.
For the clarity of presentation, we omit those formulae. Furthermore, it should be clear how to build
them using $\alpha_{a}$ as template. Moreover, the complete formulae can be found
in [12]. We present next the remaining two subformulae,
namely $\alpha_{d}$ and $\alpha_{e}$.
$\alpha_{d}\equiv\exists y\big{(}E_{\mathit{ST}}(x,y)\wedge\neg\exists z(E_{%
\mathit{ST}}(y,z))\wedge$
$\exists p_{11}p_{12}p_{11}^{\prime}p_{12}^{\prime}\big{(}M(x,f_{1},f_{2})%
\wedge M(y,p_{11}^{\prime},p_{12}^{\prime})\wedge$
$E_{C}(f_{1},f_{2},p_{11},p_{12})\wedge E_{C}(p_{11},p_{12},l_{1},l_{2})\wedge E%
_{C}(l_{1},l_{2},p_{11}^{\prime},p_{12}^{\prime})\wedge$
$\neg\exists p_{21}^{\prime}p_{22}^{\prime}(E_{C}(p_{11}^{\prime},p_{12}^{%
\prime},p_{21}^{\prime},p_{22}^{\prime}))\wedge$
$C_{(}(f_{1},f_{2})\wedge C_{)}(l_{1},l_{2})\wedge$
$((C_{1}(p_{11},p_{12})\wedge C_{1}(p_{11}^{\prime},p_{12}^{\prime}))\vee(C_{0}%
(p_{11},p_{12})\wedge C_{0}(p_{11}^{\prime},p_{12}^{\prime})))\big{)}\big{)}$
Note that the first line in $\alpha_{d}$ expresses “$x$ is the predecessor of the leaf in $E_{\mathit{ST}}$”, so that this case corresponds to the last transition (see Figure 6.11). Also note that the last transition is necessarily an instance of case (c) in Figure 6.9”
$\alpha_{e}\equiv\mathrm{A5.10}\wedge\exists p_{1}^{\prime}p_{2}^{\prime}\big{(%
}M(x,p_{1}^{\prime},p_{2}^{\prime})\wedge\mathrm{A5.11}\wedge\mathrm{A5.12}%
\big{)}$ where
•
A5.10 expresses “$x$ is the leaf in $E_{\mathit{ST}}$”.
$\mathrm{A5.10}\equiv\neg\exists y(E_{\mathit{ST}}(x,y))$
•
A5.11 expresses “$(p_{1}^{\prime},p_{2}^{\prime})$ is the leaf in $E_{C}$”.
$\mathrm{A5.11}\equiv\neg\exists y_{1}^{\prime}y_{2}^{\prime}(E_{C}(p_{1}^{%
\prime},p_{2}^{\prime},y_{1}^{\prime},y_{2}^{\prime}))$
•
A5.12 expresses “the last formula in $(C,E_{C})$ is $1$”.
$\mathrm{A5.12}\equiv C_{1}(p_{1}^{\prime},p_{2}^{\prime})$
6.3 Auxiliary Formulae
For the sake of completeness, we define next the remaining auxiliary formulae used through the previous
subsections. We assume an edge relation $E$ and a total order $\leq$.
“$x=0$” $\equiv\neg\exists y(y\neq x\wedge y\leq x)$
“$x=1$” $\equiv\exists y(y\neq x\wedge y\leq x\wedge\neg\exists z(z\neq x\wedge z\neq y%
\wedge y\leq z\wedge z\leq x)\wedge\neg\exists z(z\neq y\wedge z\leq y))$
$\mathrm{SUC}_{\leq}(x,y)\equiv x\leq y\wedge y\neq x\wedge\neg\exists z(z\neq x%
\wedge z\neq y\wedge x\leq z\wedge z\leq y)$
$\mathrm{PRED}_{\leq}(y,x)\equiv\mathrm{SUC}_{\leq}(x,y)$
$\mathrm{PATH_{E}}(v,w)$ is used to denote the following formula which is
satisfied by a given graph ${\bf G}$ iff $(v,w)$ is in the transitive closure
of the relation $E^{\bf G}$.
$\mathrm{PATH}_{E}(v,w)\equiv v=w\,\vee\exists V^{\prime}E^{\prime}\big{(}V^{%
\prime}(v)\wedge V^{\prime}(w)\wedge\mathrm{A1}\wedge\mathrm{A2}\wedge\mathrm{%
A3}\wedge\mathrm{A4}\wedge\mathrm{A5}\big{)}$
•
A1 expresses “$(V^{\prime},E^{\prime})$ is a subgraph of $(V,E)$ with no loops”.
$\mathrm{A1}\equiv\forall xy(E^{\prime}(x,y)\rightarrow(V^{\prime}(x)\wedge V^{%
\prime}(y)\wedge E(x,y)))\wedge\forall x(V^{\prime}(x)\rightarrow V(x))\wedge%
\forall x(\neg E^{\prime}(x,x))$
•
A2 expresses “$v$ is the only minimal node”.
$\mathrm{A2}\equiv\neg\exists x(E^{\prime}(x,v))\wedge\forall y((V^{\prime}(y)%
\wedge y\neq v)\rightarrow\exists x(E^{\prime}(x,y)))$
•
A3 expresses “$w$ is the only maximal node”.
$\mathrm{A3}\equiv\neg\exists x(E^{\prime}(w,x))\wedge\forall y((V^{\prime}(y)%
\wedge y\neq w)\rightarrow\exists x(E^{\prime}(y,x)))$
•
A4 expresses “all nodes except $v$ have input degree $1$”.
$\mathrm{A4}\equiv\forall z((V^{\prime}(z)\wedge z\neq v)\rightarrow\exists x(E%
^{\prime}(x,z)\wedge\forall y((V^{\prime}(y)\wedge E^{\prime}(y,z))\rightarrow
y%
=x)))$
•
A5 expresses “all nodes except $w$ have output degree $1$”.
$\mathrm{A5}\equiv\forall z((V^{\prime}(z)\wedge z\neq w)\rightarrow\exists x(E%
^{\prime}(z,x)\wedge\forall y((V^{\prime}(y)\wedge E^{\prime}(z,y))\rightarrow
y%
=x)))$
That is, $\mathrm{PATH}_{E}(v,w)$ expresses “$(V^{\prime},E^{\prime})$ is a linear subgraph of $(V,E)$, with minimal node
$v$ and maximal node $w$”. We use a similar
strategy to define the next auxiliary formula $\mathrm{LINEAR}(V,E)$ which expresses “$(V,E)$ is a linear graph”.
$\mathrm{LINEAR}(V,E)\equiv\forall xy(\mathrm{PATH}_{E}(x,y)\vee\mathrm{PATH}_{%
E}(y,x))\wedge$
$(\exists xy(x\neq y)\rightarrow\forall x(\neg E(x,x)))\wedge$
$\exists vw\big{(}V(v)\wedge V(w)\wedge$
$\neg\exists x(E(x,v))\wedge\forall y((V(y)\wedge y\neq v)\rightarrow\exists x(%
E(x,y)))\wedge$
$\neg\exists x(E(w,x))\wedge\forall y((V(y)\wedge y\neq w)\rightarrow\exists x(%
E(y,x)))\wedge$
$\forall z((V(z)\wedge z\neq v)\rightarrow$
$\exists x(E(x,z)\wedge\forall y((V(y)\wedge E(y,z))\rightarrow y=x)))$
$\forall z((V(z)\wedge z\neq w)\rightarrow$
$\exists x(E(z,x)\wedge\forall y((V(y)\wedge E(z,y))\rightarrow y=x)))\big{)}$
Note that we only allow loops in a linear graph when it has only one node.
In a similar way we can define the second-order formula $\mathrm{LINEAR}_{2}(Vâ,Eâ)$ where the free second-order variables have arity $2$ and $4$ respectively.
We also use the formula $\mathrm{PATH}_{\mathrm{E_{C}}}(x_{1},x_{2},y_{1},y_{2})$ with free first-order variables $x_{1}$, $x_{2}$, $y_{1}$, $y_{2}$, where the set of vertices is a binary relation, and the set of edges is a $4$-ary relation (see Figures 6.7 and 6.8).
7 $\mathrm{SATQBF}$ in Third-Order Logic
In this section we show how to build a formula in third-order logic that
expresses $\mathrm{SATQBF}$. We omit the tedious details of the subformulae
which can be built following the same patterns than in the detailed exposition of the
second-order formula for $\mathrm{SATQBF}_{k}$.
Roughly, we first express the existence of a third-order alternating valuation ${\bf T}_{v}$ applicable to a given $\mathrm{QBF}$ formula $\varphi$. Then we proceed
to evaluate the quantifier-free part $\varphi^{\prime}$ of $\varphi$ on each leaf
valuation ${\bf L}_{v}$ of ${\bf T}_{v}$. For this part we use the same
second-order subformulae than for $\mathrm{SATQBF}_{k}$. That is, from
$\varphi^{\prime}$ and ${\bf L}_{v}$, we build a Boolean sentence $\phi$ on $\{0,1\}$ by
replacing each occurrence of a Boolean variable $x$ in $\varphi^{\prime}$ by a constant
$0$ or $1$ according to the Boolean value assigned by ${\bf L}_{v}$ to $x$, and
then we evaluate $\phi$.
Unlike the case with $\mathrm{SATQBF}_{k}$ in which the input formulae all have a same fixed number $k$ of alternating blocks of quantifiers, in the case of $\mathrm{SATQBF}$ the number of alternating blocks $k\geq 1$ of quantifiers in the input formulae is not fixed. That is,
we need to take into account that the input formula can have any arbitrary number $k\geq 1$ of alternating blocks of quantifiers. We assume w.l.o.g. that the quantification in the input formula $\varphi$ has the form
$\exists x_{11}\cdots\exists x_{1l_{1}}\forall x_{21}\cdots\forall x_{2l_{2}}%
\exists x_{31}\cdots\exists x_{3l_{3}}\cdots Qx_{k1}\cdots Qx_{kl_{k}}($
$\varphi^{\prime}(x_{11},\ldots,x_{1l_{1}},x_{21}\ldots,x_{2l_{2}},x_{31},%
\ldots,x_{3l_{3}},\ldots,x_{k1},\ldots,x_{kl_{k}}))$
where $k\geq 1$, the formula
$\varphi^{\prime}$ is a quantifier-free Boolean formula and $Q$ is $\exists$ if $k$ is
odd, or $\forall$ if $k$ is even. To represent the formulae as relational structures, we use the same encoding based in word models as in Section 6.
We present a sketch of the third-order formula $\varphi_{\mathrm{SATQBF}}$ that expresses $\mathrm{SATQBF}$. We follow a top-down approach,
leaving most of the fine details of the formulae in the lowest level of abstraction as an exercise for the reader.
At the highest level of abstraction, we can think of $\varphi_{\mathrm{SATQBF}}$ as a third-order formula that expresses.
“There is a third-order alternating valuation $${\bf T}_{v}$$ applicable to
$$\varphi$$, which satisfies $$\varphi$$”.
At the next level of abstraction we can express $\varphi_{\mathrm{SATQBF}}$ in third-order logic as follows.
$\exists{\cal V}_{t}\,{\cal E}_{t}\,{\cal B}_{t}\,V_{t}\,E_{t}\,\Big{(}\mathrm{%
A1}\wedge\mathrm{A2}\wedge\mathrm{A3}\wedge\mathrm{A4}\wedge\mathrm{A5}\big{)}$ where
•
A1 expresses “${\cal B}_{t}:{\cal V}_{t}\rightarrow\{0,1\}$”.
•
A2 expresses “$G_{t}=(V_{t},E_{t})$ is a linear graph which represents the sequence of
quantified variables in $\varphi$”.
•
A3 expresses “$({\cal V}_{t},{\cal E}_{t})$ is a third-order binary tree with all its leaves at the
same depth, which is in turn equal to the length of $(E_{t},V_{t})$”.
•
A4 expresses “$({\cal V}_{t},{\cal E}_{t},{\cal B}_{t})$ is a third-order alternating
valuation ${\bf T}_{v}$ applicable to $\varphi$, i.e., all the nodes in $({\cal V}_{t},{\cal E}_{t})$ whose depth correspond to a universally quantified
variable in the prefix of quantifiers of $\varphi$, have exactly one sibling, and its
value under ${\cal B}_{t}$ is different than that of the given node, and
all the nodes whose depth correspond to an existentially quantified variable
in the prefix of quantifiers of $\varphi$, are either the root or have no
siblings”.
•
A5 expresses “Every leaf valuation in $({\cal V}_{t},{\cal E}_{t},{\cal B}_{t})$
satisfies $\varphi^{\prime}$”.
Recall that we use uppercase calligraphic letters for third-order variables and plain uppercase letters for second-order variables. In particular, ${\cal V}_{t}$, ${\cal E}_{t}$ and ${\cal B}_{t}$ are third-order variables while $V_{t}$ and $E_{t}$ are second-order variables.
Finally, we describe the strategies to express A2–A5 in third-order logic.
A2.
$\mathrm{LINEAR}(V_{t},E_{t})\wedge\mathrm{A2.1}$ where
–
A2.1 expresses “The length of $G_{t}$ is equal to the number of variables in the prefix of
quantifiers of $\varphi$. That is, there is a relation $V_{p}$ which is a partial bijection from the quantifier prefix of $\varphi$
(restricted to the $X$’s in the quantifier prefix) to $V_{t}$, which maps every
$X$ in the quantifier prefix to its corresponding node in $G_{t}$, and which preserves $E_{t}$
and $\leq^{{\bf G}_{\varphi}}$ in $G_{t}$ and $\varphi$ (restricted to the $X$’s in the
quantifier prefix), respectively”.
See (A) in Subsection 6.2.1 for more details.
A3.
Let ${\cal E}_{t}\restriction_{{\cal S}_{d}}$ denote the restriction of the third-order relation ${\cal E}_{t}$ to the nodes in the third-order set ${\cal S}_{d}$. We can express A3 as follows:
$\mathrm{A3.1}\wedge\mathrm{A3.2}\wedge\mathrm{A3.3}\wedge\mathrm{A3.4}$ where
–
A3.1 expresses “$({\cal V}_{t},{\cal E}_{t})$ is a third-order connected graph that has one root and one or
more leaves”.
–
A3.2 expresses “Except for the root node, all nodes in $({\cal V}_{t},{\cal E}_{t})$ have input
degree $1$”.
–
A3.3 expresses “Except for the leaf nodes, all nodes in $({\cal V}_{t},{\cal E}_{t})$ have
output degree $1$ or $2$”.
–
A3.4 expresses “All leaf nodes in $({\cal V}_{t},{\cal E}_{t})$ have the same
depth, which is in turn equal to the length of $(V_{t},E_{t})$”
A3.1, A3.2 and A3.3 can be expressed in third-order logic as follows:
$\exists R\big{(}\forall Z({\cal V}_{t}(Z)\rightarrow\mathrm{PATH}_{{\cal E}_{t%
}}(R,Z))\wedge$
$\neg\exists S_{1}({\cal E}_{t}(S_{1},R))\wedge$
$\exists S_{1}(\neg\exists S_{2}({\cal E}_{t}(S_{1},S_{2})))\wedge$
$\forall Z(({\cal V}_{t}(Z)\wedge Z\neq R)\rightarrow\exists S_{1}({\cal E}_{t}%
(S_{1},Z)\wedge\forall S_{2}({\cal E}_{t}(S_{2},Z)\rightarrow S_{1}=S_{2})))%
\big{)}\wedge$
$\forall Z\big{(}{\cal V}_{t}(Z)\rightarrow\neg\exists S_{1}S_{2}S_{3}(S_{1}%
\neq S_{2}\wedge S_{2}\neq S_{3}\wedge S_{1}\neq S_{3}\wedge{\cal E}_{t}(Z,S_{%
1})\wedge{\cal E}_{t}(Z,S_{2})\wedge$
${\cal E}_{t}(Z,S_{3}))\big{)}$
Regarding A3.4, we can express it as follows:
$\forall X\big{(}\mathrm{A3.4.1}\rightarrow\big{(}\exists{\cal S}_{d}\,{\cal D}%
(\mathrm{A3.4.2}\wedge\mathrm{A3.4.3}\wedge{\cal S}_{d}(X)\wedge\mathrm{A3.4.4%
}\wedge\mathrm{A3.4.5})\big{)}\big{)}$ where
–
A3.4.1 expresses “$X$ is a leaf node in $({\cal V}_{t},{\cal E}_{t})$”.
–
A3.4.2 expresses “${\cal S}_{d}\subseteq{\cal V}_{t}$”.
–
A3.4.3 expresses “${\cal D}:V_{t}\rightarrow{\cal S}_{d}$ is a bijection that
preserves $E_{t}$ and ${\cal E}_{t}\restriction_{{\cal S}_{d}}$”.
–
A3.4.4 expresses “${\cal D}^{-1}(X)$ is the leaf node in $G_{t}=(V_{t},E_{t})$”.
–
A3.4.5 expresses “${\cal S}_{d}$ includes the root of $({\cal V}_{t},{\cal E}_{t})$”.
A4.
We can express A4 as follows (refer to Figures 6.2 and 6.5):
$\mathrm{A4.1}\wedge\forall x\forall{\cal S}_{d}\big{(}(V_{t}(x)\wedge\mathrm{A%
4.2}\wedge\mathrm{A4.3})\rightarrow$
$\forall{\cal D}((\mathrm{A4.4}\wedge\mathrm{A4.5})\rightarrow((\mathrm{A4.6}%
\rightarrow\mathrm{A4.7})\wedge(\mathrm{A4.8}\rightarrow\mathrm{A4.9})))\big{)}$ where
–
A4.1 expresses “${\cal B}_{t}$ is a total function from ${\cal V}_{t}$ to $\{0,1\}$”.
–
A4.2 expresses “${\cal S}_{d}\subseteq{\cal V}_{t}$”.
–
A4.3 expresses “$({\cal S}_{d},{\cal E}_{t}\restriction_{{\cal S}_{d}})$ is a linear graph which includes the root of $({\cal V}_{t},{\cal E}_{t})$”.
–
A4.4 expresses “${\cal D}$ is a bijection from the initial subgraph of $G_{t}$ up to $x$, to ${\cal S}_{d}$”.
–
A4.5 expresses “${\cal D}$ preserves $E_{t}$ and ${\cal E}_{t}\restriction_{{\cal S}_{d}}$”.
–
A4.6 expresses “the predecessor of $V^{-1}_{p}(x)$ in $\leq^{{\bf G}_{\varphi}}$ is $\forall$”.
–
A4.7 expresses “${\cal D}(x)$ has exactly one sibling in $({\cal V}_{t},{\cal E}_{t})$ and ${\cal B}_{t}$ of that sibling is not equal to ${\cal B}_{t}({\cal D}(x))$”.
–
A4.8 expresses “the predecessor of $V^{-1}_{p}(x)$ in $\leq^{{\bf G}_{\varphi}}$ is $\exists$”.
–
A4.9 expresses “${\cal D}(x)$ has no siblings in $({\cal V}_{t},{\cal E}_{t})$, or ${\cal D}(x)$ is the root in $({\cal V}_{t},{\cal E}_{t})$”.
A5.
$\forall{\cal S}_{v}\big{(}(\mathrm{A5.1}\wedge\mathrm{A5.2})\rightarrow\exists%
{\cal D}\,B_{t}\big{(}\mathrm{A5.3}\wedge\mathrm{A5.4}\wedge\mathrm{A5.5}\big{%
)}\big{)}$ where
–
A5.1 expresses “${\cal S}_{v}\subseteq{\cal V}_{t}$”.
–
A5.2 expresses “$({\cal S}_{v},{\cal E}_{t}\restriction_{{\cal S}_{v}})$ is a linear graph which includes the root and a leaf of $({\cal V}_{t},{\cal E}_{t})$”.
–
A5.3 expresses “${\cal D}$ is a bijection from $V_{t}$ to ${\cal S}_{v}$ which preserves $E_{t}$ and ${\cal E}_{t}\restriction_{{\cal S}_{v}}$”.
–
A5.4 expresses “$B_{t}$ is a total function from $V_{t}$ to $\{0,1\}$ which coincides with ${\cal B}_{t}({\cal S}_{v})$ w.r.t. ${\cal D}$”.
–
A5.5 expresses “the leaf valuation represented by $(V_{t},E_{t},B_{t})$ satisfies the quantifier-free subformula $\varphi^{\prime}$ of $\varphi$”.
Note that, A5.5 can be expressed as in Subsection 6.2.2.
Remark 2
Note that while in the third-order formulae in A4 and A5 we have used universal third-order quantification (for ${\cal S}_{d}$ and ${\cal D}$ in A4, and for ${\cal S}_{v}$ in A5), it is not actually needed, and existential third-order quantification is enough. These are the only sub-formulae where we have used universal third-order quantification. Hence, we strongly believe that our third-order formula can be translated in a rather technical way into an existential third-order formula.
Let us consider the sketch for an existential third-order formula equivalent to the formula in A4 (the existential formula for A5 is easier). We can say that for every node $x$ in the graph $(V_{t},E_{t})$, and for every set $Z$ that is a node in the third-order graph $({\cal V}_{t},{\cal E}_{t})$, and such that there is a third-order set ${\cal S}_{d}$ of nodes in the third-order graph $({\cal V}_{t},{\cal E}_{t})$, such that the restriction of the edge relation ${\cal E}_{t}$ to the third-order set ${\cal S}_{d}$, together with ${\cal S}_{d}$, form a (third-order) subgraph that is a linear graph whose root is the root of the third-order graph $({\cal V}_{t},{\cal E}_{t})$, and whose leaf is the set $Z$, and such that its length is the length of the initial subgraph of the graph $(V_{t},E_{t})$, up to the node $x$, if the variable represented by $x$ in the input formula $\varphi$ is universally quantified, then the node $Z$ in the third-order graph $({\cal V}_{t},{\cal E}_{t})$ has exactly one sibling in that graph, and that sibling has a different value assigned by ${\cal B}_{t}$ than the value assigned by ${\cal B}_{t}$ to $Z$. On the other hand, if the variable represented by $x$ in the input formula $\varphi$ is existentially quantified, then the node $Z$ in the third-order graph $({\cal V}_{t},{\cal E}_{t})$ has no sibling in that graph.
To say that “the third order graph induced by the set ${\cal S}_{d}$ in the graph $({\cal V}_{t},{\cal E}_{t})$, whose leaf is the set $Z$, has the same length as the initial subgraph of the graph $(V_{t},E_{t})$, up to the node $x$”, we say that there is a binary third-order relation $\cal D$ which is a bijection between the set of nodes in the initial subgraph of the graph $(V_{t},E_{t})$, up to the node $x$, and the third-order set ${\cal S}_{d}$, and which preserves $E_{t}$ and the restriction of the edge relation ${\cal E}_{t}$ to the third-order set ${\cal S}_{d}$.
8 Final Considerations
Let $\exists\mathrm{SO}^{\leq 2}$ denote the restriction of $\exists$SO to formulae with second-order
variables of arity $\leq 2$.
As pointed out in [2], it is open whether on graphs full $\exists$SO is strictly more expressive than $\exists\mathrm{SO}^{\leq 2}$. Also as pointed out in [2], no concrete example of a graph property in PSPACE that is not in binary NP has been found yet, even though it is known that such properties exist.
Hence, it would be worthwhile to find an example of a
PSPACE query on graphs that cannot be expressed in
$\exists\mathrm{SO}^{\leq 2}$. The gained experience
on writing non-trivial queries in second-order logic, can prove to be
a valuable platform to make progress on these kind of open problems. In
particular, we used a second-order variable of arity 4 in Section 6.
We used it to represent (together with other variables) a linear digraph which, for each of the
leaf valuations, encodes a
sequence of word models corresponding to the different stages of
evaluation of the quantifier free part
of the input QBF${}_{k}$ formula. Since
the size of the Boolean formula in each stage is linear in the size of the input
QBF${}_{k}$ formula, and the number of connectives in the formula is also
linear, the length of the complete sequence of Boolean formulae is
quadratic. Therefore, we conjecture that arity $4$ is actually a lower bound,
though we have not attempted to prove it yet.
In general, the exploration of properties which force us to work
with intermediate structures of size greater than linear w.r.t. the input,
seems a reasonable way of approaching these kind of open problems.
As noted earlier, there are second-order queries that are difficult to express in the language of second-order logic,
but which have an elegant and simple characterization in third-order logic. Therefore it would be interesting to explore possible characterizations of fragments of third-order logic that admit translations of their formulae to equivalent formulae in second-order logic.
This way, those fragments of third-order logic could be assimilated to high-level programming languages, while second-order logic would be the corresponding low-level programming language. In turn, this would allow us to express complex second-order queries with greater abstraction of the low-level details, thus minimizing the probability of error.
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Received 21 December 2012. |
The Use of Generalized Information Dimension
in Measuring Fractal Dimension of Time Series
Y. Ashkenazy
Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel
(December 2, 2020)
Abstract
An algorithm for calculating generalized fractal dimension of a
time series using the
general information function is presented. The algorithm is based on a strings
sort technique and requires $O(N\log_{2}N)$ computations. A rough estimate
for the number of points needed for the fractal dimension calculation is given.
The algorithm was tested on analytic example as well as well-known examples,
such as, the Lorenz attractor, the Rossler attractor, the van der Pol
oscillator, and the
Mackey-Glass equation, and compared, successfully, with previous results
published in the literature. The computation time for the algorithm
suggested in this paper is much less then the computation time according to
other methods.
pacs: PACS numbers:05.45.+b, 47.52.+j, 47.53.+n
††preprint:
[
]
I Background
In the recent decades the study of chaos theory has gathered momentum.
The complexity that can be found in many physical and biological systems
has been analyzed by the tools of chaos theory.
Characteristic properties, such as the Lyapunov
exponent, Kolmogorov entropy and the fractal dimension (FD), have been
measured in experimental systems. It is fairly easy to calculate signs of
chaos if the system can be represented by a set of non-linear ordinary
differential equations. In many cases it is very difficult to built a
mathematical model that can represent sharply the experimental system.
It is essential, for this purpose, to reconstruct a new phase space based on
the information that one can produce from the system. A global value that is
relatively simple to compute is the FD. The FD can give
an indication of the dimensionality and complexity of the system.
Since actual living biological systems
are not stable and the system complexity varies with time,
one can distinguish between different states of the system by the FD.
The FD can also determine whether a particular system is more complex than
other systems. However, since biological systems are very complex, it is better
to use all the information and details the system can provide. In this paper
we will present
an algorithm for calculating FD based on the geometrical structure
of the system. The method can provide important information, in addition, on
the geometrical of the system (as reconstructed from a time series).
The most common way of calculating FD is through the correlation function,
$C_{q}(r)$ (eq. 6). There is also other method of FD calculation based
on Lyapunov exponents and Kaplan-Yorke conjecture [2]
(eq. 27). However, the computation of the Lyapunov exponent spectrum
from a time series is very difficult and people usually try to avoid this
method111The basic difficulty is that for high FD there are some
exponents which are very close to zero; one can easily add an extra
unnecessary exponent
that can increase the dimensionality by one; this difficulty is most
dominant in Lyapunov exponents which have been calculated from a time series..
The algorithm which is presented in this paper is important, since it gives a
comparative method for calculating FD according to the correlation function.
The need for a additional method of FD calculation is critical in some types
of time series,
such as EEG series (which are produced by brain activity), since several
different
estimations for FD were published in the literature
[3, 4, 5, 6, 7].
A comparative algorithm
can help to reach final conclusions about FD estimate for the signal.
A very simple way to reconstruct a phase space from single time series
was suggested by Takens [8]. Giving a time series ${x_{i}},i=1\ldots N_{p}$, we build a new $n$ dimensional phase space in the
following way :
$$\left.\begin{array}[]{l}\vec{y}_{0}=\{x(t_{0}),x(t_{0}+\tau),x(t_{0}+2\tau),%
\ldots,x(t_{0}+(n-1)\tau)\}\\
\vec{y}_{1}=\{x(t_{1}),x(t_{1}+\tau),x(t_{1}+2\tau),\ldots,x(t_{1}+(n-1)\tau)%
\}\\
\vdots\\
\vec{y}_{N}=\{x(t_{N}),x(t_{N}+\tau),x(t_{N}+2\tau),\ldots,x(t_{N}+(n-1)\tau)%
\}\\
\\
t_{i}=t_{0}+i\Delta t\qquad\tau=m\Delta t\\
N=N_{p}-(n-1)m\qquad m=integer,\end{array}\right.$$
(1)
where $\Delta t$ is the sampling rate, $\tau$ corresponds to the interval
on the time series
that creates the reconstructed phase space (it is usually chosen to be the
first zero of the autocorrelation function [9], or the first minimum
of mutual information [10];
in this work we will use $m$ instead of $\tau$ as an index), and $N$ is
number of reconstructed vectors.
For ideal systems (an infinite number of points without external noise)
any $\tau$ can
be chosen. Takens proved that such systems converge to the real dimension
if $n\geq 2D+1$, where $D$ is the FD of the real system. According
to Ding et al [11] if $n\geq D$ then the FD of the
reconstructed phase space is equal to the actual FD.
In this paper we will first (sec. II) present regular analytic
methods for calculating the FD of a system (generalized correlation method and
generalized information method). An efficient
algorithm for calculating the FD from a time series based on string sorting
will be described in the next section (sec. III). The following
step is to check and compare the general correlation
methods with the general information method (see eq. (2))
using known examples (sec. IV).
Finally, we summarize in sec. V.
II Generalized information dimension and
generalized correlation dimension
II.1 Generalized information dimension
The basic way to calculate an FD is with the Shannon entropy,
$I_{1}(\varepsilon)$222Generally, one has to add a superscript, $n$,
the embedding
dimension in which the general information is calculated. At this stage we
assume that the embedding dimension is known. [12], of
the system. The entropy is just a particular case of the general information
which is defined in the following way [13] :
$$I_{q}(\varepsilon)=\frac{1}{1-q}\ln(\sum_{i=1}^{M(\varepsilon)}p_{i}^{q}),$$
(2)
where we divide the phase space to $M(\varepsilon)$ hypercubes of edge
$\varepsilon$. The probability to find a point in the $i^{th}$
hyper cube is denoted by $p_{i}$. The generalization, $q$, can be any
real number; we usually
use just an integer $q$. When we increase $q$, we give more
weight to more populated
boxes, and when we decrease $q$ the less occupied boxes become dominant.
For $q=1$, by applying the l’Hospital rule, eq. (2) becomes :
$$I_{1}(\varepsilon)=-\sum_{i=1}^{M(\varepsilon)}p_{i}\ln p_{i},$$
(3)
where $I_{1}(\varepsilon)$ is referred to also as the Shannon entropy
[12] of the system.
The definition of the general information dimension (GID) is :
$$D_{q}=-\lim_{\varepsilon\to 0}\frac{I_{q}(\varepsilon)}{\ln\varepsilon},$$
(4)
for $N\to\infty$ ($N$ is the number of points).
However, in practice, the requirement ${\varepsilon\to 0}$ is not
achieved, and an average over a range of $\varepsilon$ is required.
In some cases this average is not sufficient
because several values of $I_{q}(\varepsilon)$
can be computed for the same $\varepsilon$. To illustrate this
we use a $2D$ Cantor
set, presented in Fig. 1. We start from a square. From 9 sub-squares
we erase the 5 internal squares. We continue with this evolution for each
remainder square. This procedure is continued, in principle,
to infinity. The FD, $D_{0}$, of the
$2D$ Cantor set is $\ln 4/\ln 3$ (when $q=0$, $I_{0}$ is just the number of
nonempty squares and $D_{0}$ is the logarithmic ratio between nonempty
squares and $\varepsilon$, where the square edge is normalized to one) as shown
in Fig 1. However, it is possible to locate the $2D$ grid in such
a way that there are 16 nonempty squares, giving rise to
$D_{0}=2\ln 4/\ln 3$, twice the time of real FD of the system. This
illustration shows that when $\varepsilon$ is not small, different
positions of the grid can lead to different FD’s. In this case it is clear
that we have to locate the grid in such a way
that minimum squares will be nonempty
(it is easy to show that this claim for the minimum is true for every $q$).
In general, we can say that one must locate the hyper-grid so that
the general information is minimum :
$$\tilde{I_{q}}(\varepsilon)\equiv\min\frac{1}{1-q}\ln(\sum_{i=1}^{M(\varepsilon%
)}p_{i}^{q}),$$
(5)
This proper location of the hyper-grid reduces the influence of surface
boxes that are partly contained in the attractor. The requirement
of a minimum
gives a good estimate for the GID when $\varepsilon$ is not small.
II.2 Generalized correlation dimension
In 1983 Grassberger and Procaccia presented a new method for
calculating $D_{2}$ [14]. According to this method, it is possible to
calculate dimension just from the correlation between different
points, without direct connection to the phase space, and therefore
easy to use. Some years later, a more general correlation function
was suggested by Pawelzik and Schuster [15] :
$$C_{q}(r)=\Biggl{[}\frac{1}{N}\sum_{i=1}^{N}\biggl{[}\frac{1}{N}\sum_{j=1}^{N}%
\Theta\bigl{(}r-\left|\vec{x}_{i}-\vec{x}_{j}\right|\bigr{)}\biggr{]}^{q-1}%
\Biggr{]}^{1\over q-1},$$
(6)
where $N$ is number of points, $\vec{x}_{i}$ is a point of the system
and $q$ is the generalization. $\Theta(x)$ is the Heaviside step function
$$\Theta(x)=\left\{\begin{array}[]{ll}0&{\rm when~{}}x\leq 0\\
1&{\rm when~{}}x>0.\end{array}\right.$$
(7)
According to this method we have to calculate the generalized
probability to find any two points within a distance $r$. This method is
some kind of integration on eq. (2). It is not necessary to compute
the real distance (e.g. $\left|\left|\vec{x}\right|\right|=\sqrt{x_{1}^{2}+\cdots+x_{n}^{2}}$
where n is the phase space dimension); it is equivalent to
calculating the probability to find any two points in a hyper-box where one
of them is located in the middle (e.g. $\left|\left|\vec{x}\right|\right|=\max_{1\leq i\leq n}\left|x_{i}\right|$ [16]).
It is easier to compute the last possibility. For the special case
of $q=1$, eq. (6) can be written (by applying the l’Hospital’s rule
[10]) as :
$$\ln C_{1}(r)=\frac{1}{N}\sum_{i=1}^{N}\ln\biggl{[}\frac{1}{N}\sum_{j=1}^{N}%
\Theta\bigl{(}r-\left|\vec{x}_{i}-\vec{x}_{j}\right|\bigr{)}\biggr{]}.$$
(8)
The generalized correlation dimension (GCD) has a similar definition to
the GID (eq. (4)) :
$$D_{q}=\lim_{N\to\infty;r\to 0}\frac{\ln C_{q}(r)}{\ln r}.$$
(9)
Both GID and GCD methods should give identical results.
The GCD method is easy use and gives smooth curves. On the other hand,
the method requires $O(N^{2})$ computations333If one looks just at small
$r$ values, the method requires just $O(N)$ computations [16]..
Also, the smooth curves due to
averaging over all distances are associated with a loss in information based
on the attractor structure.
As we pointed out earlier, we usually have a limited number of points,
forcing us to calculate dimension at large $r$. Thus an error enters the
calculation of FD. The minimum number of points needed for the FD
calculation has been discussed in several papers and different constraints
suggested (for example [18] and [19]). In this paper
we will use the $N_{min}$ of the Eckmann and Ruelle [19] constraint :
$$D_{2}<\frac{2\log_{10}N}{\log_{10}\Bigl{(}\frac{1}{\rho}\Bigr{)}},$$
(10)
under the following conditions :
$$\rho=\frac{r}{r_{0}}\ll 1\qquad\frac{1}{2}N^{2}\rho^{D}\gg 1,$$
(11)
with reference to Grassberger and Procaccia method. Here, $r_{0}$ is the
attractor diameter, and
$r$ is the maximum distance that gives reliable results in eq. (6) when
$q=2$. The normalized distance, $\rho$, must be small because
of the misleading influence of the hyper-surface.
A $\rho$ too large (close to 1) can cause incorrect results since we
take into account also hyper-boxes that are not
well occupied because the major volume is outside the attractor.
However, if one has a long time series, then it is not necessary to compute
all $N^{2}$ relations in eq. (6). One can compute eq. (6) for
certain reference points such that the conditions in eq. (11) will
still hold. Then, eq. (6) can be written as follows :
$$\displaystyle C_{q}(r)$$
$$\displaystyle=$$
$$\displaystyle\Biggl{[}\frac{1}{N_{ref}}\sum_{i=1}^{N_{ref}}\biggl{[}\frac{1}{N%
_{data}-2w-1}\times$$
(12)
$$\displaystyle\sum_{j=1}^{N_{data}}\Theta\bigl{(}r-\left|\vec{x}_{i\cdot s}-%
\vec{x}_{j}\right|\bigr{)}\biggr{]}^{q-1}\Biggr{]}^{1\over q-1},$$
where
$$\left|i\cdot s-j\right|>w\qquad s={\lfloor{\frac{N_{data}}{N_{ref}}}\rfloor}.$$
For each reference point we calculate the correlation function over all data
points. The step for the time series is $s$. To neglect short
time correlation one must also introduce a cutoff, $w$. Usually
$w\approx m$ ($m$ corresponds to the $\tau$ from (1))
[20]. The number of distances $\left|\vec{x}_{i}-\vec{x}_{j}\right|$ will be $P=N_{ref}(N_{data}-2w-1)$ instead of $N^{2}$. The new form
of eq. (10) is:
$$D<\frac{\log_{10}P}{\log_{10}\Bigl{(}\frac{1}{\rho}\Bigr{)}},$$
(13)
and the conditions (11) become
$$\rho=\frac{r}{r_{0}}\ll 1\qquad\frac{1}{2}P\rho^{D}\gg 1.$$
(14)
Although we discussed here a minimum number
of data points needed for the dimension
calculation, one must choose $\rho$ such that the influence from
the surface is negligible (the growth of the surface is exponential
to the embedding dimension). That gives us an upper limit for $\rho$.
On the other hand, in order to find the FD we must average over a range of
$r$, giving us a lower limit for $\rho$; therefore $N_{min}$
is determined according to the lower limit, giving rise to larger amount of
points.
III Algorithm
III.1 Background
In this section we describe an algorithm for GID method444Computer
programs are available at
http://faculty.biu.ac.il/$\sim$ashkenaz/FDprog/.
The algorithm is based on a string sort and can be useful
both for GID and GCD methods.
In previous works there have been several suggestions to compute DIG
[21, 22, 23, 24, 25, 26]. The key
idea of some of those methods [23, 24, 25] is to rescale
the coordinates of each point and to express them in binary form. Then, the
edge-box size can be initialized by the lowest value possible and then
double it each time. Those methods require $O(N\log N)$ operations. Another
method [22] uses a recursive algorithm to find the FD. The
algorithm
starts from a $d$ dimensional box which contains all data points. Then, this
box is divided into $2^{d}$ sub boxes, and so on. The empty boxes are not
considered, and those which contain only one point are marked. This procedure
requires only $O(N)$ operations (for a reasonable series length this fact does
not make a significant difference [22]).
As pointed out in ref. [22], in spite of the efficiency of the
above algorithm (speed and resolution), it is quite difficult to converge to
the FD of a high dimensional system. We were aware to this difficulty, and
suggested a smoothing term to solve it (as will be explained in this
section). Basically, we allow any choice of edge length (in contrast to power
of 2 edge size of the above methods) and optimal location of the grid is
searched. This procedure leads to convergence to the FD.
One of the works that calculates GID was done by Caswell and Yorke
[21]. They calculate FD of $2D$ maps.
They have proved that it is very efficient to
divide the phase space into circles instead of squares in spite of the
neglected area between the circles. However, this
approach is not suitable for higher phase space dimensions. The reason might
be that
the volume of the hyper-balls compared to the entire volume of the attractor
decreases according to the attractor dimension,
and in this way we lose most of the information that is included in the
attractor, since the space between the hyper-balls is not taken into account.
It is easy to show
that the ratio between the volume of the hyper-sphere with radius
$R$, $V_{S}$, and the volume of the hyper-box with edge size of
$2R$, $V_{B}$ (the sphere is in the
box), is
$$\frac{V_{S}(2n)}{V_{B}(2n)}=\frac{1}{2\cdot 4\cdot 6\cdots 2n}\Bigl{(}\frac{%
\pi}{2}\Bigr{)}^{n},$$
(15)
for even phase space dimension, and,
$$\frac{V_{S}(2n-1)}{V_{B}(2n-1)}=\frac{1}{1\cdot 3\cdot 5\cdots(2n-1)}\Bigl{(}%
\frac{\pi}{2}\Bigr{)}^{n-1},$$
(16)
for odd phase space dimension. It is clear that the ratios (15) and
(16) tend to zero for large $n$.
III.2 Algorithm
Let us call the time series $x(i)$. The form of
the reconstructed vector, (1),
depends on the jumping, $m$, that creates the phase
space. We order the time series in the following way :
$$\left.\begin{array}[]{l}x(0),x(m),x(2m),\ldots,x((l-1)m),\\
x(1),x(1+m),x(1+2m),\ldots,x(1+(l-1)m),\\
\vdots\\
x(m-1),x(m-1+m),\ldots,x(m-1+(l-1)m),\\
\end{array}\right.$$
(17)
where $l=\lfloor\frac{N_{p}}{m}\rfloor$. Let us denote the new series
as $\tilde{x}_{i}$
($i=0\ldots(lm-1)$; we lost $N_{p}-lm$ data points). If we take
$n$ consecutive numbers in each row, we create a reconstructed vector
in the $n$ dimensional embedding dimension.
At this stage, we fit a string (for each $\varepsilon$) to the
number series (17). One character is actually a number between
$0\ldots 255$,
and thus, it is possible to divide one edge of the hyper-box to 255 parts
(we need one of the 256 characters for sign). The correspondence is
applied in the following way. We search for the minimum value in $\tilde{x}_{i}$
series, and denote it as $\tilde{x}_{min}$ (similarly, we denote the maximum
values as $\tilde{x}_{max}$).
The corresponding character to $\tilde{x}_{i}$ is :
$$y_{i}=\lfloor\frac{\tilde{x}_{i}-\tilde{x}_{min}}{\varepsilon}\rfloor.$$
(18)
The value of $\varepsilon$ is in the range
$$\frac{(\tilde{x}_{max}-\tilde{x}_{min})}{255}\leq\varepsilon\leq(\tilde{x}_{%
max}-\tilde{x}_{min}).$$
(19)
If we take now $n$ consecutive characters (string), we have the “address”
of the hypercube in which the corresponding vector is found.
Let us represent the string as follows :
$$\left.\begin{array}[]{l}s_{0}=y_{0}y_{1}\ldots y_{n-1},s_{1}=y_{1}\ldots y_{n}%
,\ldots,s_{l-n}=y_{l-n}\ldots y_{l-1},\\
s_{l-n+1}=y_{l}\ldots y_{l+n-1},\ldots,s_{2(l-n)+1}=y_{2l-n}\ldots y_{2l-1},\\
\vdots\\
s_{(m-1)(l-n+1)}=y_{(m-1)l}\ldots y_{(m-1)l+n-1},\ldots,\\
\qquad\qquad\qquad\qquad\qquad s_{m(l-n+1)-1}=y_{ml-n}\ldots y_{ml-1}.\\
\end{array}\right.$$
(20)
Obviously, we do not have to keep each string $s_{i}$ in the memory; we can keep
just the pointers to the beginning of the strings in the characters series,
$y_{i}$. The number of vectors / strings is $m(l-n+1)$.
As mentioned, each string is actually the address of a hyper-box (which the
vector is in it) in a hyper-grid that cover the attractor. The first
character is the position of the first coordinate of the hyper-box on the
first edge of the hyper-grid. The second character
denotes the location on the second edge, and so on. We actually grid the
attractor with a hyper-grid so that any edge in this grid can be divided into
255 parts. Thus the maximal number of boxes is $255^{n}$, where $n$ is the
embedding dimension (there is no limit for $n$). Most of those boxes are
empty, and we keep just the occupied boxes in the hyper-grid.
The next step is to check how many vectors fall in the same box, or, in other
words, how many identical “addresses” there are. For this we have to sort
the vector that points to strings, $s_{i}$, in increasing order (one string is
less then other when the first character that is not equal is less then the
parallel character; e.g., ’abcce’$<$’abcda’ since the fourth character in the
first string, c, is less then the fourth character in the second string, d).
The above process is illustrated in Fig. 2.
The most efficient way to sort $N$ elements is the “quick sort”
[27, 28]. It requires just $O(N\log_{2}N)$ computations.
However, this sort algorithm is not suitable for our propose, since after the
vector is sorted, and we slightly increase the size of the edge of the
hyper-box, $\varepsilon$,
there are just a few changes in the vector, and most of it remains
sorted. Thus, one has to use another method which requires less computations
for this kind of situation, because the quick sort requires
$O(N\log_{2}N)$ computations independently of the initial state. The sort
that we used was a “shell sort” [27], which requires $O(N^{3/2})$
computations in worst case, around $O(N^{5/4})$ computations for random
series, and $O(N)$ operations for an almost sorted vector.
Thus, if we compute the information function for $m$ different
$\varepsilon$ values $O(N\log_{2}N+mN)$ operations will be needed.
After sorting we count how many identical vectors there are in each box.
Suppose that we want to calculate GID for embedding dimensions
$n_{min}\ldots n_{max}$, and generalizations $q_{min}\ldots q_{max}$.
We count in
the sorted pointers vector identical strings containing $n_{max}$ characters.
Once we detect a difference we know that the coordinate of the box
has changed. We detect the first different location. First
difference in the last character of the string means that it is possible
to observe the difference just in the $n_{max}^{th}$ embedding dimension, and
in lower
embedding dimension it is impossible to observe the difference. If the
first mismatch is in the $n^{th}$ character in the string, then the box
changes for embedding
dimensions higher than or equal to $n$. We hold a counter vector for the
different embedding dimensions, and we will assign zero for those embedding
dimensions in which we observed a change.
In this stage, we have to add to the results matrix the probability to fall in
a box according to eqs. (2), (3). We continue with this method
to the end of the pointer vector and then print the results matrix for
the current $\varepsilon$ and continue to the next $\varepsilon$.
It is easy to show that one does not have to worry about the range of the
generalization, $q$, because for large time series with $10^{6}$ data
points the range of computational $q$’s is -100..100.
The method of string sorting can be used also for calculating GCD. Our
algorithm is actually a generalization of the method that was suggested by
Grassberger [16]. The algorithm is specially efficient for small
$r$ ($r<\frac{1}{10}D$ where $D$ is the attractor diameter) and it requires
$O(N\log_{2}N)$ computations instead of $O(N^{2})$ computations (according to the
Grassbeger algorithm it requires $O(N)$ computations instead $O(N^{2})$
computations).
The basic idea of Grassberger was that for small $r_{max}$ values, where
$r_{max}$ is the maximum distance for which the correlation function is
computed, one does not have
to consider distances larger then $r_{max}$, distances which require most of
the computation time. Thus, it is enough to consider just the neighboring
boxes (we use the definition distance $\left|\left|\vec{x}\right|\right|=\max_{1\leq i\leq n}\left|x_{i}\right|$ in eq. (6)),
for which their edge is equal to $r_{max}$, since distances greater then
$r_{max}$ are not
considered in eq. (6). If, for example, one wants to calculate the
correlation of $r\leq r_{max}=\frac{1}{10}D$ (in accordance with
conditions
(10) (11) (13) (14)), then (in $2D$ projection)
it is enough to calculate distances in 9 squares instead 100 squares
(actually, it is enough to consider just 5 squares since eq. (6) is
symmetric). According to Grassberger, one has to build a matrix for which any
cell in it points to the list of data points located in it. It is possible to
generalize this method to a $3D$ projection matrix.
The generalization to higher dimensions can be done very easily according to
the string sort method. As a first step one has to prepare a string $y_{i}$
(18), which is the same operation as gridding the attractor by a
hyper-grid of size $r$. The next step is to sort strings $s_{i}$ (20).
For each reference point in eq. (12) the boxes adjoining the
box with the reference point in it should be found. Now, it is left to find
the distances between
the reference point to other points in neighboring boxes (we keep the pointer
vector from string $y_{i}$ to the initial series and vice versa), and calculate
the correlation according to eq. (12).
The algorithm described above is especially efficient for very complex systems
with high FD. For example, for EEG series produce by brain activity, it well
known (except for very special cases such as epilepsy [17])
that the FD is, at least, four. In this case $n_{min}=4$, and instead
of calculating distances in $l^{4}$ boxes ($l$ is the ratio between the attractor
diameter and $r$) it is sufficient to calculate distances in $3^{4}=81$ boxes.
If, for example, $l=9$, just $3^{4}/9^{4}=\frac{1}{81}$ distances must be computed
(or even less if one take into account also the symmetry of eq. (12)).
In this way, despite the $O(N\log_{2}N)$ operations needed for the initial
sort, one can reduce significantly the computation time.
III.3 Testing of GID algorithm and number of data points needed for
calculating GID
Let us test the algorithm of GID on random numbers. We create a random series
between 0 to 1 (uniform distribution). We then reconstruct the phase space
according to Takens theory (1). For any embedding dimension that we
would like to reconstruct, the phase space we will get a dimension that
is the same as the embedding dimension, since the new reconstructed vectors
are random in the new phase space, and hence fill the space
(in our case, a hyper-cube of edge 1).
For embedding dimension 1 (a simple line), if the edge length is $\varepsilon$,
then the probability to fall in any edge is also $\varepsilon$. The number of
edges $\varepsilon$ that covers the series range $[0,1)$ is $\lfloor\frac{1}{\varepsilon}\rfloor$. The probability to fall in the last edge
is $1\%\varepsilon$ (the residue of the division). For
generalization $q$ one gets :
$$\sum_{i=1}^{M(\varepsilon)}p_{i}^{q}=\lfloor 1/\varepsilon\rfloor\varepsilon^{%
q}+(1\%\varepsilon)^{q}.$$
(21)
For embedding dimension 2, one has to square (21) since the probability
distribution on different sides is equal.
For embedding dimension $n$ eq. (2) becomes :
$$I_{q}(\varepsilon)=\frac{1}{1-q}\ln[\lfloor 1/\varepsilon\rfloor\varepsilon^{q%
}+(1\%\varepsilon)^{q}]^{n}.$$
(22)
Opening eq. (22) according to the binomial formula will give all
different
combinations for the partly contained boxes. Notice that this grid location
fulfills requirement (5) for the minimum information function.
In Fig. 3 we present $-I_{2}(\varepsilon)$, both,
according to the GID algorithm, and according to eq. (22). There is full
correspondence between the curves, and, as we find,
it is possible to see that the slope
(dimension) in the embedding dimension $n$ is equal to the dimension itself.
When $-I_{2}(\varepsilon)\approx-10$, the curves separate. There is lower
convergence (when $-I_{2}(\varepsilon)\approx-12.5$) since the edge is too
small, and the number of reconstructed vectors is equal to the number of
nonempty boxes.
The “saw tooth” that seen in Fig. 3 caused by partial boxes that
fall on the edge of the hyper-box. The “saw tooth” appears when
there is an integer number of edges that covers the big edge. The slope in
the beginning of every saw tooth is equal to zero (that comes from the extremum
condition of (22)). The “saw teeth” become smaller when
$\varepsilon$ decrease, since the relative number of boxes that fall on the
surface in small compared to the entire number of boxes.
It is possible to calculate roughly the number of points needed for the
dimension calculation in a homogeneous system. The separation point of
Fig. 3 is
approximately around $-I_{2}(\varepsilon)\approx-10$, for every embedding
dimension. Thus, the number of hyper-boxes at this point is $e^{10}$
(since $-10=\ln\left(M(\varepsilon)p^{2}\right)$ where $p\sim\frac{1}{M(\varepsilon)}$ in our example). The
graph come to a saturation around $-I_{2}(\varepsilon)\approx-12.5$, giving
rise to $e^{12.5}$ data points. The number of points per box is just,
$e^{2.5}\approx 12$. Thus, generally, when there are less then 12 points per
box, the value of $I_{q}(\varepsilon)$ is unreliable. Let us denote the minimum
edges needed for the calculation as $m_{min}$ ($m_{min}$ is just
$\lfloor 1/\varepsilon\rfloor$, and thus one can define a lower value for
$\varepsilon$). The number of hyper-boxes in embedding dimension $n$ is
$m_{min}^{n}$. Thus, the minimum number of points needed for computing the
FD of an attractor with an FD $D$, is:
$$N=12m_{min}^{D}.$$
(23)
This estimate is less then what was required by Smith [18] ($42^{D}$)
and larger then the requirement of Eckmann and Ruelle [19] (if we
take, for example, $m_{min}={\frac{1}{\rho}}=10$
then according to (10)
$10^{D/2}$ points is needed and according to (23) $12\times 10^{D}$
points
is needed). However, as we pointed out earlier, (23) is a
rough estimation, and one can converge to a desired FD even with less points.
IV Examples
In this section we will present the results of both GID and GCD on well known
systems, such as the Lorenz attractor and the Rossler attractor.
The FD of those
results are well known by various methods, and we will present almost
identical results achieved by our methods. In the following examples, we
choose one of the coordinates to be the time series; we normalize the time
series to be in the range of $0\ldots 1$.
IV.1 Lorenz Attractor
The Lorenz attractor [29] is represented by a set of three
first order non-linear differential equations :
$$\displaystyle{dx\over dt}$$
$$\displaystyle=$$
$$\displaystyle\sigma(y-x)$$
$$\displaystyle{dy\over dt}$$
$$\displaystyle=$$
$$\displaystyle-xz+rx-y$$
(24)
$$\displaystyle{dz\over dt}$$
$$\displaystyle=$$
$$\displaystyle xy-bz.$$
Although it is much simpler to find the FD from the real phase-space $(x,y,z)$,
we prefer the difficult case of finding the FD just from one coordinate to
show the efficiency of Takens theorem. We choose the $z^{th}$ component to be
the time series from which we reconstruct the phase space (the parameter values
are: $\sigma=16$, $r=45.92$, $b=4$). We took $32768$ data points and $m=6$ (the
jumping on the time series which creates the reconstructed phase space; the
time step is $\Delta t=0.02$). The generalization that we used was
$q=2$. The embedding dimensions were $n=1..7$. The FD of Lorenz attractor
is $2.07$ ([30] and others).
In Fig. 4 we show the results of GCD and GID methods. As
expected from the GCD curves, in Fig. 4a we see very smooth curves,
for which the slopes
(in the center parts of the curves) converge approximately to dimension
$2.08$. In Fig. 4b we see some jumping in the GID curves. It can be
clearly seen that for small $\varepsilon$, one can not see the jumping while
for large $\varepsilon$ the non-monotonicity is much stronger. This is typical
behavior for GID curves; large $\varepsilon$ (or larger box edge) reflects more
variability of $-I_{2}(\varepsilon)$ to the hype- grid location. To smooth the
curves in order to get reliable results, it is necessary to find a proper
location of the hyper-grid that gives minimum general information
$\tilde{I}_{2}(\varepsilon)$ (or maximum $-\tilde{I}_{2}(\varepsilon)$).
We have performed 7 comparisons
to find that proper location. As for GCD, we also fitted approximate
linear curves (dashed lines) which lead to dimension $2.09$ (Fig. 4c),
and thus, agree well with previous results. To make sure of the validity of the
dimension results we add here (Fig. 4d) graphs of successive slopes of
curves in Fig. 4c, and conclude with dimension results versus
embedding dimension according to several generalizations ($q=2..5$, Fig.
4e). In both graphs, the approximate dimension is $2.03$.
IV.2 Rossler Attractor
Let us examine briefly our second example, the Rossler attractor
[31]. The
differential equations which generate the trajectory are :
$$\displaystyle\frac{dx}{dt}$$
$$\displaystyle=$$
$$\displaystyle-z-y$$
$$\displaystyle\frac{dy}{dt}$$
$$\displaystyle=$$
$$\displaystyle x+ay$$
(25)
$$\displaystyle\frac{dz}{dt}$$
$$\displaystyle=$$
$$\displaystyle b+z(x-c).$$
The parameter values are : $a=0.15$, $b=0.2$, $c=10$. The time step is $t=0.2$
and $m=8$. We use the $y$ direction to be the time series. The number of data
points is $N=16384$. In Fig. 5 we show the results of GID method
of embedding dimension $n=1..7$. There are many more discontinuities in the
unsmoothed curves (Fig. 5a) compared to the Lorenz attractor,
reflecting a sensitivity to the grid location.
The smooth curves in Fig. 5b are produced after 20 comparisons. Again,
we approximate the slopes of central linear part of the curves in two different
ways and find dimension $2.05$, which is in agreement with dimension $2.03$
calculated by the use of the GCD method, and through the Lyapunov
exponent [30].
IV.3 Van der Pol Oscillator
Another example that will be tested is the van der Pol oscillator
[32]. This system
was investigated in the beginning of the century in order to find a model for
the heart beat variability (among other uses of this equation). The behavior
of the system (with parameter values that will be used) is chaotic although it
looks quite periodic. Thus, we expect a smooth behavior of the geometrical
structure of the attractor. The equation of motion is :
$${\ddot{x}}-\alpha(1-x^{2})\dot{x}+kx=f\cos\Omega t.$$
(26)
The parameter values are : $\Omega=2.446$, $\alpha=5$, $k=1$ and $f=1$. Habib
and Ryne [33] and others [34] have found that the
Lyapunov exponents of the system are $\lambda_{1}\approx 0.098$, $\lambda_{2}=0$
and $\lambda_{3}\approx-6.84$,
and thus, according to the Kaplan and Yorke formula [2],
$$D_{L}=j+{\sum_{i=1}^{j}\lambda_{i}\over|\lambda_{j+1}|}\approx 2+{0.098\over 6%
.84}\approx 2.014$$
(27)
where $j$ is defined by the condition that $\sum_{i=1}^{j}\lambda_{i}>0$
and $\sum_{i=1}^{j+1}\lambda_{i}<0$. The fact that the system has a
“periodic” nature is reflected in this very low
FD (as known, the minimal FD of a chaotic system is 2).
In Fig. 6 we present the calculation of $D_{1}$ which we suppose to be
close to $D_{L}$. The approximate slope lead to an FD of $D_{1}\sim 2.02$, which
is in good agreement to $D_{L}$. Notice that minimization of $I_{1}(\varepsilon)$
did not succeed for large $\varepsilon$ values although we used 7 grid
comparisons; that fact can cause a small error in the slope estimation, since
usually the slope is determined from the central part of the curves.
IV.4 Mackey-Glass Equation
Up to now we examined the relation between GID and GCD on systems with finite
dimension, for which their FD can be calculated by using methods based on
the true integrated vector of the system. A class of systems which is more
close to reality are systems with infinite dimensional dynamics. A delay
differential equation of the type
$$\frac{dx(t)}{dt}=F(x(t),x(t-\tau))$$
(28)
belongs to this class. In order to solve this kind of equation, one needs to
initiate $x(t)$ in the range $0\leq t\leq\tau$, and then, by assuming that the
system can be represented by a finite set of discrete values of $x(t)$, it is
possible to evaluate the system dynamics step by step. The solution is
considered
to be accurate if one converges to the same global properties, such as,
Lyapunov exponents and FD, by different methods [35],
and without dependence on the number of intervals to which the function is
divided [14]555Note that the dynamics which is calculated
by the use of different methods, or, equivalently, by a different number of
discrete integration intervals, does not have to behave identically. In fact,
one can expect similar dynamics if the system is not chaotic, but, if the
system is chaotic, it is sensitive to initial condition and to the integration
method, and even moreover, it is sensitive to integration accuracy
[36, 37]. However, the global properties of a system
converge to the same values, by using different integration methods and using
different integration accuracy [37].. Delay equations, such as
eq. (28), describe systems in which a stimulus has a delay response.
There are many practical examples from control theory, economics, population
biology, and other fields.
One of the known examples is the model of blood cell production in patients
with leukemia, formulated by Mackey and Glass [38] :
$$\dot{x}(t)=\frac{ax(t-\tau)}{1+[x(t-\tau)]^{c}}-bx(t).$$
(29)
Following refs. [35, 14], the parameter values are :
$a=0.1$, $b=0.2$, and $c=10$. We confine ourselves to $\tau=30.0$.
As in ref. [14], we choose the time series to be
$\{x(t),x(t+\tau),x(t+2\tau),\ldots\}$666In fact, the common
procedure of reconstructing a phase space from a time series is described in
eq. (1). According to this method, one has to build the reconstructed
vectors by the use of a jumping choice $m$ which is determined by the first
zero
of the autocorrelation function, or the first minimum of the mutual information
function [9]. However, we used the same series as in
ref. [14] in order to compare results., as well as the integration
method which is described in this reference. The length of the time series is
$N=131072$.
The FD ($D_{2}$) calculation of the Mackey-Glass equation is presented in
Fig. 7. The embedding dimensions are, $n=1\ldots 9$.
In Fig. 7a, the correlation function, $C_{2}(r)$ is
shown. One notices that there are two regions in each one of which
there is convergence to a certain slope. These regions are separate by a
dashed line. In Fig.
7b, the average local slope of Fig. 7a is shown. One can
identify easily two convergences, the first in the neighborhood of
$\sim 3.0$, and the second around $\sim 2.4$. Thus, there are two
approximations for the FD, $D_{2}$, pointing to two different scales. The first
approximation is similar to the FD that was calculated in ref. [14].
However, the GID graphs which are presented in Fig. 7c and d lead to
an FD, $D_{2}\sim 2.45$, which seems to be very close to the second convergence
of Fig. 7b. Notice that the convergence to $D_{2}\sim 2.4$ appears,
in both methods, in the neighborhood of the same box size ($\sim 2.1$).
V Summary
In this work we develop a new algorithm for calculation of a general
information, $I_{q}(n)$, which is based on strings sorting (the method of string
sort can be used
also to calculate the conventional GCD method). According to our algorithm,
one can divide the phase space into
255 parts in each hyper-box edge. The algorithm requires $O(N\log_{2}N)$
computations, where $N$ is the number of reconstructed vectors. A rough
estimate for the number of points needed for the FD calculation was given.
The algorithm, which can
be used in a regular system with known equations of motion, was tested
on a reconstructed phase space (which was built according to Takens theorem).
The general information graphs have non monotonic curves, which can be smoothed
by the requirement for minimum general information. We examine our algorithm
on some well known examples, such as, the Lorenz attractor, the Rossler
attractor, the
van der Pol oscillator and others, and show that the FD that was computed by
the GID method is almost identical to the well known FD’s of those systems.
In practice, the computation time of an FD using the GID method, was much less
than for the GCD method. For a typical time series with $32768$ data points,
the computation time needed for the GCD method was about nine times greater
than the
computation time of GID method (when we do not restrict ourselves to small
$r$ values and we compute all $N^{2}$ relations). Thus, in addition to the fact
that the algorithm developed in
this paper enables the use of comparative methods (which is crucial in some
cases), the algorithm is generally faster.
The author wishes to thank J. Levitan, M. Lewkowicz and L.P. Horwitz for very
useful discussions.
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Preservation of complete Baireness
Sergey Medvedev
South Ural State University, Chelyabinsk, Russia
[email protected]
Abstract.
The main result is the following.
Let $f\colon X\rightarrow Y$ be a continuous mapping of a completely Baire space $X$ onto a hereditary weakly Preiss-Simon regular space $Y$ such that the image of every open subset of $X$ is a resolvable set in $Y$. Then $Y$ is completely Baire.
The classical Hurewicz theorem about closed embedding of the space of rational numbers into metrizable spaces is generalized to weakly Preiss-Simon regular spaces.
Key words and phrases:
resolvable set, completely Baire space, weakly Preiss-Simon space, rational numbers
2010 Mathematics Subject Classification:
54H05, 03E15
One of the main tasks considered in topology is to find the properties of topological spaces that are preserved when moving to the image of the space with this property. For example, Sierpiński [11] and Vainštain [12] have established that if $Y$ is an image of a completely metrizable space $X$ under a closed mapping or an open mapping, then $Y$ is completely metrizable.
A natural question to ask is which maps preserve complete metrizability.
Ostrovsky [10] proved that if a continuous mapping from a zero-dimensional Polish (separable, completely metrizable) space $X$ onto a metrizable space $Y$ takes any clopen set $W\subset X$ to the union of an open set and a closed set, then $Y$ is completely metrizable. He raised the question whether the same is true when the images are the intersection of an open set and a closed set.
Gao and Kieftenbeld [4] showed that if the image under $f\colon X\rightarrow Y$ of every open set or every closed set in a Polish space $X$ is a resolvable set in $Y$, then $Y$ is Polish. Holický and Pol [6] established similar results for the nonseparable case. The latter result solves Ostrovsky’s problem for metrizable spaces.
It is notorious [3, Theorem 4.3.26] that a topological space is completely metrizable if and only if it is a Čech-complete metrizable space.
Holický [5, Corollary 3.3] proved among other things that if $f\colon X\rightarrow Y$ is a continuous mapping from a Čech-complete space $X$ onto a metrizable space $Y$ such that $f$ takes open sets in $X$ to resolvable sets in $Y$, then $Y$ is completely metrizable.
Let us pass now from a Čech-complete metrizable space to a completely Baire space.
Suppose we are given a continuous mapping $f\colon X\rightarrow Y$ from a zero-dimensional metrizable space $X$ onto a metrizable space $Y$ such that $f$ takes clopen sets in $X$ to resolvable sets in $Y$. Holický and Pol [6, Corollary 4] showed that if $X$ is completely Baire, then so is $Y$.
Our main Theorem 3 generalizes this statement to regular spaces.
In order to obtain it, we strengthen the Hurewicz theorem [7] about closed embedding of the space of rational numbers into metrizable spaces, see Theorem 1.
Notation. For all undefined terms see [3].
A set $E$ in a space $X$ is said to be resolvable [8, § 12] if it can be represented as
$$E=(F_{1}\setminus F_{2})\cup(F_{3}\setminus F_{4})\cup\ldots\cup(F_{\xi}\setminus F_{\xi+1})\cup\ldots,$$
where $\langle F_{\xi}\rangle$ forms a decreasing transfinite sequence of closed sets in $X$. Equivalently, $E$ is resolvable if, for every closed set $F\subset X$, the boundary of $F\bigcap E$ relative to $F$ is nowhere dense in $F$. Resolvable sets form a field of subsets of $X$ which contains closed sets.
A topological space $X$ is called a Baire space if the intersection of countably many dense open sets in $X$ is dense. We call a space completely Baire (or hereditarily Baire cf. [6]) if every closed subspace of it is Baire. By Hurewicz’s theorem [7], a metrizable space is completely Baire if and only if it contains no closed homeomorphic copy of rational numbers.
A topological space $X$ is called Preiss-Simon at a point $x\in X$ if for any subset $A\subset X$ with $x\in\overline{A}$ there is a sequence $\langle U_{n}\colon n\in\omega\rangle$ of nonempty open subsets of $A$ that converges to $x$ in the sense that each neighborhood of $x$ contains all but finitely many sets $U_{n}$. A space $X$ is called a Preiss-Simon space if $X$ is Preiss-Simon at each point $x\in X$. It is clear that every first countable space is Preiss-Simon and each Preiss–Simon space is Frèchet–Urysohn.
We shall say that a space $X$ is weakly Preiss-Simon at a point $x\in X$ if there is a sequence $\langle U_{n}\colon n\in\omega\rangle$ of nonempty open subsets of $X$ that converges to $x$. Denote by $wPS(X)$ the set of points $x\in X$ at which $X$ is weakly Preiss-Simon. A space $X$ is called weakly Preiss-Simon space if the set $wPS(X)$ is dense in $X$. A space $X$ is called hereditary weakly Preiss-Simon space if every nonempty closed subset of $X$ is a weakly Preiss-Simon space, i. e., the set $wPS(F)$ is dense in $F$ for every nonempty closed set $F\subset X$. Clearly, every Preiss-Simon space is hereditary weakly Preiss-Simon.
Lemma 1.
Let $F$ be a nowhere dense closed subset of a regular space $X$. Suppose $X$ is weakly Preiss-Simon at a non-isolated point $x$. Then for every neighborhood $O$ of $x$ there is a sequence $\langle U_{n}\colon n\in\omega\rangle$ of nonempty open sets such that:
(a)
$\langle U_{n}\colon n\in\omega\rangle$ converges to $x$,
(b)
$\overline{U_{n}}\subset O\setminus\{x\}$ and $\overline{U_{n}}\bigcap F=\emptyset$ for every $n\in\omega$,
(c)
$\overline{U_{n}}\bigcap\overline{U_{k}}=\emptyset$ provided $n\neq k$,
(d)
$\overline{\bigcup\{U_{n}\colon n\in\omega\}}=\{x\}\cup\bigcup\{\overline{U_{n}}\colon n\in\omega\}$.
Proof.
Take a sequence $\langle V_{k}\colon k\in\omega\rangle$ of nonempty open sets in $X$ that converges to $x$. The neighborhood $O$ of $x$ contains all but finitely many sets $V_{k}$. Choose the least number $k_{0}$ with $V_{k_{0}}\subset O$. The point $x$ is non-isolated in $X$. Then there is a point $x_{0}\in V_{k_{0}}\setminus(\{x\}\bigcup F)$ and its neighborhood $U_{0}$ satisfying $\overline{U_{0}}\subset V_{k_{0}}\setminus(\{x\}\bigcup F)$. Similarly, choose the least $k_{1}$ with $V_{k_{1}}\subset O\setminus\overline{U_{0}}$. Find a point $x_{1}\in V_{k_{1}}\setminus\{x\}$ and its neighborhood $U_{1}$ satisfying $\overline{U_{1}}\subset V_{k_{1}}\setminus(\{x\}\bigcup F)$. Clearly, $\overline{U_{0}}\bigcap\overline{U_{1}}=\emptyset$. Next, choose the least $k_{2}$ with $V_{k_{2}}\subset O\setminus(\overline{U_{0}}\bigcup\overline{U_{1}})$. Find a point $x_{2}\in V_{k_{2}}\setminus\{x\}$ and its neighborhood $U_{2}$ satisfying $\overline{U_{2}}\subset V_{k_{2}}\setminus(\{x\}\bigcup F)$, and so on. One can check that all conditions (a)–(d) hold.
∎
The following statement generalizes the Hurewicz theorem [7] about closed embedding of the space of rational numbers $\mathbb{Q}$ into metrizable spaces. Note that Theorem 1 was proved in [9] for a regular space $X$ of the first category with a dense first-countable subset.
Theorem 1.
Let $X$ be a regular space of the first category such that the set $wPS(X)$ is dense in $X$. Then $X$ contains a closed copy of the space of rational numbers.
Proof.
Let $X=\bigcup\{X_{n}\colon n\in\omega\}$, where each $X_{n}$ is a closed nowhere dense subset of $X$. Without loss of generality, $X_{0}\subset X_{1}\subset\ldots$.
By induction on the tree $\omega^{<\omega}$, we shall construct points $x_{t}\in wPS(X)$, open sets $U_{t}$, and sequences $\langle U_{t\hat{\,}k}\colon k\in\omega\rangle$ of open sets such that the following conditions hold for each $t\in\omega^{<\omega}$:
(a)
$U_{t}$ is a neighborhood of the point $x_{t}$,
(b)
$\overline{U_{t}}\bigcap X_{n}=\emptyset$ whenever $t\in\omega^{n+1}$,
(c)
$\overline{U_{t\hat{\,}k}}\subset U_{t}\setminus\{x_{t}\}$ for each $k\in\omega$,
(d)
the sequence $\langle U_{t\hat{\,}k}\colon k\in\omega\rangle$ converges to $x_{t}$,
(e)
$\overline{U_{t\hat{\,}i}}\bigcap\overline{U_{t\hat{\,}k}}=\emptyset$ provided $i\neq k$,
(f)
$\overline{\bigcup\{U_{t\hat{\,}k}\colon k\in\omega\}}=\{x_{t}\}\cup\bigcup\{\overline{U_{t\hat{\,}k}}\colon k\in\omega\}$.
The space $X$ has no isolated points as a set of the first category.
For the base step of the induction, take a point $x_{\emptyset}\in wPS(X)$ and $U_{\emptyset}=X$. By Lemma 1 with $F=X_{0}$ and $O=X$, there is a sequence $\langle U_{k}\colon k\in\omega\rangle$ of nonempty open sets satisfying conditions (a)–(f) for $n=0$.
Suppose the points $x_{s}$ and the sequences $\langle U_{s\hat{\,}k}\colon k\in\omega\rangle$ of open sets have been constructed for every $s\in\omega^{j}$ with $j\leq n$. Fix $s\in\omega^{n}$. The set $X_{n+1}$ is nowhere dense in the open set $U_{s\hat{\,}i}$ for every $i\in\omega$. Then we can find a point $x_{s\hat{\,}i}\in wPS(X)\bigcap(U_{s\hat{\,}i}\setminus X_{n+1})$. By Lemma 1 with $F=X_{n+1}$ and $O=U_{s\hat{\,}i}$, there is a sequence $\langle U_{s\hat{\,}i\hat{\,}k}\colon k\in\omega\rangle$ of nonempty open sets satisfying conditions (a)–(f) for $t=s\hat{\,}i$. This completes the induction step.
Clearly, the set $Z=\bigcup\{x_{t}\colon t\in\omega^{<\omega}\}$ is countable and has no isolated points. For every $t\in\omega^{<\omega}$ from conditions (d) and (e) it follows that the sequence $\langle V_{t\hat{\,}n}\colon n\in\omega\rangle$ forms a countable base at the point $x_{t}$ with respect to $Z$, where
$$V_{t\hat{\,}n}=Z\bigcap\left(U_{t}\setminus\cup\{\overline{U_{t\hat{\,}k}}\colon k<n\}\right).$$
Hence, $Z$ is a second-countable space. The Urysohn theorem [3, Theorem 4.2.9] implies that $Z$ is metrizable. According to the Sierpiński theorem [11] (see also [3, Exercise 6.2.A]), $Z$ is homeomorphic to the space of rational numbers.
It remains to verify that $Z$ is closed in $X$. For each $n\in\omega$ define the sets
$$\displaystyle Z_{n}$$
$$\displaystyle=\bigcup\left\{x_{s}\colon s\in\omega^{j},j\leq n\right\},$$
$$\displaystyle P_{n}$$
$$\displaystyle=\bigcup\left\{\overline{U_{t\hat{\,}k}}\colon t\in\omega^{n},k\in\omega\right\}.$$
Then $Z=\bigcup\{Z_{n}\colon n\in\omega\}$, $Z_{n}\cap P_{n}=\emptyset$, and $Z\subset Z_{n}\bigcup P_{n}$ for each $n$.
By induction on $n\in\omega$, let us check that the union $Z_{n}\cup P_{n}$ is closed in $X$ for each $n$.
For $n=0$, the set $Z_{0}\cup P_{0}$ is closed in $X$ by Lemma 1. We now show that $Z_{n+1}\cup P_{n+1}$ is closed under the induction assumption on closeness of $Z_{n}\cup P_{n}$. Suppose that, on the contrary, there is a point
$$x\in\overline{(Z_{n+1}\cup P_{n+1})}\setminus(Z_{n+1}\cup P_{n+1}).$$
Since $(Z_{n+1}\setminus Z_{n})\cup P_{n+1}\subset P_{n}$, we have $x\in\overline{Z_{n}\cup P_{n}}=Z_{n}\cup P_{n}$. Then $x\in P_{n}\setminus Z_{n+1}$.
From (c) and (e) it follows that there is a unique index $t\hat{\,}k\in\omega^{n+1}$ with $x\in\overline{U_{t\hat{\,}k}}\subset P_{n}$. According to (f), we have
$$\overline{P_{n+1}}\cap\overline{U_{t\hat{\,}k}}=\{x_{t\hat{\,}k}\}\cup\bigcup\left\{\overline{U_{t\hat{\,}k\hat{\,}i}}\colon i\in\omega\right\}.$$
By (e), $x\in\overline{U_{t\hat{\,}k\hat{\,}j}}$ for some $j$. Then $x\in P_{n+1}$, a contradiction.
Thus, the set $Z_{n}\cup P_{n}$ is closed in $X$ for each $n$.
By construction, $Z\subset Z_{n}\cup P_{n}$ for each $n$. Then $\overline{Z}\subset\bigcap\{Z_{n}\cup P_{n}\colon n\in\omega\}$.
Striving for a contradiction, suppose there is a point $x\in\overline{Z}\setminus Z$. Fix the least $i$ with $x\in X_{i}$. Obviously, $x\notin Z_{i}$. Condition (b) implies that $P_{i}\cap X_{i}=\emptyset$. Then $x\notin Z_{i}\cup P_{i}$, a contradiction.
∎
Let us recall the well-known theorem of van Douwen [2]. For a first countable regular space $X$, he proved that every closed subspace of $X$ is a Baire space if and only if $X$ has no countable closed crowded subspace. Here a space is said to be crowded if it has no isolated points. Note that up to homeomorphism the space of rational numbers is the only countable first-countable crowded regular space [11]. In other words, for a first countable regular space $X$, van Douwen proved that $X$ is completely Baire if and only if the space of rational numbers does not embed as a closed subspace into $X$.
Now we obtain a generalization of this result.
Theorem 2.
A hereditary weakly Preiss-Simon regular space $X$ is completely Baire if and only if the space of rational numbers does not embed as a closed subspace into $X$.
Proof.
Clearly, if $X$ contains a closed copy of the space of rational numbers, then it cannot be completely Baire. For the other direction, aiming for a contradiction, suppose that $X$ is not completely Baire. Then $X$ contains a closed subset $F$ of the first category. It remains to apply Theorem 1 to $F$.
∎
Lemma 2.
Let $f\colon X\rightarrow\mathbb{Q}$ be a continuous mapping of a Baire space $X$ onto the space of rational numbers $\mathbb{Q}$.
Then there is an open set $U\subset X$ such that $f(U)$ is not resolvable.
Proof.
Denote by $X^{\circ}_{q}$ the interior of $f^{-1}(q)$ for every point $q\in\mathbb{Q}$. Put $\mathbb{Q}^{\circ}=\{q\in\mathbb{Q}\colon X^{\circ}_{q}\neq\emptyset\}$.
Let us verify that the open set $A=\bigcup\{X^{\circ}_{q}\colon q\in\mathbb{Q}^{\circ}\}$ is dense in $X$. Suppose the contrary and choose a nonempty open set $W\subset X$ missing $A$. For each point $q\in\mathbb{Q}\setminus\mathbb{Q}^{\circ}$ the intersection $f^{-1}(q)\cap W$ is a closed nowhere dense subset of $W$. For each point $q\in\mathbb{Q}^{\circ}$ the intersection $f^{-1}(q)\cap W=(f^{-1}(q)\setminus A)\cap W$ is the same. This implies that $W$ is a set of the first category on itself. On the other hand, $W$ is a Baire space as an open subset of the Baire space $X$, a contradiction.
Then $\mathbb{Q}^{\circ}=f(A)$ is dense in $\mathbb{Q}$ because $f$ is continuous.
Take $B\subset\mathbb{Q}^{\circ}$ such that $B$ is dense, co-dense in $\mathbb{Q}$. Clearly, $B$ is not a resolvable subset of $\mathbb{Q}$. The open set $U=\bigcup\{X^{\circ}_{q}\colon q\in B\}$ is as required.
∎
Now we are ready to give the main result of the paper.
Theorem 3.
Let $f\colon X\rightarrow Y$ be a continuous mapping of a completely Baire space $X$ onto a hereditary weakly Preiss-Simon regular space $Y$ such that the image of every open subset of $X$ is a resolvable set in $Y$. Then $Y$ is completely Baire.
Proof.
Striving for a contradiction, suppose that $Y$ is not completely Baire. By Theorem 2, $Y$ contains a closed copy $Q$ of the space of rational numbers.
The inverse image $f^{-1}(Q)$ is a Baire space as a closed subset of $X$. By Lemma 2, there is an open in $f^{-1}(Q)$ set $U\subset f^{-1}(Q)$ such that $f(U)$ is not resolvable. Take an open set $U^{*}\subset X$ with $U^{*}\bigcap f^{-1}(Q)=U$. Then $f(U^{*})$ is not resolvable because the closed subset $Q\bigcap f(U^{*})=f(U)$ of a resolvable set must be resolvable. A contradiction.
∎
References
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Resolvable maps preserve complete metrizability, Proc. Amer. Math. Soc., 138:6 (2010) 2245–2252.
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On closed mappings, Moskov. Gos. Univ. Uč. Zap. 155:5 (1952) 3–53 (in Russian). |
[
[
${}^{1}$Princeton University, Princeton, USA,
${}^{2}$Kapteyn Institute, Groningen, The Netherlands,
${}^{3}$University of Washington, Seattle, USA,
${}^{4}$Max-Planck-Inst. für Astroph., Garching, Germany
(#1; ?? and in revised form ??)
Abstract
We summarize the detection rates at wavelengths other than optical for $\sim$99,000
galaxies from the Sloan Digital Sky Survey (SDSS) Data Release 1 “main”
spectroscopic sample. The analysis is based on positional cross-correlation
with source catalogs from ROSAT, 2MASS, IRAS, GB6, FIRST, NVSS and WENSS
surveys. We find that the rest-frame UV-IR broad-band galaxy SEDs form a remarkably
uniform, nearly one parameter, family. As an example, the SDSS $u$ and $r$
band data, supplemented with redshift, can be used to predict $K$ band magnitudes
measured by 2MASS with an rms scatter of only 0.2 mag; when measurement uncertainties
are taken into account, the astrophysical scatter appears not larger than $\sim$0.1 mag.
††volume: #1\checkfont
eurm10
\checkfontmsam10
The Interplay among Black Holes, Stars and ISM in Galactic
Nuclei] Multiwavelength View of SDSS Galaxies
M. Obrić et al.]M. Obrić${}^{1,2}$
Ž. Ivezić${}^{1,3}$, R.H. Lupton${}^{1}$, G. Kauffmann${}^{4}$, G.R. Knapp${}^{1}$, J.E. Gunn${}^{1}$,
D. Schlegel${}^{1}$, M.A. Strauss${}^{1}$, S. Anderson${}^{3}$, et al.
2004
222
\pagerange1–8
\jnameThe Interplay among Black Holes, Stars and ISM
in Galactic Nuclei
\editorsTh. Storchi Bergmann, L.C. Ho & H.R. Schmitt, eds.
1. The fractions of SDSS galaxies detected at other wavelengths
We report initial results from a program aimed at the characterization of
multiwavelength properties of galaxies detected by SDSS. The details about the
positional cross-correlation method, and analysis of matched samples, will be
described elsewhere (Obrić et al. 2004); here we summarize the detection
rates and demonstrate a tight correlation between the UV and IR colors
of galaxies.
Table 1 lists the detection rates for galaxies from SDSS “main” spectroscopic
sample (a flux-limited sample, $r_{Pet}<17.77$, for details see Strauss et al.
2002) by ROSAT, 2MASS, IRAS, GB6, FIRST, NVSS and WENSS surveys. Galaxies
are separated into those with and without emission lines (using line measurements
and classification criteria from Kauffmann et al. 2003), and emission line
galaxies are further separated, using Baldwin-Phillips-Terlevich diagram,
into “star-forming”, “AGNs”, and “unclassified”. The detection rate of
SDSS galaxies is below 10% in all surveys except 2MASS.
2. Predicting 2MASS K flux from UV/optical SDSS flux
Galaxies form a nearly one-dimensional sequence in various optical
color-color diagrams (the correlation among colors is especially tight
for rest-frame colors, with a scatter of only $\sim$0.03 mag perpendicular
to the locus, Smolčić et al. 2004). We report here that this one-dimensionality
of galaxy spectral energy distributions (SEDs) extends to 2MASS wavelengths:
it is possible to predict 2MASS $K$ band flux with an rms of 0.2 mag
using SDSS $u$ and $r$ band measurements111We use UV/visual fluxes
($u$ and $r$ bands) to predict near-IR flux because this is the “hardest”
wavelength combination with most astrophysical implications (according to
“common wisdom”, such a relationship should not be very accurate due to
the effects of starbursts and dust extinction); predicting, for example,
2MASS $J$ band flux from SDSS $z$ band flux is trivial because these two
bands are adjacent in wavelength space..
The SDSS-based $K$ magnitude prediction is determined from $K_{SDSS}=r_{Pet}-(r-K)$, where $r_{Pet}$ is the SDSS $r$ band Petrosian
magnitude (Strauss et al. 2002), and $(r-K)$ is a best fit to the
observed $r-K$ colors for SDSS/2MASS galaxies.
It depends on $u-r$ color
(essentially a position along the Hubble sequence), and on redshift,
to account for
K-correction effects (we use default 2MASS magnitudes and SDSS model colors):
only2pages…
$\phantom{mala}(r-K)=A+B\,(u-r)+C\,(u-r)^{2}+D\,(u-r)^{3}+E\,z+F\,z^{2}\phantom%
{mirelajezlocesta}(2.1)\\
$
where (A, B, C, D, E, F)=(1.115, 0.940, -0.165, 0.00851, 4.92, -9.10),
and $z$ is redshift. This fit predicts 2MASS $K$ band magnitudes with an
rms scatter of 0.20 mag, which depends on neither color nor redshift, and
is nearly Gaussian (see Figure 1).
To correct for aperture and resolution effects
between SDSS and 2MASS, that presumably depend on galaxy profile, or nearly
equivalently on galaxy color (Strateva et al. 2000), we add to $(r-K)$
($0.496-0.154\,R_{50}^{z}$) for galaxies with $u-r<2.22$ and
($0.107-0.045\,R_{50}^{z}$) for redder galaxies, where $R_{50}^{z}$ is the
radius containing 50% of Petrosian flux in the $z$ band.
This correction has a negligible effect on rms scatter, and only removes a correlation
of $K_{SDSS}-K_{2MASS}$ residuals with galaxy size.
The median residuals, as a function of $u-r$ and $z$, do not exceed 0.03 mag.
The rms scatter decreases to 0.15 mag at the bright end
($K<12$). Given typical measurement errors in $u$, $r$, $R_{50}^{z}$ and
$K$, we conservatively conclude that the true astrophysical scatter of
$K$ band magnitudes predicted from the blue part of SED is less than $\sim$0.1
mag. Similarly, the relation $(J-K)=2.172\,z+0.966$, where $z$ is
redshift, predicts $J-K$ measured by 2MASS with an rms scatter of 0.11
mag (0.07 mag at the bright end), and no significant residuals with
respect to $K$, $u-r$ and redshift. These tight correlations demonstrate
remarkable one-dimensionality of galaxy spectral energy distributions from
UV to IR wavelengths.
References
[Kauffmann et al.(2003)]
Kauffmann, G., Heckman, T.M., White, S.D.M.,
and 19 co-authors 2003, MNRAS 341, 33
[Obrić et al.(2004)]
Obrić, M., Ivezić, Ž., Kauffmann, G.
and 15 co-authors 2004, MNRAS, in prep.
[Smolčić et al.(2004)]
Smolčić, V., Ivezić, Ž.,
Gaćeša, M., and 14 co-authors 2004, MNRAS, in prep.
[Strateva et al.(2001)]
Strateva, I., Ivezić, Ž., Knapp, G.R.,
and 26 co-authors 2001, AJ 122, 1861
[Strauss et al.(2002)]
Strauss, M.A., Weinberg, D., Lupton, R.H.,
and 36 co-authors 2002, AJ 124, 1810 |
\articleNo
301
\volnumber1
\hauthor
M.Yu. Skulskyy
11institutetext:
Lviv Polytechnic National University, Department of Physics, 79013, Lviv, Ukraine 11email: [email protected]
Formation of magnetized spatial structures in the Beta Lyrae system
III. Reflection of magnetically controlled matter in circumbinary structures in helium lines, in particular arising from metastable levels
M.Yu. Skulskyy
(2021; October 7, 2020; November 3, 2020; October 7, 2020)
Abstract
Spatial gaseous structures in the Beta Lyrae system have been studied with the fact of change in the longitudinal component of the donor’s magnetic field during the orbital period in mind. The investigation was based primarily on the study of the dynamics of the circumstellar structures surrounding the binary system as a whole. The special emphasis was placed on the study of complex helium lines, in particular those arising from metastable levels. A number of different observable facts from the ultraviolet to the red spectral region were analyzed. The configuration of the donor magnetic field is a factor that not only enhances mass transfer and influences the formation of spatial gas structures between stellar components but, to some extent, also affects the outflow of matter and the formation of external gas structures around this interacting binary system. Together with previous articles (Skulskyy, 2020a, b), the pieces of evidence of this work, confirming the reflection of magnetically controlled matter in circumbinary structures, define the basis for a coherent picture of the mass exchange between components and outflows of matter outwards.
keywords: binaries: individual: Beta Lyrae – emission-line: magnetic field: mass-loss
††volume: 51
1 Introduction
In previous papers, the focus was on the research of the relationship between the structure of the donor magnetic field and its reflection in the characteristic physical parameters of the visible and infrared Beta Lyrae spectra (Skulskyy, 2020a, b). Taking into account the spatial configuration of the donor magnetic field, the emphasis was put on certain facts of its reflection in the moving magnetized accretions structures between the stellar components of this binary system. The data of absolute spectrophotometry and spectral analysis, changes along with the orbital phases of the radial velocities and intensities of complex emission-absorption lines were examined, primarily based on the own spectral and spectrophotometric observations in the visual range spectrum. This was due to a series of conscious and consistent spectral observations and relevant studies conducted mainly in 1980-1995, after the discovery of the donor magnetic field. Such a method of approach has already made it possible to make certain generalizations and reflect moving magnetized structures as the original phenomenon of mass transfer, which is inherent to this binary system.
It is clear that by the time when nothing was known about the donor’s magnetic field and its possible effect on the physical conditions in the near and far gas structures of this binary system, a number of significant scientific studies had been completed. Some physical characteristics and certain parameters of radiating gas structures near both stellar components or in the common shell were often described by the known paradigms, which were later rejected or modified. This is especially true of researchers who have studied the spectral lines of the shell and the characteristics of the spectrum in the far ultraviolet region. A large number of scientific materials are available for their coverage and further rethinking. In this article, special attention is paid to the scientific works in order to investigate the influence of the magnetic field on the dynamics of the circumstellar structure surrounding this binary system as a whole. Clearly, to create a coherent picture, such studies should be conducted in comparison with the already obtained results of the investigation of the magnetized gas structures both near the donor and the gainer and between them.
The previous researches of Skulskyy (2020a, b) demonstrate an efficient collision of magnetized plasma in the phases of observation of magnetic poles on the donor surface. These shock collisions with the accretion disk in the phases of the secondary quadrature, in which the high-temperature environment and the system of formed accretion flows are observed, are especially noted. There are obvious correlations between the phase variability of the donor magnetic field and the corresponding variability of dynamic and energy characteristics of different complex lines, but above all the strongest emission lines H${}_{\alpha}$ and He I $\lambda$ 7065. Moreover, the phase boundaries of the location of the magnetic pole on the surface of the donor, above which the matter outflows are formed, were discovered. The additional loss of matter from the donor surface is observed mainly from the donor surface in the phase region round 0.855 P, i.e., when the magnetic field pole is facing the gainer. This allowed us to predict certain correlations between the behavior of the above strong lines, as well as certain helium lines formed in shell gas structures extending outward from the binary system. Therefore, in this article, closer attention is paid to such lines of helium, primarily those arising from metastable levels.
It should be recalled that the donor’s magnetic field changes significantly during the orbital period (which is close to 12.94 d); phase changes in the range from zero to one are tied to the main eclipse of this binary system in the visible spectral region when the more massive accretor obscures of the bright donor; the schematic model of the Beta Lyrae system and the picture of mass transfer are shown in Fig.1 of Skulskyy (2020a), which should be kept in mind for a better understanding of the geometry and physical properties of this interacting binary system in further analysis; as this article is a continuation of a number of ongoing comprehensive studies of the proposed topic, references to previous results are quite natural.
2 Magnetic field and gaseous structures surrounding the binary system
2.1 Magnetic field and the He I lines arising from metastable levels around the visual spectrum
The gas shell surrounding the Beta Lyrae system was discovered long ago and its structure has always been studied on the basis of helium lines originating from metastable levels. First of all, it concerned the strongest emission-absorption line of the He I $\lambda$ 3888 in the violet spectrum. Well-known traditional studies of the broadening shell in the complex line of He I $\lambda$ 3888 (e.g. Struve, 1941; Sahade et al., 1959) were subsequently extended through the study of this shell in the infrared line of the He I $\lambda$ 10830 of this triplet series (e.g. Girnyak et al., 1978). It should draw attention that the contours of these lines contain narrow and deep absorption components with negative radial velocities in all orbital phases. If they exceed the parabolic velocities of moving microparticles in the binary system, it has traditionally been believed that the structural components of this shell, in which such lines are formed, move outward from this binary system. The He I $\lambda\lambda$ 3888, 10830 lines have a metastable level of $2^{3}$S. This level is highly populated with electrons and the effects of collision are crucial for the manifestation of these lines in the spectrum. Indeed, absorption components of these triplet lines of helium are well visible in the diluted shell structure around this binary system. At the same time, these helium lines have a very strong emission component in their contours, in many respects similar to those in lines H${}_{\alpha}$ and He I $\lambda$ 7065 (see Skulskyy, 2020b), where clear intercorrelations were discovered between the phase variability of the donor’s magnetic field and dynamic and energy characteristics of these strong emission lines. This may indicate the general nature of the formation and spatial localization of all these emissions. It can be supposed that the absorption in the He I $\lambda\lambda$ 3888, 10830 lines, which cuts through their emission component, arises as self-absorption and should reflect the specificity of the matter loss from the donor surface, which, showing certain structural features of the magnetic field, reaches the Roche cavity. The advanced attempt to re-study the characteristic spectral features of these and other He I lines, which arise from metastable levels, in combination with the corresponding manifestation in the spectrum of the binary system of the donor magnetic field configuration presents new opportunities in this study.
2.2 Magnetic field and He I lines arising from metastable levels in the violet spectrum
2.2.1 The above question and known article by Sahade et al. (1959)
Based on the concept of the existing configuration of the donor’s magnetic field, the careful work of Sahade et al. (1959) can be considered as one of the most consistent and informative studies of the He I $\lambda$ 3888 line. The results of this investigation are based on studies of 195 plates with a dispersion of 10 Å/mm obtained from observations of Beta Lyrae on the Mount Wilson 100-inch reflector over 12 orbital cycles in the 1955 season (the article presents an atlas of photographic spectrograms in the region $\lambda\lambda$ 3680-4580). However, only in 4-7 cycles in June-July they were performed most fully to be suitable for a detailed analysis. In Fig. 1, which is a copy of Fig. 13 from the article of Sahade et al. (1959), “radial velocity measurements from the shell line at He I $\lambda$ 3888” are presented. As it is seen at the bottom of Fig. 1, where the two absorption components “with very definite” velocities of about -170 km/s and -130 km/s are shown, they converge to one absorption component near the 0.85 P phase of the visibility of the magnetic pole on the donor surface. These absorption components demonstrate very clear sequences over the orbital phases, and, according to the understanding of Sahade et al. (1959), their formation “must occur in a shell that surrounds the whole system”. Their radial velocities are the largest and differ in curves shape from such curves of radial velocities of the absorption components that are at the top of Fig. 1. The latter more resemble the absorption components superimposed on emissions in lines H${}_{\alpha}$ and He I $\lambda$ 7065 (reflecting the gas flows between the donor and the gainer; see Section 2.3 in Skulskyy (2020b)).
It is well known that the He I $\lambda$ 3888 line also has an emission component, which consists only of the very high red peak in contrast to the two-peak emission, with superimposed absorption, at strong lines H${}_{\alpha}$ and He I $\lambda$ 7065 (the latter are also above the continuous spectrum in all orbital phases). The violet peak in the He I $\lambda$ 3888 line is virtually absented owing to deep violet absorption, showing appreciable changes in observations of different investigators (see, for example, Skulsky & Kos, 2011). Fig. 2, which is a copy of Fig. 18 from the article of Sahade et al. (1959), illustrates the radial velocities curve from the red emission peak at the He I $\lambda$ 3888 line. One sees two maxima at the 0.35 P and 0.85 P phases, i.e., in the phases of visibility of the magnetic field poles on the donor surface (the small local maximum in the 0.6 P phase may reflect the collision of the gas flow with the accretion disk). The similar two maxima are observed on the radial velocity curve of the red peak at the He I $\lambda$ 4472 line (see Fig. 19 in Sahade et al. (1959)). The red peaks of these helium lines of the violet spectrum exhibit the same behavior as red peaks at the H${}_{\alpha}$ and He I $\lambda$ 7065 lines. The spectral features of all components at the H${}_{\alpha}$ and He I $\lambda$ 7065 lines were considered in detail on the basis of Figures 5 and 6 in Skulskyy (2020b). These maxima at the 0.35 P and 0.85 P phases match clearly the phases of the two maxima on the curve of the effective magnetic field strength of the donor (see also Fig. 1 in Skulskyy, 2020b). Being synchronized in orbital phases, all these emission components have clear similarities, their physical nature and spatial interconnection seem indisputable. It can be assumed that the formation of red peaks in these lines, including such for the line He I $\lambda$ 3888, should be associated with the moving radiation near the donor surface, close to the poles of the magnetic field on this surface. This corresponds to the conclusions of Skulskyy (2020b) on the study of dynamics of emission components in the H${}_{\alpha}$ and He I $\lambda$ 7065 lines.
The dynamic behavior of both the above absorptions of the He I $\lambda$ 3888 line in Sahade et al. (1959), which are shown at the bottom in Fig. 1, is even more interesting. Their double nature has not been so clearly defined or repeated by investigators in other studies (see, for example Harmanec & Scholz (1993); Skulsky & Kos (2011), which analyzed such studies from the early 20th century). It should be noted that the radial velocity of the center of mass of this binary system, as determined from the interstellar Ca II lines by Sahade et al. (1959), is -15.2 km/s. The $\gamma$ - velocity determined in Table 9, for the set of orbital elements of Beta Lyrae computed on the basis of all 192 spectral plates in the summer of 1955, is -16.0 km/s. The set of orbital elements, calculated from observations taken between 0.85 P and 0.15 P phases, i.e., with the data discarded during the main (or primary) eclipse, showed -15.0 km/s for the -velocity. That is, both absorption components with radial velocities of approximately -170 and -130 km/s, should be considered as components with average velocities of about -155 and -115 km/s. Modern stellar masses of this binary system and the inclination of the orbit (see in Skulskyy, 2020a) allow affirming that the parabolic velocity of the moving gas particles reaches a value slightly smaller, but close to -115 km/s, which is sufficient to reach the Roche cavity with the Lagrange point L2. A simple calculation also shows (Burnashev & Skulskij, 1991)
that the velocity of the particles, which is sufficient to reach the Roche cavity with the Lagrange point L3, is near to -160 km/s. That is, the absorption component with an average radial velocity of approximately -155 km/s, or its possible greater values, may indicate a mass loss in a wider range of directions, in particular in the direction of the Lagrange points L4 and L5.
Here it is important to emphasize that Sahade et al. (1959) themselves noted: “there are striking changes with phase and cycle in the spectrum” and “the great complexity of the spectrum of Beta Lyrae makes it impossible to give a description which would do justice to the wealth of information contained in the material”. Indeed, at some points, this spectral material is presented as a general description. Now it is possible to consider the data of the observation in the light of modern ideas and concepts, in a more detailed manner. The behavior of its identified two absorption components of the He I $\lambda$ 3888 line should be investigated as carefully as possible, supplementing the analysis of radial velocities, given primarily in Table 3 from Sahade et al. (1959). But here there are also several other lines of helium, in particular those arising from metastable levels, which on the basis of tabular data and graphs will help to significantly expand the scope of research.
Based on Table 3, it is possible to estimate the behavior of the absorption of the He I $\lambda$ 3888 line with a smaller averaged radial velocity. In most orbital phases, Table 3 shows measurements of its radial velocities in the range of 125-135 km/s. This corresponds to a matter outflow velocity of about -115 km/s relative to the center of gravity of the binary system, i.e., with the first parabolic velocity. The most “striking changes with phase and cycle in the spectrum” are visible in the phases (0.95-0.13) P within a wide primary eclipse (see Fig. 1), when the donor is in the process of the eclipse by the gainer hidden in the accretion disk. At the time, according to the data of Table 3, the radial velocity of this absorption, during several days of observations at the boundary from 6 to 7 and from 11 to 12 orbital cycles, is observed in the range from -95 to -105 km/s, i.e., smaller than the parabolic velocity. The significant decrease in the radial velocity of this absorption component at the boundary of these orbital cycles indicates that the gas, flowing in the direction of the gainer in its Roche cavity, can only reach nearer structures of the accretion disk. But this is not observed in the same phases of the zero cycle and at the boundary of 4 to 5 orbital cycles, where the averaged radial velocity is approximately -115 km/s relative to the center of gravity. This indicates that the flows of matter during mass transfer are not laminar and have a variable nature within several cycles. Because in both events the motion of matter is visible above the accretion disk, i.e., the observer behind the gainer records a significant elevation of the moving matter above the plane of the orbit, this determines a considerable height of the rim of this disk from the side of the donor. As shown in Skulskyy (2020b), the vertical component of the gaseous flow in the direction of the gainer can be formed due to the configuration of the donor’s magnetic field, in which the magnetic axis is directed at a significant angle to the plane of the orbit of this binary system.
It should be noted that in phases (0.95-0.13) P of the main eclipse, the more negative absorption component in the He I $\lambda$ 3888 line also shows the various radial velocities in different cycles. In phases (0.95-0.99) P on June 8 of the 4th cycle and on July 4 of the 6th cycle this absorption component shows the average radial velocity of -173 km/s (or -158 km/s relative to the center of gravity), reflecting the motion of matter over the accretion disk, which approaches the velocity of its exit outside the Roche cavity with the Lagrange point L3. The dynamics of gaseous flows recorded in phases (0.03-0.13) P of the 5th cycle when measuring the radial velocity of this absorption component on 8 spectrograms on June 9 and 16 spectrograms on June 10? is most intriguing. In phases (0.03-0.05) P of the first date, the average radial velocity was -170 km/s with its growth within 5 km/s, however, in phases (0.11-0.13) P on June 10, its average the velocity was -180 km/s (on the first 4 plates it was -175 km/s, on the next 4 plates in phase near 0.12 P has reached the average velocity of -185 km/s, then it was -181 km/s and -180 km/s, respectively). Confident achievement of the radial velocity of -185 km (-170 km/s relative to the center of gravity) indicates that in this direction, the velocity of the particles was sufficient for the free escaping of the matter from the binary system with the intersection here of the boundary of the Roche cavity with the Lagrange point L3. But in the phases (0.03-0.05) P on July 5 and (0.12-0.14) P on July 6 of the 7th cycle, the average radial velocity was -171 km/s, which is much smaller than the velocity of -185 km/s in the 0.12 P phase on June 10. Moreover, for example, in the phases (0.987-0.002) P at the boundary of cycles 11 and 12, this absorption component of the He I $\lambda$ 3888 line revealed a significantly lower average radial velocity of -163 km/s (or only -148 km/s relative to the center of gravity). That is, passing over the center of the accretion disk, the moving matter with such velocity, most likely, must be captured by the outer side of the revolving accretion disk. Therefore, the more negative absorption component of the He I $\lambda$ 3888 line also indicates that the matter flows during mass transfer are not laminar and have a variable character during one or several cycles.
For comparison of the behavior of the He I $\lambda$ 3888 line during phases of the main eclipse, it makes sense to evaluate the behavior of other helium absorption lines arising from metastable levels. Sahade et al. (1959) measured the radial velocities from He I $\lambda\lambda$ 3614, 3965 lines of the singlet series. The radial velocities of their absorption components, as well as for the He I $\lambda$ 3888 line, are listed in Table 3 and shown in Fig. 3 (as a copy of Fig. 14 from the article Sahade et al. (1959)). In almost all phases, the more negative component of these singlet lines forms a successive curve with an average radial velocity close to -115 km/s relative to the center of gravity (i.e., close to the first parabolic velocity) with complex deviations from it in phases (0.95-0.13) P. This also applies to phases (0.95-0.05) P around the middle of the main eclipse both for the zero’s cycle (April 5 as the first observation date in 1955) and at the boundary of the 11th and 12th cycles (September 7 as the last observation date). But an important difference in the variability of these lines around the middle of the main eclipse concerns the observation periods of June 8-10 at the boundary of the 4th and 5th cycles and July 4-5 at the boundary of the 6th and 7th cycles. It is in these June and July observations that two narrower absorption features in these singlet helium lines are clearly distinguished. The development of events should be considered from the end of the 4th cycle, noting that on June 7 in the phases (0.88-0.90) P of direct visibility of the magnetic pole on the donor surface there was the average radial velocity of this absorption of about -115 km/s relative to the center of gravity. A clear rise of the radial velocity of the above absorption at its increment from -130 km/s to -135 km/s for 8 spectrograms occurred on June 8 within the phases (0.96-0.98) P. On June 9, within the phases (0.03-0.05) P, this absorption component during 8 spectrograms reached the radial velocity from -146 km/s for the first 4 spectrograms to -156 km/s for the next 4. Such velocity of moving matter is close to the second parabolic velocity. However, on June 10, within the phases (0.11-0.13) P, this average radial velocity on all 16 spectrograms fell slightly to -148 km/s with respect to the center of gravity. For the first time, these spectrograms clearly manifested the second component of absorption that had an average radial velocity of -97 km/s, somewhat smaller than the parabolic velocity (near -115 km/s relative to the center of gravity). On July 4 of the 6th cycle, 8 spectrograms within the phases (0.96-0.99) P recorded an average radial velocity of -155 km/s for the first absorption component and -98 km/s for the secondary one. On July 5, 7 spectrograms within phases (0.04-0.06) P of the 7th cycle recorded an average radial velocity of -151 km/s for the first absorption component and -98 km/s for the secondary one. However, the next night, July 6, unexpected changes were recorded: all 12 spectrograms in phases of (0.12-0.14) P showed that the radial velocity of the first absorption component dropped to -105 km/s, and the second absorption component disappeared regarding the measurement of radial velocities.
Thus, in the June and July observations, these singlet helium lines, becoming double in certain phases of the main eclipse, reveal asynchronous differences. Between the two orbital rotations from June 10 to July 6, in the same phases (0.11-0.14) P, the spectrum of the binary system represents very different expressed sets of both absorption components of these lines. If on June 10 both components of these lines became clearly double at radial velocities of -148 km/s and -98 km/s, then after two orbital cycles on July 6 the first of the two absorption components diminished its velocity to -105 km/s, i.e., slightly less than the parabolic velocity, but the second absorption component was absent. Moreover, if in the 5th cycle in June both components of these lines with velocities of -148 km/s and -98 km/s became visible only in the phases (0.11-0.13) P, then in the 6th cycle similar radial velocities -155 km/s and -98 km/s of both these components were measured in phases (0.98-0.99) P on July 4 and close to such velocities -151 km/s and -98 km/s on July 5 (in the 7th cycle) in phases (0.04-0.06) P (the next night on July 6, in phases (0.11-0.14) P the first absorption component had the radial velocity only -105 km/s, and the second absorption component unexpectedly disappeared and was invisible on the following nights of this 7th cycle). Hence, it should be noted that within the main eclipse the He I $\lambda\lambda$ 3614, 3964 lines show two phase segments of significant changes in their intensity and dynamics, which may indicate disordered changes in the density of the accretion disk. They are grouped close to the 0.95 P and 0.05 P phases, which, reflecting the formed dynamically changing structures of the disk, correspond simultaneously to the phases of the maximum manifestation of satellite lines (see Skulskyy (2020a) and more fully Skulskij (1992)). The third region of such changes is formed by the denser matter flow in the phases (0.10-013) P, which opens after the center of the eclipse.
Comparing the changes in the radial velocities in the absorption components of singlet and triplet helium lines arising from metastable levels, it should be especially noted the observation on June 10 in the phase near 0.12 P. In the He I $\lambda$ 3888 line, there was recorded the radial velocity -170 km/s relative to the center of gravity, which is sufficient for the motion matter from the binary system in any direction, including its exit outside the Roche cavity with the Lagrange point L3. At the same time, for the first time, a clear bifurcation of He I $\lambda\lambda$ 3614, 3964 lines (of the singlet series) into two absorption components with radial velocities comparable to two parabolic velocities was recorded. It is also important that Table 4 and Fig. 15 in Sahade et al. (1959) show that these observations on June 10 recorded on all 16 spectrograms the strongly shifted radial velocities for absorptions, which cut through emissions of H${}_{\gamma}$ and He I $\lambda\lambda$ 4472, 4026 lines. Shifts in the position of these absorptions make up -151 km/s (as the average on the first eight spectrograms) and -141 km/s (on the last eight of them) relative to the center of gravity of the binary system. A similar phenomenon was recorded only during the zero’s cycle on April 6 in phase 0.10 P, when measurements of these lines showed on the obtained spectrograms the radial velocity of -157 km/s relative to the center of gravity of the binary system, at clear excess of the second parabolic velocity, sufficient to an outflow matter via stellar wind from the Lagrange point L3.
The events described above, recorded in the behavior of the radial velocities of the absorption components of helium and hydrogen lines in the violet spectrum, show, within the main eclipse, not laminar but to some extent an unpredictable sharply variable mass transfer between the components of the binary system. In close phases of different dates, the observer behind the mass gainer registers the approach of the matter from the donor to the gainer at different radial velocities above the surface of the accretion disk. It is clear that when such velocities of a matter are slightly smaller than the first parabolic velocity, this matter settles down on the surface of the accretion disk, from the donor side. When the radial velocities of the moving substance exceed the first parabolic velocity or are a significant fraction of the second parabolic velocity, such gas flows can be captured directly by the gainer or the outer parts of the accretion disk. Repeatedly recorded velocities of matter surpassing the second parabolic velocity confirm the direct exit of matter beyond the close circular structures of the binary system and the possibility of forming a distant shell. This agrees with the observations of the radio-nebula, which surrounds the Beta Lyrae system, indicating the substantial non-conservative mass-loss during the evolution of this interacting system (Umana et al., 2000). The obtained generalized picture does not yet have data that would allow us to make specific predictions of the above events. They can be caused by a holistic system of secondary periods and resonances that change synchronously with the growth of the orbital period, variable substructures of the outer layers of the accretion disk, the peculiar spatial configuration of the donor’s magnetic field, or other reasons considered in Skulskyy (2020a, b). In addition, the findings on the variable dynamic behavior of the absorption components of the above helium lines may to some extent confirm the results of disk-forming motions of matter in gas-dynamic modeling of stellar structures and accretion disk, given by Bisikalo et al. (2000).
Analyzing the behavior of the radial velocities of the absorption components in complex helium and hydrogen lines, the previous points in this section focused on the details of the transfer of matter from the donor to the gainer within the main eclipse, i.e., in the direction of the gravitational axis of the binary system. It is equally important to evaluate their behavior in the phases associated with the transfer of moving matter in the direction of the axis of the donor’s magnetic field. It should be noted (Skulskyy, 2020a) that a dipole magnetic field with the axis in the direction of the orbital phases (0.355-0.855) P has a maximum magnetic field on the donor surface at the 0.855P phase, which reflects the location of this magnetic pole close to the gainer. In addition, the magnetic dipole axis inclined to the orbit plane of the binary system by 28\degr, and therefore, this magnetic pole on the surface of the donor in phase 0.855P is also above the plane of the orbit. This means that the ionized gas directed by the donor’s magnetic field moves in the direction of the dipole axis from the donor’s surface and, deviating along the magnetic field lines towards the accretion disk, can rise above the front edge of the disk without much energy loss.
The tables and graphs in Sahade et al. (1959) allow of stand out certain aspects related to the moving matter in the direction of the axis of the magnetic field of the donor. It is reasonable to estimate first the behavior of radial velocities of absorption components of the above lines in orbital phases near 0.85 P, i.e. in the phases of direct visibility on the donor surface of the magnetic pole facing the gainer. From Fig. 1 it is seen that in phases around 0.8 P both absorption components in the He I $\lambda$ 3888 line form the single absorption with the radial velocity near -135 km/s. This is sufficient for a direct entering of matter in to an external medium in this direction through the Roche cavity with the Lagrange point L2 (the first parabolic velocity relative to the center of gravity of the binary system is near -115 km/s). In these phases, the more negative absorption component of the He I $\lambda\lambda$ 3614, 3964 singlet lines shows a single average radial velocity close to -115 km/s (not showing any bifurcation, see Fig. 3). Several absorption groups with much smaller radial velocities, which are present in other phases, are completely absent in phases before 0.855 P, i.e., the flow of magnetized plasma is manifested as a single whole. In the phases (0.88-0.90) P after 0.855 P, the physical conditions changed. The more negative absorption component in the He I $\lambda$ 3888 line reached the average radial velocity of -156 km/s, and the absorption of the He I $\lambda$ 3888 line with a lower average radial velocity approached to a radial velocity near -115 km/s, i.e., to the second and first parabolic velocities, respectively. Evidencing the certain stratification of their formation, the He I lines of both series arising from metastable levels emphasize the average direction of plasma flows in the direction of the donor’s magnetic field axis and localization in phases around 0.855 P on the donor surface of the magnetic pole close to the gainer. This indicates that in the phases of direct visibility on the donor surface of this magnetic pole there is an outflow of magnetized matter outwards from the donor surface, as one of the important directions of non-conservative loss of matter during the evolution of this interacting system.
Based on Sahade et al. (1959) observations in 1955 it can be stated that the radial velocities in the phases near both poles of the magnetic field in the He I $\lambda$ 3888 absorption line showed a radial velocity much higher than the first parabolic velocity (see Fig. 1). However, the stellar wind behaves somewhat differently in the phases of the first quadrature, in particular in the vicinity of the 0.355P phase of the magnetic field pole on the donor surface. This applies, for example, to both the phase interval (0.25-0.5) P in the He I $\lambda$ 3888 line (see Fig. 1) and phases of (0.25-0.45) P in the H${}_{\alpha}$ and He I $\lambda\lambda$ 4472, 4026 lines (see Figures 13, 15, and 16 and Tables 3 and 4 in Sahade et al. (1959)). Several groups of radial velocities inherent to the corresponding absorption components are visible. The group of absorptions with positive radial velocities around +20 km/s, i.e., +35 km/s relative to the center of gravity of the binary system is important. It indicates that within the (0.35$\pm$ 0.1) P phases of the visibility of the magnetic pole on the donor surface, a runoff of magnetized matter from the observer is formed in the direction of the axis of the donor’s magnetic field (0.35-0.85) P. Reflecting the thermal velocity of matter from the surface of the donor, which reaches its Roche cavity, the radial velocity in these phases indicates the direction of the stellar wind from the surface of the donor. It is directed from the observer along the magnetic axis of the donor to the visibility phases of 0.855P on the donor surface of the magnetic pole facing the gainer, where the magnetic field of the donor reaches its maximum value. As it can be seen from Fig. 3, such a group of radial velocities is not clearly detected in the lines He I $\lambda\lambda$ 3614, 3964, indicating some stratification of physical conditions in the moving matter. But these lines, like the lines He I $\lambda$ 3888, H${}_{\alpha}$ and He I $\lambda\lambda$ 4472, 4026 have absorption components, which clearly are changed in radial velocities from -60 km/s to -40 km/s in phases of (0.30-0.45) P. This exhibits certain variable events near this pole of the donor’s magnetic field that are related, for example, to variations in these phase ranges of the total flux of polarization recorded in Hoffman et al. (1998) and Lomax et al. (2012). To some extent, these changes were studied on the basis of helium lines arising from metastable levels by Skulsky & Kos (2011). Now, this requires detailed coverage and comparison with the evidence of Sahade et al. (1959).
2.2.2 Helium lines arising from metastable levels in more modern studies
As it was previously pointed out, the issues of the dynamics of circumstellar gaseous structures using lines He I $\lambda\lambda$ 3888, 3964, 5015, which arise from metastable levels, were studied by Skulsky & Kos (2011). During 2008-2010, they obtained almost 100 echelle spectrograms with a resolution of about 45 000 on the 2-m reflector of the Peak Terskol Observatory. The narrow absorption components of the He I $\lambda\lambda$ 3964, 5015 showed that the variability of radial velocities and intensities with the phase of the orbital period is practically correlated: both lines formed at approximately the same distance from the center of the system and participate in the same processes. The intensities of these lines reach a minimum in the phases near (0.45-0.60) P, when the donor closes the gap between it and the gainer, simultaneously highlighting the phase direction (0.6-0.1) P perpendicular to the axis of the donor’s magnetic field. The intensity of these lines also decreases sharply within less than 0.1 P in the phases of 0.35 P and 0.85 P of polar regions on the donor’s surface (more obviously in the He I $\lambda$ 3964 line, see Fig. 1 in Skulsky & Kos (2011)). But particularly sharp changes are clearly seen in the negative radial velocities of the He I $\lambda\lambda$ 3964, 5015 lines, which indicates complex processes in the orbital structures of this binary system (see Fig. 4, which is a copy of Fig. 2 from Skulsky & Kos (2011)). It can be seen that within 0.1 P in the 0.35 P and 0.85 P phases, which correspond to the location of the magnetic poles on the donor surface, there is a sharp drop in the radial velocity from -55 km/s to -125 km/s, i.e., by 70 km/s relative to the center of gravity of the binary system ($\gamma$ = -18 km/s according to measurements of the radial velocities of the sharp interstellar line Ca II $\lambda$ 3933).
The behavior of the radial velocities curve of these helium lines is actually clearly associated with the magnetic field curve obtained in the course of measurements of the Zeeman splitting at Si II $\lambda\lambda$ 6347, 6371 lines by Skulskij & Plachinda (1993). The effective intensity of the longitudinal component of the magnetic field varies during the orbital period within $\pm$ 200 G, but as it can be seen from Figures 5 and 6 of Skulskyy (2020a), rapid changes in the magnetic field curve are clearly visible only in the phases near the poles of the magnetic field. In both cases, the polarity of the magnetic field varies within 0.1 P in the 0.355 P phase of the first quadrature and in the 0.855 P phase of the second quadrature. The behavior of the photographic curve of the magnetic field is also characterized by rapid changes in the magnetic field curve around the 0.855 P phase (i.e. the phase of observation of the magnetic pole facing the gainer) and in the phases around the main eclipse.
Note that the sharp peak of radial velocities with a rapid change in their values is observed in Fig. 4 in the phase of about 0.67 P (one can assume that the opening gas flow is in collision with the accretion disk). Similarly formed two narrow peaks of radial velocities are observed in the same phases of the main eclipse, in which the radial velocities of the opposite sign are observed in the satellite lines, characterizing the outer edges of the accretion disk. It should be concluded that due to different physical parameters at the edges of the accretion disk a complex structure of plasma flows is formed, which may participate primarily in mass transfer between components of the binary system, as well as in the outflow of matter outside it. In addition, note that the rapid changes of radial velocities near the main eclipse in the He I $\lambda\lambda$ 3965, 5015 lines (see Fig. 4) are to some extent related to such events recorded in the similar phases (0.88-0.90) P and (0.96-0.98) P of radial velocities from the He I $\lambda\lambda$ 3614, 3965 lines according to their measurements in Sahade et al. (1959) (their behavior was discussed above since the end of the 4th cycle on June 8, 1955). Such a coincidence is not accidental.
Based on observations of the Peak Terskol Observatory in 2008-2010, Skulsky & Kos (2011) also investigated the behavior of radial velocities of the He I $\lambda$ 3888 line for 1900-2010. Some dynamic characteristics are shown in Fig. 5 as a copy of Fig. 3 from their paper. Without commenting on the nature of the peculiar understatement of radial velocities on the Potsdam 1900-1909 spectra (one can agree with the thorough analysis of this problem conducted by Harmanec & Scholz (1993)), one should note certain points (given the average radial velocity of the center of gravity of this binary system $\gamma$ = -18 km/s, which is determined within 2 km/s on the spectrographs of the Crimean, Ondrejov and Peak Terskol Observatories). The range of measurements of radial velocities according to our Crimean observations in 1966-68 and 1972, our more recent observations in 2008-2010 at Peak Terskol, as well as observations in 1991 in Ondrejov, lies in a range from -118 km/s to -142 km/s at the average radial velocity near -130 km/s. There are no such sharp jumps in radial velocities inherent in the curve of radial velocities of the He I $\lambda\lambda$ 3964, 5015 sharp lines shown above in Fig. 4. Well-provided observations of the 1991 season at the Ondrejov Observatory show the average radial velocity in the He I $\lambda$ 3888 line, which is only about -127 km/s, but the radial velocity curve based on the data in Table 4 and Fig. 3 by Harmanec & Scholz (1993) shows a hump with a value of -122 km/s in the phase range (0.70-0.85) P of the visibility of the magnetic pole on the donor surface (and -132 km/s in the phase of 0.05 P). This is consistent with the convergence of both absorption dependences on He I $\lambda$ 3888 (see Fig. 1) to a single absorption component, which yields a reduced radial velocity to -(130-135) km/s in the phase range (0.75-0.85) P of the visibility of this magnetic pole on the donor surface as shown in Sahade et al. (1959). In both papers, the flow of matter in the direction of the axis of the donor’s magnetic field is significantly greater than the first parabolic velocity. If the difference of 10 km/s in radial velocities between two distant seasons of active observation is real, then this could be a confirmation of long-term changes in the non-laminar loss of moving matter out of the circumstellar structures of this binary system. It should be noted that none of the researchers of the He I $\lambda$ 3888 line observed this type of bifurcation of its absorption into two different components during the almost entire orbital period. In this aspect, the article by Sahade et al. (1959) remains original and its consideration required special research. This is one of the reasons why this issue was not considered in the report of Skulsky & Kos (2011). Note that the He I $\lambda$ 3888 line behaves somewhat differently in the known article by Flora & Hack (1975). This thorough article also deserves a detailed review, which for some reason has not been conducted for a long time.
2.2.3 Magnetic field and behavior of shell lines in the known article by Flora & Hack (1975)
In this study, it is necessary to consider the radial velocity curve of the He I $\lambda$ 3888 sharp absorption line in the paper of Flora & Hack (1975), the variability of which seems somewhat unusual. In most orbital phases these radial velocities should now be considered much smaller than the first parabolic velocity for this binary system (perhaps this is one of the reasons for the inactive use of this work). Flora & Hack (1975) gave the results of the radial velocity (RV) measurements and the profiles and intensities of the most significant lines, which were taken during the international campaign in the period of July 18 - August 1, 1971. The rich spectrographic material (lacking only one day at the orbital period that is about 13 days) consisted of 70 spectrograms of the enough high-dispersion (7 and 12 Å/mm) obtained at the 1.52-m telescope of the Haute Provance Observatory. Flora & Hack (1975) indicate “an asymmetrical distribution of matter in the envelope surrounding the whole system”. They conclude that the behavior of the RV curves for the Balmer and He I shell absorption cores must be real because these lines are virtually independent of blending with absorption lines of the B9 star (or now the donor). It is also noted that “none various absorption or emission features show any correlation with the invisible companion” (or now the gainer). This motivates to consider their rich observational material, drawing on modern concepts in the approach to its interpretation.
Fig. 6, along with other spectral lines, shows the data of measurements of the radial velocities of the He I $\lambda$ 3888 line, taken directly from the tables of the article of Flora & Hack (1975). If the average velocity of the center of gravity of the binary system is -17 km/s (according to Table 3 in interstellar components line Ca II $\lambda$ 3933 it is -15 km/s, and in lines of Na I $\lambda$ 5899 and 5895 it is -19 km/s), all applied radial velocities should in Fig. 6 shift up 17 km/s, in the direction of reducing negative velocities. Then, in phases (0.65-0.85) P, the He I $\lambda$ 3888 line will show the radial velocities of about -80 km/s relative to the center of gravity of the binary system (which is actually much smaller than the current first parabolic velocity for this binary system). This result immediately revealed several important points in relation to further research. First, the radial velocity -80 km/s is comparable to the average radial velocity obtained on Potsdam spectrograms in the early 20th century, the reality of which has been repeatedly discussed and questioned (see, for example, Harmanec & Scholz (1993)). Second, the stability in profiles, intensity, and radial velocities of the He I $\lambda$ 3888 line in orbital cycles during the 1971 season can be questioned, based, for example, on significant changes in these spectral characteristics in the observations of Sahade et al. (1959). The question arises of changing the directions of active loss of matter both between stellar components and in relation to circumstellar medium, as well as the reasons for the variable density of such outflows.
Regarding the direction of interpretation of Fig. 6, it is necessary to pay attention to its three parts, which, according to the classification of Flora & Hack (1975), present the radial velocity curves of two of the non-metastable He I $\lambda\lambda$ 5875, 6678 lines of the highest intensities, the metastable lines of He I $\lambda\lambda$ 3888, 5015, 3447, and the shell components of the Na I $\lambda\lambda$ 5889, 5895 lines. One can consider a central part of Fig. 6, which shows three of the five lines measured in Flora & Hack (1975) arising from metastable levels. The absorption components of these lines form dome-shaped curves of radial velocities in the phases of the second quadrature. The maxima of these dome-shaped curves were recorded in consecutive days of observations in the phases (0.78 - 0.77) P and (0.84 - 0.86) P. Similar curves show two more such lines He I $\lambda$ 3964 and He I $\lambda$ 6313, the radial velocities of which in these phases are located mainly between the curves of radial velocities of the He I $\lambda$ 5015 and He I $\lambda$ 3447 lines (in Fig. 6 there were not introduced so as not to overload the general picture). All five lines of these helium lines with certain potential difference between the upper and lower levels demonstrate clear stratification in the radial velocities, but the matter outflows have velocity much smaller than the parabolic velocity.
Since the observer in phases (0.75 - 0.85) P looks perpendicularly to the line of centers of stellar components and along the axis of the dipole magnetic field of the donor (the motion of matter to the observer), the most adequate explanation of this picture can be considered as the consequences of loss of matter directly from the donor surface near its magnetic pole in phase 0.855 P and its subsequent transfer to the gainer. It should be noted that the He I $\lambda$ 3888 line shows the single absorption, the radial velocity of which confidently reaches the current value of the first parabolic velocity only in the vicinity of phases (0.0 - 0.1) P, i.e. in the directions of motion of the matter to the gainer, as well as in phases near 0.45 P. It is known that in phases (0.42 - 0.49) P a hot spot is registered on the accretion disk, according to Burnashev & Skulskij (1991) and Lomax et al. (2012). Here, the interpretation of such velocities can be related to the outflow of matter from the donor surface or, conversely, to this surface, reflecting the specifics of the location of the pole of the magnetic field on this surface. Indeed, Burnashev & Skulskij (1991) showed that the observer begins to see excess radiation (hot surface on the donor) only from the 0.355P phase, i.e., along the axis of the magnetic field when its pole opens on the donor surface (see Fig. 1 in Skulskyy (2020b)). This is more likely when the pole of the magnetic field is significantly shifted to the upper hemisphere on the surface of the donor (the magnetic dipole axis is inclined to the orbit plane of the binary system by 28\degr (Skulskij, 1985). Indeed, the hot surface on the donor continues to be observed almost to the 0.50P phase, demonstrating during the orbital phases (0.38-0.49) P the clear variability of the absolute radiation flux in the H${}_{\alpha}$ emission line and the rapid variability of the spectrum in the H${}_{\alpha}$ emission region (see also Alexeev & Skulskij, 1989). This may be a reflection of the collisions of the hot plasma with the donor surface, directed (to the observer with the radial velocity close to the first parabolic velocity) along the lines of force of the magnetic field. Note that in these phases, there is a bifurcation of the absorption component so that both shallow absorption components are superimposed on the strong emission component of lines H${}_{\alpha}$, He I $\lambda$ 6678, and He I $\lambda$5875, respectively, in Fig. 3d, 6a, and 7a in Flora & Hack (1975). The more violet-shifted component of this absorption shows that the tabular radial velocity approaches in the phase of 0.45 P to current values of parabolic velocity (see Fig. 6). Their occurrence can also be explained by the motion of the magnetic plasma along the lines of force of the magnetic field. In the He I $\lambda\lambda$ 5875, 6678 lines the red-shifted component of this absorption shows much smaller radial velocities close to the velocities in the lines of the donor atmosphere. Interestingly, Flora & Hack (1975) suggested that the violet-shifted absorption in phase 0.45 P may reflect the escape of gas matter through the Lagrange point in front of the donor. Such a hypothesis requires more observational evidence.
The radial velocity curve of the sodium lines resonates with the previous remark. The bottom of Fig. 6 does not so clearly show that the observation phase of 0.375 P (near the phase of 0.355 P of direct vision of the pole of the magnetic field on the surface of the donor) was missed. In the phase range (0.30 - 0.45) P one can assume both a certain minimum of this curve near the 0.355 P phase and a small local hump. In each case, the curve of radial velocities of the Na I $\lambda\lambda$ 5889, 5895 absorption lines clearly manifests itself as the wide maximum of negative velocities near the 0.355 P phase and their clear narrow minimum in the phase 0.855 P, i.e., in the phases of visibility of the magnetic field poles according to the photographic curve of the effective magnetic field strength of the donor (see Fig. 1 in Skulskyy (2020b)). The presence of narrow interstellar components of the Na I lines allowed us to check the tabular data and to clarify the radial velocities of the shell components of these sodium lines directly from Figs. 7a and 7b of Flora & Hack (1975). Fig. 6 shows that the observation in phases (0.8 - 0.9) P is especially interesting. Since the interstellar components in Na I $\lambda\lambda$ 5889, 5895 lines are -19 km/s, the radial velocity in the direction of the observer in these phases is the smallest one and is only of about -70 km/s, which is much smaller than the first parabolic velocity. Here the observer looks perpendicular to the donor surface along the axis of the magnetic field. Important is the fact of rapid dynamic changes in the moving plasma in the ranges of 0.1 P in phases near 0.855 P, i.e., of the center of the magnetic pole on the donor surface. Within 0.1 P, the movement of the plasma clearly slows down, stops at a certain level, and accelerates again. Hence, the observations of Flora & Hack (1975) clearly state the fact of actual variations in the sign of the RV progression curves of both helium and sodium lines reflecting the direction of motion of matter around the 0.855 P phase. It can be argued that the observations of Flora & Hack (1975) during the 1971 international campaign clearly identified the region of the donor surface that is responsible for the location of the pole of the magnetic field facing the gainer. This is consistent with studies of Zeilik et al. (1982) and Skulsky & Kos (2011), confirming CCD research of the magnetic field by Skulskij & Plachinda (1993).
Radial velocities in the Na I $\lambda\lambda$ 5889, 5895 lines in Fig. 6 do not show a detailed picture of their possible variability in phases (0.95 - 0.05) P of eclipsing of the donor by the gainer wrapped in the accretion disk. The observer sees in phases (0.60 - 0.85) P a gradual decrease in the radial velocities by 30 km/s, which in phase 0.855 P immediately changed to their such gradual increase in phases (0.85 - 0.10) P. One should think that the spatial configuration of the donor magnetic field reflects the direction (0.6 - 0.1) P, perpendicular to the direction (0.35 - 0.85) P of the donor magnetic axis. This is the direction of movement of the plasma perpendicular to the lines of force of the magnetic field in the space between the stellar components under the action of gravity of a massive gainer.
It should also be noted that the radial velocity curves of the He I $\lambda\lambda$ 5875, 6678 absorption lines show in certain respects similar behavior.
Their average curve at the top in Fig. 6 is made using measurements of the radial velocities of the line He I $\lambda\lambda$ 5875 as a line, the theoretical intensity of which is seven times greater than that of the He I $\lambda$ 6678 line.
The radial velocities of the He I $\lambda$ 6678 line are located slightly higher than the velocities of the He I $\lambda$ 5875 lines, but in general, the behavior of these lines with the orbital phase is similar.
The extreme values of their radial velocity curves correspond to phases of approximately 0.6 P and 0.1 P, reflecting the direction (0.6 - 0.1) P, perpendicular to the magnetic axis of the donor with the direction (0.35 - 0.85) P.
These curves are shifted relative to the 0.855 P phase by about a quarter of the orbital period, as it is seen in comparison, for example, with the RV-curves in Fig. 6 for the Na I $\lambda\lambda$ 5889, 5895 lines.
The absorption component with radial velocities, which changes rapidly in the phases near (0.60 - 0.65) P, is well seen in the higher members of the singlet lines of helium, in particular, as shown in Fig. 6 in the He I $\lambda$ 3447 line.
Starting from the phases (0.60-0.65) P, this may indicate the occurrence of gas flows that are directed perpendicularly to the axis of the donor’s magnetic field in the direction of the gainer.
Similar behavior in the phases of (0.1 - 0.6) P is demonstrated in the central depth variations of the He I $\lambda\lambda$ 3888, 3964 shell lines at the maximum value in the phase of 0.1 P, and its minimum in the phase of 0.6 P (see Fig. 14 of Flora & Hack (1975)).
It should be noted that all this is consistent with the picture of the directions of the plasma motion of magnetically controlled circumstellar matter in the helium stars according to the study of Shore & Brown (1990) (see Figure 11 in the paper of Shore & Brown (1990), and its representation as Fig. 8 in Skulskyy (2020b)).
Their model can be illustrative for the donor as an oblique magnetic rotator of intermediate obliquity because the dipole axis of its magnetic field deviates relative to the orbital plane of the binary system, and the center of the dipole of the donor’s magnetic field is significantly shifted in the direction of the gainer. The outflows of matter from the donor surface in the mass transfer picture are possible both in the direction of the donor’s magnetic axis and in the direction perpendicular to this axis (see Skulskyy, 2020b). Some proof of this type of phenomenon may be the behavior of a complex line H${}_{\alpha}$ along the surface of the donor. In particular, Fig. 15 of Flora & Hack (1975) shows that the equivalent width of the absorption, measured with respect to the total area of the emission in the H${}_{\alpha}$ line, is halved in the phase direction (0.1 - 0.6) P, and the total equivalent width of emission plus the absorption of the H${}_{\alpha}$ line shows two deep minima in the phases (0.25-0.45) P and about 0.855 P, i.e. near the phases of visibility of both magnetic poles on the donor surface. Of course, to some extent, this is the result of reflecting the characteristics of the donor that fills its Roche cavity: the loss of matter along the axis of the magnetic field at its pole and the loss of matter at the Lagrange point L1, amplified by the massive gainer and deflected by Coriolis forces to the 0.1 P phase, are significantly different. At the same time, the variable motions of the plasma are reflected more clearly in the phases around 0.855 P of the visibility of the magnetic pole, facing the gainer, in accordance with the changes in the polarity of the magnetic field in the observations of Skulskij & Plachinda (1993).
2.3 Magnetic field and the He I $\lambda$ 10830 line arising from metastable level
If the structure and behavior of the He I $\lambda$ 3888 line as the second member of the principal series of triplet helium were studied during the first half of the XX century, then to some extent successful observations and studies of the He I $\lambda$ 10830 line, as the first member of this series, was carried out by Alduseva & Esipov (1969) only in 1961-1962. They used a contact image-converter as an image receiver mounted on a 12-inch reflector. The spectral resolution was 4.8 Å(11 days of observations in 1961) and 2.8 Å at 9 days of observations in 1962. This highlighted the complex structure of the line contour and gave some initial data on the change with the orbital period of the shell radial velocities and equivalent widths of the emission in the He I $\lambda$ 10380 line during the orbital period. They indicated that “the line appears in the very external parts of the shell, surrounding Beta Lyrae as a whole”. Morgan et al. (1974) made the next attempt in the study of the He I $\lambda$ 10380 line. They observed on the rapid-scan interferometer at the coudé focus of the 2.7-m telescope at McDonald Observatory in April 1973, but received only two spectra in the 0.90 P and 0.13 P phases, respectively. The best resolution of 4.7 Å in the 0.90 P phase allowed them to record the value of the absorption component -140 km/s with an error of 60 km/s.
Our more complete observations of the He I $\lambda$ 10830 line by Girnyak et al. (1978) were conducted in April-November 1974. There are no data on the following observations, and these studies need more attention. The observations were performed using an electron-optical image-converter of type FKT-1A (S1) mounted on the 50-inch telescope of the Crimean Astrophysical Observatory. During 26 nights of this season, 59 spectrograms with a dispersion of 48 Å/mm were obtained, which allowed measuring radial velocities with an error of 10 km/s. As it can be seen from Fig. 7, which is a copy of Fig. 1 from Girnyak et al. (1978), this line in the Beta Lyrae spectrum represents the strong emission that is several times higher with respect to the continuum than such emission in the He I $\lambda$ 3888 line. The considerable width of the He I $\lambda$ 10830 line indicates the movements in the gas plasma structures with velocities up to 800 km/s. The deep absorption component, which cuts through the emission, is shifted to the short-wave side. Girnyak et al. (1978) noted that the total intensity and ratio of its short-wavelength and long-wavelength emission components change significantly with the orbital phase. This is a new important factor that requires additional attention and further research.
Girnyak et al. (1978) proposed processed physical parameters of the He I $\lambda$ 10830 line, introduced in a general table and five figures. Fig. 8 reproduces, on the basis of tabular values, two of these five figures,
which are presented below as the radial velocity curve of the absorption component of the He I $\lambda$ 10830 line and the sum of its absolute fluxes from violet and red emissions components given in erg s${}^{-1}$ A${}^{-1}$ km/s, multiplied by $10^{33}$ (contours in the He I $\lambda$ 10830 line were built in absolute energy units based on six-color colorimetry (Burnashev & Skulskij, 1978)). Fig. 8 indicates some new interesting points. They concern the upper part of Fig. 8, which illustrates the variability in the orbital phase of the ratio of the intensity of the short-wave and long-wave components of emission, i.e. the dependence of $I_{v}/I_{r}=f(P)$, which we built here on the basis of the tabular data. In Fig. 8, bottom, it is seen that the $V_{r}=f(P)$ curve of the absorption component, which separates these emission components, remains approximately the same in the orbital phases at an average value of about -130 km/s (taking into account the radial velocity of -18 km/s for the center of gravity of this binary system). The largest scattering of data and rapid changes in radial velocities are observed in the range of phases (0.7-0.9) P, and a particularly sharp velocity change is shown in the phases around 0.855 P of the visibility of the magnetic pole, facing the gainer. Similar behavior in these phases there is in changes of contours in the He I $\lambda$ 10830 line (see the middle part of Fig.8). Despite some sharp changes in the total emission, which can occur at times from tens of seconds to tens of minutes (see, e.g., Alexeev & Skulskij (1989) and Skulskyy (2020b)), some trend is observed to a sinusoidal curve of the absolute flux of the total emission with extrema in the phases of (0.35-0.45) P and 0.85 P. It is known that these phases correspond to the phases of both extrema of the donor’s magnetic field (Burnashev & Skulskij, 1991; Skulskij & Plachinda, 1993). This picture is more clearly manifested in the dependence of $I_{v}/I_{r}=f(P)$ (the upper part of Fig. 8), characterizing the expected variability of shell structures in this binary system. The dependence of $I_{v}/I_{r}=f(P)$ showed a clear decrease in the $I_{v}/I_{r}$ ratio after the 0.855 P phase (of the direct visibility of the magnetic field pole on the donor surface) to the minimum in the main eclipse phase and its rapid rise to the phase 0.15 P, when the main flow of matter from the donor to the gainer, deflected by the Coriolis forces, begins after the main eclipse. It should be noted that these observations do not provide enough data for more definite conclusions in the phases (0.95-0.05) P of the main eclipse.
Nevertheless, the radial velocity curve of the He I $\lambda$ 10830 line looks similar to that of the He I $\lambda$ 3888 line defined by Skulsky & Kos (2011) over five different seasons (see Fig. 5 above). Figs. 5 and 8 show that the average values of the radial velocity of the absorption components of the first two triplet helium lines are close to -130 km/s relative to the center of gravity of the binary system, which significantly exceeds the first parabolic velocity. It could be assumed that the sharp absorption components of both lines are formed in a dynamically identical medium moving in all directions relative to the center of gravity of the binary system. Moreover, the strong emission of the He I $\lambda$ 10830 line does not explicitly distort the position of the absorption component, and the radial velocity of the emission center as a whole turned out to be close to the orbital velocity of the gainer, based on the previously established mass ratio of the stellar components (Skulskij, 1975). However, Figs. 1 and 6, which respectively relate to the studies of Sahade et al. (1959) and Flora & Hack (1975), showed that the behavior of the He I $\lambda$ 3888 line at different seasons and epochs can significantly differ. Such facts are real, they required a detailed consideration of some, unnoticed by Girnyak et al. (1978), dynamical characteristics of the He I $\lambda$ 10830 line at certain parallels in the study of the lines He I $\lambda$ 3888 and He I $\lambda$ 10830.
First of all, this applies to the phase interval (0.75-0.9) P around the 0.855 P phase of the direct visibility of the magnetic field pole on the donor surface close to the gainer. Note that the observations of 1971 by Flora & Hack (1975) and our observations of 1974 by Girnyak et al. (1978) can be attributed to fairly close seasons and the dynamic behavior of the shell lines He I $\lambda$ 3888 and He I $\lambda$ 10830 may have some similarity. As it can be seen from the bottom of Fig. 8, our observations were most frequent in the (0.7-0.9) P phases, showing rapid changes in the radial velocity at large amplitudes of their variability. The observation data at these phases were rebuilt on a larger scale in Fig. 9. It can be seen that the radial velocity curve of the He I $\lambda$ 10830 line in these phases, like the domed-shaped curve in Fig. 6 for the He I $\lambda$ 3888 line, also has a domed-shaped peak at a clear rise in negative radial velocity immediately after the phase 0.855 P of direct visibility of the magnetic pole on the donor surface. This peak of the radial velocities of the He I $\lambda$ 10830 line that changes in the ranges of 0.1 P within 20 km/s is twice smaller than that for the He I $\lambda$ 3888 in Fig. 6. At least, this points to some analogies in the dynamic parameters of gas structures in which these triplet helium lines were formed in the first half of the 1970s. The decrease of negative radial velocities at approaching the 0.855 P phase and their subsequent rapid increase after this phase in the He I $\lambda$ 10830 line (as well as in metastable lines of He I $\lambda\lambda$ 3888, 5015, 3447, and the shell components of the Na I $\lambda\lambda$ 5889, 5895 lines - see Fig. 6 built on the data of Flora & Hack (1975)) indicates the similarity of physical processes along the donor surface near this magnetic field pole.
At the same time, the He I $\lambda$ 3888 and He I $\lambda$ 10830 lines show a physically significant difference in radial velocities. It is clearly seen that if in the He I $\lambda$ 10830 line in Fig. 9 in all phases, including phases (0.6-0.8) P, the radial velocities are greater than the parabolic velocity, then in the He I $\lambda$ 3888 line according to Fig.6 in phases (0.6-0.8) P, they are much smaller than the parabolic velocity. This indicates the nonstationarity of the outflow of matter and the variable flow density in the direction of the dipole axis of the magnetic field. The clear changes, which were noticed in the absolute radiation flux (the central part of Fig. 8) within (0.8-0.9) P, also reflect the complex structure of the emission formation near 0.855 P in this short phase interval with a width of approximately 0.1 P. This emphasizes that rapid changes are inherent in these phases of both energy and dynamic parameters of the magnetized plasma. The rapid change in radial velocities near 0.855 P may indicate that the vector components of the magnetized plasma motion at the edges of the magnetic field pole are superimposed on the total plasma motion in the direction of the donor magnetic field axis. This one can also see from Fig.4, which demonstrates similar variations in the radial velocities of the He I $\lambda\lambda$ 3964, 5015 lines in the short phase interval with a width of approximately 0.1 P in phases of the visibility of both magnetic field poles on the donor surface. The observations of Girnyak et al. (1978) in phases (0.25-0.45) P are not dense enough to conduct a detailed analysis of the behavior of the He I $\lambda$ 10830 line in the phases of visibility of the magnetic field pole on the donor surface in the first quadrature, although, in these phases, the radial velocities of this line are also subject to abrupt changes similar to Fig.4. However, as a whole, the above research facts confirm the general picture of the behavior of energetics and dynamic parameters of magnetized plasma caused by a certain configuration of the donor’s magnetic field. The study of the spectral characteristics of the He I $\lambda$ 10830 line has led to certain clarifications in the formation of the picture of the transfer matter between stellar components and its movement into the medium around the Beta Lyrae system.
3 Conclusions and discussion
The study of shell structures in the Beta Lyrae system seemed natural on the basis of previous studies by Skulskyy (2020a, b), which involved a continuous interaction of the outer shell with a complex system of plasma flows between the donor and the gainer. It is clear that the spectral lines formed in external structures, in particular the helium lines arising from metastable levels, were studied. Scientific articles were used for the study, which, on the basis of qualitative observations, covered the orbital period well. There are few similar works, in particular Sahade et al. (1959); Flora & Hack (1975); Girnyak et al. (1978), which are half a century old. It is to some extent well that the authors of these works presented their results without the influence of modern interpretations, say, without information about the magnetic field of the donor or ideas about bipolar jet structures associated with the accretion disk. However, these articles have a lot of important spectral material in the form of published tables and graphs. As a whole, this material, together with data from more recent publications of Harmanec & Scholz (1993) and Skulsky & Kos (2011), have not been analyzed in more detail. Here an attempt was made to overcome this shortcoming.
All researched articles are distinctive. They differ significantly in the energy and dynamic parameters of the moving plasma in the medium near each component and between stellar components of the binary system. First, significant long-term differences were found within half a century in the movements of matter to the gainer and in external structures. Second, there are significant differences in the physical parameters of this plasma within one observation season and their variations from cycle to cycle (especially in Sahade et al. (1959)). It was established (see Fig. 1) only in Sahade et al. (1959) that: “He I $\lambda$ 3888 has two components with very definite and fairly constant velocities” of about -155 km/s and -115 km/s (in the 1955 season), i.e., close to the second and first parabolic velocities. In Flora & Hack (1975), the absorption component of the He I $\lambda$ 3888 line in certain phases of the 1971 season is approximately -80 km/s, which is much less than the parabolic velocity (see Fig. 6). The averaged radial velocity of the absorption component in the He I $\lambda$ 3888 line during the 1991 season is -127 km/s (see also Fig.3 in Harmanec & Scholz (1993)), which is very different from that in the previous two articles. The He I $\lambda\lambda$ 3964, 5015 lines (Fig. 4) show rapid changes in radial velocities in the range from -55 km/s to -125 km/s in short phase intervals up to 0.1 P, in particular close to the magnetic field poles, which were recorded in the seasons 2008-2010 in Skulsky & Kos (2011). It seems that all these differences are not periodically ordered. It should be emphasized that all researched articles show the averaged radial velocity in the He I $\lambda\lambda$ 3888, 10830, i.e., the velocity of the outflowed plasma in external structures, to be of about -130 km/s, except for one orbital cycle in 1971, in which observations by Flora & Hack (1975) were made.
The article gradually reveals the analyzed facts from the available spectral material, and since there is quite a substantial amount of them, they are further presented as touches of the obtained research picture in a possibly concise form. It turned out that this picture is based on fragments of a generalized investigation during the orbital period. They can be grouped into certain phase intervals as a reflection of slightly different physical processes occurring in them. Formally, they can be divided into five interconnected intervals. The phase range (0.95-0.05) P reflects nonstationary processes that occur during the transfer of mass to the gainer at the donor eclipse by the accretion disk. The phase range (0.07-0.14) P reflects the motion of the main gas flow deflected by Coriolis forces outside the accretion disk. It is necessary to distinguish the behavior of plasma structures near the phases of visibility of both poles of the magnetic field (0.30-0.45) P and (0.75-0.90) P, as well as within the range of phases (0.4-0.6) P eclipse of the accretion disk by the donor.
There are a number of effects and events associated with the eclipsing of the donor surface by the accretion disk in the phases (0.9-0.1) P of the main eclipse of the binary system. It can be noted that within phases (0.95-0.05) P of the deeper eclipse, the He I $\lambda\lambda$ 3614, 3964 and He I $\lambda$ 3888 absorption lines show, according to Sahade et al. (1959), two main regions of changes in their intensities and radial velocities. They are grouped near the 0.95 P and 0.05 P phases, which corresponds to the phases of maximum accretion disk density. This follows from many studies, in particular from phase changes of the equivalent widths of satellite lines in 1991-1992 (see Fig. 2 in Skulskij (1993c), or from the light curve based on observations by the BRITE Satellites in May-October 2016 (see Fig. 2 in Rucinski et al. (2018)). In these phases, there is the greatest variability of the radial velocities of both absorption components of the He I $\lambda\lambda$ 3614, 3965 and He I $\lambda$ 3888 lines. Their more negative shifted component varies in the range from -140 km/s to -160 km/s, sometimes reaching the second parabolic velocity (near -160 km/s) without a certain regularity in orbital cycles. This also applies to the center of the eclipse in different orbital cycles in phases (0.98-0.02) P, where the radial velocities also show variation within the same limits (for example, in phases (0.98-0.99) P at the end of the 6th cycle on July 4, 1955, their radial velocity was -(155-160) km/s, that is sufficient for an outflow matter from the Lagrange point L3). It is clear that in the range (0.95-0.05) P the dynamic behavior of moving matter reflected in the more negative absorption component in the He I $\lambda\lambda$ 3614, 3964 and He I $\lambda$ 3888 lines can be the product of instability processes of the matter transfer between the stellar components. The less negative absorption component of the He I $\lambda$ 3888 line in these phases reaches only the first parabolic velocity, and in the He I $\lambda\lambda$ 3614, 3964 lines they only approach it or are much smaller, also demonstrating the fact of non-laminar, but clearly variable events during the mass transfer. It should be noted that the observations of Sahade et al. (1959) were carried out in the 1955 season with significant phase gaps during the day (its time is 0.077 P), which does not allow to clarify the structure of the accretion disk. The important fact of the detection of disk-shaped structures in phases (0.95-0.05) P are the observations of Skulsky & Kos (2011) during 2008-2010, which clearly show rapid changes of radial velocities from -55 km/s to -125 km/s in He I $\lambda\lambda$ 3965, 5015 singlet lines (see Fig. 4) in two short phase intervals.
Thus, the variable nature of the flows of matter during mass transfer in phases of the main eclipse indicates eruption processes on the donor, the appearance of matter clots over the rims of the disk from the side of the donor and their movement in the direction to the observer over the surface of the accretion disk accelerated by the massive gainer. Incidentally, Skulskyy (2020b) showed that the vertical component of the plasma flows in the direction of the gainer can be formed due to the configuration of the donor’s magnetic field, in which the magnetic axis is inclined at a significant angle to the orbit of the binary system. The structure of gas flows above the surface of the accretion disk significantly depends on the location of the magnetic pole on the donor surface relative to its distance to the gainer and relative to the plane of the orbit. These remarks also apply to the third region of active events in the phases (0.9-0.1) P reflected in the changeable and denser flows of matter in the phases (0.07-014) P, which start after the center of the main eclipse of the binary system. The dynamic behavior of the moving matter in the He I $\lambda\lambda$ 3614, 3964 and He I $\lambda$ 3888 lines is similar to that in the phase range (0.95-0.05) P. Interestingly, the highest radial velocity for the observation season in Sahade et al. (1959) was recorded in phases (0.11-0.13) P of the 5th cycle on June 10, 1955. The more negative absorption component of the He I $\lambda$ 3888 line showed the velocity of -(160-170) km/s, which is more than enough for the outflow of the matter from the Lagrange point L3. These more dense gas flows, when deflected by Coriolis forces outside the accretion disk in the Roche cavity of the gainer, push the outer part of the disk to the gainer surface (see Skulskij, 1992; Skulskyy, 2020a).
The range of phases (0.4-0.6) P eclipse of the accretion disk by the donor, as a specific part of the orbital period, deserves a separate study (by analogy with the studies in phases (0.9-0.1) P of the primary eclipse). The preliminary study of existing observations in this phase range leads to certain conclusions and ideas about the behavior of physical parameters in the vicinity of both stellar components. Although they require additional observations, they were worth paying attention to. Analysis of Tables 3 and 4 together with their illustrations 13-16 in the article Sahade et al. (1959) (see also Fig. 1-3 above), as well as the study of Skulskij (1992, 1993b, 1993a, 1993c); Ak et al. (2007) and others, leads to the conclusion that in these phases, in addition to complex shell lines formed near the accretion disk, clear changes of the curve of radial velocities of the gainer lines are visible. Their behavior can be explained by the Rossiter effect in the passage of the donor in front of the accretion disk. In phases (0.38-0.50) P the right part of the disk is observed (its left part being closed by the donor); in phases (0.45-0.55) P the central parts of the disk, closed by the donor, pass in front of the observer; in phases (0.5-0.6) P dominates in these lines the left part of the accretion disk (the donor obscures the right part of the disk). Approximately at 0.62 P, this effect disappears (further there are opened phases of the collision of the main flow into the accretion disk and the region of the magnetic field pole on the donor surface). In particular, this effect is well observed in phases (0.38-0.62) P in the behavior of the gainer radial velocities of the Si II $\lambda\lambda$ 6347, 6371 lines, for example, in Fig. 3 in Skulskij (1992), where it was revealed, and, especially in Fig. 5 in Ak et al. (2007). It is more likely that these lines are formed near the central part of the disk.
The visibility phases of both poles of the magnetic field (0.30-0.45) P and (0.75-0.90) P as specific parts of the orbital period are important for obtaining a general picture of the influence of the donor’s magnetic field on moving magnetized gas structures in near and farther medium regarding the donor. Some of the key facts identified in this study are, first of all, illustrated by the behavior of the radial velocities of helium lines arising from metastable levels. Fig. 1 shows that only within phases of about 0.1 P near the phases 0.855 P, both absorption components of the He I $\lambda$ 3888 line in Sahade et al. (1959) converge into a single absorption with radial velocities in the range -(125-135) km/s, i.e., with a radial velocity much greater than the first parabolic velocity. This indicates that in the phase of direct visibility of the pole of the magnetic field on the surface of the donor facing the gainer, the matter outflows in the direction of the axis of the donor magnetic field, perpendicular to the surface of the donor. According to Fig. 3, the radial velocities from the shells of the He I $\lambda\lambda$ 3614, 3964 singlet lines, although showing some stratification of their formation relative to the line He I $\lambda$ 3888, indicate that the He I lines of both series in phases close to 0.855 P behave in these phases in a similar way. The particularly sharp peaks of radial velocities with a rapid change in their values are observed in Fig. 4 in the He I $\lambda\lambda$ 3964, 5015 lines according to Skulsky & Kos (2011). Within 0.1 P in the 0.355 P and 0.855 P phases, there are sharp changes in the radial velocity from -55 km/s to -125 km/s, or 70 km/s relative to the center of gravity of the binary system. This indicates complex processes in the plasma structures of this binary system, which are close to the location of the magnetic poles on the donor surface. It should be recognized that Fig. 9 also confirms that the rapid changes in the radial velocity of the He I $\lambda$ 10830 line in phases immediately close to 0.855 P (at the mean velocities over the orbital period near -130 km/s - see Fig. 8) indicates the reality of rapid changes in plasma movement and events in this phase range.
Evidencing certain stratification of their formation, the He I lines of both series show in theses phases a certain relationship between their dynamic and energy parameters. Figure 9 also shows the parallel changes in the profile of the He I $\lambda$ 10830 line expressed through the dependence of $I_{v}/I_{r}=f(P)$, i.e., the ratio of the intensity of the short-wave and long-wave components of these emission profiles. This dependence is similar to the radiation increasing near the 0.855 P phase on light curves in the range of (3.5-4.6) m in Zeilik et al. (1982), which allowed to identify (see Skulskyy, 2020b) on the donor surface the region of this magnetic field pole. Figure 2 illustrates also the radial velocity curve from the red emission peak at the He I $\lambda$ 3888 line. Such two maxima are observed on the radial velocity curve of the red peak at the H${}_{\alpha}$ line (see Figures 5 in Skulskyy (2020b). The maxima at the 0.355 P and 0.855 P phases coincide with the phases of the two maxima on the curve of the effective magnetic field strength of the donor (see Figure 1 in Skulskyy (2020b) or Figure 2 in Burnashev & Skulskij (1991)). Moreover, the radial velocity curve for the Gaussian center of emission in H${}_{\alpha}$, as well as for such centers on radial velocity curves of the He I $\lambda\lambda$ 6678, 7065, and Si II $\lambda\lambda$ 6347, 6371 lines, see Figure 6 in Skulskyy (2020b), demonstrate that both their maxima match the extrema of the effective magnetic field strength of the donor (a similar shift on the $V_{r}$-curve of the center of the He I $\lambda$ 10830 emission as a whole is discussed by Girnyak et al. (1978)). The generation of the radiation flux and the formation of the emission-absorption profile of these lines can be largely carried out under the influence of the existing configuration of the donor magnetosphere in the interconnected spatial structures.
A similar conclusion is supported by the observations of Flora & Hack (1975). Conducted within one orbital cycle, they showed unusual variability of physical parameters within the phases (0.70-0.90) P close to the 0.855 P phase of the magnetic field pole close to the gainer. Fig. 6 shows a much lower velocity of the moving plasma than the first parabolic velocity in all five helium lines arising from metastable levels. This clearly indicates nonstationarity of the outflow of matter and the variable flow density in the direction of the dipole axis of the magnetic field. The behavior of radial velocities in phases (0.8-0.9) P in the shell lines Na I $\lambda\lambda$ 5889, 5895 is also particularly important. During 0.1 P, the moving plasma slows down, stops at a certain point, and accelerates again, stating the fact of actual variations in the sign of this movement. Thus, the study of observations of Flora & Hack (1975) also confirmed that the observer, looking perpendicular to the surface of the donor along the axis of the magnetic field, reveals the region of the donor surface that is responsible for the location of the pole of the magnetic field facing the gainer and the loss of matter directly from the donor surface in phase 0.855 P with its subsequent transfer to the gainer.
However, the fact of localization of the poles of the magnetic field on the surface of the donor, reflected in many studies, needs some clarification, based on the spatial location of its magnetic axis. Let us turn to the model of the donor as a magnetic rotator, by analogy to the model of a helium star as an oblique magnetic rotator presented by Shore & Brown (1990). This model (see Fig. 8 in Skulskyy (2020b)) may be quite illustrative since the dipole axis of its magnetic field is deviated by 28\degr relative to the orbital plane of the binary system (Skulskij, 1985). The center of the donor magnetic dipole is displaced by 0.08 of the distance between the centers of gravity of both components toward the gainer’s center. It is also clear that the magnetic pole, located on the surface of the donor and observed in phases of about 0.855 P, is more effective in terms of the amount of transferred matter to the gainer, which is confirmed by all studies. The ionized gas, directed by the magnetic field of the donor in the direction of its dipole axis from the surface of the donor, is deflected along the lines of the magnetic field primarily to the accretion disk. However, a certain amount of charged particles will move along the lines of force of the magnetic field in the direction to the second pole of the magnetic field on the surface of the donor, heating its surface. And, if in the first quadrature in the phases around 0.355 P the magnetic pole on the surface of the donor is located above the plane of the orbit, then in the phases around 0.855 P the magnetic pole on the surface of the donor is below the plane of the orbit, or vice versa. The detection of these poles also depends on the inclination of the plane of the orbit. Not everything is clear in this part because the inclination of the orbit in current research is taken from i = 81\degr in Mennickent & Djurašević (2013) to i = 93.5\degr in Mourard et al. (2018). This is close to the binary orbital inclination i = 90\degr, and regardless of whether the pole of the magnetic field is in the upper or lower hemisphere on the donor surface, the observer, due to the projection of the rounded shape of the surface on the line of sight, must register certain deviations of the surface heating maximum from the 0.355 P phase. This is confirmed by the absolute spectrophotometry of Burnashev & Skulskij (1991) of 1974-1985. They showed that the observer starts to register the excess radiation (a hot spot on the donor surface) only from the 0.355 P phase directed along the axis of the magnetic field (see Figure 1 in Skulskyy (2020b)). The excess of this radiation disappears near the 0.50 P phase, demonstrating during the orbital phases (0.37-0.49) P the rapid variability of the absolute radiation flux in the H${}_{\alpha}$ emission line and the continuum around this line (see Skulskyy, 2020b; Alexeev & Skulskij, 1989). The maximum of the excess of this radiation is really shifted from the 0.355 P phase and corresponds to the phases (0.43-0.47) P. Thus, in these phases, some surface heating on the donor surface may be formed due to nonstationary shock collisions of ionized gas, directed along the magnetic field lines to the magnetic field pole, the location of which is given by the spatial configuration of the magnetic field dipole. The phase range (0.43-0.47) P corresponds to the known minimum on the polarization curves for Beta Lyrae studied by Appenzeller & Hiltner (1967) and Lomax et al. (2012), which is interpreted as formed by collisions of gas flows with the accretion disk during the scattering of radiation by free electrons. This encourages the research of the reflection of the spatial configuration of the donor’s magnetic field in the polarization observations.
Acknowledgements.The author is thankful to V.I. Kudak for consultations.
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Efficient ground state preparation in variational quantum eigensolver
with symmetry breaking layers
Chae-Yeun Park
Institute for Theoretical Physics, University of Cologne, 50937 Köln, Germany
Abstract
Variational quantum eigensolver (VQE) that solves the ground state of a Hamiltonian is a promising application of noisy intermediate-scale quantum devices.
The quantum alternating operator Ansatz (QAOA) is one of the most widely studied Ansätze for this purpose and solves many Hamiltonians reliably.
However, because of inherited symmetries from the Hamiltonian, the QAOA is not necessarily good at solving problems with symmetry broken nature.
In this paper, we propose a variational Ansatz with symmetry breaking layers for solving those systems.
Notably, our Ansatz finds a constant-depth circuit for solving a symmetry broken ground state which is impossible for the QAOA.
We also propose a simple learning technique that can choose a particular symmetry broken state among degenerate ground states.
I Introduction
Experimental progress in controlling quantum systems has allowed the first demonstration of quantum computational advantage [1] in the recent years, and a noisy quantum device with hundreds of qubits is getting visible.
Those devices, which are often dubbed as noisy intermediate-scale quantum (NISQ) [2] computers, are expected to solve a practical computational problem beyond the reach of classical computers.
Among possible applications, variational quantum eigensolver (VQE) [3, 4] that solves the ground state problem of quantum many-body Hamiltonian has gained lots of attention recently (see Ref. [5] for a recent review).
The VQE works by combining a parameterized quantum circuit and a classical optimization algorithm: A quantum circuit evaluates the expectation value of the Hamiltonian and its derivatives for the output quantum state, whereas a classical optimizer finds better parameters that minimize the energy.
As solving the ground state of quantum Hamiltonians are difficult for classical computers, one may easily get an advantage of VQEs.
Still, it is unclear which Ansatz and classical optimizer should be used to efficiently solve a given Hamiltonian.
One of the most widely studied Ansätze for solving translational invariant many-body Hamiltonians is the quantum alternating operator Ansatz (QAOA) [6] .
Inspired by a short-depth quantum algorithm for solving combinatory optimization problems [7], this Ansatz is composed of rotating gates using the terms of the Hamiltonian.
Even though the QAOA solves many different Hamiltonians reliably [8, 9, 10], however,
it is not necessarily good at solving problems with symmetry broken nature which prevail in many-body systems.
Indeed, there are local Hamiltonians whose ground states cannot be generated by this Ansatz in constant-depth albeit such a circuit exists, as the circuit obeys the same symmetry as the Hamiltonian [11].
In this paper, we devise a symmetry breaking Ansatz and explore its power for solving many-body Hamiltonians.
We first show that our Ansatz can generate a ground state within a different symmetry protected sector than the input state in constant-depth, which is impossible for the bare QAOA.
We next use our symmetry breaking Ansatz to prepare a particular symmetry broken state among degenerate ground states. This is done by simply adding a symmetry penalizing term to the loss function.
II Symmetry breaking Ansatz
The first model we study is the transverse field Ising (TFI) model for $N$ qubits defined by the Hamiltonian
$$\displaystyle H_{\rm TFI}=-\sum_{i=1}^{N}Z_{i}Z_{i+1}-h\sum_{i=1}^{N}X_{i}$$
(1)
where we impose the periodic boundary condition $Z_{N+1}=Z_{1}$. This model has two distinct (ferromagnetic and paramagnetic) phases depending on the strength of $h$ that are protected by the spin flip symmetry $P=\prod_{i}X_{i}$ [12]. The critical point of this model $h=1$ is well known.
This implies that if there is a circuit that commutes with $P$ and connects two ground states in different phases, then the circuit depth must be larger than a constant [13].
On the other hand, there is a finite-depth circuit that connects two different ground states if we do not restrict to such a symmetry, as the system is gapped unless $h=1$ [13, 14].
As a common Ansatz for preparing the ground state only utilizes terms within the Hamiltonian, it fails to represent such a circuit.
For example, the QAOA Ansatz for this Hamiltonian is given by $\prod_{k=D}^{1}L_{x}(\phi_{k})L_{zz}^{\rm odd}(\kappa_{k})L_{zz}^{\rm even}(\vartheta_{k})\ket{+}^{\otimes N}$ where $L_{x}(\phi)=\exp[-i\phi\sum_{i}X_{i}]$, $L_{zz}^{\rm odd}(\kappa)=\exp[-i\kappa\sum_{i=1}^{N/2}Z_{2i-1}Z_{2i}]$, and $L_{zz}^{\rm even}(\vartheta)=\exp[-i\vartheta\sum_{i=1}^{N/2}Z_{2i}Z_{2i+1}]$. As all gates commute with $P$, i.e. $[P,L_{zz}^{\rm odd}]=[P,L_{zz}^{\rm even}]=[P,L_{x}]=0$, and the input state $\ket{+}^{\otimes N}$ is the ground state of the Hamiltonian when $h\rightarrow\infty$, we know that preparing the ground state for $h<1$ requires circuit depth larger than a constant.
Indeed, theoretical and numerical studies have found that this type of Ansatz needs depth $D\geq N/2$ to prepare the ground state faithfully [15, 8].
We now add symmetry breaking layers in VQE Ansatz and see whether it can achieve lower circuit depth for preparing the ground state.
Our Ansatz for the TFI is given as
$$\displaystyle\ket{\psi(\boldsymbol{\theta})}=\prod_{j=D}^{1}L_{z}(\phi_{j})L_{x}(\kappa_{j})L_{zz}(\vartheta_{j})\ket{+}^{\otimes N}$$
(2)
where $L_{zz}(\theta_{j})=\exp[-i\theta_{j}\sum_{i}Z_{i}Z_{i+1}]$, $L_{x}(\kappa_{j})=\exp[-i\kappa_{j}\sum_{i}X_{i}]$, $L_{z}(\phi_{j})=\exp[-i\phi_{j}\sum_{i}Z_{i}]$, and $\boldsymbol{\theta}=\{\vartheta_{j},\kappa_{j},\phi_{j}\}_{j=1}^{D}$ is a set of all parameters. All layers in the Ansatz preserve the translational symmetry but the $L_{z}(\phi_{j})$ layers break the symmetry $P$ of the Hamiltonian.
To observe the effect of symmetry breaking layers, we first present the relative errors for different system sizes, depths, with and without the symmetry breaking $L_{z}(\phi_{j})$ layers.
We optimize parameters using the quantum natural gradient [16, 17, 18]: For each epoch $t$, we update parameters as $\boldsymbol{\theta}_{t+1}=\boldsymbol{\theta}_{t}-\eta(\mathcal{F}+\lambda_{t}\mathbb{1})^{-1}\nabla_{\theta}\braket{\psi(\boldsymbol{\theta}_{t})}{H}{\psi(\boldsymbol{\theta}_{t})}$ where $\eta$ is the learning rate, $\mathcal{F}=(\mathcal{F}_{ij})$ is the quantum Fisher matrix, and $\lambda_{t}$ is a (step dependent) regularization constant.
We choose this optimizer as it works more reliably in solving the ground problem both for classical neural networks [19] and VQEs [9].
For the quantum Fisher matrix, we use the centered one $\mathcal{F}=\mathcal{F}_{ij}^{\rm c}=\mathfrak{Re}\{\braket{\partial_{\theta_{i}}\psi(\boldsymbol{\theta}_{t})}{\partial_{\theta_{j}}\psi(\boldsymbol{\theta}_{t})}-\braket{\partial_{\theta_{i}}\psi(\boldsymbol{\theta}_{t})}{\psi(\boldsymbol{\theta}_{t})}\braket{\psi(\boldsymbol{\theta}_{t})}{\partial_{\theta_{j}}\psi(\boldsymbol{\theta}_{t})}\}$ [17] mostly (unless otherwise stated), but the uncentered one $\mathcal{F}_{ij}^{\rm nc}=\mathfrak{Re}\{\braket{\partial_{\theta_{i}}\psi(\boldsymbol{\theta}_{t})}{\partial_{\theta_{j}}\psi(\boldsymbol{\theta}_{t})}\}$ [16] is also considered when it improves the performance.
The difference between two can be understood using the notion of the projected Hilbert space [18].
For the hyperparameters, we typically use $\eta=0.01$ and $\lambda_{t}=\max(100.0\times 0.9^{t},10^{-3})$.
We show the optimized normalized energies $\widetilde{E}=(E_{\rm VQE}-E_{\rm GS})/E_{\rm GS}$ for different $N$ and $D$ in Fig. 1 when $h=0.5$.
We have used the Fisher matrix with the centering term $\mathcal{F}^{\rm c}$ and initial values of parameters $\{\vartheta_{j},\kappa_{j}\,\phi_{j}\}$ are sampled from the normal distribution $\mathcal{N}(0,\sigma^{2})$ with $\sigma=0.001$ besides $D=3,5$ at Fig. 1(b).
When $D=3$ (all $N$) and $5$ (for $N\geq 12$), this initialization does not give any better energy than the Ansatz without symmetry breaking layers.
We instead found that initializing parameters for the symmetry breaking layers $\{\phi_{j}\}$ with samples from $\mathcal{N}(2\pi/D,\sigma^{2})$ finds better optima in these cases.
However, this initialization does not change the results for $D=4$, and performs even worse when $D\geq 6$ (see Appendix A for detail comparisons).
Fig. 1(a) shows that the ground state is only found for $D\geq N/2$ when symmetry breaking layers are absent, which is consistent with Refs. [8, 9].
On the other hand, results with symmetry breaking layers [Fig. 1(b)] clearly demonstrate that converged energies are significantly improved for $5\leq D<N/2$.
Most importantly, converged normalized energies are $\leq 10^{-7}$ for all $N$ when $D\geq 9$, which implies that our Ansatz finds a constant-depth circuit for solving the ground state.
In addition, the results show that there is a finite-size effect up to $N=18$ where the accurate ground state is only obtained when $D\geq N/2$.
However, there is a huge gap between the converged energies and the true ground state when $D=4$.
In fact, the results for $D=4$ are even worse than those of $D=3$, which signals that the optimizer gets stuck in local minima. This type of convergence problem is already observed in Ref. [9].
To obtain a better convergence for $D=4$, we employ a transfer learning technique.
Instead of starting from a randomly initialized circuit, we insert a block in the middle of the converged circuit from $D=3$ and perturb all parameters by adding small numbers sampled from $\mathcal{N}(0,\sigma^{\prime 2})$ (where we typically use $\sigma^{\prime}=0.01$). We then optimize the full circuit using the quantum natural gradient.
We show converged energies from the transfer learning in Fig. 2.
Fig. 2(a) illustrates that the transfer learning succeeds to find a better optima for $D=4$. However, the result for $(N,D)=(10,5)$ is worse than that of the random initialization, which implies that the transfer learning does not necessarily find the global optimia.
We additionally plot a learning curve for $(N,D)=(18,4)$ in Fig. 2(b) where the initial energy is much higher due to an added perturbation but it eventually finds a lower energy state.
We next consider the transverse field cluster (TFC) model, the Hamiltonian of which is given as
$$\displaystyle H_{\rm TFC}=-\sum_{i=1}^{N}Z_{i}X_{i+1}Z_{i+2}-h\sum_{i=1}^{N}X_{i}.$$
(3)
It is known that the phase transition of this model also takes place at $h=1$ [20].
As the terms $Z_{i}X_{i+1}Z_{i+2}$ are mutually commuting, the ground state at $h=0$ is the common eigenvector of those operators which is also known as the cluster state.
Two relevant symmetries that determine the ground state of this model are $P_{1}=\prod_{i=1}^{N/2}X_{2i}$ and $P_{2}=\prod_{i=1}^{N/2}X_{2i-1}$ [21, 22].
We thus expect that a circuit cannot bring the product state $\ket{+}^{\otimes N}$ to the ground state of the Hamiltonian when $h<1$ in a constant depth as long as it commutes with $P_{1}$ and $P_{2}$.
We study whether a symmetry breaking layer can improve it. To see this, we construct our Ansatz
$$\displaystyle\ket{\psi(\boldsymbol{\theta})}$$
$$\displaystyle=\prod_{j=D}^{1}L_{z}^{\rm even}(\phi_{j})L_{z}^{\rm odd}(\chi_{j})$$
$$\displaystyle\qquad\qquad\times L_{x}(\kappa_{j})L_{zxz}(\theta_{j})\ket{+}^{\otimes N}$$
(4)
where $L_{zxz}(\theta_{j})=\exp[-i\theta_{j}\sum_{i=1}^{N}Z_{i}X_{i+1}Z_{i+2}]$, $L_{z}^{\rm even/odd}(\phi_{j})=\exp[-i\phi_{j}\sum_{i=1}^{N/2}Z_{2i/2i+1}]$.
Converged normalized energies with and without symmetry breaking layers for $h=0.5$ are shown in Fig. 3.
The results without symmetry breaking layers show that circuits with $D\geq\lfloor N/4\rfloor$ solve the ground state accurately.
However, the symmetry breaking layers only improve the results for $D=3$ when $N\geq 16$ for this Hamiltonian.
In addition, such an improvement is observed only when the uncentered Fisher matrix $\mathcal{F}^{\rm nc}$ and the initial value $2\pi/D$ for symmetry breaking layers are used.
We moreover see that our Ansatz does not find a constant-depth circuit for solving the ground state based upon the results up to $N=20$.
We attribute this to the finite-size effect and expect that much larger system size $N$ (which is beyond our computational capacity) is required to find the constant-depth circuit.
III Preparing a symmetry broken ground state
We next consider the cluster Hamiltonian [Eq. (3) with $h=0$] with the open boundary condition, which is given as
$$\displaystyle H_{\rm cluster}=-\sum_{i=1}^{N-2}Z_{i}X_{i+1}Z_{i+2}.$$
(5)
Ground states of this model are stabilized by $N-2$ terms ($Z_{i}X_{i+1}Z_{i+2}\ket{\psi_{\rm GS}}=\ket{\psi_{\rm GS}}$ for $i\in[1,\cdots,N-2]$), thus $4$-fold degenerate. As the operators $P_{1}$ and $P_{2}$ commute with all stabilizers, they further define the ground state manifold.
For the QAOA, the ground state must be $+1$ eigenstate of $P_{1}$ and $P_{2}$ as they commute with the circuit and $P_{1,2}\ket{+}^{\otimes N}=+1\ket{+}^{\otimes N}$.
In contrast, we show our Ansatz [Eq. (4)] (with $L_{zxz}(\theta_{j})$ in the open boundary condition) can be used to prepare a particular state within the manifold.
The main idea is minimizing the expectation value of $O=H_{\rm cluster}+\alpha_{1}P_{1}+\alpha_{2}P_{2}$ for suitable $\alpha_{1}$ and $\alpha_{2}$, instead of the Hamiltonian itself. We expect the obtained state is an eigenstate with the corresponding eigenvalue $\pm 1$ of $P_{i}$ when $\mathrm{sign}(\alpha_{i})=\mp 1$.
As an example, we study the VQE for preparing the ground state with $P_{1}=P_{2}=-1$ using $(N,D)=(14,8)$.
We have tested VQEs with the varying learning rates $\eta\in[0.01,0.025,0.05,0.1]$ and $\alpha_{1}=\alpha_{2}=\alpha\in[0.1,0.3,0.5,1.0,2.0,4.0]$, and found that the desired state can be prepared by the best for $\eta=0.025$, $\alpha=2.0$. Especially, all VQE runs have converged to $\braket{P_{1}}=\braket{P_{2}}=-1$ (accuracy within $10^{-8}$) with these values of $\eta$ and $\alpha$.
The resulting learning curve is shown in Fig. 4(a).
However, the converged energy $\widetilde{E}\approx 1.502\times 10^{-3}$ for $D=8$ may not be satisfying compared to the other models we have studied above.
Thus, we further study whether increasing $D$ helps the convergence in Fig. 4(b).
The results show that the converged energies are getting accurate as we increase $D$. Precisely, $47$ instances out of $48$ independent VQEs runs have converged to $\widetilde{E}\leq 10^{-8}$ when $D=20$.
Acknowledgement
The author thanks David Wierichs for helpful discussions.
This project was funded by the Deutsche Forschungsgemeinschaft under Germany’s Excellence Strategy - Cluster of Excellence Matter and Light for Quantum Computing (ML4Q) EXC 2004/1-390534769 and within the CRC network TR 183 (project grant 277101999) as part of project B01.
The numerical simulations were performed on the JUWELS cluster at the Forschungszentrum Juelich.
Source codes for the current paper can be found in Github repository [23].
Appendix A Optimization of the Ansatz with symmetry breaking layers
When the Ansatz contains symmetry breaking layers, an optimization algorithm easily stuck in local minima. We here compare several different set-ups for the transverse field Ising model we have studied in Sec. II.
We show results from the transverse Ising model with different initial values and using the centered and uncentered Fisher matrix in Fig. 5.
We can see that the result with $\{\phi_{j}\}$ initialized from $\mathcal{N}(0,\sigma^{2})$ and using the centered Fisher matrix is the most natural.
However, when $D=3,5$, the results with $\{\psi_{j}\}$ initialized from $\mathcal{N}(2\pi/D,\sigma^{2})$ show better convergence.
We have also found that using the uncentered Fisher matrix improves convergence for $D=4$ where the centered Fisher matrix failed to find an appropriate optima. Our results suggest that the learning landscape of the VQEs with symmetry breaking layers is rugged especially when the parameters are not sufficient to describe the ground state accurately.
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Canonical metric on the space of symplectic invariant tensors and its applications
Shigeyuki Morita
[email protected]
Graduate School of Mathematical Sciences,
University of Tokyo,
3-8-1 Komaba,
Meguro-ku, Tokyo, 153-8914, Japan
Dedicated to the memory of Professor Akio Hattori
Abstract.
Let $\Sigma_{g}$ be a closed oriented surface of genus $g$
and let $H_{\mathbb{Q}}$ denote $H_{1}(\Sigma_{g};\mathbb{Q})$ which we understand
to be the standard symplectic vector space over $\mathbb{Q}$ of dimension $2g$.
We introduce a canonical metric on the space $(H_{\mathbb{Q}}^{\otimes 2k})^{\mathrm{Sp}}$
of symplectic invariant tensors by analyzing the structure
of the vector space $\mathbb{Q}\mathcal{D}^{\ell}(2k)$ generated by linear chord diagrams
with $2k$ vertices. This space, equipped with a certain
inner product, serves as a universal model for $(H^{\otimes 2k})^{\mathrm{Sp}}$
for any $g$. We decompose
$\mathbb{Q}\mathcal{D}^{\ell}(2k)$ as an orthogonal direct sum of eigenspaces $E_{\lambda}$ where
$\lambda$ is indexed by
the set of all the Young diagrams with $k$ boxes.
We give a formula for the eigenvalue
$\mu_{\lambda}$ of $E_{\lambda}$ and thereby
we obtain a complete
description of how the spaces $(H_{\mathbb{Q}}^{\otimes 2k})^{\mathrm{Sp}}$ degenerate according
as the genus decreases
from the stable range $g\geq k$
to the last case $g=1$
with the largest eigenvalue
$2g(2g+1)\cdots(2g+k-1)$.
As an application of our canonical metric, we obtain certain relations among the
Mumford-Morita-Miller tautological classes, in a systematic way, which hold in the tautological
algebra in cohomology of the moduli space of curves. We also indicate other
possible applications
such as characteristic classes of
transversely symplectic foliations and a project with T. Sakasai and M. Suzuki [30]
where we study the structure of
the symplectic derivation Lie algebra.
Key words and phrases:symplectic tensor, symmetric group, Young diagram, tautological algebra, moduli space of curves, transversely symplectic foliation
2000 Mathematics Subject Classification: Primary 20C30;32G15 , Secondary 20J06; 55R40
1. Introduction and statements of the main results
Let $\Sigma_{g}$ be a closed oriented surface of genus $g$.
We denote $H_{1}(\Sigma_{g};\mathbb{Q})$ simply by $H_{\mathbb{Q}}$ and let
$$\mu:H_{\mathbb{Q}}\otimes H_{\mathbb{Q}}\rightarrow\mathbb{Q}$$
be the intersection pairing which is a non-degenerate skew symmetric
bilinear form. If we choose a symplectic basis of $H_{\mathbb{Q}}$, then
the automorphism group of $(H_{\mathbb{Q}},\mu)$ can be identified with the symplectic
group $\mathrm{Sp}(2g,\mathbb{Q})$ and $H_{\mathbb{Q}}$ serves as the fundamental
representation of $\mathrm{Sp}(2g,\mathbb{Q})$. Let $(H_{\mathbb{Q}}^{\otimes{2k}})^{\mathrm{Sp}}$
denote the $\mathrm{Sp}(2g,\mathbb{Q})$-invariant subspace of the tensor product
$H_{\mathbb{Q}}^{\otimes{2k}}$. Now consider the bilinear mapping
$$\mu^{\otimes 2k}:H_{\mathbb{Q}}^{\otimes{2k}}\otimes H_{\mathbb{Q}}^{\otimes{2%
k}}\rightarrow\mathbb{Q}$$
defined by
$$(u_{1}\otimes\cdots\otimes u_{2k})\otimes(v_{1}\otimes\cdots\otimes v_{2k})%
\mapsto\prod_{i=1}^{2k}\,\mu(u_{i},v_{i})\quad(u_{i},v_{i}\in H_{\mathbb{Q}}).$$
which is clearly symmetric. Hence the restriction of
$\mu^{\otimes 2k}$ to
the subspace $(H_{\mathbb{Q}}^{\otimes{2k}})^{\mathrm{Sp}}\otimes(H_{\mathbb{Q}}^{\otimes{2k%
}})^{\mathrm{Sp}}$
induces a symmetric bilinear form on $(H_{\mathbb{Q}}^{\otimes{2k}})^{\mathrm{Sp}}$.
Recall that irreducible representations of the symmetric group
$\mathfrak{S}_{k}$ is indexed by Young diagrams
$\lambda=[\lambda_{1}\lambda_{2}\cdots\lambda_{h}]$
whose number of boxes, denoted by $|\lambda|=\lambda_{1}+\cdots\lambda_{h}$,
is $k$. We denote the corresponding representation by $\lambda_{\mathfrak{S}_{k}}$.
For each Young diagram $\lambda$ as above, we define two types
of Young diagrams, denoted by $2\lambda$ and $\lambda^{\delta}$, with $2k$ boxes. One is defined as
$2\lambda=[2\lambda_{1}2\lambda_{2}\cdots 2\lambda_{h}]$ and the other is
defined to be $\lambda^{\delta}=[\lambda_{1}\lambda_{1}\lambda_{2}\lambda_{2}\cdots\lambda_{h%
}\lambda_{h}]$.
For example, if $\lambda=[431]$, then
$2\lambda=[862]$ and $\lambda^{\delta}=[443311]$. We call Young
diagrams with the former type (resp. the latter type)
of even type (resp. with multiple double floors).
Theorem 1.1.
For any $g$, the symmetric bilinear form $\mu^{\otimes 2k}$ on $(H_{\mathbb{Q}}^{\otimes 2k})^{\mathrm{Sp}}$
is positive definite so that it defines a metric on this space. Furthermore, there exists an orthogonal
direct sum decomposition
$$(H_{\mathbb{Q}}^{\otimes 2k})^{\mathrm{Sp}}\cong\bigoplus_{|\lambda|=k,\ h(%
\lambda)\leq g}U_{\lambda}$$
in terms of certain subspaces $U_{\lambda}$.
With respect to the natural action of $\mathfrak{S}_{2k}$
on $(H_{\mathbb{Q}}^{\otimes 2k})^{\mathrm{Sp}}$, each subspace $U_{\lambda}$
is an irreducible $\mathfrak{S}_{2k}$-submodule and we have an isomorphism
$$U_{\lambda}\cong(\lambda^{\delta})_{\mathfrak{S}_{2k}}.$$
In other words, the above orthogonal
direct sum decomposition gives also the irreducible decomposition of the
$\mathfrak{S}_{2k}$-module $(H_{\mathbb{Q}}^{\otimes 2k})^{\mathrm{Sp}}$.
Corollary 1.2.
The dimension of the space of invariant tensors $(H_{\mathbb{Q}}^{\otimes 2k})^{\mathrm{Sp}}$
is given by
$$\dim\,(H_{\mathbb{Q}}^{\otimes 2k})^{\mathrm{Sp}}=\sum_{|\lambda|=k,\ h(%
\lambda)\leq g}\mathrm{dim}\,(\lambda^{\delta})_{\mathfrak{S}_{2k}}.$$
Remark 1.3.
Recall here that there are a number of explicit formulas for the dimension of irreducible representations of
symmetric groups, e.g. the hook length formula, the number of standard tableaux on the
Young diagram and the formula of Frobenius (cf [4]). In particular
$$\dim\,U_{[k]}=\dim\,[k^{2}]_{\mathfrak{S}_{2k}}=\frac{1}{k+1}\binom{2k}{k}%
\quad(\text{the $k$-th Catalan number})$$
is equal to $\dim\,(H_{\mathbb{Q}}^{\otimes 2k})^{\mathrm{Sp}}$ for the case $g=1$.
We mention that, in this case, a basis of this space was given
in a classical paper [32] by Rumer, Teller and Weyl.
Furthermore Mihailovs [21] extended this work to obtain a nice
basis of $(H_{\mathbb{Q}}^{\otimes 2k})^{\mathrm{Sp}}$
for any $g$.
However, their basis is different from ours because it
depends essentially on the choice of ordering of tensors so that it is not canonical.
A linear chord diagram with $2k$ vertices is a partition of the set
$\{1,2,\cdots,2k\}$ into $k$-tuple
$$C=\{\{i_{1},j_{1}\},\cdots,\{i_{k},j_{k}\}\}\quad$$
of pairs $\{i_{\ell},j_{\ell}\}\ (\ell=1,\cdots,k)$. Here we assume
$$i_{1}<\cdots<i_{k},\ i_{\ell}<j_{\ell}\ (\ell=1,\cdots,k)$$
so that the above expression is uniquely determined.
We consider $C$ to be a graph with $2k$ vertices and $k$ edges (chords)
each of which connects $i_{\ell}$ with $j_{\ell}$ for $\ell=1,\cdots,k$.
Let
$\mathcal{D}^{\ell}(2k)$
denote the set of all the linear chord diagrams with $2k$ vertices.
It has $(2k-1)!!$ elements and let $\mathbb{Q}\mathcal{D}^{\ell}(2k)$ be the vector
space over $\mathbb{Q}$ spanned by $\mathcal{D}^{\ell}(2k)$.
The symmetric group $\mathfrak{S}_{2k}$ acts on $\mathcal{D}^{\ell}(2k)$ naturally
so that $\mathbb{Q}\mathcal{D}^{\ell}(2k)$ has a structure of an $\mathfrak{S}_{2k}$-module.
We define a symmetric bilinear mapping
$$\langle\ \,,\ \rangle:\mathbb{Q}\mathcal{D}^{\ell}(2k)\times\mathbb{Q}\mathcal%
{D}^{\ell}(2k)\rightarrow\mathbb{Q}[g]$$
by setting
$$\langle C,C^{\prime}\rangle=(-1)^{k-r}(2g)^{r}$$
for any two elements $C,C^{\prime}\in\mathcal{D}^{\ell}(2k)$, where $r$ denotes the number
of connected components of the graph $C\cup C^{\prime}$
(here we consider interiors of $2k$ edges to be mutually disjoint), and extending this bilinearly.
If we let $g$ to take a specific value, then the above can be understood
to be a symmetric bilinear pairing. It is clear that this pairing
is $\mathfrak{S}_{2k}$-invariant. Namely the equality
$$\langle\gamma(C),\gamma(C^{\prime})\rangle=\langle C,C^{\prime}\rangle\quad(C,%
C^{\prime}\in\mathcal{D}^{\ell}(2k))$$
holds for any $\gamma\in\mathfrak{S}_{2k}$. Let $M_{2k}$ denote
the corresponding intersection matrix. In other words, it is the matrix
which represents the linear mapping
$$\mathcal{M}:\mathbb{Q}\mathcal{D}^{\ell}(2k)\rightarrow\mathbb{Q}\mathcal{D}^{%
\ell}(2k)$$
defined as
$$\mathcal{M}(C)=\sum_{D\in\mathcal{D}^{\ell}(2k)}\langle C,D\rangle\,D.$$
Theorem 1.4.
The intersection matrix $M_{2k}$ is positive semi-definite for any $g$,
so that it defines a metric on
$\mathbb{Q}\mathcal{D}^{\ell}(2k)$ depending on $g$.
Furthermore, there exists an orthogonal
direct sum decomposition
$$\mathbb{Q}\mathcal{D}^{\ell}(2k)\cong\bigoplus_{|\lambda|=k}E_{\lambda}$$
in terms of eigenspaces $E_{\lambda}$ of $\mathcal{M}$ which
are defined independently of the genus $g$.
The eigenvalue of $E_{\lambda}$ , denoted by $\mu_{\lambda}$, is given by
the following formula
$$\mu_{\lambda}=\prod_{\text{b: box of $\lambda$}}(2g-2s_{b}+t_{b})$$
where $s_{b}$ denotes the number of columns of $\lambda$
which are on the left of the column containing $b$ and
$t_{b}$ denotes the number of rows which are above the
row containing $b$.
With respect to the natural action of $\mathfrak{S}_{2k}$
on $\mathbb{Q}\mathcal{D}^{\ell}(2k)$, $E_{\lambda}$
is an irreducible $\mathfrak{S}_{2k}$-submodule and we have an isomorphism
$$E_{\lambda}\cong(2\lambda)_{\mathfrak{S}_{2k}}.$$
In other words, the above orthogonal
direct sum decomposition gives also the irreducible decomposition of the
$\mathfrak{S}_{2k}$-module $\mathbb{Q}\mathcal{D}^{\ell}(2k)$.
Theorem 1.5.
The correspondence
$$\lambda\mapsto\mu_{\lambda}$$
defines an injective mapping from the set of all the Young diagrams
$\lambda$ with $k$ boxes into the space of polynomials in $g$ of degree $k$.
Example 1.6.
The largest (resp. the smallest) eigenvalue occurs for $\lambda=[1^{k}]$
(resp. $\lambda=[k]$) and the corresponding eigenvalues are
$$\mu_{[1^{k}]}=2g(2g+1)\cdots(2g+k-1),\quad\mu_{[k]}=2g(2g-2)\cdots(2g-2k+2).$$
The space $\mathbb{Q}\mathcal{D}^{\ell}(2k)$ is a universal model for
the symplectic invariant tensors because we can make the following construction.
First there exists a natural homomorphism
$$\Phi:\mathbb{Q}\mathcal{D}^{\ell}(2k)\rightarrow(H_{\mathbb{Q}}^{\otimes 2k})^%
{\mathrm{Sp}}$$
which is surjective for any $g$ and “anti” $\mathfrak{S}_{2k}$-equivariant
(see Proposition 2.1 for details).
In the dual setting, there exists also a natural homomorphism
$$\mathcal{K}:H_{\mathbb{Q}}^{\otimes 2k}\rightarrow\mathbb{Q}\mathcal{D}^{\ell}%
(2k)$$
which enumerate all the contractions and can detect $\mathrm{Sp}$-invariant
component of any given tensor
(see section $\lx@sectionsign 3$, in particular Proposition 3.2, for details).
We use the same symbol $\mathcal{K}$ for its restriction
to the subspace $(H_{\mathbb{Q}}^{\otimes 2k})^{\mathrm{Sp}}$.
Theorem 1.7.
The following diagram is commutative
$$\begin{CD}\mathbb{Q}\mathcal{D}^{\ell}(2k)@>{\mathcal{M}}>{}>\mathbb{Q}%
\mathcal{D}^{\ell}(2k)\\
@V{\Phi}V{}V\Big{\|}\\
(H_{\mathbb{Q}}^{\otimes 2k})^{\mathrm{Sp}}@>{\mathcal{K}}>{}>\mathbb{Q}%
\mathcal{D}^{\ell}(2k)\end{CD}$$
and all the homomorphisms $\Phi,\mathcal{M},\mathcal{K}$ are isomorphisms for $g\geq k$.
For any $\lambda$ with $|\lambda|=k$, $U_{\lambda}=\Phi(E_{\lambda^{\prime}})$
where $\lambda^{\prime}$ denotes the conjugate
Young diagram of $\lambda$.
Furthermore the subspaces $\mathcal{K}(U_{\lambda})=E_{\lambda^{\prime}}\subset\mathbb{Q}\mathcal{D}^{%
\ell}(2k)$ are mutually
orthogonal to each other with respect to
the ordinary Euclidean metric on $\mathbb{Q}\mathcal{D}^{\ell}(2k)$ induced by taking
$\mathcal{D}^{\ell}(2k)$ as an orthonormal basis.
This work was originally motivated by an attempt, begun some years ago, to extend
our former work [28] on the structure of
the tautological algebra of the moduli space of curves.
Although it turned out to require further works in this direction, we present in $\lx@sectionsign 6$
an explicit set of relations among the Mumford-Morita-Miller
tautological classes by making use of the main result of this
paper (see Theorem 6.3 and Conjecture 6.5). On the other hand,
in a joint work with T. Sakasai and M. Suzuki [30], where we study
the structure of the symplectic derivation Lie algebra, we
use the canonical metric introduced in this paper and found another role of it.
Because of these, we describe a detailed proof of the above results.
Acknowledgement The author would like to thank Carel Faber,
Richard Hain and Nariya Kawazumi for enlightening discussions
about the cohomology of the mapping class groups
on various occasions in the $1990$’s and early $2000$’s.
He also would like to thank
Takuya Sakasai and Masaaki Suzuki
for their helps in recent years
which greatly encouraged him to write up this paper.
2. Space of linear chord diagrams and $(H^{\otimes 2k})^{\mathrm{Sp}}$
For each linear chord diagram $C\in\mathcal{D}^{\ell}(2k)$, define
$$a_{C}\in(H_{\mathbb{Q}}^{\otimes 2k})^{\mathrm{Sp}}$$
by permuting the elements $(\omega_{0})^{\otimes 2k}$ in such a way that the $s$-th part
$(\omega_{0})_{s}$ of this tensor product goes to $(H_{\mathbb{Q}})_{i_{s}}\otimes(H_{\mathbb{Q}})_{j_{s}}$, where $(H_{\mathbb{Q}})_{i}$
denotes the $i$-th component of $H_{\mathbb{Q}}^{\otimes 2k}$, and multiplied by the factor
$$\mathrm{sgn}\,C=\mathrm{sgn}\begin{pmatrix}1&2&\cdots&2k-1&2k\\
i_{1}&j_{1}&\cdots&i_{k}&j_{k}\end{pmatrix}.$$
Here $\omega_{0}\in(H_{\mathbb{Q}}^{\otimes 2})^{\mathrm{Sp}}$ denotes the symplectic class.
Now define a linear mapping
$$\Phi:\mathbb{Q}\mathcal{D}^{\ell}(2k)\rightarrow(H_{\mathbb{Q}}^{\otimes 2k})^%
{\mathrm{Sp}}$$
by setting $\Phi(C)=a_{C}$.
Proposition 2.1.
The correspondence
$$\Phi:\mathbb{Q}\mathcal{D}^{\ell}(2k)\rightarrow(H_{\mathbb{Q}}^{\otimes 2k})^%
{\mathrm{Sp}}$$
is surjective for any $g$ and bijective for any $g\geq k$. Furthermore this correspondence
is “anti” $\mathfrak{S}_{2k}$-equivariant in the sense that
$$\Phi(\gamma(C))=\mathrm{sgn}\,\gamma\ \gamma(\Phi(C))$$
for any $C\in\mathbb{Q}\mathcal{D}^{\ell}(2k)$ and $\gamma\in\mathfrak{S}_{2k}$.
Proof.
The surjectivity follows from a classical result of Weyl on symplectic invariants.
The latter part can be shown by comparing how the signs of permutations act
on the symplectic invariant tensors and the linear chord diagrams.
∎
Proposition 2.2.
The following diagram is commutative
$$\begin{CD}\mathbb{Q}\mathcal{D}^{\ell}(2k)\otimes\mathbb{Q}\mathcal{D}^{\ell}(%
2k)@>{\langle\ ,\ \rangle}>{}>\mathbb{Q}[g]\\
@V{\Phi\otimes\Phi}V{}V@V{}V{\text{evaluation}}V\\
(H_{\mathbb{Q}}^{\otimes 2k})^{\mathrm{Sp}}\otimes(H_{\mathbb{Q}}^{\otimes 2k}%
)^{\mathrm{Sp}}@>{\mu^{\otimes 2k}}>{}>\mathbb{Q}.\end{CD}$$
To prove this, we prepare a few facts.
Lemma 2.3.
Let $x_{1},\cdots,x_{g},y_{1},\cdots,y_{g}$ be a symplectic basis of $H_{\mathbb{Q}}$ and let
$$\omega_{0}=\sum_{i=1}^{g}\{x_{i}\otimes y_{i}-y_{i}\otimes x_{i}\}\in(H_{%
\mathbb{Q}}^{\otimes 2})^{\mathrm{Sp}}$$
be the symplectic class.
$\mathrm{(i)}\ \text{For any two elements $u,v\in H_{\mathbb{Q}}$, we have the identity}$
$$\mu(u,v)=\sum_{i=1}^{g}\{\mu(u,x_{i})\mu(v,y_{i})-\mu(u,y_{i})\mu(v,x_{i})\}.$$
$\mathrm{(ii)}\ \text{For any two elements $u,v\in H_{\mathbb{Q}}$, we have the%
identity}$
$$\mu(u,v)=\mu^{\otimes 2}(u\otimes v,\omega_{0}).$$
Proof.
It is easy to verify (i) while (ii) is just a restatement of (i) in terms of the linear mapping
$$\mu^{\otimes 2}:H_{\mathbb{Q}}^{\otimes 2}\otimes H_{\mathbb{Q}}^{\otimes 2}%
\rightarrow\mathbb{Q}.$$
∎
Lemma 2.4 (Lemma 3.3 of [28]).
For any two indices $i,j$ with $1\leq i<j\leq 2k$, let
$$p_{ij}:H_{\mathbb{Q}}^{\otimes 2k}\rightarrow H_{\mathbb{Q}}^{\otimes 2k}$$
be the linear map defined by first taking the intersection number of the $i$-th entry with
the $j$-th entry and then putting the element $\omega_{0}$ there. Namely we set
$$\displaystyle p_{ij}(u_{1}\otimes\cdots\otimes u_{i}\otimes\cdots$$
$$\displaystyle\otimes u_{j}\otimes\cdots\otimes u_{2k})=$$
$$\displaystyle\mu(u_{i},u_{j})\sum_{s=1}^{g}$$
$$\displaystyle\{u_{1}\otimes\cdots\otimes x_{s}\otimes\cdots\otimes y_{s}%
\otimes\cdots\otimes u_{2k})$$
$$\displaystyle-u_{1}\otimes\cdots\otimes y_{s}\otimes\cdots\otimes x_{s}\otimes%
\cdots\otimes u_{2k}\}.$$
Then we have
$$p_{ij}(a_{C})=\begin{cases}2g\ a_{C}\quad\{i,j\}\in C\\
-a_{C^{\prime}}\quad\{i,j\}\notin C\end{cases}$$
where $C^{\prime}\in\mathcal{D}^{\ell}(2k)$ is the linear chord diagram defined as follows.
Let $i^{\prime},j^{\prime}$ be the
indices such that $\{i,j^{\prime}\},\{i^{\prime},j\}\in C$. Then
$$C^{\prime}=C\setminus\{\{i,j^{\prime}\},\{i^{\prime},j\}\}\cup\{\{i,j\},\{i^{%
\prime},j^{\prime}\}\}.$$
Proof of Proposition 2.2.
By the linearity, it is enough to prove the following. For any two
linear chord diagrams $C,C^{\prime}\in\mathcal{D}^{\ell}(2k)$, we have the identity
(1)
$$\mu^{\otimes 2k}(a_{C}\otimes a_{C^{\prime}})=\langle C,C^{\prime}\rangle=(-1)%
^{k-r}(2g)^{r}$$
where the right equality is the definition of the pairing
$\langle\ ,\ \rangle$ and $r$ denotes the number of connected components of the graph $C\cup C^{\prime}$.
To prove the above equality (1), we decompose $\mu^{\otimes 2k}$ as
$$\mu^{\otimes 2k}=\prod_{\ell=1}^{k}\mu_{i_{\ell},j_{\ell}}^{\otimes 2}$$
where $C=\{\{i_{1},j_{1}\},\cdots,\{i_{k},j_{k}\}\}$ as before and $\mu_{i_{\ell},j_{\ell}}^{\otimes 2}(u\otimes v)$
$(u,v\in H_{\mathbb{Q}}^{\otimes 2k})$ denotes
taking the product of $\mu(u_{i_{\ell}},v_{i_{\ell}})$ and $\mu(u_{j_{\ell}},v_{j_{\ell}})$.
Now we consider the first chord $\{i_{1},j_{1}\}$ of $C$
(actually $i_{1}=1$) and apply $\mu_{i_{1},j_{1}}^{\otimes 2}$ on $a_{C}\otimes a_{C^{\prime}}$. By Lemma 2.3
this can be calculated by applying the operator $p_{i_{1}j_{1}}$ on $a_{C^{\prime}}$ and by Lemma 2.4 the answer is
given as follows. Namely if $\{i_{1},j_{1}\}\in C^{\prime}$, then we obtain the number $\mu^{\otimes 2}(\omega_{0}\otimes\omega_{0})=2g$
and the linear chord diagram $C^{\prime}$ is unchanged so that $C^{\prime}_{1}=C^{\prime}$
while if $\{i_{1},j_{1}\}\notin C^{\prime}$, then we obtain the factor $-1$ and the new linear chord diagram $C^{\prime}_{1}$ is
the one obtained from $C^{\prime}$ by replacing $\{i_{1},j^{\prime}_{1}\},\{i^{\prime}_{1},j_{1}\}\in C^{\prime}$ by $\{i_{1},j_{1}\},\{i^{\prime}_{1},j^{\prime}_{1}\}$.
In the latter case, it is easy to observe that the number of connected components of the graph $C\cup C^{\prime}_{1}$
increases by one than that of the graph $C\cup C^{\prime}$ because one of the connected components of
the latter graph is split into two components one of which corresponds to the common edge $\{i_{1},j_{1}\}$
of the two graphs $C$ and $C^{\prime}_{1}$. Next we consider the second chord $\{i_{2},j_{2}\}$ of $C$
and apply $\mu_{i_{2},j_{2}}^{\otimes 2}$ on $a_{C}\otimes a_{C^{\prime}_{1}}$. By the same argument as above,
if $\{i_{2},j_{2}\}\in C^{\prime}_{1}$, then we obtain the number $2g$
and the linear chord diagram $C^{\prime}_{1}$ is unchanged so that $C^{\prime}_{2}=C^{\prime}_{1}$
while if $\{i_{2},j_{2}\}\notin C^{\prime}_{1}$, then we obtain the factor $-1$ and the new linear chord diagram $C^{\prime}_{2}$ is
the one obtained from $C^{\prime}_{1}$ by replacing $\{i_{2},j^{\prime}_{2}\},\{i^{\prime}_{2},j_{2}\}\in C^{\prime}_{1}$ by $\{i_{2},j_{2}\},\{i^{\prime}_{2},j^{\prime}_{2}\}$.
In the latter case, the number of connected components of the graph $C\cup C^{\prime}_{2}$
increases by one than that of the graph $C\cup C^{\prime}_{1}$ because one of the connected components of
the latter graph is split into two components one of which corresponds to the common edge $\{i_{2},j_{2}\}$
of the two graphs $C$ and $C^{\prime}_{2}$. Continuing in this way, we finally arrive at the last one $C^{\prime}_{k}$ which
coincides with $C$ and along the way we obtain the factor $(2g)$ $r$-times while the factor $(-1)$
$(k-r)$ times for the evaluation of $\mu^{\otimes 2k}$. This completes the proof.
∎
Proposition 2.5.
The irreducible decomposition of the $\mathfrak{S}_{2k}$-module $\mathbb{Q}\mathcal{D}^{\ell}(2k)$
is given by
$$\mathbb{Q}\mathcal{D}^{\ell}(2k)=\bigoplus_{|\lambda|=k}E_{\lambda}$$
where $E_{\lambda}$ is isomorphic to $(2\lambda)_{\mathfrak{S}_{2k}}$.
Proof.
The symmetric group $\mathfrak{S}_{2k}$ acts on $\mathcal{D}^{\ell}(2k)$ naturally and
this action is clearly transitive. If we denote by $C_{0}\in\mathcal{D}^{\ell}(2k)$ the
linear chord diagram
$$C_{0}=\{\{1,2\},\{3,4\},\cdots,\{2k-1,2k\}\}$$
then the isotropy group of it
is the subgroup $H_{k}\subset\mathfrak{S}_{2k}$
consisting of permutations which fix $C_{0}$ considered as an element
of the power set of $\{1,2,\cdots,2k\}$. $H_{k}$ is isomorphic to
$$(\mathfrak{S}_{2}\times\overset{\text{$k$-times}}{\cdots}\times\mathfrak{S}_{2%
})\rtimes\mathfrak{S}_{k}$$
and we have a bijection
(2)
$$\mathfrak{S}_{2k}/H_{k}\cong\mathcal{D}^{\ell}(2k)$$
from the set of left $H_{k}$-cosets of $\mathfrak{S}_{2k}$ to $\mathcal{D}^{\ell}(2k)$.
Note that the number of elements of both sets is equal to $(2k-1)!!$. It is easy to see that the
above bijection (2) is an isomorphism of $\mathfrak{S}_{2k}$-set.
As is well known, the $\mathfrak{S}_{2k}$-module $\mathfrak{S}_{2k}/H_{k}$
is isomorphic to the induced representation $\mathrm{Ind}^{\mathfrak{S}_{2k}}_{H_{k}}1_{H_{k}}$,
where $1_{H_{k}}$ denotes the one dimensional trivial representation of $H_{k}$.
Hence the proof of the present proposition is reduced to the following one.
∎
Proposition 2.6.
Let $\mathfrak{S}_{2k}$ be the symmetric group of order $2k$ and let $H_{k}$ denote the subgroup
defined above. Then the irreducible decomposition of the induced representation $\mathrm{Ind}^{\mathfrak{S}_{2k}}_{H_{k}}1_{H_{k}}$
is given by
$$\mathrm{Ind}^{\mathfrak{S}_{2k}}_{H_{k}}1_{H_{k}}\cong\bigoplus_{|\lambda|=k}(%
2\lambda)_{\mathfrak{S}_{2k}}.$$
This is an elementary fact about representations of symmetric groups
and can be proved by applying standard argument in the theory of representations of finite groups
(see e.g. [10]).
However, in order to motivate later arguments and also for completeness,
we give a proof along the line of the ingredient of this paper.
We first recall a few terminologies about representations of the symmetric groups from [4].
We will use them also in section $\lx@sectionsign 5$.
For a Young diagram $\lambda$ with $|\lambda|=d$, choose a Young tableau $T$ (say a standard one) on $\lambda$.
Then we set
$$\displaystyle P_{\lambda}$$
$$\displaystyle=\{\gamma\in\mathfrak{S}_{d};\text{$\gamma$ preserves each row}\}$$
$$\displaystyle Q_{\lambda}$$
$$\displaystyle=\{\gamma\in\mathfrak{S}_{d};\text{$\gamma$ preserves each column}\}$$
$$a_{\lambda}=\sum_{\gamma\in P_{\lambda}}\gamma,\quad b_{\lambda}=\sum_{\gamma%
\in Q_{\lambda}}\mathrm{sgn}\,\gamma\,\gamma$$
The element
$$c_{\lambda}=a_{\lambda}b_{\lambda}\in\mathbb{Q}[\mathfrak{S}_{d}]$$
is called a Young symmetrizer. We can also use the following element
$$c^{\prime}_{\lambda}=b_{\lambda}a_{\lambda}\in\mathbb{Q}[\mathfrak{S}_{d}]$$
which is another form of Young symmetrizer and can play the same role as above.
In this paper, we use both ones.
Next we recall the following well known result.
Theorem 2.7 (Mackey’s restriction formula, see e.g. [34]).
Let $G$ be a finite group and let $H,K$ be subgroups of $G$. Let
$\rho$ be a representation of $H$ and for each $(H,K)$-double coset
representative $g$, let $\rho_{g}$ be the representation of $H_{g}=gHg^{-1}\cap K$
defined by $\rho_{g}(x)=\rho(g^{-1}xg)\ (x\in gHg^{-1})$.
Then we have the equality
$$\mathrm{Res}^{G}_{K}\,\mathrm{Ind}^{G}_{H}\,\rho=\bigoplus_{g\in K\backslash G%
/H}\mathrm{Ind}^{K}_{H_{g}}\,\rho_{g}.$$
Proof of Proposition 2.6.
First, we show that
(3)
$$\left|H_{k}\backslash\mathfrak{S}_{2k}/H_{k}\right|=p(k).$$
Namely the number of elements of the double cosets
$H_{k}\backslash\mathfrak{S}_{2k}/H_{k}$ is equal to $p(k)$ which is the
number of partitions of $k$. Recall that the set of left $H_{k}$-cosets of $\mathfrak{S}_{2k}$
can be canonically identified with $\mathcal{D}^{\ell}(2k)$. $H_{k}$ acts on this set
from the left and it is easy to see that the space of orbits of this action can be
identified with the set of $H_{k}$-double cosets of $\mathfrak{S}_{2k}$.
Now define the $C_{0}$-relative type of an element $C\in\mathcal{D}^{\ell}(2k)$,
denoted by $\tau_{C_{0}}(C)$,
as follows. Consider the graph $\Gamma_{C}=C_{0}\cup C$. Each connected component
of $\Gamma_{C}$ contains even number of vertices from the set of all vertices $\{1,2,\cdots,2k\}$.
If we enumerate all these even numbers in the decreasing order, then we obtain
an even type Young diagram $2\lambda$ where $\lambda$ is
a partition of $k$ (equivalently a Young diagram with $k$ boxes).
We put $\tau_{C_{0}}(C)=\lambda$. For example $\tau_{C_{0}}(C_{0})=[1^{k}]$.
Then we can show the following. For any two linear chord diagrams $C,D\in\mathcal{D}^{\ell}(2k)$,
the corresponding left $H_{k}$-cosets belong to the same right $H_{k}$-orbits if and only if
$\tau_{C_{0}}(C)=\tau_{C_{0}}(D)$. Also, for any Young diagram $\lambda$ with $|\lambda|=k$,
there exists $C\in\mathcal{D}^{\ell}(2k)$ such that $\tau_{C_{0}}(C)=\lambda$.
Since there are exactly $p(k)$ Young diagrams $\lambda$ with $|\lambda|=k$, the claim
(3) is proved.
Next, by the Frobenius reciprocity theorem, for any Young diagram $\mu$ with $|\mu|=2k$ we have
(4)
$$\text{multiplicity of $\mu_{\mathfrak{S}_{2k}}$ in $\mathrm{Ind}^{\mathfrak{S}%
_{2k}}_{H_{k}}1_{H_{k}}$}=\mathrm{dim}\,(\mathrm{Res}^{\mathfrak{S}_{2k}}_{H_{%
k}}\,\mu_{\mathfrak{S}_{2k}})^{H_{k}}.$$
We show that for any Young diagram $\lambda$ with $|\lambda|=k$
(5)
$$\mathrm{dim}\,(\mathrm{Res}^{\mathfrak{S}_{2k}}_{H_{k}}\,(2\lambda)_{\mathfrak%
{S}_{2k}})^{H_{k}}\geq 1.$$
To prove this, it suffices to show that
$$g_{\lambda}=\left(\sum_{\gamma\in H_{k}}\gamma\right)c_{(2\lambda)}\not=0\ \in%
\mathbb{Q}[\mathfrak{S}_{2k}]$$
because this is clearly an $H_{k}$-invariant element.
Here we choose a standard tableau $T$ on $2\lambda$ to be the one defined as follows.
We give the numbers $1,2,\cdots,2\lambda_{1}$ to fill the first row from the left to the right
and then the numbers $2\lambda_{1}+1,\cdots,2\lambda_{1}+2\lambda_{2}$ to the second row
from the left to the right
and so on, where $\lambda=[\lambda_{1}\cdots\lambda_{h}]$. Consider
$k$ ordered pairs
$$\langle 1,2\rangle,\langle 2,3\rangle,\cdots,\langle 2k-1,2k\rangle$$
of numbers each of which is placed on some row of $2\lambda$. Now the subgroup
$H_{k}$ is isomorphic to the semi-direct product $(\mathfrak{S}_{2})^{k}\rtimes\mathfrak{S}_{k}$
and the $i$-th $\mathfrak{S}_{2}$ acts on $\langle 2i-1,2i\rangle$ naturally while $\mathfrak{S}_{k}$
acts on the above pairs by permutations (preserving each order). It follows that we can write
$$\sum_{\gamma\in H_{k}}=\sum_{\gamma\in\mathfrak{S}_{k}}\gamma\sum_{\delta\in(%
\mathfrak{S}_{2})^{k}}\delta$$
and also that $(\mathfrak{S}_{2})^{k}$ is contained in $P_{2\lambda}$ which is the subgroup of
$\mathfrak{S}_{2k}$ consisting of elements which permute the numberings
but within each row of $2\lambda$.
Therefore
$$g_{\lambda}=\left(\sum_{\gamma\in\mathfrak{S}_{k}}\gamma\right)\left(\sum_{%
\delta\in(\mathfrak{S}_{2})^{k}}\delta\right)a_{2\lambda}b_{2\lambda}=2^{k}%
\left(\sum_{\gamma\in\mathfrak{S}_{k}}\gamma\right)c_{2\lambda}$$
because $\delta\,a_{2\lambda}=a_{2\lambda}$ for any $\delta\in P_{2\lambda}$.
Recall that the coefficient of the unit element $1\in\mathfrak{S}_{2k}$ in the expression $c_{2\lambda}$ is $1$.
To prove the assertion $g_{\lambda}\not=0$, it suffices to show that the coefficient of $1$ is different from $0$.
Suppose for some elements $\gamma\in\mathfrak{S}_{k},g\in P_{2\lambda},h\in Q_{2\lambda}$, the
equality
(6)
$$\gamma gh=1$$
holds. It implies that $gh\in\mathfrak{S}_{k}$. Let $\langle 2i-1,2i\rangle$ be any one of the above ordered
pair. Then we can write
$$gh\langle 2i-1,2i\rangle=\langle gh(2i-1),gh(2i)\rangle=\langle 2j-1,2j\rangle$$
for some $j$, because the action of $gh=\gamma^{-1}\in\mathfrak{S}_{2k}$ reduces to permutations of the ordered pairs preserving
the order of each pair. Recall that $h(2i-1)$ (resp. $h(2i))$ lies in the same column as $2j-1$ (resp. $2j$).
Since $g$ preserves each row and still $gh(2i-1)=2j-1$ and $gh(2i)=2j$ belong to the same row, we can conclude that
$h(2j-1)$ and $h(2j)$ belong to the same row. It follows that the image of $\langle 2i-1,2i\rangle$ under
$h$ is again one of the ordered pairs. Since $\langle 2i-1,2i\rangle$ was an arbitrary ordered pair,
we can conclude that $h$ is contained in the subgroup $\mathfrak{S}_{k}\subset\mathfrak{S}_{2k}$.
Now observe that the $sgn$ of any element of this subgroup $\mathfrak{S}_{k}$, but considered as
an element of $\mathfrak{S}_{2k}$, is $+1$ because it acts on $\{1,2,\cdots,2k\}$
by an even permutation. Thus, whenever the equality (6) occurs, we have $\mathrm{sgn}\,h=1$.
We can now conclude that the coefficient of $1$ in $g_{\lambda}$ is different from $0$ as claimed.
Finally, we prove
(7)
$$\mathrm{dim}\,\left(\mathrm{Res}^{\mathfrak{S}_{2k}}_{H_{k}}\,\mathrm{Ind}^{%
\mathfrak{S}_{2k}}_{H_{k}}1_{H_{k}}\right)^{H_{k}}=p(k).$$
It is easy to see that the representation
$\mathrm{Res}^{\mathfrak{S}_{2k}}_{H_{k}}\,\mathrm{Ind}^{\mathfrak{S}_{2k}}_{H_%
{k}}1_{H_{k}}$
is nothing other than the one associated with the left $H_{k}$-action on the
set $\mathcal{D}^{\ell}(2k)\cong\mathfrak{S}_{2k}/H_{k}$. Therefore, by a well-known
basic fact, the multiplicity of the trivial representation occurring in it
is the same as the number of orbits of this action which is $p(k)$ by
(3).
More precisely, we can apply Theorem 2.7
to the most simple case where $G=\mathfrak{S}_{2k},H=K=H_{k}$ and $\rho$ is the
one dimensional trivial representation. Then we obtain
$$\mathrm{Res}^{\mathfrak{S}_{2k}}_{H_{k}}\,\mathrm{Ind}^{\mathfrak{S}_{2k}}_{H_%
{k}}1_{H_{k}}=\bigoplus_{g\in H_{k}\backslash{\mathfrak{S}_{2k}}/H_{k}}\mathrm%
{Ind}^{H_{k}}_{(H_{k})_{g}}\,1_{(H_{k})_{g}}.$$
Taking the $H_{k}$-trivial part, this implies
$$\left(\mathrm{Res}^{\mathfrak{S}_{2k}}_{H_{k}}\,\mathrm{Ind}^{\mathfrak{S}_{2k%
}}_{H_{k}}1_{H_{k}}\right)^{H_{k}}=\bigoplus_{g\in H_{k}\backslash{\mathfrak{S%
}_{2k}}/H_{k}}\left(\mathrm{Ind}^{H_{k}}_{(H_{k})_{g}}\,1_{(H_{k})_{g}}\right)%
^{H_{k}}.$$
Again by the Frobenius reciprocity theorem, we have
$$\mathrm{dim}\,\left(\mathrm{Ind}^{H_{k}}_{(H_{k})_{g}}\,1_{(H_{k})_{g}}\right)%
^{H_{k}}=1\ $$ for any $$g\in H_{k}\backslash{\mathfrak{S}_{2k}}/H_{k}$$
and the claim (7) follows.
Since there are exactly $p(k)$ Young diagrams $\lambda$ with $|\lambda|=k$, (5) implies
$$\mathrm{dim}\,(\mathrm{Res}^{\mathfrak{S}_{2k}}_{H_{k}}\,\mu_{\mathfrak{S}_{2k%
}})^{H_{k}}=\begin{cases}1\quad\text{if $\mu=2\lambda$ for some $\lambda$}\\
0\quad\text{otherwise}.\end{cases}$$
In view of (4), we can now conclude
$$\mathrm{Ind}^{\mathfrak{S}_{2k}}_{H_{k}}1_{H_{k}}\cong\bigoplus_{|\lambda|=k}(%
2\lambda)_{\mathfrak{S}_{2k}}$$
as required.
∎
Proposition 2.8.
A basis for the subspace $E_{\lambda}$ is given by
$$\{c_{\tau}(C_{0});\tau\in ST(\lambda)\}$$
where $ST(\lambda)$ denotes the set of all the standard tableaux on the Young
diagram $2\lambda$ and $c_{\tau}$ denotes the Young symmetrizer corresponding to
$\tau\in ST(\lambda)$.
Proposition 2.9.
The correspondence
$$\lambda\mapsto\mu_{\lambda}$$
defines an injective mapping from the set of all the Young diagrams
$\lambda$ with $k$ boxes into the space of polynomials in $g$ of degree $k$.
Proof.
It is enough to recover the shape of any Young diagram $\lambda=[\lambda_{1}\cdots\lambda_{h}]$ from the
corresponding polynomial $\mu_{\lambda}$.
Recall that $\mu_{\lambda}$ is the product of $\mu_{b}=2g-2s_{b}+t_{b}$ where $b$ runs through all the boxes of
$\lambda$. It is clear that the largest (reap. the smallest) one among $\mu_{b}$ is $(2g+h-1)$ (resp. $(2g-2\lambda_{1}+2)$).
It follows that $\mu_{\lambda}$ determines $h$, which is the number of rows of $\lambda$, and $\lambda_{1}$.
Now consider the quotient
$$\mu_{\lambda}^{(1)}=\frac{\mu_{\lambda}}{2g(2g+1)\cdots(2g+h-1)(2g-2)\cdots(2g%
-2\lambda_{1}+2)}$$
which is again a polynomial in $g$ of degree $(k-h-\lambda_{1}+1)$. Let $\lambda^{(1)}$ be the
Young diagram obtained from $\lambda$ by deleting the first row and the first column. It has
$(k-h-\lambda_{1}+1)$ boxes. For example, if $\lambda=[432]$, then $\lambda^{(1)}=[21]$.
Now $\mu_{\lambda}^{(1)}$ is expressed as the product of $\mu_{b}$ where $b$ runs through
all the boxes of $\lambda^{(1)}$ which we consider as a subset of the original $\lambda$.
Now it is easy to see that the largest (reap. the smallest) one among
$\mu_{b}\ (b\in\lambda^{(1)}$ is $(2g+h_{2}-3)$ (resp. $(2g-2\lambda_{2}+3)$)
where $h_{2}$ denotes the number of boxes of the second column of $\lambda$.
It follows that we can recover $h_{2}$ and $\lambda_{2}$. Continuing this kind of argument,
we see that we can eventually recover the shape of $\lambda$. This completes the proof.
∎
3. Orthogonal decomposition of $(H_{\mathbb{Q}}^{\otimes 2k})^{\mathrm{Sp}}$
In this section, we deduce Theorem 1.1 from Theorem 1.4 whose proof
will be given in the next section. See Table 1 for an example of the orthogonal direct
sum decomposition.
Proof of Theorem 1.1.
In view of Proposition 2.1 and Proposition 2.2, we have only to determine
the kernel of the surjective linear mapping
$$\Phi:\mathbb{Q}\mathcal{D}^{\ell}(2k)\rightarrow(H_{\mathbb{Q}}^{\otimes 2k})^%
{\mathrm{Sp}}$$
when we fix a specific genus $g$.
Let us denote the eigenvalue of $E_{\lambda}$ by $\mu_{\lambda}(g)$ to indicate its dependence
on the genus. The smallest factor in the expression of $\mu_{\lambda}(g)$,
considered as a polynomial in $g$ of degree $k$, is
$(2g-2\lambda_{1}+2).$
Therefore we have
$$\mu_{\lambda}(g)\not=0\ \Leftrightarrow\ \lambda_{1}\leq g$$
It follows that
$$\mathrm{Ker}\,\Phi=\bigoplus_{|\lambda|=k,\lambda_{1}>g}E_{\lambda}$$
and $\Phi$ induces an isometric linear isomorphism
$$\bar{\Phi}:\bigoplus_{|\lambda|=k,\lambda_{1}\leq g}E_{\lambda}\cong(H_{%
\mathbb{Q}}^{\otimes 2k})^{\mathrm{Sp}}$$
which is also an “anti” -isomorphism as $\mathfrak{S}_{2k}$-modules.
Now for each Young diagram $\lambda$ with $|\lambda|=k$ and
$h(\lambda)\leq g$, we set
$$U_{\lambda}=\bar{\Phi}(E_{\lambda^{\prime}})$$
where $\lambda^{\prime}$ denotes the conjugate Young diagram of $\lambda$ so that
$\lambda^{\prime}_{1}=h(\lambda)\leq g$. Also as an $\mathfrak{S}_{2k}$-module,
we have isomorphisms
$$U_{\lambda}\cong E_{\lambda^{\prime}}\otimes[1^{2k}]_{\mathfrak{S}_{2k}}\cong(%
2\lambda^{\prime})_{\mathfrak{S}_{2k}}\otimes[1^{2k}]_{\mathfrak{S}_{2k}}\cong%
\lambda^{\delta}_{\mathfrak{S}_{2k}}$$
where the last isomorphism follows because it is easy to see that the identity
$$(2\lambda^{\prime})^{\prime}=\lambda^{\delta}$$
holds for any Young diagram $\lambda$. Thus we obtain an orthogonal direct sum decomposition
$$(H_{\mathbb{Q}}^{\otimes 2k})^{\mathrm{Sp}}=\bigoplus_{|\lambda|=k,h(\lambda)%
\leq g}U_{\lambda}$$
completing the proof.
∎
If we are concerned only with the easier part of Theorem 1.1, namely
the irreducible decomposition of $(H_{\mathbb{Q}}^{\otimes 2k})^{\mathrm{Sp}}$ as an $\mathfrak{S}_{2k}$-module,
then there exists an alternative proof based on a classical result together with an argument
in the symplectic representation theory. We briefly sketch it here. First recall the following classical result.
Theorem 3.1 ([4],Theorem 6.3. (2)).
The $\mathrm{GL}(2g,\mathbb{Q})$-irreducible decomposition of $H_{\mathbb{Q}}^{\otimes 2k}$ is give by
$$H_{\mathbb{Q}}^{\otimes 2k}\cong\bigoplus_{|\lambda|=2k,h(\lambda)\leq 2g}\,%
\lambda_{\mathrm{GL}}^{\oplus\dim\,\lambda_{\mathfrak{S}_{2k}}}$$
where $\lambda_{\mathrm{GL}}$ denotes the irreducible representation of $\mathrm{GL}(2g,\mathbb{Q})$
corresponding to $\lambda$.
If we combine this with the following fact
$$\mathrm{dim}\,(\lambda_{\mathrm{GL}})^{\mathrm{Sp}}=\begin{cases}&1\quad(\text%
{$\lambda$: multiple double floors})\\
&0\quad(\text{otherwise})\end{cases}$$
which follows from the restriction law corresponding to
the pair $\mathrm{Sp}(2g,\mathbb{Q})\subset\mathrm{GL}(2g,\mathbb{Q})$
(see Proposition 4.1 of [29]), then we obtain a proof of
the easier part of Theorem 1.1. We mention here that,
in a recent paper [1],
Enomoto and Satoh describes the $\mathrm{Sp}$-irreducible
decomposition of $H_{\mathbb{Q}}^{\otimes 2k}$.
In a dual setting to the homomorphism
$\Phi:\mathbb{Q}\mathcal{D}^{\ell}(2k)\rightarrow(H_{\mathbb{Q}}^{\otimes 2k})^%
{\mathrm{Sp}}$
given in $\lx@sectionsign$2, we define a linear mapping
$$\mathcal{K}:H_{\mathbb{Q}}^{\otimes 2k}\rightarrow\mathbb{Q}\mathcal{D}^{\ell}%
(2k)$$
by
$$\mathcal{K}(\xi)=\sum_{C\in\mathcal{D}^{\ell}(2k)}\alpha_{C}(\xi)\,C\quad(\xi%
\in H_{\mathbb{Q}}^{\otimes 2k})$$
where $\alpha_{C}\in\mathrm{Hom}(H_{\mathbb{Q}}^{\otimes 2k},\mathbb{Q})^{\mathrm{Sp}}$ is defined as
$$\alpha_{C}(u_{1}\otimes\cdots\otimes u_{2k})=\mathrm{sgn}\,C\prod_{s=1}^{k}\mu%
(u_{i_{s}},u_{j_{s}})\ (u_{i}\in H_{\mathbb{Q}}).$$
We use the same symbol $\mathcal{K}$ to denote its
restriction to the subspace $(H_{\mathbb{Q}}^{\otimes 2k})^{\mathrm{Sp}}$.
Proposition 3.2.
$\mathrm{(i)}\ \text{
The following diagram is commutative}$
$$\begin{CD}\mathbb{Q}\mathcal{D}^{\ell}(2k)@>{\mathcal{M}}>{}>\mathbb{Q}%
\mathcal{D}^{\ell}(2k)\\
@V{\Phi}V{}V\Big{\|}\\
(H_{\mathbb{Q}}^{\otimes 2k})^{\mathrm{Sp}}@>{\mathcal{K}}>{}>\mathbb{Q}%
\mathcal{D}^{\ell}(2k)\end{CD}$$
and the lower homomorphism $\mathcal{K}$ is
injective for any $g$ and bijective for any $g\geq k$.
$\mathrm{(ii)\ }$
For any element $\xi\in E_{\lambda}\subset\mathbb{Q}\mathcal{D}^{\ell}(2k)$, we have the following equality
$$\mathcal{K}(\Phi(\xi))=\mu_{\lambda\,}\xi.$$
Proof.
First we prove $\mathrm{(i)}$. The commutativity of the diagram follows by comparing the definitions
of the homomorphisms $\mathcal{M},\Phi$ and $\mathcal{K}$. The latter claim follows from the basic
result of Weyl that any $\mathrm{Sp}$-invariant can be described in terms of various contractions.
Next we prove $\mathrm{(ii)}$. Since $\xi$ is an eigenvector of $\mathcal{M}$
corresponding to the eigenvalue $\mu_{\lambda}$, we have $\mathcal{M}(\xi)=\mu_{\lambda}\,\xi$.
If we combine this with the commutativity just proved above, then the claim follows.
∎
Proof of Theorem 1.7.
The fact that $U_{\lambda}=\Phi(E_{\lambda^{\prime}})$ follows from Proposition 2.1.
The last claim follows from $\mathrm{(ii)}$ of the above Proposition 3.2.
∎
4. Proof of Theorem 1.4
In this section, we prove Theorem 1.4.
For that, we use a few terminologies about representations of the symmetric groups from [4]
which we recalled in $\lx@sectionsign 2$.
Let
$$\hat{\ \ }:\mathbb{Q}\mathfrak{S}_{2k}\rightarrow\mathbb{Q}\mathfrak{S}_{2k}$$
denote the anti-involution induced by the anti-automorphism
$\mathfrak{S}_{2k}\ni\gamma\mapsto\gamma^{-1}\in\mathfrak{S}_{2k}$.
Lemma 4.1.
$\mathrm{(i)\ }$ For any Young diagram $\lambda$, the following equalities hold
$$\hat{a}_{\lambda}=a_{\lambda},\quad\hat{b}_{\lambda}=b_{\lambda},\quad\hat{c}_%
{\lambda}=c^{\prime}_{\lambda},\quad\hat{c^{\prime}}_{\lambda}=c_{\lambda}.$$
$\mathrm{(ii)\ }$ For any element $x\in\mathbb{Q}\mathfrak{S}_{2k}$ and $\xi,\eta\in\mathbb{Q}\mathcal{D}^{\ell}(2k)$, the following equality holds
$$\langle\xi,x\eta\rangle=\langle\hat{x}\,\xi,\eta\rangle.$$
Proof.
$\mathrm{(i)}$ follows from the definitions and $\mathrm{(ii)}$
follows from the fact that the pairing $\langle\ ,\ \rangle$ is $\mathfrak{S}_{2k}$-equivariant.
∎
Now, for each Young diagram $\lambda$ with $k$ boxes, we have a map
$$\left(\mathbb{Q}[\mathfrak{S}_{2k}]\,c^{\prime}_{2\lambda}\right)\otimes%
\mathcal{D}^{\ell}(2k)\rightarrow\mathbb{Q}\mathcal{D}^{\ell}(2k).$$
By Proposition 2.5 and the property of the Young symmetrizer,
the image of this map is precisely the subspace $E_{\lambda}\subset\mathbb{Q}\mathcal{D}^{\ell}(2k)$
which is the unique summand isomorphic to $(2\lambda)_{\mathfrak{S}_{2k}}$.
Proposition 4.2.
Two subspaces $E_{\lambda}$ and $E_{\mu}$ of $\mathbb{Q}\mathcal{D}^{\ell}(2k)$ are mutually orthogonal
to each other whenever $\lambda\not=\mu$. Namely we have
$$\langle E_{\lambda},E_{\mu}\rangle=\{0\}.$$
Proof.
It suffices to show that, for any $C,D\in\mathcal{D}^{\ell}(2k)$
$$\langle c_{2\lambda}C,c^{\prime}_{2\mu}D\rangle=0$$
because the elements of the form $c_{2\lambda}C$ generate the subspace $E_{\lambda}$ for any $\lambda$
and the same is true if we replace $c_{2\lambda}C$ by $c^{\prime}_{2\lambda}C$.
On the other hand, by Lemma 4.1 we have
$$\displaystyle\langle c_{2\lambda}C,c^{\prime}_{2\mu}D\rangle$$
$$\displaystyle=\langle\hat{c^{\prime}}_{2\mu}c_{2\lambda}C,D\rangle$$
$$\displaystyle=\langle{c}_{2\mu}c_{2\lambda}C,D\rangle$$
$$\displaystyle=0.$$
The last equality follows because, as is well known, ${c}_{\mu}c_{\lambda}=0$
whenever $\mu\not=\lambda$.
This completes the proof.
∎
Proposition 4.3.
The subspace $E_{\lambda}\subset\mathbb{Q}\mathcal{D}^{\ell}(2k)$ is an eigenspace for the linear mapping
$\mathcal{M}$ for any $\lambda$.
Proof.
First we prove that $E_{\lambda}$ is an $\mathcal{M}$-invariant subspace,
namely $\mathcal{M}(E_{\lambda})\subset E_{\lambda}$. For this, it suffices to show that
$\mathcal{M}(E_{\lambda})$ is orthogonal to $E_{\mu}$ for any $\mu\not=\lambda$.
This is because we already know that the whole space
$\mathbb{Q}\mathcal{D}^{\ell}(2k)$ can be written as
$$\mathbb{Q}\mathcal{D}^{\ell}(2k)=\bigoplus_{|\lambda|=k}E_{\lambda}\quad\text{%
(orthogonal direct sum)}$$
(see Proposition 2.5 and Proposition 4.2).
Recall that $\mathcal{M}(E_{\lambda})$ is generated by the elements of the form
$\sum_{D}\langle c_{2\lambda}C_{1},D\rangle D$
while $E_{\mu}$ is generated by the elements of the form
$c^{\prime}_{2\mu}\,C_{2}$
where $C_{1},C_{2}\in\mathcal{D}^{\ell}(2k)$.
Now we fix $C_{1},C_{2}\in\mathcal{D}^{\ell}(2k)$ and compute
$$\displaystyle\left\langle\sum_{D}\langle c_{2\lambda}C_{1},D\rangle D,c^{%
\prime}_{2\mu}\,C_{2}\right\rangle$$
$$\displaystyle=\sum_{D}\langle c_{2\lambda}C_{1},D\rangle\langle D,c^{\prime}_{%
2\mu}\,C_{2}\rangle$$
$$\displaystyle=\sum_{D}\langle c_{2\lambda}C_{1},D\rangle\langle c_{2\mu}D,C_{2}\rangle$$
$$\displaystyle=\sum_{D}\langle c_{2\lambda}C_{1},c^{\prime}_{2\mu}D\rangle%
\langle D,C_{2}\rangle$$
$$\displaystyle=\sum_{D}\langle c_{2\mu}c_{2\lambda}C_{1},D\rangle\langle D,C_{2}\rangle$$
$$\displaystyle=0$$
because $c_{2\mu}c_{2\lambda}=0$. Therefore $\mathcal{M}(E_{\lambda})\bot E_{\mu}$ as claimed.
Next observe that $\mathcal{M}$ is an $\mathfrak{S}_{2k}$-equivariant linear mapping because,
for any $\gamma\in\mathfrak{S}_{2k}$ and $C\in\mathcal{D}^{\ell}(2k)$, we have
$$\displaystyle\mathcal{M}(\gamma\,C)$$
$$\displaystyle=\sum_{D\in\mathcal{D}^{\ell}(2k)}\langle\gamma\,C,D\rangle D$$
$$\displaystyle=\sum_{D\in\mathcal{D}^{\ell}(2k)}\langle C,\gamma^{-1}\,D\rangle
D$$
$$\displaystyle=\sum_{D\in\mathcal{D}^{\ell}(2k)}\langle C,D\rangle\gamma\,D$$
$$\displaystyle=\gamma(\mathcal{M}(C)).$$
It follows that any eigenspace of $\mathcal{M}$ is an $\mathfrak{S}_{2k}$-submodule.
Thus $E_{\lambda}$ is an $\mathcal{M}$-invariant and
$\mathfrak{S}_{2k}$-irreducible subspace of $\mathbb{Q}\mathcal{D}^{\ell}(2k)$.
We can now conclude that it is an eigenspace as claimed.
∎
Remark 4.4.
It follows from Theorem 1.7 that, besides our inner product, the
subspaces $E_{\lambda}\subset\mathbb{Q}\mathcal{D}^{\ell}(2k)$ are mutually orthogonal to each other also with respect to
the Euclidean metric on $\mathbb{Q}\mathcal{D}^{\ell}(2k)$ induced by taking $\mathcal{D}^{\ell}(2k)$ as
an orthonormal basis. Indeed, this fact can be proved
by a standard argument in the theory of representations of symmetric groups
without mentioning our inner product on $\mathbb{Q}\mathcal{D}^{\ell}(2k)$.
The above argument is an adaptation of this to the case of our symmetric pairing
$\langle\ ,\ \rangle$ which depends on $g$. It follows that the answer should be given by rescaling
eigenspaces by certain factors. Thus, in addition to the positivity, the essential point of our work is the
explicit determination of these factors as below.
Proof of Theorem 1.4.
In view of Proposition 2.5 and the arguments following it, the proof for the latter part
of this theorem is finished so that it remains to prove the former
part.
We know already that each space $E_{\lambda}$ is an eigenspace (see Proposition 4.3).
Therefore for any choice of a tableau on $2\lambda$ and for any $C\in\mathcal{D}^{\ell}(2k)$,
we should have an equality
$$\mathcal{M}(c^{\prime}_{2\lambda}C)=\sum_{D\in\mathcal{D}^{\ell}(2k)}\langle c%
^{\prime}_{2\lambda}C,D\rangle\,D=\mu_{\lambda}c^{\prime}_{2\lambda}C$$
where $\mu_{\lambda}$ denotes the eigenvalue of $E_{\lambda}$. It follows that, for any $C,D$, we have
$$\langle c^{\prime}_{2\lambda}C,D\rangle=\text{$\mu_{\lambda}\cdot$ (%
coefficient of D in $c^{\prime}_{2\lambda}C$)}.$$
Our task is to determine $\mu_{\lambda}$ by the above equality and for that we may choose
$C,D$ arbitrarily. We choose $C=D=C_{0}(2k)$ where $C_{0}(2k)$ denotes the following linear chord diagram
$$C_{0}(2k)=\{\{1,2\},\cdots,\{2k-1,2k\}\}\in\mathcal{D}^{\ell}(2k).$$
We set
$$v_{2\lambda}(2k)=c^{\prime}_{2\lambda}C_{0}(2k),\quad m_{\lambda}=\text{%
coefficient of $C_{0}(2k)$ in $v_{2\lambda}(2k)$}$$
and we determine $\mu_{\lambda}$ from the identity
$$\langle v_{2\lambda}(2k),C_{0}(2k)\rangle=m_{\lambda}\mu_{\lambda}.$$
The rest of the proof is divided into three cases.
$\mathrm{(I)}\ $ The case of $\lambda=[k]$.
In this case, the group $P_{2\lambda}$ is the whole group while $Q_{2\lambda}$ is the trivial
group. Hence, for any tableau $\tau$, we have
$$v_{[2k]}=c_{[2k]}\,C_{0}(2k)=2^{k}k!\sum_{C\in\mathcal{D}^{\ell}(2k)}C$$
so that $m_{[k]}=2^{k}k!$. Furthermore it was already proved in [28] (proof of
Proposition 4.1) that
$$\left\langle\sum_{C\in\mathcal{D}^{\ell}(2k)}C,C_{0}(2k)\right\rangle=2g(2g-2)%
\cdots(2g-2k+2).$$
Therefore
$$\mu_{[k]}=2g(2g-2)\cdots(2g-2k+2)$$
as claimed.
$\mathrm{(II)}\ $ The case of $\lambda=[1^{k}]$.
The case $k=1$ was already treated in $\mathrm{(I)}\ $ so that we assume $k\geq 2$ and compute
the value of
$$\langle c^{\prime}_{[2^{k}]}C_{0},C_{0}\rangle$$
step by step as follows. We have
$$c^{\prime}_{[2^{k}]}=b_{[2^{k}]}a_{[2^{k}]}$$
and we choose a standard tableau $\tau$ on $[2^{k}]$ as below
$$\begin{bmatrix}1&2\\
3&4\\
\cdots&\cdots\\
2k-1&2k\end{bmatrix}.$$
Then the element $b_{[2^{k}]}$ can be described as
$$b_{[2^{k}]}=t_{k}t_{k-1}\cdots t_{3}t_{2}$$
where
$$\displaystyle t_{2}$$
$$\displaystyle=\{1-(13)\}\{1-(24)\},\ t_{3}=\{1-(15)-(35)\}\{1-(26)-(46)\},\cdots,$$
$$\displaystyle t_{k}$$
$$\displaystyle=\{1-(1,2k-1)-(3,2k-1)-\cdots-(2k-3,2k-1)\}\cdot$$
$$\displaystyle \{1-(2,2k)-(4,2k)-\cdots-(2k-2,2k)\}.$$
We define
$$c_{1}=a_{[2^{k}]},\quad c_{s}=t_{s}\cdots t_{2}a_{[2^{k}]}\quad(s=2,\cdots,k)$$
and compute
$$\langle c_{s}C_{0},C_{0}\rangle,\quad m_{s}=\text{coefficient of $C_{0}$ in $c%
_{s}C_{0}$}$$
inductively. Here the essential point in our computation is Lemma 2.4.
In fact, we use this lemma repeatedly in the whole of later argument and
we find a “perfect canceling” in sgns of various operations on linear chord diagrams.
In the first case $s=1$,
the group $P_{[2^{k}]}\cong\mathfrak{S}_{2}^{k}$ acts on $C_{0}$ trivially. Therefore
$$c_{1}C_{0}=a_{[2^{k}]}C_{0}=2^{k}C_{0},\quad m_{1}=2^{k}$$
and we have
$$\langle c_{1}C_{0},C_{0}\rangle=m_{1}(2g)^{k}.$$
Next we compute the case $k=2$
$$\langle c_{2}C_{0},C_{0}\rangle=2^{k}\langle t_{1}C_{0},C_{0}\rangle,\quad%
\text{$m_{2}$= coefficient of $C_{0}$ in $c_{2}C_{0}$}.$$
Among the $4$ elements in
$$t_{2}=\{1-(13)\}\{1-(24)\}=1+(13)(24)-(13)-(24)$$
exactly $2$ elements, namely $1,(13)(24)$ preserve $C_{0}$ unchanged.
Therefore $m_{2}=2m_{1}$ and if we apply Lemma 2.4, we can conclude that
$$\langle t_{2}C_{0},C_{0}\rangle=(2g)\{2\cdot 2g-2\cdot(-1)\}(2g)^{k-2}=(2g)2(2%
g+1)(2g)^{k-2}.$$
Therefore we have
$$\langle t_{2}C_{0},C_{0}\rangle=m_{2}(2g)(2g+1)(2g)^{k-2}.$$
Next we compute
$$\langle c_{3}C_{0},C_{0}\rangle=2^{k}\langle t_{3}t_{2}C_{0},C_{0}\rangle,%
\quad\text{$m_{3}$= coefficient of $C_{0}$ in $c_{3}C_{0}$}.$$
Among the $9$ elements in
$$t_{3}=\{1-(15)-(35)\}\{1-(26)-(46)\}$$
exactly $3$ elements, namely $1,(15)(26),(35)(46)$ preserve $C_{0}$ unchanged.
Therefore $m_{3}=3m_{2}$ and if we apply Lemma 2.4, we can conclude that
$$\langle t_{3}t_{2}C_{0},C_{0}\rangle=(2g)2(2g+1)\{3\cdot 2g-(9-3)\cdot(-1)\}(2%
g)^{k-3}=(2g)2(2g+1)3(2g+2)(2g)^{k-3}.$$
Therefore we have
$$\langle t_{3}t_{2}C_{0},C_{0}\rangle=m_{3}(2g)(2g+1)(2g+2)(2g)^{k-3}.$$
Continuing the computation like this, we can prove that
the equalities
$$\displaystyle\langle t_{s}\cdots t_{1}C_{0},C_{0}\rangle$$
$$\displaystyle=m_{s}(2g)(2g+1)(2g+2)\cdots(2g+s)(2g)^{k-s-1}$$
$$\displaystyle m_{s}$$
$$\displaystyle=2^{k}s!$$
hold for all $s=1,\cdots,k$.
We check the inductive argument from the $s$-th step to the next $(s+1)$-th step.
Among the $(s+1)^{2}$ elements in the expansion of
$$\displaystyle\{1-(1,2s+1)-(3,2s+1)-\cdots-(2s-1,2s+1)\}$$
$$\displaystyle\{1-(2,2s+2)-(4,2s+2)-\cdots-(2s,2s+2)\}$$
exactly the following $(s+1)$ elements preserve $C_{0}$ unchanged, namely
$$1,(1,2s+1)(2,2s+2),\cdots,(2s-1,2s+1)(2s,2s+2).$$
Therefore $m_{s+1}=(s+1)m_{s}$ and if we apply Lemma 2.4, we can conclude that
$$\displaystyle\langle t_{s+1}t_{s}\cdots t_{2}C_{0},C_{0}\rangle$$
$$\displaystyle=m_{s}(2g)(2g+1)\cdots(2g+s-1)\{(s+1)\cdot 2g-((s+1)^{2}-(s+1))%
\cdot(-1)\}(2g)^{k-s}$$
$$\displaystyle=(s+1)m_{s}2g(2g+1)\cdots(2g+s)(2g)^{k-s}.$$
Therefore we have
$$\langle t_{s+1}\cdots t_{2}C_{0},C_{0}\rangle=m_{s+1}(2g)(2g+1)\cdots(2g+s)(2g%
)^{k-s-1}$$
as required.
$\mathrm{(III)}\ $ The general case.
Finally we consider the general case. Roughly speaking, we combine the above two arguments, one is
the case $\mathrm{(I)}$ and the other is the case $\mathrm{(II)}$, to treat the general case.
Let $\lambda=[\lambda_{1}\lambda_{2}\cdots\lambda_{h}]$ be a general Young diagram with $|\lambda|=k$.
Choose a standard tableau on $2\lambda$, say $1,2,\cdots,2\lambda_{1}-1,2\lambda_{1}$ on the
first row, $2\lambda_{1}+1,2\lambda_{1}+2,\cdots,2\lambda_{1}+2\lambda_{2}-1,2\lambda_{1}+%
2\lambda_{2}$ on the
second row, and so on. Then we can write
$$P_{2\lambda}=P_{2\lambda_{1}}\times P_{2\lambda_{2}}\times\cdots\times P_{2%
\lambda_{h}}$$
where $P_{2\lambda_{i}}\ (i=1,\cdots,h)$ denotes the symmetric group consisting of permutations of
labels attached to the boxes in the $i$-th row of $2\lambda$.
Thus we have
$$a_{2\lambda}=a_{2\lambda_{1}}a_{2\lambda_{2}}\cdots a_{2\lambda_{h}}.$$
If we apply the result of case $\mathrm{(I)}\ $ to each row of $2\lambda$, then we obtain
$$\displaystyle\langle a_{2\lambda}C_{0},C_{0}\rangle=$$
$$\displaystyle 2^{\lambda_{1}}\lambda_{1}!\cdot 2g(2g-2)\cdots(2g-2\lambda_{1}+%
2)\cdot$$
$$\displaystyle 2^{\lambda_{2}}\lambda_{2}!\cdot 2g(2g-2)\cdots(2g-2\lambda_{2}+%
2)\cdot$$
$$\displaystyle\cdots$$
$$\displaystyle 2^{\lambda_{h}}\lambda_{h}!\cdot 2g(2g-2)\cdots(2g-2\lambda_{h}+2)$$
and
$$\text{coefficient of $C_{0}$ in $a_{2\lambda}C_{0}\ =\prod_{i=1}^{h}2^{\lambda%
_{i}}\lambda_{i}!=2^{k}\prod_{i=1}^{h}\lambda_{i}!$}.$$
Next we consider the element $b_{2\lambda}$. The group $Q_{2\lambda}$ can be described as
$$Q_{2\lambda}=Q_{2\lambda_{1}}\times Q_{2\lambda_{1}-1}\times\cdots\times Q_{2}%
\times Q_{1}$$
where $Q_{j}\ (j=1,2,\cdots,2\lambda_{1})$ denotes the symmetric group consisting of permutations of the
labels attached to the $j$-th column of $2\lambda$.
Here we mention that $2\lambda_{1}$ is the number of columns of $2\lambda$ and
also both of $Q_{1}$ and $Q_{2}$ are isomorphic to $\mathfrak{S}_{h}$.
Now we decompose the element $b_{2\lambda}$ as
$$b_{2\lambda}=b_{\lambda_{1}}\cdots b_{2}b_{1}$$
where
$$b_{j}=\prod_{\gamma\in Q_{2j-1}}(1+\mathrm{sgn}\,\gamma)\prod_{\delta\in Q_{2j%
}}(1+\mathrm{sgn}\,\delta).$$
First we consider the element $b_{1}a_{2\lambda}C_{0}$.
It can be seen that
we can apply the result of case $\mathrm{(II)}\ $ to the first two columns of $2\lambda$ from the left to obtain
$$\displaystyle\langle b_{1}a_{2\lambda}C_{0},C_{0}\rangle=$$
$$\displaystyle\,h!\,2^{\lambda_{1}}\lambda_{1}!\cdot 2g(2g-2)\cdots(2g-2\lambda%
_{1}+2)\cdot$$
$$\displaystyle 2^{\lambda_{2}}\lambda_{2}!\cdot[(2g+1)(2g-1)]\cdot(2g-4)\cdots(%
2g-2\lambda_{2}+2)\cdot$$
$$\displaystyle\cdots$$
$$\displaystyle 2^{\lambda_{h}}\lambda_{h}!\cdot[(2g+h-1)(2g+h-3)]\cdot(2g-4)%
\cdots(2g-2\lambda_{h}+2)$$
$$\displaystyle=$$
$$\displaystyle\,h!\,2^{k}\prod_{i=1}^{h}\lambda_{i}!\cdot 2g(2g-2)\cdots(2g-2%
\lambda_{1}+2)\cdot$$
$$\displaystyle[(2g+1)(2g-1)]\cdot(2g-4)\cdots(2g-2\lambda_{2}+2)\cdot$$
$$\displaystyle\cdots$$
$$\displaystyle[(2g+h-1)(2g+h-3)]\cdot(2g-4)\cdots(2g-2\lambda_{h}+2)$$
$$\displaystyle=$$
$$\displaystyle\,h!\,2^{k}\prod_{i=1}^{h}\lambda_{i}!\cdot[2g(2g+1)\cdots(2g+h-1%
)]\cdot[(2g-2)(2g-1)\cdots(2g+h-3)]\cdot$$
$$\displaystyle[(2g-4)(2g-6)]^{\lambda^{\prime}_{2}}\cdots[(2g-2\lambda_{1}+4)(2%
g-2\lambda_{1}+2)]^{\lambda^{\prime}_{\lambda_{1}}}$$
and
coefficient of $$C_{0}$$ in $$b_{1}a_{2\lambda}C_{0}\ =h!\,2^{k}\prod_{i=1}^{h}\lambda_{i}!$$
where $\lambda^{\prime}=[h\lambda^{\prime}_{2}\cdots\lambda^{\prime}_{\lambda_{1}}]$ denotes the conjugate Young diagram of $\lambda$.
Next we can apply the result of case $\mathrm{(II)}\ $ to the two columns,
the third and the fourth from the left of $2\lambda$, to obtain
$$\displaystyle\langle b_{2}b_{1}a_{2\lambda}C_{0},C_{0}\rangle=$$
$$\displaystyle h!\lambda^{\prime}_{2}!\,2^{k}\prod_{i=1}^{h}\lambda_{i}!\cdot 2%
g(2g-2)\cdots(2g-2\lambda_{1}+2)\cdot$$
$$\displaystyle(2g+1)(2g-1)\cdot[(2g-3)(2g-5)]\cdot(2g-8)\cdots(2g-2\lambda_{2}+%
2)\cdot$$
$$\displaystyle\cdots$$
$$\displaystyle(2g+h-1)(2g+h-3)\cdot[(2g+h-5)(2g+h-7)]\cdot(2g-8)\cdots(2g-2%
\lambda_{h}+2)$$
$$\displaystyle=$$
$$\displaystyle\,h!\lambda^{\prime}_{2}!\,2^{k}\prod_{i=1}^{h}\lambda_{i}!\cdot[%
2g(2g+1)\cdots(2g+h-1)]\cdot[(2g-2)(2g-1)\cdots(2g+h-3)]\cdot$$
$$\displaystyle\cdot[(2g-4)(2g-3)\cdots(2g+\lambda^{\prime}_{2}-1)]\cdot[(2g-6)(%
2g-5)\cdots(2g+\lambda^{\prime}_{2}-3)]\cdot$$
$$\displaystyle[(2g-8)(2g-10)]^{\lambda^{\prime}_{3}}\cdots[(2g-2\lambda_{1}+4)(%
2g-2\lambda_{1}+2)]^{\lambda^{\prime}_{\lambda_{1}}}$$
and
$$\text{coefficient of $C_{0}$ in $b_{2}b_{1}a_{2\lambda}C_{0}\ =h!\lambda^{%
\prime}_{2}!\,2^{k}\prod_{i=1}^{h}\lambda_{i}!$}.$$
Continuing in this way until the last two columns from the right of $\lambda$,
we finally obtain
$$\displaystyle\langle c^{\prime}_{2\lambda}C_{0},C_{0}\rangle=$$
$$\displaystyle m_{2\lambda}\cdot 2g(2g-2)\cdots(2g-2\lambda_{1}+2)\cdot$$
$$\displaystyle(2g+1)(2g-1)\cdots(2g-2\lambda_{2}+3)\cdot$$
$$\displaystyle\cdots$$
$$\displaystyle(2g+h-1)(2g+h-3)\cdots(2g+h-2\lambda_{h}+1)$$
$$\displaystyle m_{2\lambda}=$$
$$\displaystyle\prod_{i=1}^{h}2^{\lambda_{i}}\lambda_{i}!\prod_{j=1}^{\lambda_{1%
}}\lambda^{\prime}_{j}!=2^{k}\prod_{i=1}^{h}\lambda_{i}!\prod_{j=1}^{\lambda_{%
1}}\lambda^{\prime}_{j}!.$$
Here we would like to emphasize again that an essential use of Lemma 2.4 is made in the above computation.
This completes the proof.
∎
5. Topological study of the tautological algebra of the moduli space of curves
As already mentioned in the introduction, this work was originally motivated by
a topological study of the tautological algebra of the moduli space of curves.
Let $\mathcal{M}_{g}$ be the mapping class group of $\Sigma_{g}$
and let $\mathcal{R}^{*}(\mathcal{M}_{g})$ denote the tautological algebra in cohomology of
$\mathcal{M}_{g}$ . Namely it is the subalgebra of $H^{*}(\mathcal{M}_{g};\mathbb{Q})$
generated by the Mumford-Morita-Miller tautological classes $e_{i}\in H^{2i}(\mathcal{M}_{g};\mathbb{Q})$.
We also consider the mapping class group $\mathcal{M}_{g,*}$ of
$\Sigma_{g}$ relative to the base point $*\in\Sigma_{g}$ and the corresponding
tautological algebra $\mathcal{R}^{*}(\mathcal{M}_{g,*})$. In this case,
it is the subalgebra of $H^{*}(\mathcal{M}_{g,*};\mathbb{Q})$ generated by
$e_{i}$ and the Euler class $e\in H^{2}(\mathcal{M}_{g,*};\mathbb{Q})$.
In the context of algebraic geometry, there is a more refined form of the tautological algebra.
Namely, if we denote by $\mathbf{M}_{g}$ the moduli space of curves of genus $g$, then
its tautological algebra, denoted by $\mathcal{R}^{*}(\mathbf{M}_{g})$, is defined to be the
subalgebra of the Chow algebra $\mathcal{A}^{*}(\mathbf{M}_{g})$ of $\mathbf{M}_{g}$ generated
by the tautological classes $\kappa_{i}\in\mathcal{A}^{i}(\mathbf{M}_{g})$ introduced by Mumford.
There is a natural surjection $\mathcal{R}^{*}(\mathbf{M}_{g})\rightarrow\mathcal{R}^{*}(\mathcal{M}_{g})$
which sends $\kappa_{i}$ to $(-1)^{i+1}e_{i}$.
There have been obtained many interesting results about the tautological algebras,
Faber’s beautiful conjecture given in [2] together with his related works
being a strong motivation for them.
In particular, Faber and Zagier proposed a set of conjectural relations in
$\mathcal{R}^{*}(\mathbf{M}_{g})$ and Faber in fact proved that it
gives an affirmative answer to his conjecture including the
most difficult part, namely the perfect pairing property, up to genus $g\leq 23$.
Very recently, Pandharipande and Pixton [31] proved a remarkable
result that the Faber-Zagier relations are actual relations in $\mathcal{R}^{*}(\mathbf{M}_{g})$ for all $g$.
However the structure of $\mathcal{R}^{*}(\mathbf{M}_{g})$ for general $g$ remains mysterious.
For more details of these developments, we refer to Faber’s recent survey paper
[3] as well as the references therein.
The main tools in these works belong to algebraic geometry which are quite powerful.
A theorem of Looijenga given in [18], which is one of the earliest
results in this area, is a typical example.
On the other hand, there are a few topological approaches to this subject.
The first one was given in our papers [23][24].
Recently Randal-Williams [33] greatly generalized this
to obtain very interesting results. See also
Yin [35] for the case of $\mathcal{R}^{*}(\mathbf{M}_{g,1})$
and Grigoriev [7] for the case of higher dimensional manifold bundles.
In [28] we adopted a closely related but a bit different
approach, namely a combination of topology and representation theory.
In the next section we extend this approach and
give a systematic way of producing relations among
the $\mathrm{MMM}$ tautological classes in $\mathcal{R}^{*}(\mathcal{M}_{g})$ and $\mathcal{R}^{*}(\mathcal{M}_{g,*})$.
For that, in the remaining part of this section, we summarize former results.
By analyzing the natural action of $\mathcal{M}_{g}$ on the third nilpotent quotient of
$\pi_{1}\Sigma_{g}$,
in [27] we constructed the following commutative diagram
(8)
$$\begin{CD}\pi_{1}\Sigma_{g}@>{}>{}>[1^{2}]_{\mathrm{Sp}}\tilde{\times}H_{%
\mathbb{Q}}\\
@V{}V{}V@V{}V{}V\\
\mathcal{M}_{g,*}@>{\rho_{2}}>{}>(([1^{2}]^{\text{torelli}}_{\mathrm{Sp}}%
\oplus[2^{2}]_{\mathrm{Sp}})\tilde{\times}\wedge^{3}H_{\mathbb{Q}})\rtimes%
\mathrm{Sp}(2g,\mathbb{Q})\\
@V{}V{p}V@V{}V{}V\\
\mathcal{M}_{g}@>{\bar{\rho}_{2}}>{}>([2^{2}]_{\mathrm{Sp}}\tilde{\times}U_{%
\mathbb{Q}})\rtimes\mathrm{Sp}(2g,\mathbb{Q}).\end{CD}$$
This extends our earlier work in [25] in which we treated the case of the action on the second nilpotent quotient.
Here $U_{\mathbb{Q}}=\wedge^{3}H_{\mathbb{Q}}/H_{\mathbb{Q}}$ and $[2^{2}]_{\mathrm{Sp}}$ is the unique $\mathrm{Sp}$-representation
corresponding to the Young diagram $[2^{2}]$ which appears in both of $\wedge^{2}U^{*}_{\mathbb{Q}}$
and $\wedge^{2}(\wedge^{3}H^{*}_{\mathbb{Q}})$.
Also $([2^{2}]_{\mathrm{Sp}})$ denotes the ideal generated by $[2^{2}]_{\mathrm{Sp}}$.
$(([1^{2}]^{\text{torelli}}_{\mathrm{Sp}}\oplus[2^{2}]_{\mathrm{Sp}})$ is defined similarly
(see [27] for details).
In our joint works with Kawazumi [12][13], we determined the homomorphisms
$\rho^{*}_{2},\bar{\rho}^{*}_{2}$ in cohomology which are induced by the representations
$\rho_{2},\bar{\rho}_{2}$ in the above diagram (8). More precisely, we proved that the images of them
are precisely the tautological algebras and also that these representations are isomorphisms in the
stable range of Harer [8] (so that the images are in fact isomorphic to
the whole rational cohomology algebras of the mapping class groups
by the definitive result of Madsen and Weiss [19]).
In the proof of these results, the work of Kawazumi in [11]
played a crucial role.
Thus we have the following commutative diagram
(9)
$$\begin{CD}\left(\wedge^{*}(\wedge^{3}H^{*}_{\mathbb{Q}})/([1^{2}]^{\text{%
torelli}}_{\mathrm{Sp}}\oplus[2^{2}]_{\mathrm{Sp}})\right)^{\mathrm{Sp}}@>{%
\rho_{2}^{*}}>{\text{stably $\cong$}}>\mathcal{R}^{*}(\mathcal{M}_{g,*})\\
@A{}A{}A@A{}A{p^{*}}A\\
\left(\wedge^{*}U^{*}_{\mathbb{Q}}/([2^{2}]_{\mathrm{Sp}})\right)^{\mathrm{Sp}%
}@>{\bar{\rho}_{2}^{*}}>{\text{stably $\cong$}}>\mathcal{R}^{*}(\mathcal{M}_{g%
}).\end{CD}$$
Here $([2^{2}]_{\mathrm{Sp}})$ denotes the ideal generated by $[2^{2}]_{\mathrm{Sp}}$.
$(([1^{2}]^{\text{torelli}}_{\mathrm{Sp}}\oplus[2^{2}]_{\mathrm{Sp}})$ is defined similarly.
Furthermore we have given explicit formulas for the homomorphisms $\rho_{2}^{*},\bar{\rho}_{2}^{*}$
in terms of graphs which we now recall.
Let $\mathcal{G}$ be the set of isomorphism classes of connected trivalent graphs
and let $\mathbb{Q}[\Gamma;\Gamma\in\mathcal{G}]$ denote the polynomial algebra generated by
$\mathcal{G}$. We define the degree of a graph $\Gamma\in\mathcal{G}$ to be the number
of vertices of it. Thus the subspace $\mathbb{Q}[\Gamma;\Gamma\in\mathcal{G}]^{(2k)}$ consisting
of homogeneous polynomials of degree $2k$ can be naturally identified with the linear space
spanned by trivalent graphs with $2k$ vertices. Also let $\mathcal{G}^{0}$
be the subset of $\mathcal{G}$ consisting of connected trivalent graphs
without loops.
In [25], the following commutative diagram
was constructed
(10)
$$\begin{CD}\mathbb{Q}[\Gamma;\Gamma\in\mathcal{G}]@>{}>{}>(\wedge^{*}(\wedge^{3%
}H^{*}_{\mathbb{Q}}))^{\mathrm{Sp}}\\
@A{\iota_{g}}A{}A@A{}A{}A\\
\mathbb{Q}[\Gamma;\Gamma\in\mathcal{G}^{0}]@>{}>{}>(\wedge^{*}U^{*}_{\mathbb{Q%
}})^{\mathrm{Sp}}\end{CD}$$
where both of the horizontal homomorphisms are surjective
and bijective in a certain stable range and $\iota_{g}$ denotes some
twisted inclusion.
Then in [13], the following commutative diagram
$$\begin{CD}\mathbb{Q}[\Gamma;\Gamma\in\mathcal{G}]@>{\alpha}>{}>\mathbb{Q}[g][e%
,e_{1},e_{2},\cdots]\\
@A{\iota_{g}}A{}A@A{}A{\cup}A\\
\mathbb{Q}[\Gamma;\Gamma\in\mathcal{G}^{0}]@>{\beta}>{}>\mathbb{Q}[g][e_{1},e_%
{2},\cdots]\end{CD}$$
was constructed which realizes the composition of the commutative diagrams (10) followed by (9)
by means of explicit formulas $\alpha$ and $\beta$. In particular, for each trivalent graph
$\Gamma$ with $2k$ vertices, we have the associated characteristic class
$$\alpha_{\Gamma}\in\mathbb{Q}[g][e,e_{1},e_{2},\ldots]^{(2k)}.$$
If we fix a genus $g$, then
the commutative diagram (9) is isomorphic to the following
$$\begin{CD}\mathbb{Q}[\Gamma;\Gamma\in\mathcal{G}]/\widetilde{W}@>{\alpha_{g}}>%
{\text{stably $\cong$}}>\mathbb{Q}[e,e_{1},e_{2},\cdots]\\
@A{\iota_{g}}A{}A@A{}A{\cup}A\\
\mathbb{Q}[\Gamma;\Gamma\in\mathcal{G}^{0}]/W@>{\beta_{g}}>{\text{stably $%
\cong$}}>\mathbb{Q}[e_{1},e_{2},\cdots]\end{CD}$$
where $\widetilde{W}$ denotes certain equivalence relation in $\mathbb{Q}[\Gamma;\Gamma\in\mathcal{G}]$
induced by graphical operations on trivalent graphs including the Whitehead move
(or the $\mathrm{IH}$ move) and $W$ is a similar one.
We mention that the relation between the ideal $([2^{2}]_{\mathrm{Sp}})$ and
the $\mathrm{IH}$ move for the lower case
was first studied by Garoufalidis and Nakamura in [5].
6. A set of relations in $\mathcal{R}^{*}(\mathcal{M}_{g})$ and $\mathcal{R}^{*}(\mathcal{M}_{g,*})$
In this section, we apply Theorem 1.1 to the description of the tautological algebras
$\mathcal{R}^{*}(\mathcal{M}_{g})$ and $\mathcal{R}^{*}(\mathcal{M}_{g,*})$ in terms of certain symplectic invariant tensors
as well as trivalent graphs,
which we recall in the previous section, and obtain relations among the $\mathrm{MMM}$ tautological classes
in a systematic way. This extends our former work in [28].
The symplectic invariant tensors we are concerned is $\left(\wedge^{2k}(\wedge^{3}H_{\mathbb{Q}})\right)^{\mathrm{Sp}}$
which can be considered as both a natural quotient space as well as the
subspace of $(H_{\mathbb{Q}}^{\otimes 6k})^{\mathrm{Sp}}$. Furthermore this subspace
can be described by means of the natural action of $\mathfrak{S}_{6k}$ on the
whole space $(H_{\mathbb{Q}}^{\otimes 6k})^{\mathrm{Sp}}$. It follows that the orthogonal
direct sum decomposition given by Theorem 1.1 induces that
of the space $\left(\wedge^{2k}(\wedge^{3}H_{\mathbb{Q}})\right)^{\mathrm{Sp}}$
so that we obtain the following commutative diagram
(11)
$$\begin{CD}(H_{\mathbb{Q}}^{\otimes 6k})^{\mathrm{Sp}}@>{\text{quotient, %
contain}}>{\supset}>\left(\wedge^{2k}(\wedge^{3}H_{\mathbb{Q}})\right)^{%
\mathrm{Sp}}\\
\Big{\|}\Big{\|}\\
\bigoplus_{|\lambda|=3k}U_{\lambda}@>{\text{quotient, contain}}>{\supset}>%
\bigoplus_{|\lambda|=3k}L_{\lambda}\end{CD}$$
where $L_{\lambda}$ denotes certain subspace of $U_{\lambda}$.
The totality of the upper quotient homomorphism
in this diagram for all $k$ can be realized graphically by the following collapsing operation.
Definition 6.1 ([28]).
For each $C\in\mathcal{D}^{\ell}(6k)$, let $\Gamma_{C}$ denote the trivalent graph defined as follows.
We understand $C$ as a disjoint union of $3k$ chords each of which connects
the two points $i_{s},j_{s}\ (s=1,2,\cdots,3k)$ in the set of vertices $\{1,2,\cdots,6k\}$.
Then $\Gamma_{C}$ is the graph obtained from $C$ by joining the three vertices
in each of the sets $\{1,2,3\},$ $\{4,5,6\},$ $\cdots,$ $\{6k-2,6k-1,6k\}$ to a single point.
Thus $\Gamma_{C}$ is a (possibly disconnected) trivalent graph with $2k$ vertices.
By extending this operation linearly, we obtain a linear mapping
$$\mathbb{Q}\mathcal{D}^{\ell}(6k)\rightarrow\mathbb{Q}[\Gamma;\Gamma\in\mathcal%
{G}]^{(2k)}.$$
For each element $\xi\in\mathbb{Q}\mathcal{D}^{\ell}(6k)$, we denote its image
under the above mapping by $\Gamma_{\xi}$.
Then we have the following commutative diagram
(12)
$$\begin{CD}\mathbb{Q}\mathcal{D}^{\ell}(6k)@>{\text{collapse}}>{\supset}>%
\mathbb{Q}[\Gamma;\Gamma\in\mathcal{G}]^{(2k)}\\
\Big{\|}\Big{\|}\\
\bigoplus_{|\lambda|=3k}E_{\lambda}@>{\text{collapse}}>{\supset}>\bigoplus_{|%
\lambda|=3k}L^{\prime}_{\lambda}\end{CD}$$
where $L^{\prime}_{\lambda}$ denotes the subspace corresponding to $L_{\lambda}$
(and in fact isomorphic to $L_{\lambda^{\prime}}$).
In view of (10), this is equivalent to the diagram (11).
We need one more terminology to formulate our result and that concerns the
following direct sum decomposition
(13)
$$(\wedge^{2k}(\wedge^{3}H_{\mathbb{Q}}))^{\mathrm{Sp}}\cong\bigoplus_{p=0}^{2k}%
(\wedge^{2k-p}U_{\mathbb{Q}}\otimes\wedge^{p}H_{\mathbb{Q}})^{\mathrm{Sp}}.$$
This is based on the fact that $\wedge^{3}H_{\mathbb{Q}}$ is not irreducible as an $\mathrm{Sp}$-module
but is a sum of two irreducible summands $\wedge^{3}H_{\mathbb{Q}}\cong U_{\mathbb{Q}}\oplus H_{\mathbb{Q}}$.
Notice that the above decomposition is an unstable operation in the sense that it depends on the
genus $g$ and also that it cannot be described in terms of the action of the symmetric group.
To realize the above decomposition (13) graphically,
we recall a few terminologies from our paper [28] in suitably modified forms.
Consider the mapping
(14)
$$\begin{split}\displaystyle\wedge^{2k}&\displaystyle(\wedge^{3}H_{\mathbb{Q}})%
\ni\xi=\xi_{1}\wedge\cdots\wedge\xi_{2k}\\
&\displaystyle\mapsto\xi^{(p)}=\sum_{i_{1}<\cdots<i_{p}}\xi_{1}\wedge\cdots%
\wedge\widetilde{C}\xi_{i_{1}}\wedge\cdots\wedge\widetilde{C}\xi_{i_{p}}\wedge%
\cdots\wedge\xi_{2k}\in\wedge^{2k}(\wedge^{3}H_{\mathbb{Q}})\end{split}$$
where $\xi_{i}\in\wedge^{3}H_{\mathbb{Q}}$, $0\leq p\leq 2k$ and $\widetilde{C}:\wedge^{3}H_{\mathbb{Q}}\rightarrow\wedge^{3}H_{\mathbb{Q}}$
is the mapping defined by
$$\widetilde{C}(u\wedge v\wedge w)=\frac{1}{g-1}\,(\mu(u,v)w+\mu(v,w)u+\mu(w,u)v%
)\wedge\omega_{0}.$$
Now let $\Gamma$ be a trivalent graph with $2k$ vertices and let $V_{\Gamma}$ denote the
set of vertices. Fix a subset $F\subset V_{\Gamma}$ consisting of $p$ vertices where $0\leq p\leq 2k$.
We denote by $\mathcal{P}(F)$ the set of all ways of choices $\varpi$ where, for each vertex
$v\in F$, $\varpi(v)$ specifies $2$ edges emerging from $v$ out of the $3$ such edges.
Since there are $3$ edges emerging from each vertex, $\mathcal{P}(F)$
has $3^{p}\binom{2k}{p}$ elements.
For each $\varpi\in\mathcal{P}(F)$, we define a trivalent graph $\Gamma_{\varpi}$ and a number
$\ell(\varpi)$ as follows. At each vertex $v\in F$, cut the $2$ edges emerging from it which correspond to $\varpi(v)$
at the one third point along each edge from $v$. Then we connect the two endpoints of the shortened edges emerging
from $v$ to make a loop. At the same time, we also connect the two endpoints of the remaining
$2$ edges. After performing this operation at every vertex $v\in F$, we are left with a trivalent graph
together with certain number of disjoint circles and/or intervals. We define $\Gamma_{\varpi}$ to be this
trivalent graph throwing away the other parts. The number $\ell(\varpi)$ is defined to be that of the
disjoint circles. Namely it is the sum of the number of loops and that of double edges “occurring”
in $\varpi$.
Definition 6.2 (graphical contraction).
Let $\Gamma$ be a trivalent graph with $2k$ vertices. For each $0\leq p\leq 2k$, we define
$$\Gamma^{(p)}=\frac{1}{(2g-2)^{p}}\sum_{F\subset V_{\Gamma}}\sum_{\varpi\in%
\mathcal{P}(F)}(-1)^{p}(-2g)^{\ell(\varpi)}{\Gamma}_{\varpi}\ \in\mathbb{Q}[%
\Gamma;\Gamma\in\mathcal{G}]^{(2k)}$$
where $F$ runs through all subsets of $V_{\Gamma}$ consisting of $p$ elements.
Furthermore we set
$$\bar{\Gamma}=\Gamma-\Gamma^{(1)}+\Gamma^{(2)}-\cdots+\Gamma^{(2k)}.$$
Finally we extend these operations on the whole space $\mathbb{Q}[\Gamma;\Gamma\in\mathcal{G}]^{(2k)}$ by
linearity.
In summary, each element $\xi\in\mathbb{Q}\mathcal{D}^{\ell}(6k)$ gives rise to the associated element
$\Gamma_{\xi}\in\mathbb{Q}[\Gamma;\Gamma\in\mathcal{G}]^{(2k)}$ by Definition 6.1
and then we have the following series of elements
$$\Gamma_{\xi}=\Gamma_{\xi}^{(0)},\Gamma_{\xi}^{(1)},\cdots,\Gamma_{\xi}^{(2k)},%
\bar{\Gamma}_{\xi}$$
belonging to $\mathbb{Q}[\Gamma;\Gamma\in\mathcal{G}]^{(2k)}$ by Definition 6.2.
Then it was shown in [28] that the above graphical operations realize the decomposition
(13).
Now we are ready to state our main result in this section.
Theorem 6.3.
$\mathrm{(i)\ }$ In the tautological algebra $\mathcal{R}^{*}(\mathcal{M}_{g})$, the following relations hold.
$$\text{For all $\xi\in E_{\lambda}\subset\mathbb{Q}\mathcal{D}^{\ell}(6k)$ with%
$\lambda_{1}>g$,
we have $\alpha_{\bar{\Gamma}_{\xi}}=0\in\mathcal{R}^{2k}(\mathcal{M}_{g})$}.$$
$\mathrm{(ii)\ }$ In the tautological algebra $\mathcal{R}^{*}(\mathcal{M}_{g,*})$, the following relations hold.
$$\text{For all $\xi\in E_{\lambda}\subset\mathbb{Q}\mathcal{D}^{\ell}(6k)$ with%
$\lambda_{1}>g$,
we have $\alpha_{\Gamma^{(p)}_{\xi}}=0\in\mathcal{R}^{2k}(\mathcal{M}_{g,*})$ %
for any $0\leq p\leq 2k$}.$$
Proof.
We first prove $\mathrm{(ii)}$.
Let $\xi\in E_{\lambda}\subset\mathbb{Q}\mathcal{D}^{\ell}(6k)$ be any element. Then we have $\Phi(\xi)\in U_{\lambda^{\prime}}$
by Theorem 1.7. If we further assume that $\lambda_{1}>g$, then $h(\lambda^{\prime})=\lambda_{1}>g$
so that $\Phi(\xi)=0\in(H_{\mathbb{Q}}^{\otimes 6k})^{\mathrm{Sp}}$ by Theorem 1.1.
It follows that the projected image of $\Phi(\xi)$ in $(\wedge^{2k}(\wedge^{3}H_{\mathbb{Q}}))^{\mathrm{Sp}}$
is also trivial. Furthermore all the components of this element with respect to the direct sum
decomposition (13) are trivial because they are obtained by applying the
operations (14). Since these tensorial operations are realized by the graphical
operations given in Definition 6.1 and Definition 6.2, the claim holds.
Next we prove $\mathrm{(i)}$. The graphical operation
$$\Gamma\mapsto\bar{\Gamma}=\Gamma-\Gamma^{(1)}+\Gamma^{(2)}-\cdots+\Gamma^{(2k)}$$
given in Definition 6.2 realizes the tensorial operation of taking the
$(\wedge^{2k}U_{\mathbb{Q}})^{\mathrm{Sp}}$ part out of the whole $(\wedge^{2k}(\wedge^{3}H_{\mathbb{Q}}))^{\mathrm{Sp}}$
in the direct sum decomposition (13).
On the other hand, the
diagram (9) shows that this operation
extracts exactly the
“basic” part with respect to the projection $\mathcal{M}_{g,*}\rightarrow\mathcal{M}_{g}$,
namely the part which comes from the ”base” $\mathcal{M}_{g}$
(observe here that we have a natural
injection $\mathcal{R}^{*}(\mathcal{M}_{g})\subset\mathcal{R}^{*}(\mathcal{M}_{g,*})$).
The claim follows.
∎
Remark 6.4.
The first case $\lambda=[3k]$ in the above theorem was treated in our former paper [28] completely.
In this case $2\lambda=[6k]$ is the one dimensional trivial representation of $\mathfrak{S}_{6k}$
so that $\mathrm{dim}\,E_{\lambda}=1$. We have determined the relations
$\alpha_{\bar{\Gamma}_{\xi}}=0\in\mathcal{R}^{2k}(\mathcal{M}_{g})$
explicitly for all $k$ and $g=3k-1,3k-2,\ldots,2$, where $\xi$ denotes a non-trivial
unique (up to scalars) element of $E_{\lambda}$. In this computation, the eigenvalue
$$\mu_{[3k]}=2g(2g-2)\cdots(2g-6k+2)$$
of $E_{[3k]}$, which is the smallest one, played an important role.
As an application of this first case, we showed that $\mathcal{R}^{*}(\mathcal{M}_{g})$
is generated by the first $[g/3]$ $\mathrm{MMM}$ classes. Later Ionel [9] proved a stronger result that this fact holds
at the Chow algebra level, namely in $\mathcal{R}^{*}(\mathbf{M}_{g})$.
The second case corresponds to $\lambda=[3k-1,1]$ and $\mathrm{dim}\,E_{\lambda}=9k(2k-1)$,
the third case corresponds to $\lambda=[3k-2,2]\ (k\geq 2)$ and $\mathrm{dim}\,E_{\lambda}=\frac{1}{2}k(3k-1)(6k-1)(6k-7)$,
and so on. Schematically, we may write
$$\displaystyle[3k]\mapsto 0\quad(g\leq 3k-1)\ \Rightarrow\ \text{$\mathcal{R}^{%
*}(\mathcal{M}_{g})$ is generated by the first $[g/3]$ $\mathrm{MMM}$ classes}$$
$$\displaystyle[3k-1,1]\mapsto 0\quad(g\leq 3k-2)$$
$$\displaystyle[3k-2,2],[3k-2,1^{2}]\mapsto 0\quad(g\leq 3k-3)$$
$$\displaystyle[3k-3,3],[3k-3,21],[3k-3,1^{3}]\mapsto 0\quad(g\leq 3k-4)$$
$$\displaystyle[3k-4,4],[3k-4,31],[3k-4,2^{2}],[3k-4,21^{2}],[3k-3,1^{4}]\mapsto
0%
\quad(g\leq 3k-5)$$
$$\displaystyle\cdots$$
Although we have to consider the collapsing $E_{\lambda}\rightarrow L_{\lambda^{\prime}}$ described
in diagram (12), it seems reasonable to expect that many relations can be obtained in this way.
We would like to pursue this in future.
In view of the fact that Theorem 1.1 gives a complete description how the
$\mathrm{Sp}$-invariant tensors degenerate according as the genus decreases from the stable range to the
last case $g=1$, it would be reasonable to make the following.
Conjecture 6.5.
The relations given in Theorem 6.3 make a complete set of relations for the tautological algebras
$\mathcal{R}^{*}(\mathcal{M}_{g})$ and $\mathcal{R}^{*}(\mathcal{M}_{g,*})$.
7. Concluding remarks
As was already mentioned in the introduction, in a joint work with T. Sakasai and M. Suzuki, we found another
role of the canonical metric introduced in this paper. In this work, we are studying the structure of the
symplectic derivation Lie algebra $\mathfrak{h}_{g,1}$ which is the Lie algebra consisting of symplectic
derivations of the free Lie algebra generated by $H_{\mathbb{Q}}$. This Lie algebra is a very important subject in topology
and also in other related fields of mathematics including number theory.
In particular, the symplectic invariant part $\mathfrak{h}_{g,1}^{\mathrm{Sp}}$
of this Lie algebra concerns many deep questions in various contexts, e.g. homology cobordism invariants for homology cylinders,
the rational cohomology group of the outer automorphism group of free groups and the Galois obstructions
in the theory of arithmetic mapping class groups.
We obtain several results about $\mathfrak{h}_{g,1}^{\mathrm{Sp}}$ and
we refer to our paper [30] for details.
Finally we would like to mention another possible application of our metric.
That is concerned with “harmonic representatives” of cohomology classes
which arise in various symplectic Lie algebras, by which we mean graded Lie algebras
such that each graded piece is an $\mathrm{Sp}$-module and the bracket operation
is an $\mathrm{Sp}$-morphism.
We are keeping in mind the Lie algebra $\mathfrak{h}_{g,1}$ mentioned above
and more generally Kontsevich’s graph (co)homology theory (see [14][15]).
It might be worthwhile to investigate whether harmonic representatives of
cohomology classes of these Lie algebras would uncover some unknown geometric property of them.
In particular, the case of the Gel’fand-Fuks cohomology of formal Hamiltonian vector fields
on $H_{\mathbb{R}}=H_{\mathbb{Q}}\otimes\mathbb{R}$, both in the stable as well as the unstable contexts, seems
to be interesting. In the simplest case of formal Hamiltonian vector fields on the plane,
Gel’fand, Kalinin and Fuks found in [6] an exotic class and Metoki
found another one in [20]. The problem of determining whether these classes are non-trivial
as characteristic classes of transversely symplectic foliations remains unanswered up to the present.
In the framework due to Kontsevich [16], it was shown in our paper with Kotschick [17]
that the Gel’fand-Kalinin-Fuks class can be decomposed into a product of certain leaf cohomology class
and the transverse symplectic form and in [22] Mikami proved a similar statement for the
Metoki class. It would be worthwhile to study whether the harmonic representatives of these leaf cohomology
classes would shed light on the above difficult problem.
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AML-SVM: Adaptive Multilevel Learning with Support Vector Machines
1st Ehsan Sadrfaridpour
School of Computing
Clemson University
Clemson, USA
[email protected]
2nd Korey Palmer
School of Computing
Clemson University
Clemson, USA
[email protected]
3rd Ilya Safro
School of Computing
Clemson University
Clemson, USA
[email protected]
Abstract
The support vector machines (SVM) is one of the most widely used and practical optimization based classification models in machine learning because of its interpretability and flexibility to produce high quality results. However, the big data imposes a certain difficulty to the most sophisticated but relatively slow versions of SVM, namely, the nonlinear SVM.
The complexity of nonlinear SVM solvers and the number of elements in the kernel matrix quadratically increases with the number of samples in training data.
Therefore, both runtime and memory requirements are negatively affected.
Moreover, the parameter fitting has extra kernel parameters to tune, which exacerbate the runtime even further.
This paper proposes an adaptive multilevel learning framework for the nonlinear SVM, which addresses these challenges, improves the classification quality across the refinement process, and leverages multi-threaded parallel processing for better performance.
The integration of parameter fitting in the hierarchical learning framework and adaptive process to stop unnecessary computation significantly reduce the running time while increase the overall performance.
The experimental results demonstrate reduced variance on prediction over validation and test data across levels in the hierarchy, and significant speedup compared to state-of-the-art nonlinear SVM libraries without a decrease in the classification quality. The code is accessible at https://github.com/esadr/amlsvm.
Classification, Multilevel Computation, Large-scale learning, Support Vector Machine
\definechangesauthor
[name=”Ehsan”, color=blue]es
\definechangesauthor[name=”Korey”, color=green]kp
\definechangesauthor[name=”Ilya”, color=red]is
\bstctlcite
IEEEexample:BSTcontrol
I Introduction
Support vector machine (SVM) is a widely used family of classification methods that leverage the principle of separating hyperplane. Technically, this is achieved by solving underlying regularized optimization model adjusting which can provide highly accurate and interpretable classification.
Training linear SVM is very fast and can scale to millions of data points and features without using significant high-performance computing (HPC) resources.
For problems that are not linearly separable, the nonlinear SVM uses the kernel trick by implicitly projecting the data into the higher-dimensional space to separate it by a hyperplane.
Nonlinear SVM usually reaches higher prediction quality on complex datasets. However, it comes with a price tag of being not scalable in comparison to its linear version.
Solving the Lagrangian dual problem is typically the way to cope with regularized nonlinear SVM models with the underlying convex quadratic programming (QP) problem. In a number of libraries (such as LibSVM [1]) multiple methods have been implemented for solving both primal and dual problems.
The complexity of the convex QP solvers for nonlinear SVM often scales between $O(n^{2}f)$ to $O(n^{3}f)$ [2] , where $n$ is the number of data points, and $f$ is the number of features.
Therefore, as $n$ increases, the running time of the solver also increases, which hinders the usage of the nonlinear SVM for massive data sets.
As the size of many datasets continues to grow due to the advancements in technologies such as high-throughput sequencing, e-commerce [3] and the Internet of Things [4], more scalable machine learning algorithms are required.
Therefore, while a nonlinear SVM is fast on small datasets and can provide highly accurate prediction, more research is required to develop scalable nonlinear SVM solvers for massive datasets.
Our previous framework, Multilevel SVM (MLSVM), is a library that scales to millions of data points and exhibits up to two orders of magnitude faster running time compared to LIBSVM on reported benchmark datasets [5].
The MLSVM leverages multilevel algorithms that are inspired by the algebraic multigrid and some of its restricted versions [6]. These algorithms are known to be successful in accelerating computational optimization and modeling tasks without a loss in the quality. Examples include hypergraph partitioning [7], graph generation [8], and clustering [9]. Such multilevel methods have two essential phases, namely, coarsening and uncoarsening [6].
The coarsening phase gradually reduces the original problem size and generates a chain of smaller problems that approximate the original one. This is done by constructing
a (possibly fuzzy) hierarchy of aggregated data points.
The uncoarsening phase starts from the smallest generated problem and gradually uses generated approximations to refine the solution of the original problem. During the uncoarsening phase, a refinement leverages a solution inherited from the previous coarser problem and is performed using a local processing solver to avoid any heavy computation.
The multilevel SVM [10, 11, 5] is an approach to accelerate traditional SVM model solvers which also often improves the model quality as it exploits the geometry of data before starting the training. The key principles of multilevel SVM are as follows:
1) Learning the separating hyperplane occurs within a small proximity to support vectors. 2) Perform computationally expensive parts of the training only at the coarse levels. 3) The best final model is not necessarily the finest one.
We discussed several other principles in [5]. Among them is the one that discusses the way multilevel SVM copes with imbalanced data.
While the MLSVM library [5] provided fast runtime and highly accurate predictions on massive datasets, we observed several unexpected results on benchmark datasets that motivated us to extend that work and introduce the Adaptive MLSVM in this paper.
In multilevel algorithms [12, 6], it is expected to observe that the quality of optimized objective is improved at each next finer level during the uncoarsening. As such, we expected to see a non-decreasing quality improvement across the uncoarsening. However, on some most difficult benchmarks, we observed that while the trained coarse models have exhibited reasonable to optimal prediction quality, some of the middle levels experienced a significant drop in the quality. In some cases, these middle level drops were continuing to the fine levels and in some were gradually improved back to the high quality results.
Although the observed quality drops were rare, they often led to accumulating and increasing errors during the uncoarsening. Indeed, low quality support vectors may potentially be disaggregated into the data points that are even more distant to the optimal hyperplane which gives too much freedom to the optimizer. Other stages in the MLSVM pipeline such as the filtering of data points after too aggressive disaggregation were also affected.
Our Contribution In this paper, we propose the Adaptive Multilevel SVM, a successful approach to detect the problem of inconsistent learning quality during the uncoarsening and efficiently mitigate it. At each level of the multilevel hierarchy, we detect the problem of quality decrease by validating the model using the finest level data.
In the adaptive multilevel SVM learning framework, we adjust the training data by filling the training data gap with the misclassified validation data points, retraining the model and improving the quality with a new set of support vectors.
Our exhaustive experimental results on the benchmark datasets demonstrate the proposed method recovers the multilevel framework from such quality drops and achieves higher quality in comparison to the non-adaptive multilevel SVM as well as other state of the art solvers.
In addition, we speed up even more the runtime compared to non-adaptive multilevel SVM and reduce the prediction quality variance.
Moreover, in the new version of the MLSVM library we implement the multi-thread support for parameter fitting to speed up the model training at each level.
Our implementation is open-source and available at http://github.com/esadr/amlsvm.
II Related Works
Improving the performance of SVM has been widely studied with a common goal of improving the training time without a significant decrease in prediction accuracy such as recent researches
[13, 14, 15, 16].
The complexity of nonlinear SVM on massive datasets is a potential target to develop scalable SVMs.
A challenge with solving the quadratic programming problem for the kernel SVM is the kernel matrix of size $n^{2}$, where $n$ is the number of training data points. The matrix requires large memory, which is not feasible for massive datasets.
The main categories of accelerated solvers follow one of the following approaches: a) target the solver performance directly, b) partition the original data, or c) considering alternative representation for the training data (including the multilevel approach), and their combinations. Some of the improvements also rely on the advancement of software infrastructure, hardware, and distributed frameworks.
Solver performance:
A Sequential Minimal Optimization (SMO)[17] has been used to solve the underlying quadratic programming problem.
The ThunderSVM [15] directly works on improving the performance of the solver using multiple cores and GPUs.
Partition original training data:
Instead of solving the QP with a large number of data points, a set of smaller problems can be generated and solved independently. This reduces the running time of training and memory utilization required to store the kernel matrix. A drawback to this approach is the quality of partitioning the original data and the quality of the approach to combine all the trained models to drive a final model.
DC-SVM[18] used adaptive clustering for dividing the training data
and relied on coordinate descent method to converge to a solution from multiple models.
A disadvantage for these approaches is relying on the partitioning or clustering
that is sensitive to the order of the training data, number of clusters. Assignment of a point to a partition or cluster is strict, which limits the points to move between clusters or participate in multiple clusters.
Graph representation of training data:
There are two schemas for partitioning the data. The earlier approaches rely on the original data in the feature space, while newer approaches such as[11, 5, 13] rely on a graph representation of data. The graph representation of the training data is constructed as a preprocessing step for partitioning which is explained in detail in section III-B.
The graph representation provides the opportunity to leverage the multilevel paradigm that has been used successfully in a wide range of problems. We briefly mention some of the multilevel research related to graphs such as graph partitioning [19], graph clustering [20], and image segmentation [21].
The advantage of using a graph and multilevel paradigm such as Algebraic Multigrid (AMG) is to exploit more relaxed and less strict assignments of the data points to smaller aggregates compare to clustering, which has strict assignments and larger cluster sizes.
The position of AML-SVM:
Our proposed framework (AML-SVM) has a robust coarsening, which allows partial participation of points in multiple aggregates through gradual assignment and relaxation steps. Our results demonstrate that a small training data at the coarsest level can be used to train a model with high accuracy.
AML-SVM can directly leverage the performance improvements introduced by advanced solvers with multi-core or GPU support to achieve even faster runtime without any change in the coarsening or refinement process.
The uncoarsening (refinement) phase as a general step in multilevel methods has the potential for improvement of initial solutions and carrying essential information such as support vectors, and parameters to other levels.
The improvements proposed in this paper reduce the variance between the quality of models at various levels and reduce the sensitivity of the framework to configuration parameters.
Furthermore, these improvements, along with multiple core support for faster parameter fitting, can be used for ensemble models that we have not explored.
III Preliminaries
Consider a set $\mathcal{J}$ of input samples that contains $n$ data points denoted by $x_{i}$, where $x_{i}\in\mathbb{R}^{d}$, $1\leq i\leq n$. Each data point $x_{i}$ has a corresponding label $y_{i}\in\{-1,1\}$.
The SVM as a binary classification finds the largest margin hyperplane that separates the two classes of labeled data. The solution to the following optimization problem produces the largest margin hyperplane, which is defined by $w$, and $b$.
minimize
$$\displaystyle\frac{1}{2}\lVert w\rVert^{2}+{C}\sum_{i=1}^{n}\xi_{i}$$
(1)
subject to
$$\displaystyle y_{i}(w^{T}\phi(x_{i})+b)\geq 1-\xi_{i},$$
$$\displaystyle\quad i=1,\dots,n$$
$$\displaystyle\xi_{i}\geq 0,$$
$$\displaystyle\quad i=1,\dots,n.$$
The misclassification is penalized using slack variables $\{\xi_{i}\}_{i=1}^{n}$.
The parameter $C>0$ controls the magnitude of penalty for miscalssified data points.
The primal formulation in (1) is known as the soft margin SVM [22].
The SVM takes advantage of kernel $\phi:\mathbb{R}^{d}\rightarrow\mathbb{R}^{p}~{}(d\leq p)$ to map data points to higher-dimensional space. The kernel measures the similarity for pairs of points $x_{i}$ and $x_{j}$.
The Gaussian kernel (RBF), $\exp(-\gamma||x_{i}-x_{j}||^{2})$, which is known to be generally reliable [23], is the default kernel in this paper.
For imbalanced datasets, different weights can be assigned to classes using the weighted SVM (WSVM) to controllably penalize points in the minority class $C^{+}$ (e.g., rare events).
The set of slack variables is split into two disjoint sets $\{\xi_{i}^{+}\}_{i=1}^{n^{+}}$, and $\{\xi_{i}^{-}\}_{i=1}^{n^{-}}$, respectively.
In WSVM, the objective of (1) is changed into
minimize
$$\displaystyle\frac{1}{2}\lVert w\rVert^{2}+C\big{(}{W^{+}}\sum_{i=1}^{n^{+}}%
\xi_{i}^{+}+{W^{-}}\sum_{j=1}^{n^{-}}\xi_{j}^{-}\big{)}.$$
(2)
The imbalanced data sets contain significanlty fewer positively labeled data points. Hence, the data points in the minority class is denoted by $\mathcal{J}^{+}$, where size of minority class is $n^{+}=|\mathcal{J}^{+}|$.
The rest of the points belongs to the majority class which is denoted by $\mathcal{J}^{-}$, where $n^{-}=|\mathcal{J}^{-}|$, and $\mathcal{J}=\mathcal{J}^{+}\cup\mathcal{J}^{-}$.
We assign the class weight as the inverse of the total number of points in that class. For instance, the weight for minority class is $W^{+}=\frac{1}{n^{+}}$
In the our multilevel framework, one of the basic steps is aggregating data points. The aggregation can be performed either on full points or their fractions (i.e., a data point can be split into fractions, and different fractions can contribute themselves to different aggregates). Therefore, a coarse level data point normally contains several finer points or their fractions. Because of this, a data point can be more or less representative in comparison to other data points. We define and use the volume property for each point and calculate it as number of partial or fully participated original points in that aggregate.
We denote $v_{i}$ as volume for point $x_{i}$ which represent the internal importance of point $i$ among points in the same class.
Moreover, we consider the class importance for the final weight of each data point. The calculation for both classes are the same, hence, we only present the positive class. We calculate the class weights based on sum of the volumes in the class, i.e.,
$W^{+}=\frac{1}{\sum\limits_{i\in\mathcal{J}^{+}}v_{i}}.$ We calculate the final weight for each point as product of its volume weight times class weight
$\forall i\in\mathcal{J}^{+(-)}\quad W_{i}=W^{+(-)}\frac{v_{i}}{\sum\limits_{j%
\in\mathcal{J}^{+(-)}}v_{j}}.$
The instance weight based on volume and class, helps to improve prediction quality for the smaller class.
III-A Two-level problem
To describe the coarsening, uncoarsening and refinement algorithms, we introduce the two-level problem notation that can be extended into a full multilevel hierarchy (see Figure LABEL:fig:aml_svm). In Algorithm LABEL:alg:aml_general_refinement, we will use subscript $(\cdot)_{f}$ and $(\cdot)_{c}$ to represent fine and coarse variables, respectively. For example, the data points of two consecutive levels, namely, fine and coarse, will be denoted by $\mathcal{J}_{f}$, and $\mathcal{J}_{c}$, respectively. The sets of fine and coarse support vectors are denoted by $\textsf{sv}_{f}$, and $\textsf{sv}_{c}$, respectively. We will also use a subscript in the parentheses to indicate the level number in the hierarchy where appropriate. For example, $\mathcal{J}_{(i)}$ will denote the set of data points at level $i$.
III-B Proximity graphs
The original MLSVM framework [5] is based on the algebraic multigrid coarsening scheme for graphs (see coarsening in [24, 25]) that creates a hierarchy of data proximity graphs.
Initially, at the finest level, $\mathcal{J}$ is represented as two approximate $k$-nearest neighbor ($k$NN) graphs $G^{+}_{(0)}=({\textbf{V}}^{+},E^{+})$, and $G^{-}_{(0)}=({\textbf{V}}^{-},E^{-})$ for minority and majority classes, respectively, where each $x_{i}\in{\textbf{V}}^{+(-)}$ corresponds to a node in $G^{+(-)}_{(0)}$.
A pair of nodes in $G_{(0)}^{+(-)}$ is connected with an edge that belongs to $E^{+(-)}$ if one of these nodes belongs to a set of $k$-nearest neighbors of another.
Throughout the multilevel hierarchies, in two-level representation, the fine and coarse level graphs will be denoted by $G^{+(-)}_{f}=({\textbf{V}}^{+(-)}_{f},E^{+(-)}_{f})$, and $G^{+(-)}_{c}=({\textbf{V}}^{+(-)}_{c},E^{+(-)}_{c})$, respectively. For $k$NN, our experiments show $k=10$, for all tested datasets, provides a good trade-off between computational cost of calculating $k$NN and average node degree for $G^{+(-)}$.
III-C Learning Challenge
In the hierarchical framework, it is expected that the first solution is only a reasonable (but not necessarily the best) approximation of the optimal solution. Therefore, extra steps are required to improve coarse solutions gradually.
In the context of the classification task, the initial solution should provide a reasonable decision boundary. We can measure the quality of the initial solution with performance measures on validation data, which we cover in section IV. We can express it more accurately by
$Q_{(1)}\leq Q_{(2)}...\leq Q_{(k)},$
when $Q_{(i)}$ is the classification quality at level $i$. We denote this expected trend with a continuous increase in quality as the natural trend and Figure 1(a) demonstrates it for the Ringnorm data set.
However, this is not what we observed over exhaustive experiments with various data sets. In our experimental results, there were some cases in which the classification quality was dropped significantly during the refinement. The consequences of unexpected phenomena are as follows.
The training process for the models at the following levels of refinement will be based on sub-optimal training data. As explained earlier, the training data at each level of refinement is filtered to contain only the neighbor points of SVs of a model at the coarser level. When the SVs are sub-optimal, the filtering adds the points which are not representative of the data for the SVM to learn an optimal decision boundary.
Moreover, the number of SVs may increase dramatically without significant improvement in the classification quality. The increase in the number of SVs drastically increases the number of SV’s neighbors, which forms a larger training data at the next level. Therefore, it reduces the performance of the rest of levels in the refinement.
We recognized a pattern that the quality improves during refinement but at some levels, declines significantly. We denote it as unnatural trend and Figure 1(b) demonstrates an example of the unnatural trend for the Cod-RNA dataset.
Benefits of detecting and addressing the decrease in quality during the refinement are:
•
Improving the classification quality of the finer levels and in general
•
Improving the computational performance of the refinement process by improving the quality of training data set for the finer levels.
•
Reducing the computational cost of refinement through the Early Stopping process when there is not a significant improvement in quality by training larger models
•
Reducing the variance for the classification quality across levels
•
Preserve smaller training set during the refinement
Intuitively, the coarsening procedure gradually creates approximated (or summarized) representations of the finest level (original) data. An initial model we train at the coarsest level, provides a large margin hyperplane which can be used as the final solution.
On the one hand, at the finest level, rich data can easily lead to over-fitted models, a phenomenon frequently observed in practice [31]. On the other hand, over-compressed data representation may lead to an under-fitted model due to lack of detail information.
Our experiments confirm that more than half of the best models are obtained from the coarse (but not coarsest) and middle levels which typically prevents over- and under-fitting.
In general, if the best models were produced by the finest and middle levels, we recommend to use the model obtained at the middle levels to avoid potential over-fitting. This recommendation is based on the observation that same quality models can be generated by different hyperplanes but finest models may contain a large number of support vectors that can lead to over-fitting and slower prediction. However, this is a hypothesis that requires further exploration. In our experiments, no additional parameters or conditions are introduced to choose the final model. We simply choose the best model among those generated at different levels.
III-D Proposed Adaptive Method to Recover the Quality Drop
The drop in classification quality is equivalent to an increase in the number of points that are misclassified. Therefore, a classifier quality can be improved by training on more similar data to misclassified data points. The training data at each level is a sub-sample of data around the decision boundary (hyperplane) at the coarser level. The data is sampled from the neighborhood of SVs from the trained model at the coarser level. The points which are misclassified in the training data are not crucial for improvement since the SVM did not find a better solution for the decision boundary. However, the validation data points which are misclassified are the crucial points that need to be classified correctly in this step. It is worth to notice, this step does not rely on any information from the test data and only rely on the validation and training data.
Increasing the number of data points usually help the classifier to learn a better decision boundary. A $p$,$n$ number of neighbors of misclassified validation data points are found in the data from positive and negative class, respectively. The neighbor points are added to the training data, and the model is retrained. The $p$, $n$ can start from 1 and increase to larger values. However, as these parameters increase, the number of data points that are added to training data increase with respect to the validation data set size and affect the computational performance in the current level and overall framework.
We mention important considerations of earlier work in the MLSVM framework.
It is essential to mention that for the validation data, we used $FF$ method in MLSVM that is based on the sampling of the original training data at the finest level.
Therefore, the size of the validation data set for large data sets is significantly larger than the training data at the coarser levels. For instance, a data set with $5M$ data points such as SUSY with k-fold cross-validation using $k=5$ has $4M$ training data at the finest level. Using the validation sample rate of $10\%$, the validation size at all levels has $400k$ data points. At the coarsest level, the training size is less than $2\times M$ (600 samples with default $M=300$)
, and it continues to grow slowly over the next few levels of refinement. Suppose the accuracy is $80\%$ at a level. The maximum number of points that can be added to training data would be $400k*20\%*(p+n)$ or $80k*(p+n)$. This would be an impractical training size. Therefore, we limit the number of $p$ and $n$ to one. In our experiments, many miclassified points have common neighbors which were not in the training. We only add distinct points into training set, hence, the increase is training size is much smaller.
The focus of MLSVM is on extremely imbalanced data sets.
The majority of data points belong to the larger class in highly imbalanced data sets. Therefore, a more restrict sampling ratio helps to reduce the number of data points from the majority or larger class in the validation data. However, the minority or smaller class typically has ten times fewer data points, and its size would not cause performance problems. Therefore, a more significant sample rate, such as $50\%$ or more for minority class compares to $10\%$ or less for majority class, both reduce the number of points in the validation data and make the validation data better representative sample to evaluate the quality of the classifier. A validation set is significant for the framework, which allows us to find the best model from many models at various levels. If the validation data is not representative, the quality of the model will severely affect the test data, which is the ultimate goal of training a classifier.
The validation data size for a large data set with a sample ratio of $10\%$ is still huge. Using smaller values such as $2\%$, the validation data was not representative of test data anymore in the past. With proposed new unbalanced sampling ratios for validation, the $2\%$ validation sample ratio of majority class, and $50\%$ sample ratio of the minority class, reduce the computational challenge.
The notation used in Algorithm 1 is explained in Table I.
IV Experimental Results
First, we report the comparison between the ML-SVM algorithm and the state-of-the-art SVM algorithms such as LIBSVM and DC-SVM in terms of classification quality and computational performance. Then, we compare the proposed AML-SVM method and ML-SVM method using new set of experiments with more details.
IV-A Performance measures
Performance measures are metrics which are used to evaluate the classification quality of the model. We report the
accuracy (ACC), recall or sensitivity (SN), specificity (SP), and geometric mean (G-mean). They defined as
${\textrm{ACC}=\frac{TP+TN}{FP+TN+TP+FN}},{\textrm{SN}=\frac{TP}{TP+FN}},{%
\textrm{SP}=\frac{TN}{TN+FP}},$
${\textrm{G-mean}=\sqrt{\textrm{SP}\cdot\textrm{SN}}},$
Where $TN$, $TP$, $FP$, and $FN$ correspond to the numbers of real negative, true positive, false positive, and false negative points.
Our primary metric for comparison is G-mean, which measures the balance between classification quality on both the majority and minority classes. This metric is illuminating for imbalanced classification as a low G-mean is an indication of low-quality classification of the positive data points even if the negative points classification is of high quality. This measure indicates the over-fitting of the negative class and under-fitting of the positive class, a critical problem in imbalanced data sets.
In both ML and AML -SVM frameworks, many models are trained. We need to provide one value for prediction as to the final model performance measure. Therefore, we evaluate all the models using the validation set and select the best model and report the performance measures of the selected model over the test (hold-out) set in the results.
The detail of imbalanced data sets used in our experiment is presented in Table II. The imbalance ratio of data sets is denoted by $\epsilon$.
IV-B LIBSVM, DC-SVM and ML-SVM with AML-SVM Comparison
For self completeness of this paper and convenience of the readers, we present all the relevant results from the ML-SVM paper. The following results provides the clear comparison between the state-of-the-art methods which set the base for next section which we provide more detailed results for comparing the ML-SVM and AML-SVM methods.
IV-C Implementation
The ML-SVM was developed using C++ and PETSc library [32].
The following libraries are used for various parts of the framework. The FLANN library for finding approximate k-nearest neighbors before coarsening, and METIS library
for graph partitioning during the refinement.
The LIBSVM [1] as the underlying solver for SVM. The implementation is based on a single CPU core, and all the speedups are through algorithmic improvement without leveraging parallel processing.
The proposed framework is developed based on the ML-SVM library.
Moreover, the OpenMP library is speed up the parameter fitting at each level using multi-threading.
Since the number of parameter fitting used for experiments is 9 in the first stage and 4 in the second stage of nested uniform design, we do not evaluate the speedups by increasing the number of threads. The coarsening implementation is sequential and based on a single CPU core.
All experiments for data sets with less than 1M data points have executed on a single machine with CPU Intel Xeon E5-2665 2.4 GHz and 64 GB RAM. For larger data sets, we used one single machine with CPU Intel Xeon E5-2680v3 2.5 GHz and 128 GB RAM.
IV-D ML-SVM with AML-SVM Comparison
The same experiment setup and hardware configuration is used for the results in this section and next section. Tables III and
IV demonstrate a comparison across various libraries [5]. The C-Time and W-Time are used to report CPU time and wall time respectively.
However, the purpose of experiments and the version of ML-SVM are different. The experiments are designed to have a better understanding of the multilevel framework and refinement process on each level of the hierarchical framework, while the earlier experiments are designed to compare the overall frameworks regardless of internal details.
The CPU time is accurate metric for computational time of sequential programs such as LIBSVM, DC-SVM, and ML-SVM. Therefore, the CPU time is presented for them. However, The AML-SVM leverages multicore parallel processing which can reduce the wall time, hence, both CPU and wall times are reported.
In all experiments the data is normalized using z-score.
The computational time reported in all experiments contains generating the $k$-NN graph. The computational time is reported in seconds unless it is explicitly mentioned otherwise.
In each class, a part of the data is assigned to be the test data using $k$-fold cross validation. We experimented with $k=5$ and $10$ (no significant difference was observed). The experiments are repeated $k$ times to cover all the data as test data. The data randomly shuffled for each $k$-fold cross validation.The presented results are the averages of performance measures for all $k$ folds.
Data points which are not in the test data are used as the training data in $\mathcal{J}^{+(-)}$.
The test data is never used for any training or validation purposes.
We run 18 experiments per dataset using nested cross-validation with k-fold where $k\in\{5,10\}$. with various configurations to evaluate the performance of the proposed method (AML). For each data set, we evaluated the combination of important parameters such as interpolation order (r), validation sample ratio (v), and where we stop to partition the data in ML method, which we define as stopping criteria for the AML method. We controlled all the other variables only to consider the effect of new method for an exactly similar configuration. Each combination of configuration provide slightly different results per dataset, and therefore, we evaluate 18 combinations of parameters and present the statistical information per dataset. The goal of the experiment is to have a well-designed experiment for comparing the current and new methods. We provide the related results from the MLSVM paper for convenient comparison of AML method with other states of the arts SVM solvers such as LIBSVM and DC-SVM.
For each configuration, we present three sets of results over k-fold cross-validation: (1) Classification quality: we plot the statistics of G-mean at each level of uncoarsening; (2) Computational Performance: the average running time for both coarsening and uncoarsening framework is reported; and (3) An overall comparison only for the final results of the whole framework without details for levels are provided in Tables III to V. We only plot the statistics for Clean, Cod, and SUSY data set in figure 2. The rest of plots are available online at the GitHub repo.
The number of levels in coarsening and uncoarsening is the same, and we use them interchangeably for the rest of this section.
The Advertisement dataset has 3,279 data points and 1,558 features. It has a large number of features and a small number of data points. The number of levels in the coarsening phase, only depends on the number of data points. Therefore, there are only 3 levels in (un)coarsening phase for the Advertisement dataset.
Both ML-SVM and AML-SVM frameworks have identical results for the first and second levels. However, the AML-SVM achieved higher quality on the final level. All the qualities are reported based on G-mean performance measures.
The Buzz dataset has 140,707 data points, 77 features, and 7 levels of coarsening. The quality of both methods is comparable except for a significant drop on the 2nd level for the ML-SVM quality.
The Clean dataset has 6,598 data points, 166 features, and 5 levels of coarsening. The ML-SVM’s quality is descending from the 3rd level to the 5th level. However, while the AML-SVM has not decreased, in the 4th and 5th levels, it achieves the highest quality across all the levels.
The Cod-RNA dataset is related to Breast Cancer. It has 59,535 data points and 8 features. It has a similar decreasing trend as the Clean dataset.
The Forest (Cover Type) dataset is one of the larger data sets with 581,012 data points and 54 features. Contrary to the earlier data sets, the quality (G-mean) at the first (coarsest) level is lower than most of the levels. The ML-SVM quality is increased in the 2nd level but continues to descend til level six. The quality at 7th level is increased with a slight decrease in the following levels.
The AML-SVM recovers the descend at the third level and achieves the highest G-mean in third and fourth levels.
We designed early stopping for the AML-SVM to reduce the computational complexity. Therefore, as the size of training data reaches a threshold $\theta$ in Algorithm LABEL:alg:aml_general_refinement, the refinement process stops.
The ML-SVM has a descending trend for quality on the Letter data set similar to Clean and Cod-RNA data sets. However, AML-SVM has no decrease in quality.
The Ringnorm dataset has 7,400 data points and 20 features. The ascending quality is observed for both methods without any drop in qualities.
The Twonorm dataset has the same size and number of features as the Ringnorm.The ML-SVM quality drops in the 3rd and 4th levels. The adaptive methods achieve high-quality results on all levels.
The quality of ML-SVM for both HIGGS and SUSY data set has a similar trend of high quality at the coarsest levels following with lower quality in the middle levels. At the finer levels, there is a slight improvement, but the variance of qualities is larger than the coarser levels.
The adaptive method for the SUSY dataset achieves high quality up to level 6. In levels 7 and 8 the quality of adaptive method drops as well, and at level 8, the adaptive method stops. For the HIGGS dataset, the adaptive method has better quality compared to the normal method. At level 9, the adaptive method stops to reduce the computational complexity.
V Conclusion
In this paper, we introduced a novel adaptive recovery technique for under/over-fitting challenges in hierarchical frameworks which performs well on nonlinear support vector machines. Our results based on 18 distinct configurations of essential parameters for both coarsening and uncoarsening phases show that the adaptive approach is less sensitive to changes in hyper-parameters. A significant advantage is improved computational time.The adaptive method has reduced the prediction quality variance across various levels in the multilevel framework, which provides a more robust solution.
On massive data sets, the training size may increase significantly at the fine levels. Our proposed early stopping functionality improves the computational performance as it prevents computationally expensive fine level training. In addition, we developed multi-threading support for parameter fitting, which utilizes more processing power on single computing node and increases the overall performance.
The proposed approach can be extended to a hierarchical learning framework with a clear objective for quality.
While our exhaustive results demonstrate the ability of classifiers to learn a high-quality model at the coarse and middle levels, the model selection over many models, or training an ensemble model can be studied further.
For large-scale problems, sampling is a traditional approach to reduce the computational time for intermediate tasks such as feature selection or dimensionality reduction. The proposed framework is a useful alternative for blind sampling of data.
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\preprinttext
AUTHORS’ PREPRINT
\onlineid1105
\vgtccategoryResearch
\vgtcpapertypeVIS Full paper
\authorfooter
Jingyuan Liu and Chiew-Lan Tai are with Hong Kong University of Science and Technology. E-mail: [email protected]. Nazmus Saquib is with Tero Labs, California, United States.
Zhutian Chen is with Harvard University.
Rubaiat Habib Kazi and Li-Yi Wei are with Adobe Research.
Hongbo Fu is with City University of Hong Kong.
VCoach: A Customizable Visualization and Analysis System for Video-based Running Coaching
Jingyuan Liu
Nazmus Saquib
Zhutian Chen
Rubaiat Habib Kazi
Li-Yi Wei
Hongbo Fu
and Chiew-Lan Tai
Abstract
Videos are accessible media for analyzing sports postures and providing feedback to athletes.
Existing video-based coaching systems often present feedback on the correctness of poses by augmenting videos with visual markers either manually by a coach or automatically by computing key parameters from poses.
However, previewing and augmenting videos limit the analysis and visualization of human poses due to the fixed viewpoints, which confine the observation of captured human movements and cause ambiguity in the augmented feedback.
Besides, existing sport-specific systems with embedded bespoke pose attributes can hardly generalize to new attributes; directly overlaying two poses might not clearly visualize the key differences that viewers would like to pursue.
To address these issues,
we analyze and visualize human pose data with customizable viewpoints and attributes in the context of common biomechanics of running poses, such as joint angles and step distances.
Based on existing literature and a formative study, we have designed and implemented a system, VCoach, to provide feedback on running poses for amateurs.
VCoach provides automatic low-level comparisons of the running poses between a novice and an expert, and visualizes the pose differences as part-based 3D animations on a human model.
Meanwhile, it retains the users’ controllability and customizability in high-level functionalities, such as navigating the viewpoint for previewing feedback and defining their own pose attributes through our interface.
We conduct a user study to verify our design components and conduct expert interviews to evaluate the usefulness of the system.
keywords: Human Pose, Video Processing, Sports Data Analysis
\CCScatlist\CCScat
K.6.1Management of Computing and Information SystemsProject and People ManagementLife Cycle;
\CCScatK.7.mThe Computing ProfessionMiscellaneousEthics
\teaser
We present VCoach, a novel analysis and visualization system for providing feedback on running pose correction for amateur runners.
Our system compares two running poses (a) with respect to both common running pose attributes and specific attributes defined by users (e), and visualizes the differences with 3D animations (d).
The profile of running poses in the sample video (b) and differences in running poses (c) are previewed with glyphs.
\vgtcinsertpkg
1 Introduction
Running is a globally popular exercise, and many runners want to avoid injuries and improve their performance.
Not everyone can have access to human coaches, and thus various online materials and mobile apps have emerged to provide guidance on achieving correct running forms.
As with general sports training, an accessible means for novice sports players is to learn from pre-recorded performances of coaches or professional players by performing and comparing the same actions.
Despite the previous video-based systems for providing posture feedback [6, 11], analyzing and visualizing the differences in posture data in videos remain challenging, as discussed below.
According to the taxonomy of comparison-based visualization [15], existing visualizations for human pose comparison include displaying related poses in two videos side-by-side (juxtaposition) [48, 47, 37], overlaying one pose onto another (superposition) [11], and augmenting video with visual markers (explicit encoding) [46].
However, the main limitation of these video-based pose comparison techniques is that the appearances of observational biomechanical measurements, such as angles and distances, are often subject to changing viewpoints (see the toy example in Figure 1).
For sports coaching systems, such an ambiguity problem affects both the observation and the feedback.
When observing the actions in videos, the 3D human pose attributes might be distorted due to perspective shortening and thus fail to reflect the actual biomechanical measurements.
In visualization, the shapes of graphical annotation markers overlaid on videos are also subject to changing viewpoints, and are thus ambiguous in providing accurate corrective feedback to be perceived by amateur runners.
To promote spatial awareness, prior studies have attempted to analyze reconstructed 3D poses [13], fuse videos in multiviews [46], and use situated AR [28] and immersive visualization [10, 24].
Thanks to the emerging methods in monocular human reconstruction in computer vision [9, 16], reconstructing 3D poses has become an effective and accessible solution for videos.
Besides the ambiguity problem, another consideration is the data attributes for comparison, which can be classified as parametric and non-parametric.
Parametric pose features (e.g., knee angle) are sport-specific and pre-defined by domain experts [6].
The embedded bespoke knowledge makes sport-specific systems hard to scale and support users’ needs for individual customization.
Alternatively, non-parametric comparison avoids embedding bespoke knowledge by comparing the transferred and overlaid human poses [11].
Novices would need to infer the corrective feedback based on their perceptions.
To address the above-mentioned issues, we aim to develop an interactive system to analyze and visualize differences in human biomechanical data.
Our system, VCoach, provides intuitive and customizable corrective feedback for amateur runners.
To achieve this goal, we worked closely with experts in Sports Science to identify its designs based on the coaching process in practice.
As shown in VCoach: A Customizable Visualization and Analysis System for Video-based Running Coaching, our system takes as input a sample user video from an amateur runner and an exemplar video from an expert runner, and automatically performs pose analysis tasks, such as reconstructing 3D poses from videos and computing pose differences.
The differences are then visualized as short animations on a 3D human body model (VCoach: A Customizable Visualization and Analysis System for Video-based Running Coaching(d)) to resemble the dynamic demonstration of human coaches in practice.
To reduce the ambiguity of visualization, we propose to augment 3D visual markers onto the 3D body model instead of the video, such that users can either preview under our suggested viewpoints or manually navigate through viewpoints for better perception.
VCoach embeds pre-defined biomechanical attributes that are commonly used for analyzing running poses (e.g., leaning angle and foot landing position).
To support the analysis of attributes users are interested in but not embedded in the system (e.g., vertical bend angle of knees and height of feet),
we also provide an interface (VCoach: A Customizable Visualization and Analysis System for Video-based Running Coaching(e)) that allows users (advanced amateur runners or coaches) to manually label biomechanics.
The user-customized attribute will then be retrieved from both the sample and exemplar videos for comparison in the same way as those pre-defined attributes.
This attribute generalization is facilitated by a design of mappings for biomechanical data that unifies the representations of attributes, their differences and users’ interactions to label the attributes.
Specifically, we make use of the semantic model definition of SMPL 3D human mesh model [31].
Users annotate and define measurements on a 3D SMPL body model in T-pose, such that the defined attributes can be retrieved across multiple videos using model correspondence.
We design a user study and expert interviews to evaluate the design components and the overall effectiveness of our system.
For the scope of the user study we focus on adults in moderate-speed running (jogging), since this is the most common type and demography for running exercises.
The human pose analysis model in VCoach can generalize to other user groups than adult amateur runners, such as children and professional runners, with adapted visualizations of pose differences (e.g., cartoons for children and infographics with detailed figures for professional runners).
By replacing the current pre-defined attributes with key attributes of other sports, VCoach can also be generalized to support the posture analysis of new techniques, such as in skating [12] and high jump [38].
2 Related Work
Pose Coaching Systems
Previous research work on video-based running pose analysis is limited, partly because in-the-wild running poses contain larger variations in appearance than other sports with more confined locomotion ranges, such as yoga [6] and golf [35].
Running dynamics, such as ground contact time and vertical oscillation, require specific combinations of hardware to capture (e.g., [52]).
In the following, we review posture coaching systems in general sports.
According to how the bespoke knowledge of a specific sport is introduced into the system, existing coaching tools span the spectrum from fully-manual to fully-automatic, as illustrated in Figure 2.
The other dimension is whether the poses are captured in 2D (videos) or in 3D (MoCap or Kinect).
The fully-manual coaching tools require human coaches to either manually annotate on video playbacks to suggest improvements [48, 37], or analyze data of running gaits captured by MoCap [45].
MotionPro [35] supports manual selection of keypoints on each of the video frames such that some quantities, such as ball trajectory and 2D angles, can be obtained to facilitate analysis.
Kinovea [22] and OnForm [37] further simplify the manual tracking by providing basic processing of videos (e.g., automatically track objects and estimate 2D human poses).
On the automatic (right) side of the spectrum, a few video-based coaching tools assess the movements based on the reconstructed 2D poses from videos using embedded rules for a specific type of sports, such as skiing (AI Coach) [50] and yoga [6].
Such systems would require extensive domain knowledge to design.
To avoid bespoke knowledge, some systems compute suggestions based on the comparisons between novices’ actions with experts’ reference actions.
For example, MotionMA [49] and ReactiveVideo [11] align the experts’ poses captured by Kinect onto the novices’ poses in videos to visualize the difference in postures.
AIFit [13] mines and highlights the most significantly different features from the comparisons of reconstructed 3D poses from videos.
Even though AIFit is fully automatic, the dominant differences might not reflect informative feedback to the sport.
VCoach closes the gap in both dimensions in this spectrum:
the input is monocular videos such that it removes the constraint of indoor controlled environments, but it analyzes and visualizes in 3D to ensure spatial awareness.
It automatically performs low-level tasks but allows users the controllability to introduce high-level bespoke knowledge to the system.
Video-based Sports Data Analysis
A previous work [8] has classified general video-based sports data analysis into four levels: image level, object level, event level, and tactic level.
We adopt the same taxonomy as that in [8] and review w.r.t. video-based human pose data in sports.
Image-level analysis mainly includes video effects, such as slow-motion playback and displaying frames side-by-side [48].
Image-level analysis does not involve further image understanding from video frames, and thus the image contents would need to be analyzed manually (e.g., by a human coach).
Object-level mainly includes obtaining parameters of a single human instance, such as human pose estimation [50] and motion tracking [29, 22].
In sports videos object-level analysis is often more challenging than that in ordinary videos due to motion blurs, large subject displacements and complex sports poses (e.g., high diving).
Prior studies addressing these challenges include adopting sports motion priors [7], collecting sports motion datasets [40], and capturing human motions with multi-modal references [18].
Event-level analysis mainly includes recognition tasks from video streams, such as action recognition [40], action quality assessment [27], and key frame detection [54].
Tactic-level is mainly involved in ball games, such as soccer [43], table tennis [8], and basketball [3], by parsing the movements of athletes and objects from videos.
VCoach performs object-level analysis, but it focuses on local pose attributes rather than whole-body poses.
User-Customizability in UI
The goal of promoting user-customizability is to generalize to new instances other than those embedded in the systems, without requiring end-users’ explicit programming.
For example, in gesture recognition, a few systems, such as KinectScript [36] and Visual Gesture Builder [33] allow users to interactively define gestures by recording a few repetitions.
MotionMA [49] and YouMove [1] allow users to define movements via Programming by Demonstration (PbD).
Besides gestures and movements instances, other finer analysis tasks involve users’ specification of which body part(s) to analyze.
A medical research analysis tool, DeepLabCut [32], allows manual labeling of body parts across animal species for training data-driven models.
Kinovea [22] and RealitySketch [44] allow users to manually select points to track on top of videos, and customized joint angles can be further computed from the tracked points.
While such keypoint definitions apply to a specific video, in this work we develop a systematic set of mappings for users to customize reusable human pose biomechanics across videos.
3 Formative Study
At the beginning of this project we set out to decide the directions and the scope of a sports coaching system suitable for amateurs, which include but are not limited to runners.
We conducted a survey on potential target users to understand their usual ways of obtaining feedback on posture correctness in practising sports (subsection 3.1).
We also interviewed three experts on human locomotion to inform our design (subsection 3.2).
The results of this formative study form a set of design requirements for our system (subsection 3.3).
3.1 Target User Survey
To investigate the demands of potential target users (amateur sports players), we conducted a survey via the Amazon Mechanical Turk (MTurk).
We designed a questionnaire with three questions: (1) “What sport(s) do you frequently practise?” (2) “Have you paid attention to the correctness of your body postures while practising the sport(s)?” (3) “If yes, please describe how you get feedback on the correctness of your postures; if not, please explain why not.”
We distributed 120 questionnaires in total, and filtered out obvious spam responses according to the quality of the short answers to question (3).
Eventually 70 effective answers were collected.
Figure 3 shows the summaries of responses.
Among the responses, jogging/running accounts for the most, followed by football.
Other mentioned sports include those involving posture correctness, such as yoga and swimming.
24.3% of the subjects said they only depended on learned instructions of the actions but obtained no feedback; 21.4% of respondents stated that they got feedback from a coach or peers.
Other main feedback includes:
5.7% used outcome (e.g., score) as an indicator of posture correctness,
15.7% used feeling (e.g., tense on lower back) as an indicator,
and 8.6% adopted extra training on postures.
One respondent said he/she video-recorded the actions when practising gymnastics, and two responses explicitly said that they did not get any feedback since no one was watching.
Through this survey we learned that the public has the awareness of the importance of maintaining good postures, and there is a need for accessible posture analysis tools.
Based on the survey results, we set the focus of our system to jogging, due to its popularity and the requirement on correct postures to avoid injuries, without needing to consider ball/racket trajectories for instrument sports or tactics for team sports.
3.2 Expert Interviews
In order to understand the process and the key factors of human movement analysis, we conducted semi-structured interviews with three experts, two were medical doctors in Sports Medicine working in a hospital (E1, E2), and the other one (E3) was a researcher in Sports Science in a startup company studying performance analysis in sport.
During the interviews we first invited the participants to describe a representative case in which human movement analysis is involved in their daily practice.
During the description, they were asked to identify what is the routine they analyze human movements, what are the key factors they focus on, and what is the decision process based on their observations.
Then we raised open questions such as difficulties in human movement analysis, and the role of video-based analysis in practice.
All of the three experts mentioned that human movement analysis is based on gold standards, i.e., comparisons with the normal values in rehabilitation exercises or with top athletes’ postures and performances in sports.
Even for a full-body movement only a few key factors are concerned in evaluation (deterministic models [17]).
For example, E1 described a case of imbalance testing, where the key factors were movement accuracy and time required for completion.
E3 emphasized the advantage of externally-focused training over internally-focused training [53].
He pointed out that even though real-time feedback provides direct guidance, it would distract a subject during the action by interfering the subject’s intention of movements.
He also mentioned that since a coach’s attention is limited, he/she often can only focus on a specific body part during instruction, and that it would be ideal to analyze other parts during playback.
Since our system is focused on running, throughout the project we closely worked with E3 and another expert (E4), a third-year postgraduate student in Sports Science, who was involved after this formative study.
We initiated discussions with them as needed via remote chats.
3.3 Design Requirements
From the expert interviews on the human movement analysis, as well as the limitations of existing systems, we identify the following design requirements:
R1 - The tool should be accessible to users without an expert.
The potential users of our system might have no domain knowledge to determine the posture correctness directly from their videos.
This can be mitigated by comparing their videos with another video involving standard running poses from a professional runner and learning from the differences.
Our system should not only include key factors to running, but should also allow users to easily introduce other key factor(s) in case needs arise, instead of embedding redundant bespoke knowledge of running in the system.
Our system should be as easy to use as possible for novice users.
R2 - The comparison should adapt to variations.
The videos input by users may contain large variations on the running poses, due to viewpoints and subjects’ physical characteristics.
The comparison should be able to factor out these interferences and focus on only factors that indicate running posture incorrectness.
R3 - The visualization should highlight the part-based differences.
As pointed out by E3, the attention of both coaches and athletes is limited, they are often advised to correct one part at a time.
Thus instead of showing all the mistakes at the same time, our system should show the differences in each body part separately.
E3 also mentioned that for both coaches and athletes the quantitative figures do not make sense; they desire a direct corrective suggestion.
Thus instead of presenting analysis results as infographics, we need to design an intuitive way to demonstrate the differences.
R4 - The system should enable user interactivity.
As suggested by E4 in a later discussion, when a coach corrects an action, he/she usually first points out the mistakes, and then shows the correct action.
Our system should also follow this routine.
Following the design requirement R1, since there is no remote coach explaining the results, our system should allow users to explore the feedback to make the most sense out of it.
4 System Overview
We design our system VCoach based on the aforementioned requirements.
Since we target novice users, the overall system workflow follows the “overview first, details-on-demand” principle [41].
Users input videos and preview suggestions through the user interface (VCoach: A Customizable Visualization and Analysis System for Video-based Running Coaching).
The input to our system contains two videos (VCoach: A Customizable Visualization and Analysis System for Video-based Running Coaching(a)): a sample running video to be analyzed, and an exemplar running video for comparison (R1).
Upon loading the two videos, our system automatically processes the videos to reconstruct 3D human poses, normalizes the motions (R2), and segments the videos into running cycles.
Our system then performs the pose analysis by aligning the sample and exemplar running pose sequences based on 3D pose similarity, and retrieves the pre-defined key attributes to conduct comparisons (R1).
The suggestions for correction are generated based on the part-based differences from the comparison (R3), and directly reflect on a timeline tailored for running pose sequences (VCoach: A Customizable Visualization and Analysis System for Video-based Running Coaching(c)).
Those attributes that require improvement are represented with glyphs.
By clicking on each glyph on the timeline (R4), a detailed instruction for improving the corresponding attribute is shown as a short 3D animation of a body part on a human model in the suggestion preview window (VCoach: A Customizable Visualization and Analysis System for Video-based Running Coaching(d)).
Users can rotate the body model to navigate through viewpoints for better perception (R4).
For other pose attributes that are not embedded in our system as pre-defined attributes, the users can interactively label (R4) on a 3D body model via the query editor (VCoach: A Customizable Visualization and Analysis System for Video-based Running Coaching(e)).
The labeled attributes will then be retrieved and analyzed from the videos in the same way as the pre-defined attributes.
Our system contains five modules, as shown in Figure 4.
5 Visual Design
In this section we first describe the formulation of the data (attributes of running poses) we study to design our system.
Then we propose three mappings based on the data, i.e., the representation of data, the visualization of their differences, and the user operations to interactively define the attribute of each type.
5.1 Data Attributes
The data attributes in our system include both pre-defined attributes that are commonly used for evaluating running poses, and user-defined attributes for their customized analysis.
To determine the common data attributes for running pose correction, we collected a corpus of running pose tutorials by searching with key words “running pose tutorials”, “running pose corrections”, “running techniques”, “running form”, etc., from Google and YouTube.
The current corpus contains 55 items (including 37 videos and 18 articles).
The data attributes are summarized from the corpus into four types, as shown in Figure 5.
We conducted another interview with E4 to verify the coverage of these attributes in running pose evaluation in practice.
The fourth type “categorical data” is different from the previous three in that they are not computed from comparison with exemplar poses, but computed directly based on the other three classes (i.e., first compute a value and then discretize it into a category by a certain threshold).
Thus we focus on the design for the first three types, but support the visualization of the categorical data for commonly evaluated attributes in running.
5.2 Representing attributes
In this section we summarize the visual encoding of the positional, angular, and temporal attributes.
Positional attributes (Figure 6(a)) are defined as the relative distance between two points (classified as type P1), or the position of a point from a specific axis (P2).
For example, the trajectory of the wrist is its relative distance to the body center (P1).
Another example is the knee lift, which is a vertical distance from the knee joint to the body center (P2).
Angular attributes (Figure 6(b)) are defined as either the angle formed by three endpoints (classified as type A1), or the orientation of a vector formed by two joints with respect to an axis (A2).
For example, the elbow angle (A1) is an angle formed by the shoulder, the elbow and the wrist joint.
The leaning of the upper body (A2) is the orientation of the vector pointing from the root joint to the neck joint w.r.t. the z-axis.
Temporal attributes are defined as either a single moment (T1) or a time range within a running cycle (T2).
We use a temporal axis to show the temporal context.
The temporal axis (Figure 6(c)) is a fixed full running cycle, with the three dots from left to right respectively corresponding to the states of right foot landing ($RL$), left foot landing ($LL$), and right foot landing for the next cycle.
The positioning of the human center on the temporal axis reflects the state of the current pose within the running cycle.
5.3 Visualizing attribute differences
This section introduces the representation of the differences in data attributes.
Such differences are mainly used for presenting feedback, i.e., from an incorrect configuration to a correct one.
We define a set of visuals for attribute differences (Figure 6(d)), which are unified with the attribute representation.
Positional difference is shown by two points and an arrow pointing from the wrong position to the correct position.
Angular difference is shown by two vectors forming a wedge to show an angular difference.
Temporal difference is represented by a red marker segment on the temporal axis showing a temporal offset.
For example, the red segment along the forward temporal axis direction indicates the current event should appear later.
5.4 User operations
In this section we introduce the user operations (Figure 6(e)) for defining their own data attributes under the three data attribute classes.
Specifically, the query editor in our user interface (VCoach: A Customizable Visualization and Analysis System for Video-based Running Coaching(e)) contains a 3D viewer presenting the 3D human body model in T-pose, radio buttons for specifying properties and two draggable cursors (red lines) on top of a running cycle diagram for specifying timings.
A user may either refer to the mesh or skeleton of the body model and directly mouse-click on the body model to select joints; our system will snap the mouse click to the nearest joint.
A user first selects the attribute type by selecting either the angle button or distance button for angular and positional attributes, respectively, or directly dragging the temporal cursors for a temporal attribute.
To edit a positional attribute, a user first specifies the joint to track, and then specifies the base point (P1).
When the user further selects an axis, only the component of the selected dimension will be recorded (P2).
To edit an angular attribute, a user either selects three endpoints in order on the body model (A1), or two points and one axis (A2).
To edit a temporal attribute, the user either moves one cursor to specify a moment (T1), or both cursors to specify a time range (T2).
Our system will record a phase or a phase range accordingly.
When the positional and angular attributes are associated with an event, the user also moves the temporal cursor to specify the timing.
Please refer to the demo video for the authoring process of “left foot landing position” example.
5.5 Design of Attributes Overview
In this section we discuss the design of the overview for the problems reflected from the comparison.
The overview should show which attributes appear in question in the sample video and their timings.
We thus propose to use glyphs for representing attributes and a timeline tailored for running to organize them temporally.
Glyphs
We designed two types of glyphs for the four classes of the attributes, namely suggestion glyphs and profile glyphs.
Suggestion glyphs are icons for each of the three classes of attributes in Figure 6, i.e., positional, angular and temporal attributes in the collected corpus, whose values are continuous variables and are compared with those in the exemplars.
As shown in Figure 7(a-c), the suggestion glyphs are designed based on the idea of traffic signs that augment markers to symbols, such that users do not need to memorize the encoding, but can easily get familiar with the meaning of the icons and can interpret the meaning by intuition.
The profile glyphs are used to represent categorical attributes which do not need comparison with the exemplar.
We adopt the idea from the dance notations [34] to discretize complex human movements into reference planes (sagittal,
frontal and horizontal).
As shown in Figure 7(d), we use three transverse planes that capture the joints with a large degree of freedom, i.e., foot, knee, and shoulder.
Then the motions of these joints in relation to the body center are reflected by their projections into the three planes.
For example, by referring to the projection of wrists, users gain an intuitive profile of whether the wrists cross the body’s middle line in front of the chest.
In the transverse plane for feet, beyond showing the relative landing position to the body center, the triplet stacked squares further show the strike mode (fore-foot, mid-foot or rear-foot strike) of each foot by highlighting one of the blocks at the corresponding position.
Timeline
A characteristic of a running pose attribute sequence is that it is temporally periodical, and each period can be divided into a right-phase and a left-phase.
Based on this characteristic, we propose to design a timeline that transforms the temporal space into a running event space.
As shown in VCoach: A Customizable Visualization and Analysis System for Video-based Running Coaching(c), the horizontal axis is a complete running cycle, and the vertical axes correspond to the attributes of the left side of the body, right side of the body, and middle, respectively.
All the data attributes are summarized among cycles to be shown on the timeline.
Our system will automatically select significant errors, with the sizes of the glyphs proportional to the significance of the errors of a particular type.
We have conducted a pilot study to verify the above design against their alternatives.
For glyph design, the alternatives include a set of simplified icons highlighting the body parts in question, and color and shape encoding.
For timeline design, the alternatives are an ordinary linear timeline of a video that is not segmented into running cycles, and a spiral timeline displaying all running cycles without summarizations.
We invited two users, both of them are novices to running, and one of them with design background.
We introduced the overall function of our system along with the two sets of designs, and then let them vote on which representation they prefer.
Both of them chose the semantic glyph and the aggregated timeline, because they thought the semantic icons are intuitive and can be easily remembered.
As novice users they do not desire all the occurrences of the problems, but rather what kinds of problems appear in their running; thus the aggregated timeline is more preferable.
6 Data Analysis Model
In this section we introduce the methods of the backend modules in VCoach (Figure 4): video processing, pose analysis, and feedback.
6.1 Video Processing
3D Pose Reconstruction and Normalization
When the sample and the exemplar videos are loaded into the system, the pose at each frame is retargeted onto the SMPL models, denoted as $M_{s}$ for the sample video and $M_{e}$ for the exemplar video.
The retargeting (reconstruction) is implemented with TCMR [9], which is a monocular pose reconstruction method achieving state-of-the-art accuracy on challenging outdoor video datasets.
$M_{s}$ and $M_{e}$ are then rotated to a unified global orientation to facilitate comparison (VCoach: A Customizable Visualization and Analysis System for Video-based Running Coaching(a)).
The video frames are cropped to maximize their preview in windows.
Temporal Segmentation
The running pose sequences in both the sample and exemplar videos are segmented by the key frames of foot landing and foot extension.
Since the action of running is periodical, we adopt the phase variable of human locomotion, as in [20].
A full running cycle thus contains four key phases, in “right foot landing” ($phase=0$), “right foot extension” ($phase=0.25$), “left foot landing” ($phase=0.5$), and “left foot extension” ($phase=0.75$) order.
These four key phases are detected from the local extrema of the foot trajectories.
6.2 Pose Analysis
Sequence Alignment
Given the detected key phases, the running pose sequences in the sample and exemplar videos are first temporarily aligned at key phases, and then aligned at a finer level between each two key phases using the dynamic time warping technique [2].
We use joint rotations to measure human pose similarity [30].
Attributes Retrieval
Each data attribute is represented as a meta data tuple: $\left[name,type,J_{A},J_{o},J_{B},axis,side,phase\right]$,
where $J_{A},J_{o},J_{B}$ are the joint IDs of the attribute endpoints in the body model (as shown in Figure 6);
$side$ is one of the “left”, “neutral” and “right”;
$axis$ and $phase$ are the related axis and timing of the attribute; they are left empty if not applicable.
For the attributes embedded in VCoach (Figure 5) the meta tuples are pre-defined.
For customized attributes, the meta is formed from users’ input from the query editor.
Our attribute retrieval program parses the meta tuple and outputs retrieved values from the videos.
The retrieved values are then used for comparison.
Comparison
Since different attributes have different scales and units, we normalize the attribute values to the range $\left[0,1\right]$.
Then the differences in the attribute values are computed as the relative errors between the attributes from the sample video and those from the exemplar video.
We set a threshold of $25\%$ to select the significantly different attributes and scale the sizes of the suggestion icons according to the relative errors.
6.3 Feedback
Animation-based Demonstration
The corrective suggestion from pose comparison is conveyed by animating a 3D human model.
To make the demo easily understandable, the animation follows the design guideline as data-GIF [42].
The animation contains two key frames corresponding to the wrong pose and the same pose with a specific body part in the position as the exemplar pose, respectively.
Specifically, we use the joint rotations to drive the model: for angular attributes, the intermediate frames are interpolated with the joint rotations of $J_{o}$; while for positional attributes, the animation is interpolated with the joint rotations of the parent joint of $J_{o}$ along the kinematics tree.
The 3D animations are augmented with visual markers to highlight differences, as in Figure 6(b).
Suggestive Viewpoint
Since the animation of corrective suggestion is in 3D, we would like to demonstrate it at the most informative viewpoint.
While there are prior studies on the automatic selection of viewpoints for previewing a 3D mesh, the definition and criteria of the optimal viewpoints are often dependent on the purpose, such as to demonstrate region visual saliency [26], to set man-made models in upright orientation [14], and to incorporate modelers’ creation processes [5].
Previous studies on optimal viewpoints for human poses mainly include reducing prediction uncertainty in estimating 3D pose [21] and metrics defined over body part visibility [25].
In VCoach, since we would like to provide suggestions w.r.t. specific 3D local pose attributes, we develop a set of schemes to suggest viewpoints according to the geometry of the attributes.
The main idea is to minimize the ambiguity in the attributes due to camera projection, while preserving the human model as the spatial context.
Based on this goal, we make use of the normal vector formed by the 3D attributes to decide the orientation of the viewpoint (see Figure 9).
We further use the side of the body to determine whether to revert a normal to its opposite direction.
For example, to present an attribute on the right side of the body, the camera should also be placed to the right facing the body model.
The up direction of the viewpoint is along the average of the two vectors.
We also determine whether to revert the up direction according to whether it keeps the human model heading upwards.
Even though we present the 3D animation in the suggested viewpoint, users can still manually change the viewpoint to explore the corrective suggestion.
7 Results
In this section, we show the results of a user study evaluating the visualizations of posture correction feedback in VCoach and the baseline methods (subsection 7.1) for novices, and expert interviews (subsection 7.2) to evaluate the overall effectiveness of the system in pose correction.
7.1 User Evaluations
The main purpose of the user study is to evaluate the improvement of VCoach in promoting novices’ perception of running pose differences over existing methods (see Baselines).
It also evaluates the effectiveness of other components (e.g., viewpoint navigation and summarization of feedback) in assisting novices’ perceptions of running pose improvements.
Apparatus
VCoach was implemented with PyQt5 on a PC running Win10 (Intel x64 i5 [email protected], 8.00GB RAM).
Due to the current local COVID-19 regulation, the user study was conducted via Zoom with remote screen control.
Baselines
The baseline methods are visualizations of pose differences via juxtaposition and superposition, as shown in Figure 10.
We implement the baselines as follows.
For juxtaposition, we used the setup in [48] and put two running poses side-by-side.
To facilitate the preview, the two poses are cropped with subjects’ bounding boxes in videos, and the two videos are temporally synchronized using joint rotations.
For superposition, we adopted the method in [11].
Since [11] is based on Kinect, we transformed the 3D pose in a temporally correspondent exemplar frame and aligned it to the pose in the sample video frame at the body center, such that the temporally synchronized exemplar pose is overlaid on the sample video frame for comparison.
Participants
12 members from a local university were invited to participate in the user study (a1$\sim$a12, aged 23$\sim$32, 3 female).
Except for a1 and a7, all the other participants practise running more than once a week, but do not have access to professional coaches.
a12 stated that he was once curious about the correctness of his running poses and searched for mobile apps providing running pose checking functions but could not find a suitable one.
a2 focused on foot landing during running to avoid injuries; a6 used body senses after running as feedback.
a3, a10 and a11 said that they used mirrors during fitness workout, but obtained no feedback on pose correctness during running.
Task
We prepared 9 sample videos (V1$\sim$V9) covering all of the ten pre-defined attributes.
They were collected from running tutorial videos such that the ground-truth of the mistakes in running poses was known from the coaches’ comments in the videos, such as foot landing in front of the body (the braking position) and insufficient knee lift.
The difficulty level of the videos was controlled by containing only one main problem.
The general task for the participants was to explore the corrective feedback from videos using either VCoach or the baseline methods in a think-aloud manner, and complete a questionnaire afterwards.
The user study contained three sessions: two sessions using our system with and without the suggestive viewpoints, and one session using the baseline methods.
The order of the three sessions was counter-balanced, and the order of the nine videos was randomized among the three sessions (three videos for each session).
During training, we first gave a detailed tutorial on the operations of VCoach as well as the baseline system.
The participants then tried freely to get familiar with both systems.
In the session using VCoach without suggestive viewpoints (denoted as “VCoach-w/o”), we disabled the suggestive viewpoint function, and the participants would need to manually navigate the viewpoints to preview the 3D animations.
The system recorded the participants’ navigation activities in the suggestion preview window, parameterized by viewpoint azimuth and elevation, and the duration of each viewpoint.
In another session using VCoach (denoted as “VCoach”), the suggestive viewpoint function was enabled; the participants could also manually navigate, and their navigation activities were also recorded.
In the session using the baseline methods (denoted as “Baseline”), the participants explored the corrective feedback by comparing running poses in videos in either juxtaposition or superposition visualization.
After the sessions, the participants completed a designed questionnaire (Table 1) in a 7-point Likert Scale (1 is Strongly Disagree and 7 is Strongly Agree), and a standard System Usability Scale (SUS) [4].
The user study with each participant took about 90 minutes.
Effectiveness of Pose Difference Visualization
We first investigate the effectiveness of VCoach in presenting feedback compared with the baseline system.
Q10 explicitly asked the comparison between VCoach and the baseline methods, where 10 out of 12 participants strongly agreed that VCoach was more effective in conveying feedback than the baselines.
We recorded the time required to explore the running pose problem(s) in each video, as shown in Figure 12(a).
Paired t-tests on exploration time required for each video between sessions “VCoach” and “Baseline” showed that using VCoach with the suggestive viewpoint significantly requires less time to obtain the desired feedback ($p=0.019$).
However, there is no significance on exploration time between sessions “VCoach-w/o” and “Baseline” ($p=0.519$).
We evaluated the accuracy via the successful rate of the participants’ discovered mistakes matched the ground-truth mistakes as commented by the coaches in videos.
In sessions “VCoach-w/o” and “VCoach” the successful rate was $100\%$.
In other words, all the participants could figure out the problem(s) in the running poses with the visualization provided by VCoach.
In contrast, the successful rate was $77.8\%$ in session “Baseline”.
From the participants’ think-aloud in session “Baseline”, they often referred to the superposition visualization more than the juxtaposition visualization, especially when the subjects in the sample and exemplar videos are running in different directions.
For superposition in the baseline system, a6 and a8 said that they would refer to the lower limbs more often than upper limbs, since upper limbs were often occluded and misaligned due to differences in limb lengths.
Effectiveness of System Components
We then investigate the influence of specific design components on users’ perception of feedback on running pose correction.
Q6 asked the participants to rate the key component in VCoach, which visualizes pose differences via animations of local body parts on a human model.
8 out of 12 participants strongly agreed that such visualization was helpful for understanding, and the other four chose agreed.
The component that received the most disagreement is the preview of normalized poses from the sample and exemplar videos shown in juxtaposition (VCoach: A Customizable Visualization and Analysis System for Video-based Running Coaching(a) middle).
Since their orientations are often different from those in the original videos, the participants stated that referring to them increased the cognitive load by having to imagine the transformation to understand.
Thus even though normalized poses are crucial to computing pose differences, they do not necessarily contribute to users’ visual comparison.
During the participants’ think-aloud in sessions “VCoach-w/o” and “VCoach”, they often directly moved on to check the glyphs on the timeline after loading both videos.
After watching the animation, they sometimes checked the sample video frame to verify the problem.
At first they sometimes also referred to the exemplar frame to verify the animation, but many of them skipped the exemplar frame later because they found the corrective feedback illustrated by the animation was trust-worthy.
We also evaluated the usefulness of the design component of suggestive viewpoint.
We would like to figure out the following two questions: (1) do users find previewing the animations of pose correction under a certain viewpoint yields better perception?
(2) If yes, do our suggestive viewpoints match the preferred viewpoints selected by users?
We thus analyze the usage of viewpoint selection during the user study.
In session “VCoach-w/o”, the average number of times the participants manually changed the viewpoint was 7.36 times per video, compared with 2.05 times per video in session “VCoach”.
A paired t-test on the numbers of manual navigation between sessions “VCoach-w/o” and “VCoach” shows that enabling the suggestive viewpoint function significantly reduces users’ manual navigation ($p=0.00059$).
To answer question (2), we further analyze the relevance of the participants’ manually-selected viewpoints with the suggested viewpoints computed by our system in session “VCoach-w/o”.
We analyzed previewing viewpoints that lasted more than one second and considered those with a duration less than one second as the navigation process.
The average errors of azimuth and elevation relative to $360^{\circ}$ were $3.19\%$ and $4.52\%$, respectively, indicating a good match between our suggestive viewpoints and preferred viewpoints by the participants.
In the rating of the usefulness of suggestive viewpoint, seven participants chose “strongly agree”, and four of them explicitly stated during exploration that this function was very convenient.
a2 in session S1 asked whether the suggestive viewpoint function could be enabled, because she found this function especially useful when she was comparing the magnitudes of corrections on foot landing position.
a4 found the suggestive viewpoint more useful in observing upper limbs because they often suffer from heavier occlusions by the body torso than lower limbs.
Interestingly, a12 rated “Neutral” in Q9.
He explained that since he studied exoskeleton robotics, he was more used to imagining the attributes using the sagittal, coronal and transverse planes as reference, rather than using the human body as a spatial context.
Since VCoach targets at novice users without human movement analysis background, and most participants found the suggestive viewpoint function convenient, it can serve as a helpful option in VCoach.
System Usability
In the training session, all the participants could get familiar with VCoach within 5 minutes by completing a pipeline of operations, including loading videos, previewing frames and poses, and navigating on the timeline to preview animations of suggestions.
The SUS score for all the ten questions in the SUS questionnaire was 83.125 on average (SD: 10.56), out of a scale of
100, indicating the good usability of VCoach.
In post-study interviews with the participants, they commented favorably towards VCoach.
For example, a3: “Besides clarity, the summarization in VCoach helps me form a better impression of frequent mistakes. With VCoach I don’t even have to browse the entire video, but only need to refer to the frames the system has highlighted for me.”
The participants also commented on the potential generalization of VCoach in other scenarios.
Specifically, a11: “This tool is solving a very practical problem. I can see how it is useful in running and can imagine it generalizes to many other sports.”
a12 (from exoskeleton robotics background): “… current rehabilitation training often relies on wearable sensors to detect patients’ biomechanics, such as joint angular velocities and accelerations.
Such a video-based tool is promising in providing a non-invasive means to analyze patients’ movements.”
Evaluation of Query Editor
From the user study we also evaluate the easiness of use of the query editor, specifically, how efficiently and accurately users can edit a pose data attribute.
There is no baseline method for this task.
We chose three frequently used data attributes from each of the classes in the pre-defined attributes, and asked the participants to edit the attributes using the query editor in our interface.
The three attributes were: “foot landing position” (P2), “elbow angle” (A1) and “foot contact time” (T2).
They covered all the operations on the query editor.
The participants were given sample running video clips as references.
As shown in Figure 12(b), the average editing time for the three attributes were 95.36s ($SD=37.71$), 39.91s ($SD=10.11$) and 38.64s ($SD=14.03$).
On average the editing of the foot landing position took the longest time, since it required the most operations covering all the components on the query editor.
The successful rates that the participants can implement the same attribute as our pre-defined was $83.3\%$, $100\%$, and $91.7\%$, respectively.
In the failure cases, a3 failed the temporal attribute, because he misunderstood the question and labeled the time between two consecutive foot landings instead.
a4 and a10 both correctly annotated the positional attribute on the human model, but forgot to associate with the timing for foot landing by dragging the timeline cursor.
Through this experiment we verified that novice users could easily understand and implement the representative attributes with minimal training.
Even though for most amateur runners the pre-defined attributes would suffice, they can annotate their interested attributes via the query editor with reasonable efforts.
7.2 Expert Interviews
We conducted expert interviews to evaluate the overall usefulness of our system in helping amateur runners correct running poses.
Two experts with running backgrounds were invited: one was a licensed running coach (E5); the other was a professional marathon runner (E6).
The two interview sessions were conducted separately, and each session lasted 50 minutes.
During the interviews we provided a detailed introduction of functions in VCoach with three demonstrations of usage scenarios, and then invited them to try the system freely.
Both experts strongly agreed that VCoach would benefit a lot of runners.
E5: “Not only beginners, but experienced runners are also often bothered by the problems of running pose correctness. I can expect this tool serves a lot of runners.”
They also appreciated that the design rationale of VCoach is very reasonable for practical usage.
E5 said that coaching is a highly personalized process; and thus there is no absolute “correct” running pose regulated by numbers, such as the legal range of elbow angle in degree.
A significant advantage of the design of VCoach is that it does not directly classify a runner as right or wrong, but retains the flexibility to compare with various running poses to show the differences.
E5 thus finds VCoach especially useful for novices to iteratively adjust to different exemplars to find their most suitable poses.
E6 commented that the design of VCoach is similar to the idea of the “champion model” for elite athletes, such as
Su Bingtian, who was trained by shortening the gaps (on both poses and capabilities) with elite exemplars.
This comment is consistent with E3’s advice in the formative study.
We also invited experts to comment on the positioning of VCoach in training in real life.
E5: “It is suitable for the majority of ordinary runners.
But for severely over-weight people, asking them to resemble the running of ordinary people might cause injury instead of reducing it; they should seek for professional advice instead.”
E6 suggested that if the athletes’ parameters (mainly including height, leg lengths and training years) in the videos are accessible, it would be helpful to also suggest exemplars to users according to the similarity in these parameters, since runners with similar body configurations are more likely to have similar suitable running poses.
8 Conclusion and Future Work
We have presented a novel system, VCoach, for assisting amateur runners in improving their running poses.
We designed the system based on the design requirements formed from the literature research and expert interviews.
VCoach embeds common running pose attributes based on a collected corpus, and also provides an interface for users to customize attributes.
VCoach analyzes the poses from a sample video and an exemplar video in 3D, and visualizes the pose differences via 3D animations on a human body model.
Our user study showed that demonstrating pose corrective feedback via 3D animations is more effective than displaying frames side-by-side or overlaying the correct poses onto the sample frames.
There are several limitations and possible future work directions for VCoach.
In the current setting the running pose attributes are analyzed and visualized independently.
But there are certain correlations among the attributes, e.g., a higher knee lift might yield a larger stride.
A potential improvement is to incorporate human body harmonics [23, 19] to further summarize the problematic attributes.
Besides, in our user study we mainly evaluated the effectiveness of the visualization in VCoach in providing intuitive pose correction feedback.
It would be meaningful to conduct a long-term user study with participants from running backgrounds to further evaluate the effectiveness of VCoach in promoting running forms in practice.
Finally, currently VCoach focuses on the kinematics measurements (e.g., angles and positions).
However, more professional analysis [51] would require kinetics measurements, such as ground reaction force (braking force) [55] and muscle elastic energy [39].
Since the measure of kinetics parameters is currently limited to biomechanics laboratories, developing methods that recover the kinetics from videos would increase accessibility to many fields, including but not limited to sports posture analysis.
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Pathways for virus assembly around nucleic acids
Jason D Perlmutter
Matthew R Perkett
Michael F Hagan
[email protected]
Martin Fisher School of Physics, Brandeis University,
Waltham, MA, USA.
\PrependGraphicsExtensions
*.pdf,.PDF
I Abstract
Understanding the pathways by which viral capsid proteins assemble around their genomes could identify key intermediates as potential drug targets. In this work we use computer simulations to characterize assembly over a wide range of capsid protein-protein interaction strengths and solution ionic strengths. We find that assembly pathways can be categorized into two classes, in which intermediates are either predominantly ordered or disordered. Our results suggest that estimating the protein-protein and the protein-genome binding affinities may be sufficient to predict which pathway occurs. Furthermore, the calculated phase diagrams suggest that knowledge of the dominant assembly pathway and its relationship to control parameters could identify optimal strategies to thwart or redirect assembly to block infection. Finally, analysis of simulation trajectories suggests that the two classes of assembly pathways can be distinguished in single molecule fluorescence correlation spectroscopy or bulk time resolved small angle x-ray scattering experiments.
II Introduction
In many virus families, the spontaneous assembly of a protein shell (capsid) around the viral genome is an essential step in the viral life cycle Hagan2014 . These families include most viruses with single stranded RNA (ssRNA) genomes, as well as the Hepadnaviridae (e.g. hepatitis B virus, HBV). Understanding the mechanisms which underlie this cooperative assembly process could facilitate efforts to develop anti-viral drugs that block or derail the formation of infectious particles (for reviews see Zlotnick2011 ; Prevelige2011 ) and promote efforts to reengineer them for biomedical delivery. In this article, we explore how the interactions between the molecular components determine the mechanism of assembly, and how these interactions can be altered by changing solution conditions or mutagenesis to modulate assembly pathways.
The most detailed knowledge of capsid-RNA interactions comes from structural analysis of assembled viral particles. Atomic resolution structures of capsids assembled around ssRNA have been obtained by x-ray crystallography and/or cryo-electron microscopy (cryo-EM) (e.g. Fox1994 ; Valegard1997 ; Johnson2004 ; Tihova2004 ; Krol1999 ; Stockley2007 ; Toropova2008 ; Lucas2002 ; Valegard1994 ; Worm1998 ; Grahn2001 ; Valegard1997 ; Helgstrand2002 ; Schneemann2006 ).
The packaged NAs are less ordered than their protein containers and thus have been more difficult to characterize. However cryo-EM experiments have identified that the nucleotide densities are nonuniform, with a peak near the inner capsid surface and relatively low densities in the interiorTihova2004 ; Zlotnick1997 ; Conway1999 . While atomistic detail has not been possible in these experiments, all-atom models have been derived from equilibrium simulations Freddolino2006 ; Devkota2009 ; Zeng2012 . In some cases, striking image reconstructions reveal that the packaged RNA adopts the symmetry of the overlying capsid (e.g. Tihova2004 ; Toropova2008 ; Schneemann2006 ; Bakker2012 ; Ford2013 ). While it has been proposed that this order arises as a function of the assembly mechanism for several viruses Larson2001 ; Dykeman2011 ; Dykeman2013 , computational analysis of polyelectrolyte configurations inside capsids also indicate that capsid-polymer interactions can generically drive spatial organization of the packaged polymer Siber2008 ; Hu2008c ; Devkota2009 ; Forrey2009 ; Harvey2009 ; Belyi2006 ; Angelescu2006 ; Zhang2004a ; Lee2008 ; Angelescu2008 ; Jiang2009 ; Elrad2010 ; Perlmutter2013 . Theoretical works have also characterized the relationship between the NA charge and structure and the length which is optimal for packaging Perlmutter2013 ; Schoot2013 ; Hu2008a ; Belyi2006 ; Schoot2005 ; Angelescu2006 ; Siber2008 ; Ting2011 ; Ni2012 ; Siber2010 ; Siber2012 .
In addition to this structural data on assembled capsids, an extensive combination of mass spectrometry, assembly kinetics experiments, constraints from assembled capsid structures, and mathematical modeling has delineated assembly pathways for several viruses, with a particular focus on the role of interactions between CPs and specific RNA sequences called packaging signals. Recent single molecule fluorescence correlation spectroscopy (smFCS) experiments indicate that, for these viruses, assembly around the viral genome is more robust and proceeds by a different mechanism as compared to around heterologous RNA Borodavka2012 . Yet, in other cases capsid proteins show no preference for genomic RNA over heterologous RNA (e.g. HBV Porterfield2010 ), and cowpea chlorotic mottle virus (CCMV) proteins preferentially encapsidate heterologous RNA (from BMV) over the genomic CCMV RNA with equivalent length Comas-Garcia2012 . Furthermore, experimental model systems in which capsid proteins assemble into icosahedral capsids around synthetic polyelectrolytes or other polyanions Hiebert1968 ; Bancroft1969 ; Dixit2006 ; Loo2006 ; Goicochea2007 ; Huang2007 ; Loo2007 ; Sikkema2007 ; Sun2007 ; Hu2008 ; Comellas-Aragones2009 ; Crist2009 ; Chang2008 ) demonstrate that specific RNA sequences are not required for capsid formation or cargo packaging.
Thus, a complete picture of capsid assembly mechanisms requires understanding how assembly pathways depend on those features which are generic to polyelectrolytes, as well as those which are specific to viral RNAs.
In previous work on assembly around a simple model for a polymer, Elrad and Hagan Elrad2010 proposed that mechanisms for assembly around a cargo (i.e. RNA, polymer, or nanoparticle) can be classified on the basis of two extreme limits. In the first (originally proposed by McPherson McPherson2005 and then Refs Hagan2008 ; Devkota2009 ), strong protein-cargo interactions drive proteins to adsorb ‘en masse’ onto the cargo in a disordered manner, meaning there are few protein-protein interactions. Once enough subunits are bound, subunits undergo cooperative rearrangements (potentially including dissociation of excess subunits) to form an ordered capsid. This mechanism has been observed in recent simulations Mahalik2012 ; Perlmutter2013 ; Elrad2010 ; Hagan2008 ; Perkett2014 . In the second limit, where protein-protein interactions dominate, a small partial capsid nucleates on the cargo, followed by a growth phase in which individual proteins or small oligomers sequentially add to the growing capsid. This class of pathways resembles the nucleation-and-growth mechanism by which empty capsids assemble Endres2002 , except that the polymer plays an active role by stabilizing protein-protein interactions and by enhancing the flux of proteins to the assembling capsid Hu2007 ; Kivenson2010 ; Elrad2010 .
It is difficult to determine assembly mechanisms directly from experiments, due to the small size ($\lesssim 10$ of nm) and transience ($\sim$ms) of most intermediates. Observations in vitro suggest that both mechanisms may be viable. Kler et al. Kler2012 used time-resolved X-ray scattering (trSAXS) to monitor SV40 capsid proteins assembling around ssRNA. The scattering profiles at all time points during assembly could be decomposed into unassembled components (RNA + protein subunits) and complete capsid; the absence of any signal corresponding to a large disordered intermediate suggests this assembly follows the nucleation-and-growth (ordered) assembly mechanism Kler2012 .
Other observations suggest that viruses can assemble through the en masse mechanism. Refs. Garmann2013 ; Cadena-Nava2012 found that in vitro assembly CCMV assembly was most productive when performed in two steps: (1) at low salt (strong protein-RNA interactions) and neutral pH (weak protein-protein interactions) the proteins undergo extensive adsorption onto RNA, then (2) pH is reduced to activate binding of protein-protein binding Garmann2013 . Similarly, a recent observation of capsid protein assembly around charge-functionalized nanoparticles found that assembly initially proceeded through nonspecific aggregation of proteins and nanoparticles, followed by the gradual extrusion of nanoparticles within completed capsids Malyutin2013 . These experiments used viral proteins with relatively weak protein-protein interactions (CCMV and BMV) Zlotnick2013 and moderate salt concentrations ($100-150$mM). The Kler et al. Kler2012 ; Kler2013 experiments considered SV40 proteins, which have strong protein-protein interactions Zlotnick2013 , and high salt ($250$mM) Together, these in vitro experiments suggest that productive assembly could proceed by either the en masse or nucleation-and-growth mechanism.
In this work, we use dynamical simulations to investigate the extent to which the assembly mechanism can be controlled by tuning solution ionic strength and protein-protein attractions. We extend a model that was recently used to calculate the thermostability and assembly yields of viral particles as a function of protein charge and nucleic acid (NA) length and structure. Those previous simulations found quantitative agreement between predicted NA lengths that optimize capsid thermostability and viral genome length for seven viruses Perlmutter2013 . Here, we perform extensive new simulations of assembly, in which protein-protein interactions, the sequence of charges in capsid protein-NA binding domains, and the solution ionic strength are varied. We find that, by varying these control parameters, the assembly mechanism can be systematically varied between the two extreme limits described above. Our results suggest that knowledge of protein-protein and protein-NA binding affinities may be sufficient to predict which assembly mechanism will occur, and we estimate relative protein-NA binding interactions for three viruses (based on non-specific electrostatic interactions). These findings suggest that assembly mechanisms can be rationally designed through choice of solution conditions and mutagenesis of capsid protein-protein interfaces and protein-NA binding domains.
Finally, by calculating hydrodynamic radii and SAXS profiles associated with assembly intermediates, we show that assembly mechanisms can be distinguished by experimental techniques recently applied to virus assembly, respectively smFCS Borodavka2012 ) and trSAXS Kler2012 . While the NA is represented by a linear polyelectrolyte in most of the simulations, we obtain qualitatively similar results when considering a model for base-paired NAs developed in Ref. Perlmutter2013 .
III Results
To study how capsid assembly around a polyelectrolyte depends on the strength of protein subunit-subunit and subunit-polyelectrolyte interactions, we performed Brownian dynamics simulations with a recently developed model Perlmutter2013 (Figure 1).
This model was motivated by the observation Kler2012 ; Kler2013 that purified simian virus 40 (SV40) capsid proteins assemble around ssRNA molecules to form capsids composed of 12 homopentamer subunits. The capsid is modeled as a dodecahedron composed of 12 pentagonal subunits (each of which represents a rapidly forming and stable pentameric intermediate, which then more slowly assembles into the complete capsid, as is the case for SV40 in vitro around ssRNA Li2002 ; Kler2012 ; Kler2013 ). Subunits are attracted to each other via attractive pseudoatoms at the vertices (type ‘A’) and driven toward a preferred subunit-subunit angle by repulsive ‘Top’ pseudoatoms (type ‘T’) and ‘Bottom’ pseudoatoms (type ‘B’) (see Fig. 1 and the Methods section). These attractions represent the interactions between capsid protein subunits that arise from hydrophobic, van der Waals, and electrostatic interactions as well as hydrogen bonding Hagan2014 ). The magnitude of these interactions varies between viral species Zlotnick2013 and can be experimentally tuned by pH and salt concentration Ceres2002 ; Kegel2004 ; Hagan2014 . Here, the attraction strength is controlled by the model parameter $\varepsilon_{\text{ss}}$. In order to relate this potential energy to the free energy of dimerization, we have run a separate series of calculations, where in the absence of cargo, we find that the free energy of subunit of dimerization is $g_{\text{ss}}{=}5.0-1.5*\varepsilon_{\text{ss}}$ (see SI section 12). Throughout this article energies are given in units of the thermal energy, $k_{\mathrm{B}}T$.
Capsid assembly around nucleic acids and other polyelectrolytes is primarily driven by electrostatic interactions between negative charges on the encapsulated polyelectrolyte and positive charges on the inner surfaces of capsid proteins Hagan2014 . We consider a linear bead-spring polyelectrolyte, with a charge of $âe$ per bead and a persistence length comparable to that of ssRNA in the absence of base pairing. Positive charges on capsid proteins are located in flexible polymers affixed to the inner surface of the model subunit, which represent the highly charged, flexible terminal tails known as arginine rich motifs that are typical of positive-sense ssRNA virus capsid proteins (e.g., Schneemann2006 ).
Electrostatics are modeled using Debye-Huckel interactions, where the Debye screening length ($\lambda_{\text{D}}$) is determined by the ionic strength $I$, equivalent to the concentration $C_{\text{salt}}$ of a monovalent salt, as $\lambda_{\text{D}}\approx 0.3/I^{1/2}$ with $\lambda_{\text{D}}$ in nm and $I$ in molar units. Perlmutter et al. Perlmutter2013 showed that Debye-Huckel interactions compare well to simulations with explicit counterions for the parameter values under consideration; furthermore, inclusion of multivalent ions at experimentally relevant concentrations had only a small effect on properties such as the optimal genome length. The effect of base-pairing on the optimal length was also considered in that reference Perlmutter2013 .
III.1 Kinetic phase diagram
We first consider the predominant assembly products (Fig. 2) and the packaging efficiency (Fig. 3) as a function of Debye length $\lambda_{\text{D}}$ and subunit-subunit interaction strength $\varepsilon_{\text{ss}}$. The packaging efficiency is defined as the fraction of trajectories in which the polyelectrolyte is completely encapsulated by a well-formed capsid, which contains 12 subunits each of which interact with 5 neighbors. We refer to this as a kinetic phase diagram Hagan2006 ; Hagan2014 because we characterize products at a finite observation time of $t_{\text{f}}{=}2\times 10^{8}$ time steps, which is long enough that assemblages do not vary significantly with time (except for under weak interactions, see below), but is not sufficient to equilibrate kinetic traps if there are large activation barriers Elrad2010 ; Hagan2014 . We see that for the range of simulated ionic strengths ($1\leq C_{\text{salt}}\leq 500$ mM or $10\geq\lambda_{\text{D}}\geq 0.4$ nm) assembly yields are highest for $\varepsilon_{\text{ss}}{=}5k_{\mathrm{B}}T$ and $C_{\text{salt}}{=}100$ mM (the parameter values focused on in Perlmutter2013 ), and that for moderate subunit-subunit interaction strengths ($4\leq\varepsilon_{\text{ss}}\leq 6,k_{\mathrm{B}}T$) yield to remain high as the ionic strength is increased to about 300 mM ($\lambda_{\text{D}}\approx 0.6$ nm). For higher salt concentrations, yields are depressed by the appearance of long-lived on-pathway intermediates. As will be discussed further below, weakening the electrostatic interaction between the polymer and protein limits the ability of the polymer to promote assembly. Although we expect that these simulations would eventually result in complete capsids, the low yield at our finite measurement time reflects the fact that assembly is less efficient than for lower salt concentrations. At the highest ionic strength considered ($C_{\text{salt}}{=}500$ mM), the most prevalent outcome is that no nucleation occurs. At lower salt concentrations ($C_{\text{salt}}\geq 10$ mM), rapid adsorption of a super-stroichiometric quantity of subunits results in malformed capsids.
At larger-than-optimal protein-protein interaction strengths ($\varepsilon_{\text{ss}}>6$) assembly yields are lower for two reasons. The first, and more frequent outcome, is long-lived malformed structures with strained interactions. This kinetic trap arises in a wide variety of assembly systems when interactions become strong in comparison to the thermal energy, because strained interactions are unable to anneal before additional subunits lock them in place Hagan2014 ; Hagan2011 ; Grant2011 . In our simulations, we found that these structures frequently result from an incorrect merger between two partial capsid intermediates; even when each individual interaction is too weak to lock-in the non-optimal structure, multiple erroneous interactions formed by two partial capsids are effectively irreversible on our timescale. The tendency for oligomer binding to lead to malformed structures was seen previously in the context of empty capsid assembly Hagan2006 ; Whitelam2009 . Here, the polymer helps to bring oligomers together, and thus this trap arises when nucleation on the polymer is faster than growth of a nucleus into a complete capsid. This trap resembles Geminivirus particles, which are composed of a union of two nearly complete capsids Bottcher2004 .
The second obstacle to polyelectrolyte encapsulation arises at the highest protein-protein interaction strengths studied ($\varepsilon_{\text{ss}}\geq 8k_{\mathrm{B}}T$), for which subunits not associated with the polyelectrolyte undergo spontaneous assembly. The resulting off-polyelectrolyte assembly depletes the pool of available monomers and small oligomers available for assembly on the polyelectrolyte, leading to a form of the monomer-starvation kinetic trap previously discussed for empty capsid assembly Hagan2014 ; Endres2002 . Triggering formation of empty capsids and thus preventing nucleic acid encapsidation by strengthening subunit-subunit interactions has been suggested as a mode of action for a putative antiviral drug for hepatitis B virus Katen2010 ; Katen2013 .
At smaller-than-optimal protein-protein interaction strengths ($\varepsilon_{\text{ss}}<4$) assembly is unsuccessful for two reasons, depending upon the ionic strength. At smaller values ($C_{\text{salt}}\leq 300$ mM), electrostatic interactions are relatively strong, and many proteins adsorb onto to the polymer. However, because of the weak protein-protein interaction strength, these proteins do not form stable capsids, predominantly because nucleation is slow in comparison to $t_{\text{f}}$ (the final observation time, $t_{\text{f}}{=}2\times 10^{8}$ time steps). In a minority of cases, a nucleus will form, but completion is prevented by the excess number of subunits adsorbed to the polyelectrolyte.
We refer to the resulting configurations as disordered, due to the lack of ordered binding between protein subunits. At larger ionic strengths ($C_{\text{salt}}>300$ mM) electrostatic interactions are relatively weak, and individual subunits rapidly desorb from the polyelectrolyte. In this regime, assembly requires a fluctuation in the number of adsorbed subunits which leads to nucleation of a partial capsid intermediate which has enough subunit-polyelectrolyte interactions to be stable against rapid desorption. The nucleation rate decreases exponentially with subunit-subunit interaction strength Hagan2014 ; Kivenson2010 , and thus most simulations at high salt and weak subunit-subunit interactions never undergo nucleation. (We categorize simulations with fewer than 3 subunits adsorbed to the polyelectrolyte and no subunit-subunit interactions (i.e. no progress towards assembly) as ‘unnucleated’.)
Importantly, we expect that trajectories in this region of parameter space will eventually undergo nucleation. Thus, as the finite assembly time $t_{\text{f}}$ is increased the region of successful assembly will expand to lower values of $\varepsilon_{\text{ss}}$ and higher ionic strength, until eventually reaching values below which capsid assembly is thermodynamically unstable (see Hagan2006 , Fig. 7). To confirm this possibility, we used the Markov state model (MSM) approach described in Ref. Perkett2014 to characterize assembly with $\varepsilon_{\text{ss}}{=}5,6$ and $C_{\text{salt}}{=}500$ mM (see Figs. 4C and S4 (in the SI)). In contrast, the malformed capsids encountered under large parameter values typically will not resolve on any relevant timescales, since numerous strong interactions would need to be broken Hagan2014 . Thus, the boundaries of successful assembly at large $\varepsilon_{\text{ss}}$ values are insensitive to $t_{\text{f}}$.
Several additional conclusions can be drawn from the variation of packaging efficiency (Fig. 3). Firstly, the yield of complete virus-like particles is relatively high for moderate subunit-subunit interaction strengths $\varepsilon_{\text{ss}}\in[4,6]$ across a range of ionic strengths ($C_{\text{salt}}\in[100,300]$ mM). Even above ($C_{\text{salt}}>300$ mM) and below ($C_{\text{salt}}{=}10$ mM) this range we observe moderate yields of complete particles. Secondly, as electrostatic interactions are weakened (moving to the left), the subunit-subunit interaction strength which optimizes the yield increases (i.e. from $\varepsilon_{\text{ss}}=5k_{\mathrm{B}}T$ at $C_{\text{salt}}=100mM$ to $\varepsilon_{\text{ss}}=6k_{\mathrm{B}}T$ at $C_{\text{salt}}:150-400mM$ to $\varepsilon_{\text{ss}}=7k_{\mathrm{B}}T$ at $C_{\text{salt}}=500mM$). This result suggests that one interaction type can compensate for the other within a limited range. However, though all successful capsids contain the same subunit geometry, the mechanism by which they form depends on the relative interaction strengths, as discussed next.
III.2 Assembly mechanisms
As noted previously Hagan2008 ; Elrad2010 ; Hagan2014 , pathways for assembly around a central core such as a polyelectrolyte can be roughly separated into two classes. In the first class (Fig. 4a), which we refer to as the ‘en masse’ mechanism, subunits first adsorb onto the polyelectrolyte in a disordered manner, followed by cooperative rearrangements to form an ordered capsid. In the second class (Fig. 4b), referred to as the nucleation-and-growth mechanism, a small partial capsid nucleates on the polyelectrolyte followed by the sequential, reversible addition of subunits or small oligomers until assembly completes. In contrast to the earlier models which considered a qualitative subunit-polyelectrolyte interaction, we study here how assembly pathways depend on the ionic strength.
To quantify the degree of order along assembly pathways, we record the total number of subunits adsorbed to the polyelectrolyte $n_{\text{ad}}$ and the number of subunits in the largest cluster $n$. Trajectories that pass through configurations with a large value of $n_{\text{free}}=n_{\text{ad}}-n$ are disordered, with many adsorbed subunits not participating in ordered assemblies. In Fig. 4, these quantities are shown as a function of time averaged over all simulation trajectories (leading to successful assembly or not), for parameter sets that respectively lead to the en masse mechanism (Fig. 4A) and nucleation-and-growth mechanism (Fig. 4B). In the first case, there are strong subunit-polyelectrolyte interactions (low ionic strength, $C_{\text{salt}}{=}100$ mM) and weak subunit-subunit interactions ($\varepsilon_{\text{ss}}{=}3k_{\mathrm{B}}T$). Subunits therefore initially adsorb nonspecifically and form only transient subunit-subunit interactions, leading to a rapid rise in $n_{\text{ad}}$ with $n\approx 0$. Once enough subunits are adsorbed ($\sim 12$ around an optimal-length polyelectrolyte for this model with a 12-subunit capsid), a cooperative fluctuation in subunit configurations eventually leads to a stable nucleus and then rapid completion of the ordered capsid geometry. Since this nucleation process is stochastic, there is a distribution of waiting times and thus a more gradual increase in the average cluster size $n$ (see Fig. S2). In the nucleation-and-growth case, on the other hand, the subunit-polyelectrolyte interactions are weak ($C_{\text{salt}}{=}300$ mM) and the subunit-subunit interactions are strong ($\varepsilon_{\text{ss}}{=}6k_{\mathrm{B}}T$). There is limited nonspecific subunit adsorption onto the polyelectrolyte, adsorbed subunits form relatively strong associations, and thus $n_{\text{ad}}$ and $n$ increase nearly in lockstep. Snapshots from typical trajectories for each of these parameter sets are shown in Fig. 4A.
To visualize the degree of order as a function of parameter values, we define a trajectory-averaged order parameter $\bar{n}_{\text{free}}$, which is $n_{\text{free}}$ averaged over all configurations with $4\leq n\leq 6$ and over all trajectories at a given parameter setElrad2008 . Large values of this parameter ($\bar{n}_{\text{free}}\gtrsim 5$) indicate the en masse mechanism, while small values ($\bar{n}_{\text{free}}\lesssim 2$) indicate the nucleation-and-growth mechanism. As shown in Fig. 5, the degree of order generally increases with ionic strength and subunit-subunit interaction strength, with the most ordered assembly occurring at $C_{\text{salt}}=500$ mM (where on average fewer than one subunit is adsorbed nonspecifically) and $\varepsilon_{\text{ss}}\geq 6k_{\mathrm{B}}T$. However, notice that for $\varepsilon_{\text{ss}}=3$ assembly is always disordered; for such weak subunit-subunit interactions the critical nucleus size is large and a high density of adsorbed subunits is required to achieve nucleation. On the other hand, for moderate subunit-subunit interactions we do not observe the extreme limit of the en masse mechanism even for low ionic strength. Though a low ionic strength drives strong nonspecific subunit adsorption, absorbed subunits collide frequently due to cooperative polymer motions and subunit sliding along the polymer Hu2007 ; Elrad2010 . For $\varepsilon_{\text{ss}}>3$ absorbed subunits therefore achieve nucleation before nonspecific absorption has time to saturate.
The nucleation-and-growth trajectories can be further classified based on the relative timescales of nucleation and growth. When nucleation is slow compared to growth (Figs. 4C and S4), the reaction is effectively two-state — each critical nucleus rapidly proceeds to completion, leading to low concentrations of intermediates which are essentially undetectable in a bulk experiment. When nucleation and growth timescales are comparable, multiple capsids within a spatial region can be assembling simultaneously, and thus potentially could be detected in bulk experiments. Below, we consider whether it is possible to experimentally distinguish between the latter case, ordered assembly with rapid nucleation, and en masse assembly pathways characterized by disordered intermediates.
Biological ARM sequences. The assembly simulations described above consider a simplified peptide ARM, which contains only 5 positive ARM charges (Fig. 1). However, our previous work demonstrated that the ARM amino acid sequence (specifically the charges) can significantly affect binding to a polyelectrolyte Perlmutter2013 . Furthermore, results from a more simplified model suggested that the polyelectrolyte binding affinity is a determining factor for the assembly mechanism Elrad2010 . To test this hypothesis and to place our results in the context of specific viruses, we calculated the polyelectrolyte binding affinity for the simplified ARM and three ARM sequences from biological viruses (SV40, BMV, PC2), with each amino acid classified as neutral, cationic, or anionic. These calculations were performed by setting the subunit-subunit interaction strength to zero and recording the average number of adsorbed subunits at varying $C_{\text{salt}}$ (or Debye length $\lambda_{\text{D}}$).
The measured equilibrium linear densities of adsorbed subunits, $c_{\text{eq}}$, are shown as a function of $C_{\text{salt}}$ in Fig. 6A-C. In all cases the linear density increases monotonically with decreasing $C_{\text{salt}}$; saturating at a maximum density.
The simplified ARM has the largest binding affinity, despite having the lowest net positive charge ($+5$, 0 neutral segments) of the four species considered. Comparison with SV40 ($+6$, 14 neutral segments) and BMV ($+9$, 33 neutral segments) illustrates the expected result that that neutral segments decrease the binding affinity, particularly at high salt. The PC2 subunits, with net charge $+22$ demonstrate markedly different behavior, with significant subunit absorption at the highest salt concentration simulated (500 mM), but saturating at about 300 mM ionic strength due to subunit-subunit charge repulsions. Variations in adsorption density with subunit concentration are shown in Fig. S1.
In Fig. 6D we connect these measures of adsorption with assembly mechanism by plotting the assembly order parameter $\bar{n}_{\text{free}}$ as a function of $c_{\text{eq}}$ (controlled by varying $C_{\text{salt}}$), for several values of the subunit dimerization free energy (determined by $\varepsilon_{\text{ss}}$ and $C_{\text{salt}}$) for the 5-ARM and SV40 models. Plotting against $c_{\text{eq}}$ rather than $C_{\text{salt}}$ ($\lambda_{\text{D}}$) allows us to overlay data from these two models while accounting for the differences in affinity due to ARM sequence described above. In support of the proposed link between binding affinity and assembly mechanism, we find rough agreement in the measured assembly order parameters between the two models.
The results indicate that ARM sequence can significantly influence the assembly mechanism. For example, if we define $\bar{n}_{\text{free}}\leq 2$ as the nucleation-growth mechanism, Fig. 6D indicates that nucleation-growth occurs for $c_{\text{eq}}\leq c_{\text{eq}}^{*}$ with the threshold value $c_{\text{eq}}^{*}=0.0375$ for $\varepsilon_{\text{ss}}\geq 2.5k_{\mathrm{B}}T$. From Fig. 6A-C, we can then identify the threshold values of ionic strength $C_{\text{salt}}^{*}$, above which the nucleation-growth mechanism will occur: $C_{\text{salt}}^{*}\approx 300$ mM for the 5-ARM and $C_{\text{salt}}^{*}\approx 175$ mM for the BMV and SV40 models; while PC2 is below the threshold value for all salt concentrations considered. This allows us to predict, for example, that recent experiments on SV40 assembly (at $C_{\text{salt}}{=}250$mM and observed strong subunit-subunit attraction) would have a very low $\bar{n}_{\text{free}}$ ($\sim 1$), which is consistent with SAXS observations Kler2012 .
III.3 Experimental Observables
We now seek to provide experimental signatures through which the capsid assembly pathways discussed above can be distinguished. We focus on two experimental techniques which have recently been applied to respectively probe the formation of individual capsids and bulk assembly kinetics.
smFCS measurements on individual capsids can distinguish assembly mechanisms. Borodavka et al. Borodavka2012 used single molecule fluorescence correlation spectroscopy (smFCS) to monitor the hydrodynamic radii $R_{\text{H}}$ of nucleocapsid complexes during assembly of MS2 and STNV capsid proteins around cognate RNA or non-cognate RNA. Assembly around cognate RNA was characterized by either constant $R_{\text{H}}$ or, in some trajectories, a collapsed complex followed by gradual increase until reaching the size of an assembled capsid. In contrast, assembly around non-cognate RNA led to an increase in $R_{\text{H}}$ before eventually decreasing to the size of the capsid. The difference between these two behaviors can be attributed to sequence-specific ‘packaging signals’ on cognate RNA that interact with the capsid proteins. In this article we do not consider the effect of packaging signals (these will be considered in a subsequent article); instead, we consider whether the pathways described in the previous section can be distinguished by this experimental technique.
We estimated the hydrodynamic radii $R_{\text{H}}$ for polymer-subunit intermediates using the program HYDROPRO, which has been shown to accurately predict $R_{\text{H}}$ for large protein complexes Ortega2011 . The resulting $R_{\text{H}}$ are shown during representative en masse and ordered assembly trajectories in Fig. 7E. We see that the complex $R_{\text{H}}$ first increases as subunits adsorb onto the complex and then decreases as subunits assemble (Fig. 7E). However, the en masse mechanism leads to a much larger and longer duration increase in $R_{\text{H}}$, due to the extensive and long-lived disordered adsorption of unassembled subunits. The difference in $R_{\text{H}}$ between en masse and ordered trajectories is conserved across their respective parameter ranges (see SI Fig. S3 for other trajectories), and also occurs for assembly trajectories around the model nucleic acid with intramolecular base- pairing developed in Ref. Perlmutter2013 (see SI Fig. S3C,F). These results suggest that smFCS can distinguish among the classes of assembly pathways observed in our simulations. They are consistent with an interpretation of the Borodavka et al. Borodavka2012 results in which assembly around the non-cognate RNA proceeds via the disordered mechanism while packaging signals lead to an ordered mechanism.
To further characterize the differences in polymer conformations between disordered and ordered assembly pathways, we show the polymer radius of gyration, $R_{\text{g}}$, during assembly trajectories in Fig. 7. In contrast to $R_{\text{H}}$, contributions of the capsid proteins are not included in $R_{\text{g}}$. While the results in Fig. 7 are averaged over multiple trajectories, example individual trajectories are shown in SI Fig. S2. In all cases of successful assembly, the polymer is gradually compacted from its free size into its encapsidated size. However, at fixed $C_{\text{salt}}{=}100$mM, the average rate of compaction increases with $\varepsilon_{\text{ss}}$, with a dramatic increase in rate for $\varepsilon_{\text{ss}}>3k_{\mathrm{B}}T$ (Fig. 7A). Similarly, decreasing $C_{\text{salt}}$ increases the rate of compaction (Fig. 7B). Notice that the rate of polymer compaction is not determined by the assembly mechanism — increased order correlates with faster compaction in Fig. 7A but with slower compaction in Figure 7B. When the $R_{\text{g}}$ is plotted as a function of number of adsorbed subunits ($n_{\text{ad}}$), the en masse and ordered pathways clearly split into two groups (Fig. 7C). However, this distinction disappears when $R_{\text{g}}$ is plotted as a function of the number of subunits in the largest cluster ($n$, Fig. 7D). Taken together, these data demonstrate that polymer compaction is driven by adsorbed subunits forming ordered intermediates, with the rate of compaction consequently mirroring the rate of capsid assembly.
trSAXS measurments of bulk assembly kinetics can distinguish assembly mechanisms. While smFCS can detect individual assembly intermediates, Kler et al. Kler2012 ; Kler2013 recently used time resolved small angle x-ray scattering (trSAXS) to elucidate structures of assembling capsids in bulk. They found that the SAXS profiles at all time points could be decomposed into scattering contributions from the unassembled and complete components, suggesting that assembly proceeded by an effectively two-state reaction with undetectable levels of intermediates (i.e. the nucleation-and-growth pathway with relatively slow nucleation). While it is evident that profiles from profiles from a two-state reaction can be decomposed in this way, we investigated the extent to which SAXS profiles from the other pathway classes (en masse or nucleatation-and-growth with rapid nucleation) can be distinguished from the two-state case.
First, Fig. 8A,B shows SAXS profiles calculated (using CRYSOL Svergun1995 ) from simulation snapshots along ensembles of en masse and ordered assembly trajectories. For each parameter set, SAXS profiles are averaged over 6 time windows. In both cases, the first profile is dominated by scattering from free subunits (Fig. 8C) and the final time segment shows clear minima and maxima corresponding to the complete capsid (Fig. 8C). For comparison, Fig. 8C presents the SAXS profiles for ordered subunit clusters ranging in size from a single subunit (black) to a complete capsid (yellow). As the capsid grows distinct minima and maxima appear and become more pronounced. We note that the positions of the minima and maxima in the complete model capsid are similar to those observed experimentally for SV40 Kler2012 .
To test the extent to which these trajectories can be distinguished from the two-state case, we attempted to fit SAXS profiles using a linear combination of the profiles for unassembled polymer and subunits and the complete capsid. The resulting fits for the second fifth of the trajectory (where intermediates are most plentiful) are shown in Figs. 8D–F. At this stage, the ordered systems contain mostly incomplete subunit clusters (6-11 subunits), while the disordered simulations contain mostly adsorbed but unassembled subunits. For the ordered simulations, we find that the fit reproduces all of the features of the intermediates, except at low q. In contrast, the intermediates in the disordered trajectory display a shoulder at $q\sim 0.3nm^{-1}$ that is not observed in any of the unassembled or assembled components. This shoulder is a distinct feature of the disordered intermediate, and thus could be used to identify this class of pathways. Finally, as expected, SAXS profiles from trajectories at parameter sets (Fig. 4C) which lead to two-state kinetics are very well fit by a linear combination of polymer/subunit and complete capsid (Fig. 8F).
A significant distinction between the SAXS profiles is that the ordered pathways lead to an isosbestic point at $q\sim 0.3nm^{-1}$ (as seen in SV40 experimentsKler2012 ), whereas the disordered pathways do not. The presence of an isosbestic point is frequently assumed to indicate two-state behaviour; however, it occurs in our data for ordered trajectories even when the reaction is far from two-state due to rapid nucleation. In these cases the isosbestic point appears due to due to the similarity in scattering from the ordered intermediates and the complete capsid. This suggests that an isosbestic point may distinguish ordered from disordered assemblies, but is less sensitive to the extent to which the reaction kinetics can be approximated as two-state (i.e., how undetectable the intermediate concentrations are).
IV Discussion
Our simulations demonstrate that capsid assembly around a NA or other cargo can proceed through two mechanisms, and that which mechanism occurs depends on the relative strengths of protein-protein and protein-cargo interactions. The assembly mechanism can be rationally designed by tuning these interactions through solution conditions or mutagenesis of capsid protein-NA binding domains. However, because productive assembly requires weak interactions, the parameters must be tuned within relatively narrow ranges, and alterations which increase (decrease) the strength of one type of interaction must be compensated by a decrease (increase) in the other. Our results suggest that the subunit-cargo dissociation constant is an important parameter, whose value might be used to map specific viruses onto our phase diagrams (Figs. 2, 3 ,5), although experimental tests of this capability are required. Finally, we have shown that the nature of assembly pathways can be inferred from the results of recently developed techniques to monitor the assembly of individual capsids or bulk assembly kinetics.
Our simulations predict that a single viral species can be driven to assemble via different mechanisms in vitro by changing solution conditions. In particular, under a constant subunit-subunit binding energy ($\varepsilon_{\text{ss}}\sim 5-6k_{\mathrm{B}}T$) robust assembly occurs for a range of solvent conditions, with highly ordered or disordered assemblies occurring depending on salt concentration ($C_{\text{salt}}\in[50,400]$ mM). To our knowledge, this prediction has not yet been realized experimentally, although the signatures of the two classes of assembly pathways have been seen in experiments on different viruses and/or different cargoes Kler2012 ; Garmann2013 ; Cadena-Nava2012 ; Malyutin2013 ; Borodavka2012 .
One recent experimental study sought to test the role of of $C_{\text{salt}}$ and subunit-subunit attractions (controlled by solution $p$H) on in vitro assembly of CCMV around RNA Garmann2013 . In some regards, these experiments mirror our observations, with malformed capsids observed for strong subunit-subunit attractions, disordered products for weak subunit-subunit attractions, and well-formed capsids at intermediate conditions. However, robust assembly was only observed for a two-step process: first $C_{\text{salt}}$ was reduced (from $1M$) to drive RNA-subunit interactions, and then secondly $p$H was reduced to activate subunit-subunit attractions. The resulting assembly pathways resemble the en masse mechanism described here. On the other hand, one-step assembly led to malformed particles even at moderate $p$H, suggesting an inability to assemble through a nucleation-and-growth mechanism in these experiments. In our simulations, this outcome would only occur at high salt concentrations (e.g. $C_{\text{salt}}\sim 400-500$mM, see Figs. 2, 3), where the narrow range of $\varepsilon_{\text{ss}}$ leading to successful assembly indicates that parameters must be finely tuned. Reproducing such a lack of successful assembly at moderate salt concentrations would require a reduction in the orientation-dependence of the subunit-subunit interaction potential (see Methods), or introduction of additional factors as discussed below. Experiments in which solution conditions are changed for other viruses which do undergo one step assembly (e.g. SV40 Kler2012 ; Kler2013 ), may elucidate which of these possibilities to pursue and would directly test our prediction that the assembly mechanism can be controlled by solution conditions.
Understanding capsid assembly mechanisms and their location within the assembly phase diagram has important implications for the design of antiviral agents. As one example, we consider the recently developed class of HBV inhibitors based on phenylpropenamides Katen2010 ; Katen2013 , which act by increasing the strength of subunit-subunit interactions, driving subunits to assemble in the absence of their genome and thus increasing the generation of empty, non-infective capsids Katen2010 ; Katen2013 . Comparing Figs. 3 and 5 shows that a virus that undergoes ordered assembly (e.g. $\varepsilon_{\text{ss}}\sim 6-7$, $C_{\text{salt}}\sim 300$ mM), sits close to parameters that support empty capsid assembly, which are demarcated by dashed lines in Fig. 3. Thus only a small increase in subunit-subunit interactions is required to trigger unproductive RNA-free assembly. In contrast, a much larger perturbation would be required to achieve empty capsid assembly for a virus that assembles via the en masse mechanism.
Outlook.
We have described two classes of assembly pathways and several distinct failure modes (Fig. 2) that arise when assembly is driven by nonspecific electrostatic subunit-cargo interactions. Our phase diagrams can serve as a starting point to understand how virus-specific features, such as packaging signals Stockley2013 , allosteric NA-induced Kler2013 ; Stockley2007 or ‘autosteric’ protein-induced conformational changes Caspar1980 , base pairing-induced NA-structure Perlmutter2013 ; Yoffe2008 ; Borodavka2012 , or subcellular localization Bamunusinghe2011 can robustly avoid failure modes amidst the crowded intracellular milieu while enabling selective assembly around the viral genome Routh2012 ; Ford2013 ; Rao2006 . For example, allosteric or autosteric conformational changes may allow for strong subunit-subunit interactions on the NA while avoiding the off-cargo assembly we observe at large $\varepsilon_{\text{ss}}$. Systematically studying how these additional factors expand (or contract) regions of parameter space that lead to successful assembly will ultimately reveal how viruses have optimized their structures and interactions for robust assembly in vivo and how their assembly in vivo or in vitro can be manipulated for biomedical or materials science applications.
V Methods
Model. We have recently a presented a complete description of our model system, which we summarize here briefly Perlmutter2013 and in the SI. Our model subunits are based upon that previous used to simulate assembly of empty capsids Wales2005 ; Fejer2009 ; Johnston2010 which we extended previously to model assembly around cargo Perlmutter2013 . The pseudoatoms in the capsid subunit model are illustrated in Fig. 1. Subunit assembly is mediated through an attractive Morse potential between Attractor (‘A’) pseudoatoms located at each subunit vertex. The Top (‘T’) pseudoatoms interact with other ‘T’ psuedoatoms through a potential consisting of the repulsive term of the Lennard-Jones (LJ) potential, the radius of which is chosen to favor a subunit-subunit angle consistent with a dodecahedron (116 degrees). The Bottom (‘B’) pseudoatom has a repulsive LJ interaction with ‘T’ pseudoatoms, intended to prevent ‘upside-down’ assembly. The ‘T’, ‘B’, and ‘A’ pseudoatoms form a rigid body Wales2005 ; Fejer2009 ; Johnston2010 . See Refs. Schwartz1998 ; Hagan2006 ; Hicks2006 ; Nguyen2007 ; Wilber2007 ; Nguyen2008 ; Nguyen2009 ; Johnston2010 ; Wilber2009a ; Wilber2009 ; Rapaport1999 ; Rapaport2004 ; Rapaport2008 ; Hagan2008 ; Elrad2010 ; Hagan2011 ; Mahalik2012 ; Levandovsky2009 for related models.
To model electrostatic interaction with a negatively charged NA or polyelectrolyte we extend the model as follows. Firstly, to better represent the capsid shell we add a layer of ‘Excluder’ pseudoatoms which have a repulsive LJ interaction with the polyelectrolyte and the ARMs. Each ARM is modeled as a bead-spring polymer, with one bead per amino acid. The ‘Excluders’ and first ARM segment are part of the subunit rigid body. ARM beads interact through repulsive LJ interactions and, if charged, electrostatic interactions modelled by a Debye-Huckel potential. We note that repulsion between subunits due to the positive charges does affect the magnitude of the subunit-subunit interaction. Previously, we estimated this repulsion contributes $1k_{\mathrm{B}}T$ to the dimerization free energy Perlmutter2013 . This contribution (and entropic terms) are not included in $\varepsilon_{\text{ss}}$, which is the magnitude of the Morse potential depth. See SI section VII.2) for a discussion of binding free energies.
Simulations and units. Simulations were performed with the Brownian Dynamics algorithm of HOOMD, which uses the Langevin equation to evolve positions and rigid body orientations in time Anderson2008 ; nguyen2011 ; LeBard2012 . Simulations were run using a set of fundamental units. The fundamental energy unit is selected to be $E_{\text{u}}\equiv 1k_{\mathrm{B}}T$. The unit of length $D_{\text{u}}$ is set to the circumradius of a pentagonal subunit, which is taken to be $1D_{\text{u}}\equiv 5$ nm so that the dodecahedron inradius of $1.46D_{\text{u}}=7.3$ nm gives an interior volume consistent with that of the smallest $T{=}1$ capsids. Assembly simulations were run at least 10 times for each set of parameters, each of which were concluded at either completion, persistent malformation, or $2\times 10^{8}$ time steps. For all dynamics simulations there were 60 subunits with $\mbox{box size}{=}200\times 200\times 200$ nm, resulting in a concentration of $12\mu$M.
SAXS profile and hydrodynamic radius estimations. Small Angle X-ray Scattering (SAXS) analysis was performed using CRYSOL Svergun1995 . For this analysis the all-atom structure of an SV40 pentameric subunit Stehle1996 was aligned into the position of each coarse-grained subunit and the polymer was replaced with a carbon chain. We note that this entails significant simplification: segments of the protein which were not resolved in the crystal structure were not reconstructed and there is no optimization of structure at the all-atom resolution. We believe this approximation is suitable, given that our analysis is limited to the X-ray scattering profile in the small angle regime, which reflects $\sim$nm scale structural features. Fitting of the scattering profiles was performed using least squares fitting. Hydrodynamic radius analysis was performed using HYDROPRO Ortega2011 . This program is capable of calculating the hydrodynamic radius of large protein complexes. In order to perform this analysis, we treated the synthetic polymer as an amino acid chain. Though this is a gross approximation, it has a negligible effect: removing the polymer entirely does not change the $R_{\text{H}}$ trend and only alters the magnitude by $\sim 1\%$. Simulations were visualized using VMD Humphrey1996 .
VI Acknowledgments
We gratefully acknowledge Adam Zlotnick for insightful discussion and critical reading of the manuscript. This work was supported by Award Number R01GM108021 from the National Institute Of General Medical Sciences. Computational resources were provided by the NSF through XSEDE computing resources (Longhorn, Maverick, Condor, and Keeneland) and the Brandeis HPCC which is partially supported by NSF-MRSEC-0820492.
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VII Supporting Information
VII.1 Model potentials and parameters
The model details are described in Ref. [89]; we summarize them here for the convenience of the reader. In our model, all potentials can be decomposed into pairwise interactions. Potentials involving capsomer subunits further decompose into pairwise interactions between their constituent building blocks – the excluders, attractors, ‘Top’ and ‘Bottom’, and ARM pseudoatoms. It is convenient to write the total energy of the system as the sum of 6 terms: a capsomer-capsomer $U_{\mathrm{cc}}$ part (which does not include interactions between ARM pseudoatoms), capsomer-polymer $U_{\mathrm{cp}}$, capsomer-ARM $U_{\mathrm{ca}}$, polymer-polymer $U_{\mathrm{pp}}$, polymer-ARM $U_{\mathrm{pa}}$, and ARM-ARM $U_{\mathrm{aa}}$ parts. Each is summed over all pairs of the appropriate type:
$$\displaystyle U=$$
$$\displaystyle\sum_{\mathrm{cap\ }{i}}\sum_{\mathrm{cap\ }{j<i}}U_{\mathrm{cc}}%
+\sum_{\mathrm{cap\ }{i}}\sum_{\mathrm{poly\ }{j}}U_{\mathrm{cp}}+\sum_{%
\mathrm{cap\ }{i}}\sum_{\mathrm{ARM\ }{j}}U_{\mathrm{ca}}+$$
$$\displaystyle\sum_{\mathrm{poly\ }{i}}\sum_{\mathrm{poly\ }{j<i}}U_{\mathrm{pp%
}}+\sum_{\mathrm{poly\ }{i}}\sum_{\mathrm{ARM\ }{j}}U_{\mathrm{pa}}+\sum_{%
\mathrm{tail\ }{i}}\sum_{\mathrm{ARM\ }{j<i}}U_{\mathrm{aa}}$$
(1)
where $\sum_{\mathrm{cap\ }{i}}\sum_{\mathrm{cap\ }{j<i}}$ is the sum over all distinct pairs of capsomers in the system, $\sum_{\mathrm{cap\ }{i}}\sum_{\mathrm{poly\ }{j}}$ is the sum over all capsomer-polymer pairs, etc.
The capsomer-capsomer potential $U_{\mathrm{cc}}$ is the sum of the attractive interactions between complementary attractors, and geometry guiding repulsive interactions between ‘Top’ - ‘Top’ pairs and ‘Top’ - ‘Bottom’ pairs. There are no interactions between members of the same rigid body, but ARMs are not rigid and thus there are intra-subunit ARM-ARM interactions. Thus, for notational clarity, we index rigid bodies and non-rigid pseudoatoms in Roman, while the pseudoatoms comprising a particular rigid body are indexed in Greek. E.g., for capsomer $i$ we denote its attractor positions as $\{\bm{a}_{i\alpha}\}$ with the set comprising all attractors $\alpha$, its ‘Top’ positions $\{\bm{t}_{i\alpha}\}$, and its ‘Bottom’ positions $\{\bm{b}_{i\alpha}\}$. The capsomer-capsomer interaction potential between two capsomers $i$ and $j$ is then defined as:
$$\displaystyle U_{\mathrm{cc}}$$
$$\displaystyle(\{\bm{a}_{i\alpha}\},\{\bm{t}_{i\alpha}\},\{\bm{b}_{i\alpha}\},%
\{\bm{a}_{j\beta},\{\bm{t}_{j\beta}\},\{\bm{b}_{j\beta}\})=$$
$$\displaystyle\sum_{\alpha,\beta}^{N_{\mathrm{t}}}\varepsilon\mathcal{L}\left(%
\left|\bm{t}_{i\alpha}-\bm{t}_{j\beta}\right|,\ \sigma_{\mathrm{t}}\right)+$$
$$\displaystyle\sum_{\alpha,\beta}^{N_{\mathrm{b}},N_{\mathrm{t}}}\varepsilon%
\mathcal{L}\left(\left|\bm{b}_{i\alpha}-\bm{t}_{j\beta}\right|,\ \sigma_{%
\mathrm{b}}\right)+$$
$$\displaystyle\sum_{\alpha,\beta}^{N_{\mathrm{a}}}\varepsilon\mathcal{M}\left(%
\left|\bm{a}_{i\alpha}-\bm{a}_{j\beta}\right|,\ r_{\mathrm{0}},\varrho,r_{%
\text{cut}}\right)$$
(2)
where $\varepsilon$ is an adjustable parameter which both sets the strength of the capsomer-capsomer attraction at each attractor site and scales the repulsive interactions which enforce the dodecahedral geometry. $N_{\mathrm{t}}$, $N_{\mathrm{b}}$, and $N_{\mathrm{a}}$ are the number of ‘Top’, ‘Bottom’, and attractor pseudoatoms respectively in one capsomer, $\sigma_{\mathrm{t}}$ and $\sigma_{\mathrm{b}}$ are the effective diameters of the ‘Top’ – ‘Top’ interaction and ‘Bottom’ – ‘Top’ interactions, which are set to 10.5 nm and 9.0 nm respectively throughout this work, $r_{\mathrm{0}}$ is the minimum energy attractor distance, set to 1 nm, $\varrho$ is a parameter determining the width of the attractive interaction, set to 2.5, and $r_{\text{cut}}$ is the cutoff distance for the attractor potential, set to 10.0 nm.
The function $\mathcal{L}$ is defined as the repulsive component of the Lennard-Jones potential shifted to zero at the interaction diameter:
$$\mathcal{L}(x,\sigma)\equiv\left\{\begin{array}[]{ll}\left(\frac{\sigma}{x}%
\right)^{12}-1&:x<\sigma\\
0&:\mathrm{otherwise}\end{array}\right.$$
(3)
The function $\mathcal{M}$ is a Morse potential:
$$\mathcal{M}(x,r_{\mathrm{0}},\varrho)\equiv\left\{\begin{array}[]{ll}\left(e^{%
\varrho\left(1-\frac{x}{r_{\mathrm{0}}}\right)}-2\right)e^{\varrho\left(1-%
\frac{x}{r_{\mathrm{0}}}\right)}&:x<r_{\text{cut}}\\
0&:\mathrm{otherwise}\end{array}\right.$$
(4)
The capsomer-polymer interaction is a short-range repulsion that accounts for excluded-volume. For capsomer $i$ with excluder positions $\{\bm{x}_{i\alpha}\}$ and polymer subunit $j$ with position $\bm{R}_{j}$, the potential is:
$$\displaystyle U_{\mathrm{cp}}(\{\bm{x}_{i\alpha}\},\bm{R}_{j})$$
$$\displaystyle=$$
$$\displaystyle\sum_{\alpha}^{N_{\mathrm{x}}}\mathcal{L}\left(|\bm{x}_{i\alpha}-%
\bm{R}_{j}|,\sigma_{\mathrm{xp}}\right)$$
(5)
where $N_{\text{x}}$ is the number of excluders on a capsomer and $\sigma_{\mathrm{xp}}=0.5(\sigma_{\text{x}}+\sigma_{\text{p}})$ is the effective diameter of the excluder – polymer repulsion. The diameter of the polymer bead is $\sigma_{\text{p}}=0.5$ nm and the diameter for the excluder beads is $\sigma_{\text{x}}=3.0$ nm for the $T{=}1$ model and $\sigma_{\text{x}}=5.25$ nm for the $T{=}3$ model.
The capsomer-ARM interaction is a short-range repulsion that accounts for excluded-volume. For capsomer $i$ with excluder positions $\{\bm{x}_{i\alpha}\}$ and ARM subunit $j$ with position $\bm{R}_{j}$, the potential is:
$$\displaystyle U_{\mathrm{cA}}(\{\bm{x}_{i\alpha}\},\bm{R}_{j})$$
$$\displaystyle=$$
$$\displaystyle\sum_{\alpha}^{N_{\mathrm{x}}}\mathcal{L}\left(|\bm{x}_{i\alpha}-%
\bm{R}_{j}|,\sigma_{\mathrm{xA}}\right)$$
(6)
with $\sigma_{\mathrm{xA}}=0.5(\sigma_{\text{x}}+\sigma_{\text{A}})$ as the effective diameter of the excluder - ARM repulsion with $\sigma_{\text{A}}=0.5$ nm the diameter of an ARM bead.
The polymer-polymer non-bonded interaction is composed of electrostatic repulsions and short-ranged excluded-volume interactions. These polymers also contain bonded interactions which are only evaluated for segments occupying adjacent positions along the polymer chain and angular interactions which are only evaluated for three sequential polymer segments. As noted in the main text, electrostatics are represented by Debye Huckel interactions.
$$\displaystyle U_{\mathrm{pp}}(\bm{R}_{i},\bm{R}_{j},\bm{R}_{k})$$
$$\displaystyle=$$
$$\displaystyle\left\{\begin{array}[]{l}\mathcal{K}_{\mathrm{bond}}(R_{ij},%
\sigma_{\mathrm{p}},k_{\mathrm{bond}})\\
\quad:\{i,j\}\ \mathrm{bonded}\\
\mathcal{K}_{\mathrm{angle}}(R_{ijk},k_{\mathrm{angle}})\\
\quad:\{i,j,k\}\ \mathrm{angle}\\
\mathcal{L}(R_{ij},\sigma_{\mathrm{p}})+\mathcal{U}_{\mathrm{DH}}(R_{ij},q_{%
\text{p}},q_{\text{p}},\sigma_{\mathrm{p}})\\
\quad:\{i,j\}\ \mathrm{nonbonded}\\
\end{array}\right.$$
(7)
where $R_{ij}\equiv|\bm{R}_{i}-\bm{R}_{j}|$ is the center-to-center distance between the polymer subunits, $q_{\text{p}}=-1$ is the valence of charge on each polymer segment, and $\mathcal{U_{\mathrm{DH}}}$ is a Debye-Huckel potential smoothly shifted to zero at the cutoff:
$$\displaystyle\mathcal{U}_{\mathrm{DH}}$$
$$\displaystyle(r,q_{1},q_{2},\sigma{})\equiv$$
(8)
$$\displaystyle\left\{\begin{array}[]{l}\frac{q_{1}q_{2}l_{\mathrm{b}}\ e^{%
\sigma{}/\lambda_{\mathrm{D}}}}{\lambda_{\mathrm{D}}+\sigma{}}\left(\frac{e^{-%
r/\lambda_{\mathrm{D}}}}{r}\right)\\
\quad:x<2\lambda_{\mathrm{D}}\\
\frac{(r_{cut}^{2}-r^{2})^{2}(r_{cut}^{2}+2r^{2}-3r_{on}^{2}))}{(r_{cut}^{2}-2%
r_{on}^{2})^{3}}\frac{q_{1}q_{2}l_{\mathrm{b}}\ e^{\sigma{}/\lambda_{\mathrm{D%
}}}}{\lambda_{\mathrm{D}}+\sigma{}}\left(\frac{e^{-r/\lambda_{\mathrm{D}}}}{r}%
\right)\\
\quad:2\lambda_{\mathrm{D}}<x<3\lambda_{\mathrm{D}}\\
0\\
\quad:\mathrm{otherwise}\end{array}\right.$$
(9)
$\lambda_{\mathrm{D}}$ is the Debye length, $l_{\mathrm{b}}$ is the Bjerrum length, and $q_{1}$ and $q_{2}$ are the valences of the interacting charges.
Bonds are represented by a harmonic potential:
$$\mathcal{K}_{\mathrm{bond}}(R_{ij},\sigma,k_{\mathrm{bond}})\equiv\frac{k_{%
\mathrm{bond}}}{2}(R_{ij}-\sigma)^{2}.$$
(10)
Angles are also represented by a harmonic potential:
$$\mathcal{K}_{\mathrm{angle}}(R_{ijk},k_{\mathrm{angle}})\equiv\frac{k_{\mathrm%
{angle}}}{2}(\vartheta_{ijk})^{2}$$
(11)
where $\vartheta_{ijk}$ is the angle formed by the sequential polymer units $i,j,k$.
The ARM-ARM interaction is similar to the polymer-polymer interaction, consisting of non-bonded interactions composed of electrostatic repulsions and short-ranged excluded-volume interactions. These ARMs also contain bonded interactions which are only evaluated for segments occupying adjacent positions along the polymer chain:
$$\displaystyle U_{\mathrm{aa}}(\bm{R}_{i},\bm{R}_{j})$$
$$\displaystyle=$$
$$\displaystyle\left\{\begin{array}[]{l}\mathcal{K}_{\mathrm{bond}}(R_{ij},%
\sigma_{\mathrm{a}},k_{\mathrm{bond}})\\
\quad:\{i,j\}\ \mathrm{bonded}\\
\mathcal{L}(R_{ij},\sigma_{\mathrm{a}})+\mathcal{U}_{\mathrm{DH}}(R_{ij},q_{i}%
,q_{j},\sigma_{\mathrm{a}})\\
\quad:\{i,j\}\ \mathrm{nonbonded}\\
\end{array}\right.$$
(12)
where $R_{ij}\equiv|\bm{R}_{i}-\bm{R}_{j}|$ is the center-to-center distance between the ARM subunits and $q_{i}$ is the valence of charge on ARM segment $i$.
Finally, the ARM-Polymer interaction is the sum of short-ranged excluded-volume interactions and electrostatic interactions:
$$U_{\mathrm{pa}}(\bm{R}_{i},\bm{R}_{j})=\mathcal{L}(R_{ij},\sigma_{\mathrm{ap}}%
)+\mathcal{U}_{\mathrm{DH}}(R_{ij},q_{i},q_{j},\sigma_{\mathrm{ap}})$$
(13)
VII.2 Calculation of binding free energy estimates
Subunit-Subunit binding free energy. Our method of calculating the subunit-subunit binding free energy was described previously [89] and is similar to that presented in our previous work [29, 42]. Briefly, subunits were modified such that only one edge formed attractive bonds, limiting complex formation to dimers. We measured the relative concentration of dimers and monomers for a range of attraction strengths ($\varepsilon_{\text{ss}}$). The free energy of binding along that interface is then $g_{\text{ss}}/k_{\mathrm{B}}T=-\ln(c_{\text{ss}}/K_{\mathrm{d}})$ with standard state concentration $c_{\text{ss}}=1$ M and $K_{\mathrm{d}}$ in molar units. We can then correct for the multiplicity of dimer conformations, by adding in the additional term $-T\Delta s_{\mathrm{c}}=\ln(25/2)k_{\mathrm{B}}T$, where the five pentagonal edges are assumed to be distinguishable, but complex orientations which differ only through global rotation are not. For subunits which do not contain ARMs, the free energy is well fit by the linear expression $g_{\text{ss}}/k_{\mathrm{B}}T=-1.5\varepsilon_{\text{ss}}-Ts_{\mathrm{b}}$ where $Ts_{\mathrm{b}}=-5.0k_{\mathrm{B}}T$, and these values are used in Figures 3 and 5. Calculations were also performed with ARMs at varying $C_{\text{salt}}$. For $C_{\text{salt}}{=}100$ mM, $g_{\text{ss}}$ increases by $\sim 0.5k_{\mathrm{B}}T$ due to ARM-ARM repulsion for the simple 5-ARM model. For the SV40 ARM, $g_{\text{ss}}$ increases by $\sim 2k_{\mathrm{B}}T$ at $C_{\text{salt}}{=}100$ mM.
Subunit-Polymer interaction. Our method of estimating the subunit-polymer binding free energy is also similar to that presented in our previous work [29, 42]. In these simulations, the subunit-subunit attraction is eliminated (i.e. $\varepsilon_{\text{ss}}{=}0k_{\mathrm{B}}T$), and the average number of subunits adsorbed to the polymer is measured as a function of the subunit concentration and salt concentration (Fig. S1A). With sufficient data, the dissociation constant and free energy as a function of salt concentration could be determined using the McGhee-von Hippel formulation, wherein a bound ligand occupies multiple binding sites on the polymer [77]. In our previous simulations the number of binding sites occupied by a subunit was determined by the model [29, 42], whereas here the number of polymer sites occupied per subunit emerges from collective interactions. As our estimate, we use the number of subunits per polymer length for the maximal saturation observed, which for our subunits containing 25 ARM charges (5 ARMs of length 5 each) results in $p=22$. This leads to estimates in the free energy of subunit-polymer binding of $\sim 3-6k_{\mathrm{B}}T$ for $C_{\text{salt}}=500-100$ mM. However, we found systematic variations of our estimate for changing total subunit concentration, suggesting that our sampling was incomplete.
VII.3 Constructing the Markov state model
Markov state models (MSMs) have been used extensively to study protein folding [12, 11, 10, 111, 110, 92, 87, 86, 84, 66, 55, 18, 48, 17, 23], and are one of the few methods that can describe out-of-equilibrium dynamical processes that include rare events, such as a nucleation barrier crossing. Our strategy for using MSMs to study capsid self-assembly is described in detail in Ref. [88]; we summarize the key points here. We construct an MSM by running many short (relative to the assembly timescale) simulations using a ratcheting procedure, which starts simulations from configurations based on the most poorly sampled states. System configurations are partitioned into states such that configurations which interconvert rapidly are collected in the same state while those which interconvert slowly are in separate states. The separation of timescales provided by such decomposition ensures that the system behaves Markovian beyond the relatively short intra-state relaxation timescales. Inter-state transition probabilities are then measured after a ‘lag time’ which is longer than intra-state relaxation times.
We follow the simplest approach described in Ref. [88], in which states are defined by the largest cluster size $n$ and the total number of subunits adsorbed to the polyelectrolyte $n_{\text{ad}}$. Using this state decomposition produced 74 states for the slow nucleation parameters ($\varepsilon_{\text{ss}}=5k_{\mathrm{B}}T$, $C_{\text{salt}}=500$mM) with a lag time of $4\times 10^{5}$ simulation steps sufficient to maintain the Markov property.
MSM Calculations. The transition probability matrix $\mathbf{T}(\tau)$ is calculated by column-normalizing the count matrix $\mathbf{C}(\tau)$, in which each element $C_{ji}$ gives the total number of transitions from state $i$ to state $j$ measured at a lag time $\tau$. The count matrix is calculated from the many, relatively short, trajectories run in parallel using the ratcheting procedure described below. After diagonalizing $\mathbf{T}(\tau)$, the time-dependent state probabilities can be written as
$$\displaystyle\vec{P}(t;\tau)$$
$$\displaystyle=\sum_{i=1}^{N}\ket{i}\bra{i}\ket{\vec{P}(0)}e^{-\lambda_{i}t}$$
$$\displaystyle\lambda_{i}$$
$$\displaystyle=-\log(\omega_{i})/\tau$$
(14)
where $\omega_{i}$ is the $i^{\text{th}}$ eigenvalue of $\mathbf{T}(\tau)$ and $\bra{i}$ and $\ket{i}$ are the corresponding left/right eigenvectors, which are assumed to be normalized. Since $\mathbf{T}(\tau)$ is generally not Hermitian the left and right eigenvectors are not equivalent. There is only one unit eigenvalue, whose associated right eigenvector corresponds to the equilibrium distribution, while all other eigenvalues are positive and real [110]. The average value of an order parameter, $Q$, can then be calculated as a function of time from
$$\bar{Q}(t)=\sum_{i=1}^{N}P_{i}(t;\tau)Q_{i}$$
(15)
with $Q_{i}$ as the order parameter value for state $i$ and $P_{i}(t;\tau)$ as the time dependent probability of state $i$ (from Eq. 14). Eq. 15 was used to calculate the mean cluster size $n$ and mean number adsorbed $n_{\text{ad}}$ in SI Fig. 4C. In order to calculate the probability of a given cluster size as a function of time in SI Fig. S4, the same procedure was followed except that all states for a given $n$ were lumped together to give a single curve for each $n$.
Ratcheting Procedure. Because of the Markov property, starting coordinates for simulations can be chosen to efficiently generate good statistics for all of the relevant transition elements. Many simulations are run in parallel for a time $t_{\text{s}}$ , which must be longer than the lag time $\tau$, but can be much shorter than the longest relaxation timescale. For the slow nucleation calculation in this paper, we continuously ran $100$ simulations in parallel starting from an unassembled bath of subunits. New simulations were seeded from the states with the fewest number of simulation starts, biasing sampling to unexplored regions of state space. Our ratcheting procedure initially grouped states only by the number of subunits adsorbed to polymer $n_{\text{ad}}$. However, after discovering states for each cluster size on the pathway to assembly, we ran a large number of simulations in parallel seeded from a random selection of existing states. This crude, but effective approach could be improved by using adaptive sampling [11], which explicitly considers the error in the MSM when seeding new simulations. A total of $6.3\times 10^{9}$ simulation steps were used to build the MSM. |
AEI-2000-078 OHSTPY-HEP-T-00-032 hep-th/0012080
Tachyon condensation and universality of DBI action
G. Arutyunov${}^{*,**}$ 111emails: agleb, theisen @aei-potsdam.mpg.de; frolov,
tseytlin @mps.ohio-state.edu
${}^{**}$On leave of absence from Steklov Mathematical Institute,
Gubkin str. 8,
117966, Moscow, Russia,
S. Frolov${}^{\dagger,**}$,
S. Theisen${}^{*}$
and A.A. Tseytlin${}^{\dagger}$
222Also at Imperial College, London and Lebedev Institute, Moscow
${}^{*}$
Max-Planck-Institut für Gravitationsphysik,
Albert-Einstein-Institut,
Am Mühlenberg 1, D-14476 Golm, Germany
${}^{\dagger}$ Department of Physics,
The Ohio State University,
Columbus, OH 43210-1106, USA
Abstract
We show that a low-energy action for
massless fluctuations around a tachyonic soliton background
representing a codimension one D-brane coincides with the
Dirac-Born-Infeld action. The scalar modes which
describe transverse oscillations of the D-brane
are translational collective coordinates of the soliton.
The appearance of the DBI action is a universal
feature independent of details of a tachyon effective action,
provided it has the structure implied by the open
string sigma model partition function.
1 Introduction
In the original perturbative
string-theory description
D-branes [1] are specified by boundary conditions
on the open strings ending on them. Their
collective coordinates
are identified with massless modes of the
open strings, and the “acceleration-independent”
part of their action – the DBI action [2] –
can be derived
(using T-duality)
as a reduction of Born-Infeld action
(directly from the open string sigma model
partition function [3]
or from conformal invariance condition [4],
see [2, 5, 6]).
Supersymmetric
D-branes appear also in another guise
as black-hole type solitons of $D=10$ type II
supergravity
[7, 8] which carry Ramond-Ramond charges [9].
Their collective coordinates,
determined by the
massless fluctuation modes in these
backgrounds,
can be related to
the parameters of spontaneously broken gauge
symmetries [10]. With
an appropriate non-linear parametrization
of the supergravity fields in terms of the collective
coordinates, one should be able to derive the
corresponding DBI actions directly from the
type II supergravity action.111To a large extent
this should follow
essentially from reparametrization
invariance,
implying that the action of a boosted 11-d Schwarzschild
black hole should be $M\int dt\sqrt{1-v^{2}}$, and
T- and S- dualities of type II supergravities.
Recently, a new, third,
description of D-branes as
solitons was suggested [11, 12].
Arguments supporting the proposal
that D-branes can be interpreted as tachyonic solitons
were given
using a variety of approaches:
boundary conformal field theory (see, e.g., [13]),
Witten’s open string field theory [14]
(see, e.g., [15]), non-commutative
field theory obtained in large $B_{\mu\nu}$ limit
(see, e.g., [16]) and simple
low-energy effective Lagrangian models
with specific tachyon potentials [17, 18, 19].
Remarkably, the
latter models were shown to follow
from the boundary string field theory (BSFT)
[20, 21] in [22, 23, 24].
The BSFT approach, consistently restricted to the lowest-level
(renormalizable) tachyon
and vector couplings, is essentially the same
as the off-shell sigma model approach [25, 3, 26]
being based on the disc partition function
of the open string sigma model.
The partition function encodes the information not only about
the beta-functions but also about the field space
metric which relates them
to field equations of motion, and thus
allows to smoothly interpolate between the standard perturbative
tachyon
vacuum and a new non-trivial vacuum at the minimum
of the tachyon potential.222A simple
consequence of the sigma model
approach to the tachyon condensation
is the background independence of
the tachyon potentials discussed in the
framework of Witten’s open string field
theory in [27]: the zero mode
of the tachyon field which is the only argument
of the potential
does not feel any closed string background.
One finds that
there are solitonic solutions corresponding to D-branes of
lower dimensions
[17, 18],
and that the descent relations between D-brane tensions
hold exactly [23, 24].
These results
have been generalized to include a background
gauge field
[28, 29, 30, 31, 32, 26].333Since a constant
field strength $F_{\mu\nu}$ enters the
actions in
the same way as a constant $B_{\mu\nu}$-field,
the sigma model approach
should provide a
natural explanation for some
of the results for noncommutative tachyon
condensation
in [16].
From the boundary sigma model point of view, one starts with a
conformal theory with $d$ Neumann boundary conditions in the UV
and adds relevant (tachyon) perturbations driving the theory
to an IR fixed point that corresponds to a new (stable or unstable)
vacuum with $(d-1-p)$ Dirichlet boundary conditions. The IR fixed
point is then interpreted as a closed string vacuum with a Dp-brane.
Given that the tension of a Dp-brane is correctly
reproduced [23, 24],
a further crucial test
is to find the spectrum and the effective action
for light modes on
a Dp-brane obtained as a result of tachyon condensation.
The aim of the present paper is to show that
(the “acceleration-independent” part of) the action
for the massless scalar modes
$\Phi$ on the tachyonic soliton and the massless
$U(1)$ vector field $A$ is indeed the standard DBI action.
In the context of the Sen’s proposal to describe D-branes
as open string field theory solitons
the massless modes representing transverse D-brane fluctuations
appear as collective coordinates.
The presence of these massless scalars is
a general phenomenon independent of details of an
effective field theory action, and is a consequence
of spontaneous breaking of the
translational symmetry [33].
The existence of a massless vector mode on the soliton
is related to
the fact that the tachyon is coupled to the (abelian)
open string
vector field in a non-minimal
way, i.e. only through the field strength $F_{\mu\nu}$.444Note that
one does not find massless vector modes and DBI actions
for the collective coordinates for brane
solitons in familiar scalar-vector field-theory systems
where
scalars are coupled to vector fields in a
minimal way.
The general structure
of the dependence of the effective action on $F_{\mu\nu}$
and its coupling to the tachyon $T$ is dictated
by the open string sigma model [3, 4, 26]:
the action is an integral of
a product of the Born-Infeld Lagrangian
for the vector field and some function of $T$ and its derivatives
contracted with functions of $F_{\mu\nu}$.
This structure is
the only essential assumption one needs
to show that the scalar and vector zero modes always
combine into the standard DBI action,
irrespective of all other details of the effective action.555The precise
value of the tension
of the resulting
Dp-brane does depend of course on details of an
effective string field theory action.
To reproduce the expected tension
one should compute the disc partition
at the proper conformal point, or minimize
the “trial actions” depending on a
finite number of parameters
obtained in the BSFT framework
in [23, 24].
While the universality of the DBI action holds irrespective of the
presence of a tachyonic mode in the spectrum of fluctuations around
the soliton, the form of an action for the tachyonic
mode666The
tachyonic mode is always present in open bosonic string theory
context, and is
absent on the kink solution describing condensation of an
unstable D9-brane tachyon to produce a D8-brane, which
is a stable BPS-object in type IIA theory. does depend on
details of an effective string field theory action (the
action for the tachyonic mode
on D24-brane was shown [34] to have the same form as the
one on the D25-brane).
We should add that while finding the DBI action
for the D-branes described by tachyonic solitons was
expected – after all, the required information is
contained in the open string sigma model which gives
both, the effective action for the original brane
and the action for massless fluctuations of the solitonic brane –
to see how this derivation works in detail seems
quite instructive. In particular, this clarifies
how the brane collective coordinates should appear
in the string world sheet action which may be relevant
for off-shell aspects of D-branes
(cf. [35]).
In Section 2 we shall
present a derivation of the DBI action from an
effective field theory action with the structure implied by
the open string sigma model. In Section 3 we shall
discuss how one can see the same
directly in the world-sheet
approach, identifying the boundary couplings of the D-brane
collective coordinates.
2 Effective field theory picture
2.1 Tachyon–vector actions
The string sigma model approach to
field theory
describing dynamics of an unstable D25-brane
in bosonic string theory or
a non-BPS D9-brane in type IIA string theory
leads to an effective action for the tachyon
and vector gauge fields
of the following general form (for a review, see [26])
777In the units we shall use $\alpha^{\prime}=2$, and our
normalizations for $T$ and $A_{\mu}$ are given in Section 3.
$$\displaystyle S=\int d^{d}x\sqrt{\det(\delta_{\mu\nu}+F_{\mu\nu})}\ {\cal L}(T%
,F,\partial T,\partial F,\partial\partial T,\ldots).$$
(1)
The tachyon field couples to the vector
field through its strength, i.e. through the combinations
(and their derivatives)888The case when the starting point is
a Dp-brane with a non-zero number of transverse dimensions
represented by massless scalars is obtained by “dimensionally reducing”
$F_{\mu\nu}$ matrix.
the
$$\displaystyle G^{\mu\nu}=\left(\frac{1}{1+F}\right)^{(\mu\nu)}=\left(\frac{1}{%
1-F^{2}}\right)^{\mu\nu}\ ,\ \ \ \ \ \ \ H^{\mu\nu}=\left(\frac{1}{1+F}\right)%
^{[\mu\nu]}=-\left(\frac{F}{1-F^{2}}\right)^{\mu\nu}\ ,$$
(2)
where $G^{\mu\nu}$ plays the role of an effective metric
as seen by open string excitations in
the presence of a constant gauge field background [4].
We will be interested in
deriving an effective action for massless modes on
a soliton solution up to “acceleration-dependent” (i.e.
second and higher derivative) terms.
For that reason we may restrict our consideration to
the terms in $S$ which do not
depend on derivatives of $F_{\mu\nu}$.
We will study only codimension one D-branes, i.e.
soliton solutions represented by the tachyon field
depending only on one
coordinate $x_{1}$ and the vector field having
constant field strength $F_{\mu\nu}$.
As will be shown below, such backgrounds
satisfy the equations of motion for
the vector field only if
$F_{1i}=0$ ($i=2,...10$), i.e., if the matrices in (2)
obey
$G^{1i}=H^{1i}=0$.
Since in deriving the effective action for massless modes we drop
their
second and higher derivatives, none of the
possible $H^{\mu\nu}$
dependent terms in (1)
contribute to the effective action,
i.e. we are allowed to use
an action which depends only on $G^{\mu\nu}$ and $T$ and its
derivatives of
any order, i.e.
$$\displaystyle S=\int d^{d}x\sqrt{\det(\delta_{\mu\nu}+F_{\mu\nu})}\ {\cal L}(G%
^{\mu\nu}(F),T,\partial T,\partial\partial T,\ldots)\ .$$
(3)
Though any
action of the form (3) will
lead to the DBI action for the massless soliton modes,
it is useful to recall several explicit
actions of that type
which were previously discussed in the
context of tachyon condensation.
The low-energy two-derivative effective
action for a D25-brane in open bosonic string theory is (3)
with ${\cal L}$ given by [32, 26]
$$\displaystyle{\cal L}_{25}=T_{25}\ {\rm e}^{-T}\ (1+T)\bigl{(}1+G^{\mu\nu}%
\partial_{\mu}T\partial_{\nu}T\bigr{)}.$$
(4)
The corresponding expression for the
non-BPS D9-brane in type IIA theory is [24, 26]
$$\displaystyle{\cal L}_{9}=T_{9}\ {\rm e}^{-\frac{1}{4}T^{2}}\left(1+G^{\mu\nu}%
\partial_{\mu}T\partial_{\nu}T+{1\over 2}\log(\frac{4}{{\rm e}})\ G^{\mu\nu}T^%
{2}\partial_{\mu}T\partial_{\nu}T\right).$$
(5)
The actions (4) and (5)
are obtained in special schemes chosen to
reproduce the standard
values of the tachyon masses around the perturbative $T=0$ vacua.
The actions (4) and (5)
(or similar model actions in [17, 18])
have soliton solutions representing codimension one D-branes, but they
do not reproduce the expected D-brane tensions.
However, one can find
“interpolating” actions which
coincide with (4) and (5) in the
two-derivative approximation and lead to
the correct D-brane tensions.
In particular, for a non-BPS D9-brane such action
has [24, 31, 26]
$$\displaystyle{\cal L}_{9}=T_{9}\ {\rm e}^{-\frac{1}{4}T^{2}}\left(\sqrt{\pi}%
\frac{\Gamma(1+G^{\mu\nu}\partial_{\mu}T\partial_{\nu}T)}{\Gamma(\frac{1}{2}+G%
^{\mu\nu}\partial_{\mu}T\partial_{\nu}T)}+\log(\frac{4}{{\rm e}})\ G^{\mu\nu}T%
\partial_{\mu}\partial_{\nu}T\right).$$
(6)
Here the first term can be shown to coincide with the action
found in [24, 31], and the scheme-dependent coefficient
of the last
term is chosen to reproduce the
correct value of the tachyon mass at
$T=0$ [26].
The action (3),(6)
is given by the open superstring partition function
on a disc.
In the sigma model approach
the tachyon coupling at the boundary of the disc
can be represented as
$$\displaystyle T(x(\tau))=T(x)+\xi^{\mu}(\tau)\partial_{\mu}T(x)+\frac{1}{2}\xi%
^{\mu}(\tau)\xi^{\nu}(\tau)\partial_{\mu}\partial_{\nu}T(x)+\cdots,$$
(7)
where $\tau$ is the angular coordinate on the boundary of the disc,
and $x(\tau)=x+\xi(\tau)$, $\int d\tau\xi(\tau)=0$.
The derivatives of the tachyon field are then treated as
independent, and the
partition function is computed exactly in $\partial_{\mu}T$, and up to
the first order in $\partial_{\mu}\partial_{\nu}T$.
2.2 Collective coordinates and DBI action
We shall assume that the equations for $T$ and $A_{\mu}$
following from (3)
have a soliton solution with
$$\displaystyle T=T(x_{1})\ ,\ \ \ \ \ \ F_{\mu\nu}={\rm const}\ .$$
(8)
Following Sen [11],
we shall identify the solution with a D(d-2)-brane.
For example, for model two-derivative actions
one finds the kink
$T(x)=ux_{1}$ [18, 24] in the case of the type IIA
D9-brane, and the lump $T(x)=ux_{1}^{2}$
[17, 23] in the case of the bosonic D25-brane.
These solutions represent exact conformal field theories in the limit
$u\rightarrow\infty$ [13, 23, 24].
It is straightforward to see
that for any solution for $T$ found for $F_{\mu\nu}=0$
there is a corresponding solution for $F_{\mu\nu}=const$.
Since the vector potential $A_{\mu}$ is a dynamical field,
one is still to check that its equation
is satisfied for the above background.
To do that
we shall restrict ourselves to
the simplest case when the action can be put in the form
$$\displaystyle S=\int d^{d}x\sqrt{\det(\delta_{\mu\nu}+F_{\mu\nu})}\ {\cal L}(G%
^{\mu\nu}(F)\partial_{\mu}T\partial_{\nu}T,T)\ .$$
(9)
The explicit examples of the
actions considered above are all of that type.
Then varying (9) with respect to
$A_{\mu}$ one gets (assuming (8))
$$\displaystyle\partial_{\mu}\Biggl{[}(I+F)^{-1}_{[\mu\nu]}\ {\cal L}+2(I+F)^{-1%
}_{1[\mu}(I+F)^{-1}_{\nu]1}(\partial_{1}T)^{2}{\cal L}^{\prime}\Biggr{]}=0.$$
(10)
Here ${\cal L}={\cal L}(K,T),\ K\equiv G^{\mu\nu}\partial_{\mu}T\partial_{\nu}T$
and the derivative ${\cal L}^{\prime}$
is with respect to the first argument $K$.
Since ${\cal L}$ depends only on $x_{1}$
this equation reduces to
$$\displaystyle(I+F)^{-1}_{[\nu 1]}\ \partial_{1}\bigl{[}{\cal L}-2G^{11}(%
\partial_{1}T)^{2}{\cal L}^{\prime}\bigr{]}=0\ .$$
(11)
This equation is satisfied if
$$\displaystyle(I+F)^{-1}_{[\nu 1]}=\bigl{(}\frac{F}{1-F^{2}}\bigr{)}_{1\nu}=0\ ,$$
(12)
i.e. if $F_{1\nu}=0$, or if
$$\displaystyle{\cal L}-2G^{11}(\partial_{1}T)^{2}{\cal L}^{\prime}={\rm const}\,.$$
(13)
The latter condition does not hold
for a generic Lagrangian,
so we will always require that $F_{1\nu}=0$.
Then
the effective metric $G^{\mu\nu}$ simplifies, so that
$$\displaystyle G^{11}=1,\ \ \ \quad G^{1i}=0.$$
(14)
Note also that in the axial gauge, $A_{1}(x_{1},x_{i})=0$,
the condition
$F_{1\nu}=0$ implies that $A_{i}(x_{i})$ does
not depend on $x_{1}$, as one would
expect for a massless
vector field on a D(d-2)-brane.
It follows from the translational invariance
of the action (3)
that fluctuations around
the soliton solution contain a zero mode $\Phi(x_{i})$, which
satisfies the equation of motion for a massless scalar
in $d-1$ dimensions.
As usual (see, e.g., [33]),
it is natural to identify this zero mode
with a collective coordinate
describing fluctuations of the D(d-2)-brane
in the transverse $x_{1}$ direction.
For small fluctuations and in the semiclassical approximation,
the effective action
for the zero mode is obtained by substituting
$T\rightarrow T(x_{1})-\partial_{1}T(x_{1})\Phi(x_{i})$
into (3),
and integrating over $x_{1}$.
To describe arbitrary fluctuations in the semiclassical
approximation one may then consider the standard ansatz
$T\rightarrow T(x_{1}-\Phi(x_{i}))$.
However, in general, one should take into account
the global $SO(d)_{G}$
invariance of the action
(3) implied by the open string sigma model.
The subscript
$G$ means that the general linear
transformations of the $d$
coordinates $x\rightarrow\Lambda x$ should
preserve the constant metric $G^{\mu\nu}$, i.e.
$$\Lambda G\Lambda^{T}=G.$$
Assuming that $\Lambda$ has the form
$$\displaystyle\Lambda_{\mu\nu}=\frac{1}{\beta}\tilde{\Lambda}_{\mu\nu},\quad\ %
\ \tilde{\Lambda}_{11}=1,\ \ \quad\tilde{\Lambda}_{1i}=-V_{i}\ ,$$
(15)
we get
$$\displaystyle\beta=\sqrt{1+G^{ij}V_{i}V_{j}}\ .$$
(16)
Consider now the following ansatz for the
tachyon in terms of the zero mode $\Phi(x_{i})$
$$\displaystyle T=T(y_{1})\ ,\ \ \ \ \ \ \ \ y_{1}\equiv\frac{x_{1}-\Phi(x_{i})}%
{\beta(\partial\Phi,F)}\ ,$$
(17)
$$\displaystyle\beta(\partial\Phi,F)=\sqrt{1+G^{ij}(F)\partial_{i}\Phi\partial_{%
j}\Phi}\ .$$
(18)
Note that if $\Phi(x_{i})$ is
of the form $\Phi(x_{i})=V_{i}x^{i}$
(i.e. brane has “constant velocity”),
then $y_{1}=(\Lambda x)_{1}$ and $\Lambda$ belongs to the
invariance group $SO(d)_{G}$.
Taking into account
that the full tachyon field in (17)
depends only on one combination $y_{1}$
of coordinates, we can rewrite (3) as follows
(ignoring higher-derivative $\partial^{n}\Phi,\,n>1$ terms)
$$\displaystyle S=T_{d-2}\int d^{d-1}x\ \beta(\partial\Phi,F)\sqrt{\det(\delta_{%
\mu\nu}+F_{\mu\nu})}\ ,$$
(19)
where the integral over $y_{1}$
$$\displaystyle T_{d-2}\equiv\int dy_{1}{\cal L}(T(y_{1}),\partial_{1}T(y_{1}),%
\partial_{1}\partial_{1}T(y_{1}),\ldots)$$
(20)
does not depend on the field strength999As follows from
the sigma model considerations,
$G^{\mu\nu}$ can be
contracted only with the tachyon derivative terms.
Thus only $G^{11}$ could
appear in (19), but it is equal to 1 according to
(14).
and thus
defines the D(d-2)-brane tension.
Taking into account that (for $F_{i1}=0$)
$$\displaystyle\beta(\partial\Phi,F)\sqrt{\det(\delta_{\mu\nu}+F_{\mu\nu})}=%
\sqrt{\det(\delta_{ij}+F_{ij}+\partial_{i}\Phi\partial_{j}\Phi)}\ ,$$
(21)
we arrive at the standard DBI action for a D(d-2)-brane
$$\displaystyle S=T_{d-2}\int d^{d-1}x\sqrt{\det(\delta_{ij}+F_{ij}+\partial_{i}%
\Phi\partial_{j}\Phi)}\ .$$
(22)
To get the DBI action it was essential to
introduce the collective coordinate dependence as
in (17).
Indeed, the above derivation is a generalization of the argument
used to show that a point-like soliton of a relativistically
invariant (e.g. 2-d) scalar field
theory is described by the standard particle
action $M\int dt\sqrt{1-v^{2}}$: one starts with the static
solution and applies a boost $x\rightarrow{x-vt\over\sqrt{1-v^{2}}}$.
The ansatz (17) of course reduces to
$T(x_{1})-\partial_{1}T(x_{1})\Phi(x_{i})$ for small fluctuations,
but the real reason for the specific non-linear structure
of (17) is to ensure the decoupling
of massive solitonic modes from the massless mode,
so that there is no term linear in the massive modes in the action
(in the approximation where
we neglect second and higher derivatives of $\Phi$).
Thus one can consistently put all massive modes to zero, and consider only
the
$\Phi$-dependent terms.101010This is somewhat
similar to what happens
in consistent Kaluza-Klein reductions of compactified gravity models.
If one would use the standard form of the
collective coordinate dependence $T=T(x_{1}-\Phi(x_{i}))$
then to recover
the DBI action one would have to take into account
the massive mode contributions
to the low-energy effective action and at the end
to make a proper non-linear
redefinition of the collective coordinate $\Phi$.
3 World-sheet picture
Here we shall relate the discussion of the
DBI action from the effective field theory point of view
in the previous section to the
world-sheet sigma model considerations,
with the aim to identify the boundary couplings
corresponding to the tachyonic soliton collective coordinate.
3.1 Standard boundary couplings
The starting point in the usual perturbative description
of D-branes in flat space
is an exact free 2-d conformal field
theory in the bulk with
$p+1$ Neumann and $d-p-1$ Dirichlet boundary conditions.
The marginal boundary interactions are then represented by
the boundary sigma model action [2, 1]111111For the sake
of clarity here we shall
consider only the bosonic string theory
but will ignore tachyonic coupling on the resulting brane.
$$\displaystyle I_{\partial\Sigma}=\frac{1}{4\pi}\int_{\partial\Sigma}d\tau%
\biggl{[}\sum_{i=1}^{p+1}A_{i}(x^{1},...,x^{p+1})\dot{x}^{i}+\sum_{a=p+2}^{d}%
\Phi_{a}(x^{1},...,x^{p+1})\partial_{n}{x}^{a}\biggr{]}\,,$$
(23)
where the couplings
$A_{i}$ and $\Phi_{a}$ are the gauge field and the scalars on
the world-volume of the $Dp$-brane, respectively ($\partial_{n}$
is the normal derivative to the boundary $\partial\Sigma$).
The open string sigma model
partition function on the disc is then the DBI
action for $A_{i}$ and $\Phi_{a}$ (ignoring terms
higher than first derivative in the fields).
On the other hand, to describe tachyon condensation
from a space-filling brane to a
lower-dimensional brane one should start
with the following boundary theory
$$\displaystyle I_{\partial\Sigma}=\frac{1}{4\pi}\int_{\partial\Sigma}d\tau\bigl%
{[}T(x)+A_{\mu}(x)\dot{x}^{\mu}\bigr{]}\ .$$
(24)
In the presence of this interaction the standard Neumann boundary
conditions for the open string coordinate
fields $x^{\mu}$ are modified
to
$$\displaystyle\partial\Sigma:\ \ \ \ \ \ \ \ \partial_{n}x_{\mu}+F_{\mu\nu}\dot%
{x}^{\nu}+\partial_{\mu}T=0\ .$$
(25)
A particularly simple case for which
the open bosonic world-sheet theory remains solvable is obtained by taking
$A_{\mu}$ to have a constant field strength and switching on
the tachyon with a
quadratic profile in one direction, i.e.
$T(x)=a+\frac{1}{2}u\,x^{2}_{1}$. In this case the boundary conditions
are
$$\displaystyle\partial\Sigma:\ \ \ \ \ \partial_{n}x_{1}+F_{1i}\dot{x}_{i}+u\,x%
_{1}=0,~{}~{}~{}~{}~{}~{}~{}~{}~{}\partial_{n}x_{i}+F_{i1}\dot{x}_{1}+F_{ij}%
\dot{x}_{j}=0\,.$$
(26)
For $u=0$ these
are the usual Neumann conditions modified by the
presence of the constant strength $F_{\mu\nu}$ (and thus the
corresponding
partition function is the BI action for $F_{\mu\nu}$ with
$\mu=1,\ldots,d$).
On the other hand, when the tachyon condenses
into the vacuum $a,u\rightarrow\infty$, the field $x_{1}$ becomes
constrained to vanish at the boundary, i.e.
is subject to
the Dirichlet boundary condition ($x_{1}|_{\partial\Sigma}=0$). This
leads to a complete decoupling of $F_{1i}$ components
in (24),(26).
Thus, in the IR fixed point one gets the BI action for $F_{ij}$
components only.
Following this approach, it appears
that we are missing the dependence
on the massless scalar fields describing
the transverse motion of the brane.
To account for them, one should
understand how the normal derivative coupling of the type
introduced from the beginning in the standard D-brane description
(23) is effectively induced
in the process of tachyon condensation.
This is what we are going to explain below.
3.2 Generalized boundary sigma model and collective coordinate
coupling
From the sigma model viewpoint,
the role of the tachyonic field is to control
a number of “free” (Neumann)
space-time dimensions, and in this respect
the condensation may be compared with
dimensional reduction. However,
in the process of reduction
the BI action in $p+1$ dimensions
does lead to
the DBI action in $p$ dimensions with $A_{1}\rightarrow\Phi$.
The procedure of dimensional
reduction is of course formally related
to T-duality transformation
along one world-volume dimension $x_{1}$ (implying that $A_{1}$
becomes a scalar field describing the transverse fluctuations).
In the world-sheet description,
the T-duality transformation simply exchanges the
normal and tangential derivatives in the boundary vertex
operators [1].
This suggests that
in order to make contact with the D-brane description
based on (23)
one may try to generalize the boundary action
(24) by adding from the beginning an
additional
normal derivative coupling (with a new coefficient
function $V_{\mu}$)121212Such sigma model
and its renormalization was discussed in detail in [36]
and references there.
$$\displaystyle I_{\partial\Sigma}=\frac{1}{4\pi}\int_{\partial\Sigma}d{\tau}%
\biggl{[}T(x)+A_{\mu}(x)\dot{x}^{\mu}+V_{\mu}(x)\partial_{n}x^{\mu}\biggr{]}\,.$$
(27)
Adding $V_{\mu}$ term may seem
irrelevant – in
the standard perturbative vacuum where all $x^{\mu}$
are subject to
the Neumann boundary conditions this term
decouples: computing correlation functions or
the disc partition function in perturbation theory
in powers of couplings one finds that they do not
depend on $V_{\mu}$.131313There exists a (point-splitting)
regularization
preserving the Neumann boundary conditions,
and, therefore, any correlation
function involving a normal-derivative term vanishes.
Note also that lifted to the bulk of the world sheet,
this coupling becomes $\int_{\Sigma}\partial^{a}(V_{\mu}(x)\partial_{a}x^{\mu})$,
and thus (modulo a term proportional to the $x^{\mu}$ equation of motion)
redefines the target space metric by a diffeomorphism-type
term.
More precisely, one
should determine the boundary conditions
dynamically,
by minimizing the total string action $I=I_{\Sigma}+I_{\partial\Sigma}$.
Then (depending on a calculation procedure)
the normal-derivative term may contribute
to the partition
function, but this still does not imply
the appearance
of a new physical degree of freedom associated with $V_{\mu}$.
Indeed, one may interpret such normal derivative couplings
as additional pure gauge modes
of open string theory, which can be removed
by a gauge choice.
Indeed, the general form of the
boundary conditions should be
$$\displaystyle\partial\Sigma:\ \ \ \ \ \ \partial_{n}x_{\mu}+N_{\mu\nu}\dot{x}^%
{\nu}+N_{\mu}=0,$$
(28)
where $N_{\mu\nu}$ and $N_{\mu}$ should be determined
from the requirement that
the variation of the total string action ${1\over 8\pi}\int_{\Sigma}(\partial x)^{2}+I_{\partial\Sigma}$ vanishes.
One finds then the following equations141414These
boundary conditions differ from the ones used in [36]
and references there.
$$\displaystyle N_{\mu}=\partial_{\mu}\tilde{T}\ ,\ \ \ \ \ \tilde{T}\equiv T-N_%
{\nu}V^{\nu}\ ,$$
(29)
$$\displaystyle N_{\mu\nu}=\tilde{F}_{\mu\nu}\ ,\ \ \ \ \ \ \tilde{F}_{\mu\nu}%
\equiv F_{\mu\nu}-\partial_{\mu}(N_{\rho\nu}V^{\rho})+\partial_{\nu}(N_{\rho%
\mu}V^{\rho})\,.$$
(30)
It is then easy to see (cf. (28),(25)) that
one can replace (27) by the usual
boundary term (24) without $V_{\mu}$ but with transformed
tachyon
$T\rightarrow\tilde{T}$ and vector $A_{\mu}\rightarrow\tilde{A}_{\mu}$ fields:
$$\displaystyle T=(1+V^{\mu}\partial_{\mu})\tilde{T}\ ,\ \ \ \ \ \ \ \ \ \ A_{%
\mu}=\tilde{A}_{\mu}-\tilde{F}_{\mu\nu}V^{\nu}\,.$$
(31)
Thus adding the normal-derivative term
amounts simply to a redefinition of the
original open string sigma model couplings in (24).
The central point, however, is that
in the presence of a nontrivial
tachyon condensate the normal derivative coupling
becomes relevant – it provides an adequate
description of dynamics of the soliton translational zero modes,
allowing one to recover the usual
D-brane description in the IR.
Assuming that there is
a tachyonic condensate breaking
translational invariance in $x_{1}$-direction,
and that $V_{1}$ depends only on
$x_{i}$, one sees that
$V_{1}$ can be identified with
the translational collective coordinate $\Phi$.
Indeed, for small $V_{1}$
$$\displaystyle\tilde{T}\approx T-V_{1}\partial_{1}T\approx T(x_{1}-V_{1}),\quad%
\ \ \ \ \tilde{A_{i}}\approx A_{i}-V_{1}\partial_{1}A_{i}\approx A_{i}(x_{1}-V%
_{1}),$$
(32)
which is the standard collective coordinate dependence.
Another argument for this identification can be given by
analyzing the boundary conditions (28) in the IR fixed point
$a,u\rightarrow\infty$ for the quadratic tachyon coupling
$T(x_{1})=a+\frac{1}{2}ux_{1}^{2}$.
At this point the boundary condition (28) reduces to the
following Dirichlet condition for $x_{1}$
$$\displaystyle\partial\Sigma:\ \ \ \ \ \ \ \ \ x_{1}=\Phi(x_{i})\,,\ \ \ \ \ \ %
\Phi\equiv V_{1}(x_{i})\ ,$$
(33)
and the modified Neumann condition for $x_{i}$ (cf. (26))
$$\displaystyle\partial\Sigma:\ \ \ \ \ \ \ \ \partial_{n}x_{i}+\partial_{i}\Phi%
\partial_{n}x_{1}+F_{ij}\dot{x}_{j}=0\,.$$
(34)
The boundary conditions (33) and (34) define the mixed
Dirichlet-Neumann sigma-model.
The vanishing of the one-loop $\beta$-functions
for this model is known [2]
to be equivalent to the equations of motion
for space-time fields $A_{i}$ and $\Phi$ following from DBI.
One may ask then why in order
to derive the DBI action in Section 2 we needed to
use the non-linear ansatz for the tachyon (17)
instead of the
transformed tachyon and gauge fields in (32).
The redefined tachyon coupling $\tilde{T}$
with the $V_{1}=\Phi$ -dependence as in (17)
can be obtained by starting with the boundary action
(27)
with
normal-derivative couplings of an appropriate
non-linear form $f(V,\partial_{n}x,\dot{x})$.
This would modify the boundary conditions (28)
and the $V_{\mu}$-dependence of the transformed tachyon and vector fields.
From the sigma model point of view the
difference between the linearized
collective coordinate dependence
in (32) and the non-linear
one in (17) should be in contact terms only.
Such terms are crucial for implementing the symmetries of
string theory in a manifest way (see, e.g., [37]),
but they do not
contribute to the (scheme-independent)
one-loop $\beta$-functions.
This explains why one can
reproduce correct $\beta$-functions corresponding to the
DBI action in the approach of [2]
using the linear expressions
(32).
To derive the DBI action as the open string sigma model
partition function (which is sensitive to a choice of a scheme
preserving underlying symmetries of the theory)
one needs, however, to use
the non-linear ansatz (17)
for the collective coordinate dependence,
substituting it into the tachyon expansion
(7) and integrating over the fluctuations
$\xi^{\mu}$.
Acknowledgements
G.A. was supported by the DFG and by the European Commission
RTN programme HPRN-CT-2000-00131 in which G.A. and S.T.
are associated to U. Bonn,
and in part by RFBI grant N99-01-00166 and by INTAS-99-1782.
S.T. also acknowledges support from GIF,
the German-Israeli foundation
for Scientific Research.
S.F. and A.A.T are supported
by the U.S. Department of Energy under grant
No. DE-FG02-91ER-40690. S.F. is also supported
in part by RFBI grant N99-01-00190, and
A.A.T. – by the
INTAS project 991590
and PPARC SPG grant PPA/G/S/1998/00613.
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Coherent hard x-rays from attosecond
pulse train-assisted harmonic generation
Michael Klaiber${}^{1,2}$, Karen Z. Hatsagortsyan${}^{1}$, Carsten Müller${}^{1}$ and Christoph H. Keitel${}^{1,*}$
Abstract
High-order harmonic generation from atomic systems is considered in the crossed fields of a relativistically strong infrared laser and a weak attosecond-pulse train of soft x-rays. Due to one-photon ionization by the x-ray pulse, the ionized electron obtains a starting momentum that compensates the relativistic drift which is induced by the laser magnetic field, and allows the electron to efficiently emit harmonic radiation upon recombination with the atomic core in the relativistic regime. In this way, short pulses of coherent hard x-rays of up to 40 keV energy and 10 as duration can be brought about.
\address
${}^{1}$Max-Planck Institut für Kernphysik, Saupfercheckweg 1, D-69117 Heidelberg, Germany
${}^{2}$Theoretische Quantendynamik, Physikalisches Institut der Universität, H.-Herder-Str. 3, D-79104 Freiburg, Germany
${}^{*}$Corresponding author: [email protected]
\ocis
020.2649 Strong field laser physics; 020.4180 Multiphoton processes; 320.7120 Ultrafast phenomena
Coherent, short x-ray pulses are auspicious both for fundamental physics and applied science [1]. One of the successful ways to produce coherent short-wavelength radiation is based on high-order harmonic generation (HHG) [2].
With this state-of-the-art technique, coherent x-ray photons of about 1 keV [3] and short XUV pulses of about $100$ as [4] are generated via HHG in atomic gas jets. A further increase of the emitted photon energies cannot be achieved straightforwardly by an enhancement of the driving laser intensity as the interaction regime becomes relativistic and the returning electron misses the ionic core due to the relativistic drift that suppresses HHG [5]. Various methods to counteract the relativistic drift have been proposed, which have either limited scope of applicability, or are challenging to realize [6, 7]. On a different front, the attosecond laser technique is developing [2]. In particular, attosecond pulses superimposed onto an infrared laser field have recently been used to realize an attosecond streak camera [8] and to control HHG spectra in the nonrelativistic regime [9].
In this letter we investigate HHG in the relativistic regime which is driven by an infrared laser field crossed with an attosecond pulse train (APT) of soft x-ray radiation. In this case, the APT not only fixes the ionization time [9], but also induces an initial electron momentum that can compensate the subsequent relativistic drift in the infrared laser field. This permits electron recollisions with the atomic core and efficient HHG in the relativistic domain. We employ a crossed-beam setup where the polarization direction of the APT is parallel to the propagation direction of the laser field. Then the electron emission probability in the direction along or opposite the laser propagation is the largest and, in the second case, the compensation of the relativistic drift is most efficient (see Fig. 1). We calculate the HHG yield in this setup in the moderately relativistic regime and show a large increase in the HHG yield as compared to the conventional case of a driving laser field only.
Let us see how large the required energy $\Omega$ of the x-ray photons (equivalent to the carrier frequency of the APT)
has to be in order to compensate the relativistic drift. The kinetic energy of the electron directly
after one-photon ionization by the x-ray pulse is
$$\displaystyle\varepsilon_{0}=\Omega-I_{p},$$
(1)
with the ionization potential $I_{p}$ (atomic units are used throughout). For the drift compensation, the initial energy has to be of the same order of magnitude as the drift energy: $\varepsilon_{0}\approx p_{d}^{2}/2\approx c^{2}\xi^{4}/32$, with the drift momentum $p_{d}\approx c\xi^{2}/4$ [5], the relativistic field strength parameter $\xi=E_{0}/c\omega$, the electric field of the laser $E_{0}$, the laser frequency $\omega$, and the speed of light $c$. The regime is relativistic when the electron drift distance is larger than the wave-packet spreading: $\sqrt{2I_{p}}(\xi^{3}/16)(c/\omega)>1$ [5]. The intensity should be restricted, though, to prevent over-the-barrier ionization: $\xi<I_{p}^{3/2}/(4c\omega)$ [5]. These conditions are satisfied in the relativistic tunneling regime with $\xi>0.14$ and $I_{p}>2.4$ a.u. We employ a moderately relativistic infrared laser field of $\xi\approx 0.33$ (ponderomotive energy $U_{p}=500$ a.u.) and $I_{p}=5.5$ a.u. (e.g., Be${}^{2+}$ ions). Then the drift compensation demands soft x-rays of $\Omega\approx 9$ a.u. ($245$ eV) [10]. Under these conditions, a HHG cutoff $\omega_{c}\approx 3.17U_{p}$ of $43$ keV is attained.
We consider the interaction of an atomic system with the following superposition of a laser field and an APT.
The laser field is propagating in $z$-direction and defined by the vector potential $\mathbf{A}(\eta)=\mathbf{e}_{x}(cE_{0}/\omega)\cos\eta$, with $\eta=\omega(t-z/c)$
and the unit vector $\mathbf{e}_{x}$ in polarization direction.
The pulses in the APT are Gaussian-shaped, propagate in $y$-direction, and have the electric field
$\mathbf{E}^{a}(y,t)=\mathbf{e}_{z}E^{a}_{0}g(y,t)\sin\Omega(t-y/c)$, with the unit vector $\mathbf{e}_{z}$ in polarization direction,
the envelope $g(y,t)=\exp(-2(t-y/c-t_{0})^{2}/\tau^{2})$, the time delay $t_{0}$ between the laser field and the APT, and the width $\tau$ of the pulses. The repetition rate of the attosecond pulses is twice the laser frequency.
The electron dynamics can be described by the Klein-Gordon equation in the single-active-electron approximation and the radiation gauge:
$$\displaystyle\left(\partial^{\mu}\partial_{\mu}+c^{2}\right)\Psi({\rm x})=(V_{%
L}+V_{X}+V_{AI}+V_{H})\Psi({\rm x}),$$
(2)
with the electron-ion interaction operator $V_{AI}=2iV/c^{2}\partial_{t}+V^{2}/c^{2}$,
the atomic potential $V$, the time-space coordinate ${\rm x}=(ct,\mathbf{x})$, the electron-laser interaction operator $V_{L}=2i\mathbf{A}(\eta)\cdot\boldsymbol{\nabla}/c-\mathbf{A}(\eta)^{2}/c^{2}$,
the electron-APT interaction operator $V_{X}=2i\mathbf{A}^{a}(\eta)\cdot\boldsymbol{\nabla}/c-\mathbf{A}^{a}(\eta)^{2%
}/c^{2}$,
the APT vector potential $\mathbf{A}^{a}$, and the interaction operator between the electron and the quantized harmonic field $V_{H}({\rm x})=2\mathbf{A}_{H}({\rm x})/c\cdot(i\boldsymbol{\nabla}/c-\mathbf{%
A}_{tot}({\rm x})/c)$ (the $\mathbf{A}_{H}^{2}$ term is omitted as the harmonic field is a perturbation);
$\mathbf{A}_{H}({\rm x})=c\sqrt{2\pi/\omega_{H}}{\mathbf{e}}_{H}^{*}b^{\dagger}%
\exp(i\omega_{H}t-i{\bf k}_{H}\cdot{\bf x})$ is the vector potential of the harmonic field in second quantization,
with the unit vector ${\mathbf{e}}_{H}$ in polarization direction and the harmonic-photon creation operator $b^{\dagger}$, $\mathbf{A}_{tot}=\mathbf{A}+\mathbf{A}^{a}$,
$\omega_{H}$ is the harmonic frequency, and $\mathbf{k}_{H}$ the wave vector.
The differential HHG rate for the $n^{th}$ harmonic is given by $dw_{n}/d\Omega=n(\omega/c)^{3}|M_{n}|^{2}$ [7], where the HHG amplitude in the strong-field approximation reads
$$\displaystyle M_{n}$$
$$\displaystyle=$$
$$\displaystyle-i\int d^{4}{\rm x}^{\prime}\int d^{4}{\rm x}^{\prime\prime}\left%
\{\Phi({\rm x}^{\prime})^{*}V_{H}({\rm x}^{\prime})\right.$$
(3)
$$\displaystyle\times$$
$$\displaystyle\!\!\!\!\!\left.G^{V}_{L}({\rm x}^{\prime},{\rm x}^{\prime\prime}%
)\,\mathbf{x}^{\prime\prime}\cdot\left[\mathbf{E}({\rm x}^{\prime\prime})+%
\mathbf{E}^{a}({\rm x}^{\prime\prime})\right]\Phi({\rm x}^{\prime\prime})%
\right\},$$
with $\mathbf{E}=-\partial_{t}\mathbf{A}/c$. Here the APT can be treated in the dipole approximation since $E^{a}_{0}/\Omega c\ll 1$, and $c/\Omega\gg a_{B}$, with the Bohr radius $a_{B}$.
Taking into account the shortness of the pulses in the APT ($\tau=4$ a.u.),
we further assume that the dynamics of the electron wave-packet after the ionization is governed by the laser field only. Therefore, the Klein-Gordon Green function in the total field can be replaced by the Volkov-Klein-Gordon Green function in the laser field $G^{V}_{L}({\rm x}^{\prime},{\rm x}^{\prime\prime})$ [11]. Note that the wave function of the ground state $\Phi({\rm x})$ is an eigenstate of the energy operator in the radiation gauge [12].
The integral in Eq. (3) is evaluated with the saddle-point method.
Two ionization scenarios are described by Eq. (3). In the first scenario, the electron is both tunnel-ionized and propagated in the continuum by the laser field. In the other scenario, the process is initiated by the APT via single-photon ionization and the laser field takes over the electron in the continuum. In the relativistic regime, HHG due to the first scenario is heavily damped because of the relativistic drift. Therefore, we will consider the second scenario of the process. In this case, the saddle point equation reflects the energy conservation of Eq.(1) at the moment of ionization: $\varepsilon_{\mathbf{p}}(\eta^{\prime\prime})-c^{2}=\Omega-I_{p}$.
In Fig. 2 the HHG spectrum is displayed via Eq. (3).
A moderately relativistic infrared laser field of $2\times 10^{17}$ W/cm${}^{2}$ ($\xi\approx 0.33$) and an APT of soft x-rays where each pulse has a duration of $100$ as and an energy of $300$ pJ is employed.
Two cases are considered: 1) one-photon ionization by the APT
with ensuing propagation by the laser field and HHG in the APT propagation direction, and 2) tunneling ionization and propagation in a single laser field with HHG in the laser propagation direction. In the first case,
we use an APT with Gaussian pulses, or with delta pulses as in Ref. [9]. For the second case, two calculations are presented: the first one is a fully relativistic calculation via the Klein-Gordon equation, whereas the second one treats the laser field in dipole approximation (DA). The latter serves as an indicator for ideal HHG via the tunneling-recombination mechanism without the relativistic drift. Due to the drift, in the sole laser wave the HHG signal is strongly suppressed by more than 6 orders of magnitude compared to the DA results.
By superimposing a weak APT onto the infrared field, the HHG yield is enhanced by nearly 5 orders of magnitude and almost reaches the DA level. Moreover, the HHG rate can be increased proportionally to the APT intensity. Thus, the APT assistance enables recollisions with energies of 40 keV at significant rates. The description of the APT via delta-pulses is not justified for the parameters applied, since $\Omega\tau\ll 1$ [13], but gives a qualitatively correct picture. Note that the different HHG emission directions considered in 1) and 2) ensure phase-matching between the radiation of different atoms.
We now investigate the stability of the HHG enhancement against variation of the APT parameters.
Firstly, we vary the phase delay between the two fields in Fig. 3a. Phase delay changes of about 0.15 rad have only a minor influence on the near-cutoff HHG rates, whereas changes of 0.3 rad reduce them significantly. This is due to the attosecond pulse duration of $\omega\tau=0.2$ rad: a phase delay variation
of this order or larger reduces the electric field in the pulse whenever trajectories leading to high final energies are launched.
This condition also determines the limited spatial area, where
the APT assistance can strongly enhance the HHG yield.
Secondly, we vary the APT carrier frequency $\Omega$ at the optimal phase delay of the fields.
Fig. 3b shows that for small carrier frequencies the rate for high-energy harmonics is strongly reduced, since the energy supplied by the APT is too small to compensate the relativistic drift in the laser field.
For larger values of the carrier frequency starting from $\Omega\approx 9$ a.u., efficient recombination with the atomic core becomes possible.
From the spectra shown, a slow decrease in the rates for higher carrier frequencies can be deduced. This is due to the scaling of the dipole-matrix element on the electron energy.
The optimal carrier frequency thus amounts to about 9 a.u.; higher values up to 11 a.u. slightly reduce the process efficiency.
A rough estimation, taking into account the spatial restriction of the phase matched emitters, shows that $100$ harmonic photons of $42$ keV photon energy can be emitted within a $1$ keV spectral window at an ion density of $10^{16}$ cm${}^{-3}$ (the electron excursion length is less than the interatomic distance), a laser pulse duration of $100$ fs, and an interaction volume of $10$ mm${}^{3}$.
The HHG in the relativistic regime considered in this paper can also be used for generating ultra-short attosecond pulses via an appropriate filtering of harmonics.
An analysis of the harmonic chirp shows that the optimal spectral window is about 150 eV. Here, two trajectories are responsible for the HHG, which does not allow for a broad window.
As a result, short pulses of 10 as duration can be produced.
In conclusion, we have shown that the newly emerging field of attosecond physics can
be applied for the benefit of modifying relativistic electron recollision dynamics in laser fields.
With the proposed setup HHG with energies of 40 keV and significant rates as well as the generation of ultra-short pulses with a duration of 10 as is possible. Limiting factors for the respective macroscopic yields are the restricted areas which allow for phase-matching among the different ions and a high ion density.
This research was supported in part by the National Science Foundation under Grant No. PHY99-07949. We acknowledge the hospitality of the Kavli Institute for Theoretical Physics.
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Anharmonic oscillators in the complex plane, $\mathcal{PT}$-symmetry, and real eigenvalues.
Kwang C. Shin
Department of Mathematics, University of West Georgia, Carrollton, GA 30118 USA
(Date:: December 5, 2020)
Abstract.
For integers $m\geq 3$ and $1\leq\ell\leq m-1$, we study the eigenvalue problems
$-u^{\prime\prime}(z)+[(-1)^{\ell}(iz)^{m}-P(iz)]u(z)=\lambda u(z)$ with the boundary conditions that $u(z)$ decays to zero as
$z$ tends to infinity along the rays $\arg z=-\frac{\pi}{2}\pm\frac{(\ell+1)\pi}{m+2}$ in the complex plane,
where $P$ is a polynomial of degree at most $m-1$. We provide asymptotic expansions of the eigenvalues
$\lambda_{n}$. Then we show that if the eigenvalue problem is $\mathcal{PT}$-symmetric, then the eigenvalues are
all real and positive with at most finitely many exceptions. Moreover, we show that when $\gcd(m,\ell)=1$, the
eigenvalue problem has infinitely many real eigenvalues if and only if its translation or itself is
$\mathcal{PT}$-symmetric. Also, we will prove some other interesting direct and inverse spectral results.
2010 Mathematics Subject Classification: 34L40, 34L20, 81Q12
Key words: Anharmonic oscillators, asymptotics of the eigenvalues, $\mathcal{PT}$-symmetry
1. Introduction
In this paper, we study Schrödinger eigenvalue problems with real and complex polynomial potentials in the complex plane under various decaying boundary conditions. We provide explicit asymptotic formulas relating the index $n$ to a series of fractional powers of the eigenvalue $\lambda_{n}$ (see Theorem 1.1). Also, we recover the polynomial potentials from asymptotic formula of the eigenvalues (see Theorem 1.4 and Corollary 2.3) as well as applications to the so-called $\mathcal{PT}$-symmetric Hamiltonians (see Theorems 1.2 and 1.3).
For integers $m\geq 3$ and $1\leq\ell\leq m-1$, we consider the Schrödinger eigenvalue
problem
(1.1)
$$\left(H_{\ell}u\right)(z):=\left[-\frac{d^{2}}{dz^{2}}+(-1)^{\ell}(iz)^{m}-P(%
iz)\right]u(z)=\lambda u(z),\quad\text{for
some $\lambda\in\mathbb{C}$},$$
with the boundary condition that
(1.2)
$$\text{$u(z)\rightarrow 0$ as $z\rightarrow\infty$ along the two rays}\quad\arg
z%
=-\frac{\pi}{2}\pm\frac{(\ell+1)\pi}{m+2},$$
where $P$ is a polynomial of degree at most $m-1$ of the form
$$P(z)=a_{1}z^{m-1}+a_{2}z^{m-2}+\cdots+a_{m-1}z+a_{m},\quad a_{j}\in\mathbb{C}%
\,\,\text{\,for $1\leq j\leq m$}.$$
If a nonconstant function $u$ satisfies (1.1) with some $\lambda\in\mathbb{C}$ and the boundary condition
(1.2), then we call $\lambda$ an eigenvalue of $H_{\ell}$ and $u$ an eigenfunction of
$H_{\ell}$ associated with the eigenvalue $\lambda$. Sibuya [27] showed that the eigenvalues of $H_{\ell}$ are the zeros of an entire function of order
$\rho:=\frac{1}{2}+\frac{1}{m}\in(0,1)$ and hence, by the Hadamard factorization theorem (see, e.g., [7, p. 291]),
there are infinitely many eigenvalues. We call the entire function the Stokes multiplier (or the spectral
determinant), and the algebraic multiplicity of an eigenvalue $\lambda$ is the order of the zero $\lambda$ of the
Stokes multiplier. Also, the geometric multiplicity of an eigenvalue $\lambda$ is the number of linearly
independent eigenfunctions associated with the eigenvalue $\lambda$, that is $1$ for every eigenvalue $\lambda$ [16, §7.4].
We number the eigenvalues $\{\lambda_{n}\}_{n\geq N_{0}}$ in the
order of nondecreasing magnitudes, counting their algebraic
multiplicities. We will show that the magnitude of large
eigenvalues is strictly increasing (see, Lemma 2.2) and
hence, there is a unique way of ordering large eigenvalues, but
this is not guaranteed for small eigenvalues. However, how we
order these small eigenvalues will not affect results in this
paper. Throughout this paper, we will use $\lambda_{n}$ to denote
the eigenvalues of $H_{\ell}$ without explicitly indicating their
dependence on the potential and the boundary condition. Also, we
let
$$a:=(a_{1},a_{2},\ldots,a_{m})\in\mathbb{C}^{m}$$
be the coefficient vector
of $P(z)$.
The anharmonic oscillators $H_{\ell}$ with the various boundary conditions (1.2) are considered in
[4, 24]. When $m$ is even and $\ell=\frac{m}{2}$, $H_{\frac{m}{2}}$ is a Schrödinger operator in
$L^{2}(\mathbb{R})$ (see, e.g.,
[1, 2, 5, 11, 12, 15, 18, 19]). This is self-adjoint if the potential $V(z)=(-1)^{\ell}(iz)^{m}-P(iz)$ is real on the real line, and
non-self-adjoint if the potential is non-real.
Some particular classes of $H_{1}$ have been studied extensively in recent years in the context of theory of
$\mathcal{PT}$-symmetry [3, 6, 10, 14, 20, 26, 28, 29]. The $H_{\ell}$ is
$\mathcal{PT}$-symmetric if the potential $V$ satisfies $\overline{V(-\overline{z})}=V(z)$, $z\in\mathbb{C}$, that is equivalent to $a\in\mathbb{R}^{m}$. In this paper, we will generalize results in [26] (where $H_{1}$ is studied) to $1\leq\ell\leq m-1$ and introduce some new results.
These results are consequences of the following asymptotic expansion of the eigenvalues.
Theorem 1.1.
For each integer $m\geq 3$ and $1\leq\ell\leq m-1$, there exists an integer $N_{0}=N_{0}(m,\ell)$ such that the
eigenvalues $\{\lambda_{n}\}_{n\geq N_{0}}$ of $H_{\ell}$ satisfy
(1.3)
$$n+\frac{1}{2}=\sum_{j=0}^{m+1}c_{\ell,j}(a)\lambda_{n}^{\frac{1}{2}+\frac{1-j}%
{m}}+\eta_{\ell}(a)+O\left(\lambda_{n}^{-\rho}\right)\quad\text{as $n\to\infty%
$},$$
where $c_{\ell,j}(a)$ and $\eta_{\ell}(a)$ are defined in (2.4) and (2.5), respectively.
Also, we obtain the partial reality of the eigenvalues for $\mathcal{PT}$-symmetric $H_{\ell}$.
Theorem 1.2.
If $H_{\ell}$ is $\mathcal{PT}$-symmetric, then eigenvalues are all real with at most finitely many exceptions.
Proof.
If $u(z)$ is an eigenfunction associated with the eigenvalue $\lambda$ of a $\mathcal{PT}$-symmetric $H_{\ell}$,
then $\overline{u(-\overline{z})}$ is also an eigenfunction associated with $\overline{\lambda}$. In Corollary
2.2, we will show that $|\lambda_{n}|<|\lambda_{n+1}|$ for all large $n$ and $\arg(\lambda_{n})\to 0$.
Thus, $\lambda_{n}$ are real and positive for all large $n$ since $|\lambda|=|\overline{\lambda}|$.
∎
Many $\mathcal{PT}$-symmetric operators have real eigenvalues only [10, 22, 23]. However, there are some $\mathcal{PT}$-symmetric $H_{\ell}$ that produce a finite number of non-real eigenvalues [2, 8, 9, 14]. When $H_{\ell}$ is self-adjoint, the spectrum is real. Conversely, when the spectrum is real, what can we conclude about $H_{\ell}$? The next theorem provides a necessary and sufficient condition for $H_{\ell}$ to have infinitely many real eigenvalues.
Theorem 1.3.
Suppose that $\gcd(m,\ell)=1$. Then $H_{\ell}$ with the potential $V(z)=-(iz)^{m}+P(iz)$ has infinitely many real
eigenvalues if and only if $H_{\ell}$ with the potential $V(z-z_{0})$ for some $z_{0}\in\mathbb{C}$ is
$\mathcal{PT}$-symmetric.
The next theorem reveals an interesting feature of the
eigenvalues as a sequence:
Theorem 1.4.
Suppose that $\gcd(m,\ell)=1$. Let $\{\lambda_{n}\}_{n\geq N_{0}}$ and
$\{\widetilde{\lambda}_{n}\}_{n\geq N_{0}}$ be the eigenvalues of
$H_{\ell}$ with the potentials $V$ and $\widetilde{V}$,
respectively. Suppose that
$\lambda_{n}-\widetilde{\lambda}_{n}=o\left(1\right)$ as $n\to\infty$.
Then $\widetilde{V}(z)=V(z-z_{0})$ for some $z_{0}\in\mathbb{C}$ and $\lambda_{n}=\widetilde{\lambda}_{n}$ for all $n\geq N_{0}$ after, if needed, small eigenvalues are reordered.
The asymptotic expansions of the eigenvalues of $H_{\ell}$ with $\ell=\lfloor\frac{m}{2}\rfloor$ have been studied
in, for example, [13, 15, 18].
Maslov [18] computed the first three terms of asymptotic expansions of $\lambda_{n}^{\frac{3}{4}}$,
where $\lambda_{n}$ are the eigenvalues of $-\frac{d^{2}}{dx^{2}}u+x^{4}u=\lambda u,\,\,u\in L^{2}(\mathbb{R}).$
Helffer and Robert [15] considered
$$-\frac{d^{2k}}{dx^{2k}}u+(x^{2m}+p(x))u=\lambda u,\quad u\in L^{2}(\mathbb{R}),$$
where $k,\,m$ are positive integers and where $p(\cdot)$ is a real polynomial of degree at most $2m-1$.
They obtained the existence of asymptotic expansions of the eigenvalues to all orders, and suggested an explicit
way of computing the coefficients of the asymptotic expansion. In particular, for the case when the potential is
$\varepsilon x^{4}+x^{2}$, $\varepsilon>0$, Helffer and Robert [15] computed the first nine terms of the
asymptotic expansion of $\lambda_{n}^{\frac{3}{4}}$. Also, Fedoryuk [13, §3.3] considered (1.1)
with complex polynomial potentials and with (1.2) for $\ell=\lfloor\frac{m}{2}\rfloor$ and computed the
first term in the asymptotic expansion. Also, Sibuya [27] computed the first term in the asymptotic
expansion for $\ell=1$.
This paper is organized as follows. In Section 2, we define $c_{\ell,j}(a)$ and some other notations. Also, we invert (1.3), expressing $\lambda_{n}$ as a series of fractional powers of the index $n$ and prove Theorems 1.3 and 1.4, and other interesting direct and inverse spectral results. In Section 3, we introduce some properties of the solutions of the differential equation in (1.1), due to Hille [16] and Sibuya [27]. We study the asymptotic of the Stokes multiplier associated with $H_{1}$ in Section 4 and treat the general case $H_{\ell}$ in Section 5. In Section 6, we relate the eigenvalues of $H_{\ell}$ with the zeros of the Stokes multiplier. We prove Theorem 1.1 for $1\leq\ell<\frac{m}{2}$ in Section 7 and for $\ell\geq\frac{m}{2}$ in Section 8.
2. Notations and corollaries
In this section, we will define $c_{\ell,j}(a)$ and some other notations and introduce some corollaries of Theorem 1.1.
We define, for nonnegative integers $k,j$,
(2.1)
$$b_{j,k}({a})$$ is the coefficient of $$\frac{1}{z^{j}}$$ in
$${\frac{1}{2}\choose{k}}\left(\frac{a_{1}}{z}+\frac{a_{2}}{z^{2}}+\dots+\frac{a%
_{m}}{z^{m}}\right)^{k}$$ and
(2.2)
$$b_{j}({a})=\sum_{k=0}^{j}b_{j,k}({a}),\quad j\in\mathbb{N}.$$
Notice that $b_{0,0}(a)=b_{0}(a)=1$, $b_{j,0}(a)=0$ if $j\geq 1$, and $b_{j,k}(a)=0$ if $j<k$ or $k<\frac{j}{m}$.
We will also use
(2.3)
$$K_{m,j}({a})=\sum_{k=0}^{j}K_{m,j,k}\,b_{j,k}({a})\,\,\,\text{for $j\geq 0$,}$$
where for $k<\frac{j}{m}$, $K_{m,j,k}=0$ and for $\frac{j}{m}\leq k\leq j$,
$$\displaystyle K_{m,j,k}$$
$$\displaystyle:=\left\{\begin{array}[]{cl}\frac{B\left(\frac{1}{2},1+\frac{1}{m%
}\right)}{2\cos\left(\frac{\pi}{m}\right)}&\text{if $j=k=0$},\\
&\\
-\frac{2}{m}&\text{if $j=k=1$},\\
&\\
\frac{2}{m}\left(\ln 2-\sum_{s=1}^{k-1}\frac{1}{2s-1}\right)&\text{if $j=\frac%
{m}{2}+1$, $m$ even,}\\
&\\
\frac{1}{m}B\left(k-\frac{j-1}{m},\,\frac{j-1}{m}-\frac{1}{2}\right)&\text{if %
$j\not=1$ or $j\not=\frac{m}{2}+1$.}\end{array}\right.$$
Now we are ready to define $c_{\ell,j}(a)$ as follows: for $1\leq j\leq m+1$,
(2.4)
$$c_{\ell,j}(a)=-\frac{2}{\pi}\sum_{k=0}^{j}(-1)^{(\ell+1)k}K_{m,j,k}b_{j,k}(a)%
\sin\left(\frac{(j-1)\ell\pi}{m}\right)\cos\left(\frac{(j-1)\pi}{m}\right),$$
and we also define
(2.5)
$$\eta_{\ell}(a)=\left\{\begin{array}[]{rl}(-1)^{\frac{\ell-1}{2}}\frac{2\nu(a)}%
{m}&\text{if $\ell$ is odd,}\\
&\\
0&\text{if otherwise}\end{array}\right.\,\,\text{and }\,\,\mu({a})=\frac{m}{4}%
-\nu({a}),$$
where
$$\displaystyle\nu({a})=\left\{\begin{array}[]{rl}b_{\frac{m}{2}+1}({a})&\text{%
if $m$ is even,}\\
&\\
0&\text{if $m$ is odd.}\end{array}\right.$$
2.1. Further direct spectral results
One can invert the asymptotic formulas (1.3) to obtain formulas for $\lambda_{n}$ in terms of $n$.
Corollary 2.1.
One can compute $d_{\ell,j}({a})$ explicitly such that
(2.6)
$$\lambda_{n}=\sum_{j=0}^{m+1}d_{\ell,j}({a})\left(n+\frac{1}{2}\right)^{\frac{2%
m}{m+2}\left(1-\frac{j}{m}\right)}+O\left(n^{-\frac{4}{m+2}}\right),$$
where $d_{\ell,0}(a)=\left(\pi^{-1}B\left(\frac{1}{2},1+\frac{1}{m}\right)\sin\left(%
\frac{\ell\pi}{m}\right)\right)^{-\frac{2m}{m+2}}>0$.
Proof.
Equations (1.3) is an asymptotic equation and it can be solved for $\lambda_{n}$, resulting in
(2.6). For details, see, for example, [24, §5].
∎
In the next corollary, we provide an asymptotic formula for the nearest neighbor spacing of the eigenvalues. And
the large eigenvalues increase monotonically in magnitude and have their argument approaching zero.
Corollary 2.2.
The space between successive eigenvalues is
(2.7)
$$\lambda_{n+1}-\lambda_{n}\underset{n\to+\infty}{=}\frac{2m}{m+2}\,d_{\ell,0}%
\cdot\left(n+\frac{1}{2}\right)^{\frac{m-2}{m+2}}+o\left(n^{\frac{m-2}{m+2}}%
\right).$$
In particular, $\lim_{n\to\infty}\left|\lambda_{n+1}-\lambda_{n}\right|=\infty$ and
$\lim_{n\to+\infty}\arg(\lambda_{n})=0.$ Hence:
$$\left|\lambda_{n}\right|<\left|\lambda_{n+1}\right|\,\,\,\text{for all large $%
n$}.$$
Proof.
These claims are consequences of (2.6) and the generalized binomial expansion. For details, see, for example, [24, §6].
∎
Suppose that $\lambda_{n}$ are eigenvalues of $H_{\ell}$ for some $P$. Then the degree $m$ of the polynomial
potential can be recovered by
$$\frac{2m}{m+2}=\lim_{n\to\infty}\frac{\ln(|\lambda_{n}|)}{\ln n}<2.$$
The next corollary shows that one can recover the polynomial potential from the asymptotic formula for the
eigenvalues.
Corollary 2.3.
Suppose that $\gcd(m,\ell)=1$ and that $\{\lambda_{n}\}_{n\geq N_{0}}$ are the eigenvalues of $H_{\ell}$ with the
potential $V$. If
(2.8)
$$\sum_{j=0}^{m+1}c_{j}^{*}\lambda_{n}^{\frac{1}{2}+\frac{1-j}{m}}+O\left(%
\lambda_{n}^{-\frac{1}{2}-\frac{1}{m}}\right){=}\left(n+\frac{1}{2}\right)$$
as $n\to\infty$ for some $c_{j}^{*}\in\mathbb{C}$, or if
(2.9)
$$\lambda_{n}=\sum_{j=0}^{m+1}d_{j}^{*}\left(n+\frac{1}{2}\right)^{\frac{2m}{m+2%
}\left(1-\frac{j}{m}\right)}+O\left(n^{-\frac{4}{m+2}}\right),$$
for some $d_{j}^{*}\in\mathbb{C}$, then there exists $V_{0}$ such that for each polynomial $V$ with which $H_{\ell}$ generates
$\{\lambda_{n}\}_{n\geq N_{0}}$, $V(z)=V_{0}(z-z_{0})$ for some $z_{0}\in\mathbb{C}$.
Proof.
The coefficients $c_{\ell,j}({a})$ and $d_{\ell,j}({a})$ have the following properties.
(i) The $c_{\ell,j}({a})$ and $d_{\ell,j}({a})$ are all real polynomials in terms of the coefficients ${a}$ of $P(x)$.
(ii) The coefficients $c_{\ell,0}({a})$ and $d_{\ell,0}({a})$ do not depend on ${a}$ (they are constants).
(iii) For $1\leq j\leq m$, the polynomials $c_{\ell,j}({a})$ and $d_{\ell,j}({a})$ depend only on $a_{1},\,a_{2},\dots,a_{j}$.
Furthermore, if $\gcd(m,\ell)=1$ and if $j\not=1,\,\frac{m}{2}+1$, then $c_{\ell,j}({a})$ and $d_{\ell,j}({a})$ are
non-constant linear functions of $a_{j}$.
Here we will sketch the proof when (2.8) is assumed. First, notice that $c_{1}^{*}=0$. Otherwise, $\lambda_{n}$
are not the eigenvalues of $H_{\ell}$, according to (2.4). Choose $a_{1}=0$. Then since $c_{\ell,2}({a})$ is a non-constant linear function of $a_{2}$ and depends on $a_{1}$ and $a_{2}$, $c_{\ell,2}({a})=c_{2}^{*}$
determines $a_{2}$. If $m$ is even, $c_{\ell,\frac{m}{2}+1}(a)=0$ and we use $\eta_{\ell}(a)$ instead of
$c_{\ell,\frac{m}{2}+1}(a)$. Suppose that for $2\leq j\leq m-1$, the $c_{2}^{*},\dots,c_{j}^{*}$ uniquely determine
$a_{2},\,a_{3},\dots,a_{j}$. Then by (iii), $c_{\ell,j+1}({a})=c_{j+1}^{*}$ determines $a_{j+1}$ uniquely. So by
induction, one can determine all $a_{j}$ for $2\leq j\leq m$ and $V_{0}$ is the potential with these $a_{j}$’s. Then
$V(z)=V_{0}(z-z_{0})$ for some $z_{0}\in\mathbb{C}$. Otherwise, the eigenvalues $\{\lambda_{n}\}$ do not satisfy (2.8).
The case when (2.9) holds can be handled similarly.
∎
Next, we will prove Theorems 1.3 and 1.4.
Proof of Theorems 1.3.
If $V(z-z_{0})$ for some $z_{0}\in\mathbb{C}$ is $\mathcal{PT}$-symmetric, then by Theorem 1.2, all but finitely many eigenvalues are real and hence, there are infinitely many real eigenvalues.
Suppose that $H_{\ell}$ with the potential $V(z)$ has infinitely many real eigenvalues. Then we can always find $z_{0}\in\mathbb{C}$ so that $V(z-z_{0})$ has no $z^{m-1}$-term, that is, $a_{1}=0$. We will show that $V(z-z_{0})$ is $\mathcal{PT}$-symmetric. Since there are infinitely many real eigenvalues, $c_{\ell,j}(a)$, $0\leq j\leq m+1$, and $\eta_{\ell}$ in (1.3) are real. Also, the sine term in (2.4) does not vanish except for $j=1$. And the cosine term does not vanish except for $j=\frac{m}{2}+1$ when $m$ is even in which $\eta_{\ell}$ replaces the role of $c_{\ell,\frac{m}{2}}$.
Like we did for proof of Corollary 2.3, since $a_{1}=0$, from the above properties of $c_{\ell,j}$ and $\eta_{\ell}$, by the induction, we can show that $a_{j}$ for $2\leq j\leq m$ are all real and hence, $V(z-z_{0})$ is $\mathcal{PT}$-symmetric.
∎
Proof of Theorems 1.4.
Suppose that
$\lambda_{n}-\widetilde{\lambda}_{n}=o\left(1\right)$ as $n\to\infty$. Then $d_{\ell,j}(a)=d_{\ell,j}(\widetilde{a})$ for all $1\leq j\leq m+1$. Then Corollary 2.3 completes proof.
∎
3. Properties of the solutions
In this section, we introduce work of Hille [16] and Sibuya [27] about properties of the solutions of (1.1).
First, we scale equation (1.1) because many facts that we
need later are stated for the scaled equation. Let $u$ be a
solution of (1.1) and let $v(z)=u(-iz)$.
Then $v$ solves
(3.1)
$$-v^{\prime\prime}(z)+[(-1)^{\ell+1}z^{m}+P(z)+\lambda]v(z)=0.$$
When $\ell$ is odd, (3.1) becomes
(3.2)
$$-v^{\prime\prime}(z)+[z^{m}+P(z)+\lambda]v(z)=0.$$
Later we will handle the case when $\ell$ is even.
Since we scaled the argument of $u$, we must rotate the boundary conditions. We state them in a more general
context by using the following definition.
Definition .
The Stokes sectors $S_{k}$ of the equation (3.2)
are
$$S_{k}=\left\{z\in\mathbb{C}:\left|\arg(z)-\frac{2k\pi}{m+2}\right|<\frac{\pi}{%
m+2}\right\}\quad\text{for}\quad k\in\mathbb{Z}.$$
See Figure 1.
It is known from Hille [16, §7.4] that every nonconstant solution of (3.2) either decays to
zero or blows up exponentially, in each Stokes sector $S_{k}$.
Lemma 3.1 ([16, §7.4]).
(i)
For each $k\in\mathbb{Z}$, every solution $v$ of (3.2) is asymptotic to
(3.3)
$$(const.)z^{-\frac{m}{4}}\exp\left[\pm\int^{z}\left[\xi^{m}+P(\xi)+\lambda%
\right]^{\frac{1}{2}}\,d\xi\right]$$
as $z\rightarrow\infty$ in every closed subsector of $S_{k}$.
(ii)
If a nonconstant solution $v$ of (3.2) decays in $S_{k}$, it must blow up
in $S_{k-1}\cup S_{k+1}$. However, when $v$ blows up in $S_{k}$, $v$ need not be decaying in $S_{k-1}$ or in $S_{k+1}$.
Lemma 3.1 (i) implies that if $v$ decays along one ray in $S_{k}$, then it decays along all rays in $S_{k}$.
Also, if $v$ blows up along one ray in $S_{k}$, then it blows up along all rays in $S_{k}$. Thus, the boundary
conditions (1.2) with $1\leq\ell\leq m-1$ represent all decaying boundary conditions.
Still with $\ell$ odd, the two rays in (1.2) map, by $z\mapsto-iz$, to the rays
$\arg(z)=\pm\frac{(\ell+1)\pi}{m+2}$ which are the center rays of the Stokes sectors $S_{\frac{\ell+1}{2}}$ and
$S_{-\frac{\ell+1}{2}}$ and the boundary conditions (1.2) on $u$ become
$$v$$ decays to zero in the Stokes sector $$S_{\frac{\ell+1}{2}}$$ and $$S_{-\frac{\ell+1}{2}}$$.
When $\ell$ is even, we let
$y(z)=v(\omega^{-\frac{1}{2}}z)$ so that (3.1) becomes
(3.4)
$$-y^{\prime\prime}(z)+[z^{m}+\omega^{-1}P(\omega^{-\frac{1}{2}}z)+\omega^{-1}%
\lambda]y(z)=0,$$
where
$$\omega=\exp\left[\frac{2\pi i}{m+2}\right]$$
and hence, $\omega^{-{\frac{m}{2}+1}}=-1$. For these cases, the boundary conditions (1.2) become
$$y$$ decays to zero in the Stokes sector $$S_{\frac{\ell+2}{2}}$$ and $$S_{-\frac{\ell}{2}}$$.
The following theorem is a special case of Theorems 6.1, 7.2, 19.1 and 20.1 of Sibuya [27] that is the
main ingredient of the proofs of the main results in this paper. For this we will use $r_{m}=-\frac{m}{4}$ if $m$ is
odd, and $r_{m}=-\frac{m}{4}-b_{\frac{m}{2}+1}(a)$ if $m$ is even.
Theorem 3.2.
Equation (3.2), with $a\in\mathbb{C}^{m}$, admits a solution $f(z,a,\lambda)$ with the following properties.
(i)
$f(z,a,\lambda)$ is an entire function of $z,a$ and $\lambda$.
(ii)
$f(z,a,\lambda)$ and $f^{\prime}(z,a,\lambda)=\frac{\partial}{\partial z}f(z,a,\lambda)$ admit
the following asymptotic expansions. Let $\varepsilon>0$. Then
$$\displaystyle f(z,a,\lambda)=$$
$$\displaystyle\qquad z^{r_{m}}(1+O(z^{-1/2}))\exp\left[-F(z,a,\lambda)\right],$$
$$\displaystyle f^{\prime}(z,a,\lambda)=$$
$$\displaystyle-z^{r_{m}+\frac{m}{2}}(1+O(z^{-1/2}))\exp\left[-F(z,a,\lambda)%
\right],$$
as $z$ tends to infinity in the sector $|\arg z|\leq\frac{3\pi}{m+2}-\varepsilon$, uniformly on each compact set of $(a,\lambda)$-values .
Here
$$F(z,a,\lambda)=\frac{2}{m+2}z^{\frac{m}{2}+1}+\sum_{1\leq j<\frac{m}{2}+1}%
\frac{2}{m+2-2j}b_{j}(a)z^{\frac{1}{2}(m+2-2j)}.$$
(iii)
Properties (i) and (ii) uniquely determine the solution $f(z,a,\lambda)$ of (3.2).
(iv)
For each fixed $a\in\mathbb{C}^{m}$ and $\delta>0$, $f$ and $f^{\prime}$ also admit the asymptotic expansions,
(3.5)
$$\displaystyle f(0,a,\lambda)=$$
$$\displaystyle[1+O\left(\lambda^{-\rho}\right)]\lambda^{-1/4}\exp\left[L(a,%
\lambda)\right],$$
(3.6)
$$\displaystyle f^{\prime}(0,a,\lambda)=$$
$$\displaystyle-[1+O\left(\lambda^{-\rho}\right)]\lambda^{1/4}\exp\left[L(a,%
\lambda)\right],$$
as $\lambda\to\infty$ in the sector $|\arg(\lambda)|\leq\pi-\delta$, uniformly on each compact set of
$a\in\mathbb{C}^{m}$, where
$$\displaystyle L(a,\lambda)=\left\{\begin{array}[]{rl}&\int_{0}^{+\infty}\left(%
\sqrt{t^{m}+P(t)+\lambda}-t^{\frac{m}{2}}-\sum_{j=1}^{\frac{m+1}{2}}b_{j}(a)t^%
{\frac{m}{2}-j}\right)\,dt\quad\text{if $m$ is odd,}\\
&\int_{0}^{+\infty}\left(\sqrt{t^{m}+P(t)+\lambda}-t^{\frac{m}{2}}-\sum_{j=1}^%
{\frac{m}{2}}b_{j}(a)t^{\frac{m}{2}-j}-\frac{b_{\frac{m}{2}+1}}{t+1}\right)\,%
dt\quad\text{if $m$ is even.}\end{array}\right.$$
(v)
The entire functions $\lambda\mapsto f(0,a,\lambda)$ and $\lambda\mapsto f^{\prime}(0,a,\lambda)$ have orders $\frac{1}{2}+\frac{1}{m}$.
Proof.
In Sibuya’s book [27], see Theorem 6.1 for a proof of (i) and (ii); Theorem 7.2 for a proof of (iii) with the error terms $o(1)$;
and Theorem 19.1 for a proof of (iv). Moreover, (v) is a consequence of (iv) along with Theorem 20.1 in
[27]. The error terms in (iii) are improved from $o(1)$ to $O\left(\lambda^{-\rho}\right)$ in [25]. Note that properties (i), (ii) and (iv) are summarized on pages 112–113 of Sibuya [27].
∎
Remarks.
Throughout this paper, we will deal with numbers like
$\left(\omega^{\alpha}\lambda\right)^{s}$ for some $s\in\mathbb{R}$ and $\alpha\in\mathbb{C}$. As usual, we will use
$$\omega^{\alpha}=\exp\left[\alpha\frac{2\pi i}{m+2}\right]$$
and if $\arg(\lambda)$ is specified, then
$$\arg\left(\left(\omega^{\alpha}\lambda\right)^{s}\right)=s\left[\arg(\omega^{%
\alpha})+\arg(\lambda)\right]=s\left[{\rm Re}\,(\alpha)\frac{2\pi}{m+2}+\arg(%
\lambda)\right],\quad s\in\mathbb{R}.$$
Lemma 3.3.
Let $m\geq 3$ and $a\in\mathbb{C}^{m}$ be fixed. Then
(3.7)
$$L(a,\lambda)=\sum_{j=0}^{\infty}K_{m,j}(a)\lambda^{\frac{1}{2}+\frac{1-j}{m}}-%
\frac{\nu(a)}{m}\ln(\lambda)$$
as $\lambda\to\infty$ in the sector $|\arg(\lambda)|\leq\pi-\delta$, uniformly on each compact set of
$a\in\mathbb{C}^{m}$.
Proof.
See [24] for a proof.
∎
Sibuya [27] proved the following corollary, directly from Theorem 3.2, that will be used later in
Sections 4 and 5.
Corollary 3.4.
Let $a\in\mathbb{C}^{m}$ be fixed. Then $L(a,\lambda)=K_{m}\lambda^{\frac{1}{2}+\frac{1}{m}}(1+o(1))$ as $\lambda$ tends
to infinity in the sector $|\arg\lambda|\leq\pi-\delta$, and hence
(3.8)
$${\rm Re}\,\left(L(a,\lambda)\right)=K_{m}\cos\left(\frac{m+2}{2m}\arg(\lambda)%
\right)|\lambda|^{\frac{1}{2}+\frac{1}{m}}(1+o(1))$$
as $\lambda\to\infty$ in the sector $|\arg(\lambda)|\leq\pi-\delta$.
In particular, ${\rm Re}\,\left(L(a,\lambda)\right)\to+\infty$ as $\lambda\to\infty$ in any closed subsector of the
sector $|\arg(\lambda)|<\frac{m\pi}{m+2}$. In addition, ${\rm Re}\,\left(L(a,\lambda)\right)\to-\infty$ as
$\lambda\to\infty$ in any closed subsector of the sectors $\frac{m\pi}{m+2}<|\arg(\lambda)|<\pi-\delta$.
Based on Corollary 3.4, Sibuya [27, Thm. 29.1] also computed the leading term in
(1.3) for $\ell=1$. Also, Sibuya [27] constructed solutions of (3.2) that decays
in $S_{k}$, $k\in\mathbb{Z}$. Before we introduce this, we let
(3.9)
$$G^{\ell}(a):=(\omega^{(m+1)\ell}a_{1},\omega^{m\ell}a_{2},\ldots,\omega^{2\ell%
}a_{m})\quad\text{for}\quad\ell\in\frac{1}{2}\mathbb{Z}.$$
Then we have the following lemma, regarding some properties of $G^{\ell}(\cdot)$.
Lemma 3.5.
For $a\in\mathbb{C}^{m}$ fixed, and $\ell_{1},\ell_{2},\ell\in\frac{1}{2}\mathbb{Z}$,
$G^{\ell_{1}}(G^{\ell_{2}}(a))=G^{\ell_{1}+\ell_{2}}(a)$, and
$$b_{j,k}(G^{\ell}(a))=\omega^{((m+2)k-j)\ell}b_{j,k}(a),\quad\ell\in\frac{1}{2}%
\mathbb{Z}.$$
In particular,
$$b_{j}(G^{\ell}(a))=\omega^{-j\ell}b_{j}(a),\,\,\ell\in\mathbb{Z}.$$
Next, recall that the function
$f(z,a,\lambda)$ in Theorem 3.2 solves (3.2) and decays to zero exponentially as $z\rightarrow\infty$ in
$S_{0}$, and blows up in $S_{-1}\cup S_{1}$. One can check that the function
$$f_{k}(z,a,\lambda):=f(\omega^{-k}z,G^{k}(a),\omega^{2k}\lambda),\quad k\in%
\mathbb{Z},$$
which is obtained by scaling $f(z,G^{k}(a),\omega^{2k}\lambda)$ in the $z$-variable, also solves (3.2). It is clear
that $f_{0}(z,a,\lambda)=f(z,a,\lambda)$, and that $f_{k}(z,a,\lambda)$ decays in $S_{k}$ and blows up in $S_{k-1}\cup S_{k+1}$ since
$f(z,G^{k}(a),\omega^{2k}\lambda)$ decays in $S_{0}$. Since no nonconstant solution decays in two consecutive Stokes sectors (see Lemma 3.1
(ii)), $f_{k}$ and $f_{k+1}$ are linearly independent and hence any solution of (3.2) can be expressed as a linear combination of these two.
Especially, for each $k\in\mathbb{Z}$ there exist some coefficients $C_{k}(a,\lambda)$ and $\widetilde{C}_{k}(a,\lambda)$ such that
(3.10)
$$f_{k}(z,a,\lambda)=C_{k}(a,\lambda)f_{0}(z,a,\lambda)+\widetilde{C}_{k}(a,%
\lambda)f_{-1}(z,a,\lambda).$$
We then see that
(3.11)
$$C_{k}(a,\lambda)=-\frac{\mathcal{W}_{k,-1}(a,\lambda)}{\mathcal{W}_{-1,0}(a,%
\lambda)}\quad\text{and}\quad\widetilde{C}_{k}(a,\lambda)=\frac{\mathcal{W}_{k%
,0}(a,\lambda)}{\mathcal{W}_{-1,0}(a,\lambda)},$$
where $\mathcal{W}_{j,\ell}=f_{j}f_{\ell}^{\prime}-f_{j}^{\prime}f_{\ell}$ is the Wronskian of $f_{j}$ and $f_{\ell}$. Since both
$f_{j},\,f_{\ell}$ are solutions of the same linear equation (3.2), we know that the Wronskians are
constant functions of $z$. Also, $f_{k}$ and $f_{k+1}$ are linearly independent, and hence
$\mathcal{W}_{k,k+1}\not=0$ for all $k\in\mathbb{Z}$.
Also, the following is an easy consequence of (3.10) and (3.11). For each $k,\ell\in\mathbb{Z}$ we have
$$\displaystyle\mathcal{W}_{\ell,k}(a,\lambda)$$
$$\displaystyle=C_{k}(a,\lambda)\mathcal{W}_{\ell,0}(a,\lambda)+\widetilde{C}_{k%
}(a,\lambda)\mathcal{W}_{\ell,-1}(a,\lambda)$$
(3.12)
$$\displaystyle=-\frac{\mathcal{W}_{k,-1}(a,\lambda)\mathcal{W}_{\ell,0}(a,%
\lambda)}{\mathcal{W}_{-1,0}(a,\lambda)}+\frac{\mathcal{W}_{k,0}(a,\lambda)%
\mathcal{W}_{\ell,-1}(a,\lambda)}{\mathcal{W}_{-1,0}(a,\lambda)}.$$
Moreover, we have the following lemma that is useful later on.
Lemma 3.6.
Suppose $k,\,j\in\mathbb{Z}$. Then
(3.13)
$$\mathcal{W}_{k+1,j+1}(a,\lambda)=\omega^{-1}\mathcal{W}_{k,j}(G(a),\omega^{2}%
\lambda),$$
and $\mathcal{W}_{0,1}(a,\lambda)=2\omega^{\mu(a)}$, where $\mu(a)=\frac{m}{4}-\nu(a)$.
Proof.
See Sibuya [27, pp. 116-118].
∎
4. Asymptotics of $\mathcal{W}_{-1,1}(a,\lambda)$
In this section, we introduce asymptotic expansions of $\mathcal{W}_{-1,1}(a,\lambda)$ as $\lambda\to\infty$ along
the rays in the complex plane [24].
First, we provide an asymptotic expansion of the Wronskian $\mathcal{W}_{0,j}(a,\lambda)$ of $f_{0}$ and $f_{j}$
that will be frequently used later.
Lemma 4.1.
Suppose that $1\leq j\leq\frac{m}{2}+1$. Then for each $a\in\mathbb{C}^{m}$,
(4.1)
$$\mathcal{W}_{0,j}(a,\lambda)=[2i\omega^{-\frac{j}{2}}+O\left(\lambda^{-\rho}%
\right)]\exp\left[L(G^{j}(a),\omega^{2j-m-2}\lambda)+L(a,\lambda)\right],$$
as $\lambda\to\infty$ in the sector
(4.2)
$$-\pi+\delta\leq\pi-\frac{4j\pi}{m+2}+\delta\leq\arg(\lambda)\leq\pi-\delta.$$
Next, we provide an asymptotic expansion of $\mathcal{W}_{-1,1}(a,\lambda)$ as $\lambda\to\infty$ in the sector
near the negative real axis.
Theorem 4.2.
Let $m\geq 3$, $a\in\mathbb{C}^{m}$ and $0<\delta<\frac{\pi}{m+2}$ be fixed. Then
(4.3)
$$\mathcal{W}_{-1,1}(a,\lambda)=[2i+O\left(\lambda^{-\rho}\right)]\exp\left[L(G^%
{-1}(a),\omega^{-2}\lambda)+L(G(a),\omega^{-m}\lambda)\right],$$
as $\lambda\to\infty$ along the rays in the sector
(4.4)
$$\pi-\frac{4\pi}{m+2}+\delta\leq\arg(\lambda)\leq\pi+\frac{4\pi}{m+2}-\delta.$$
Proof.
This is an easy consequence of Lemma 4.1 with $j=2$ and (3.13).
∎
Also, for integers $m\geq 4$ we provide an asymptotic expansion of $\mathcal{W}_{-1,1}(a,\lambda)$ as
$\lambda\to\infty$ in the sector $|\arg(\lambda)|\leq\pi-\delta$.
Theorem 4.3.
Let $a\in\mathbb{C}^{m}$ and $0<\delta<\frac{\pi}{2(m+2)}$ be fixed. If $m\geq 4$ then
$$\displaystyle\mathcal{W}_{-1,1}(a,\lambda)=$$
$$\displaystyle[2\omega^{\frac{1}{2}+\mu(a)}+O\left(\lambda^{-\rho}\right)]\exp%
\left[L(G^{-1}(a),\omega^{-2}\lambda)-L(a,\lambda)\right]$$
(4.5)
$$\displaystyle+[2\omega^{\frac{1}{2}+\mu(a)+2\nu(a)}+O\left(\lambda^{-\rho}%
\right)]\exp\left[L(G(a),\omega^{2}\lambda)-L(a,\lambda)\right],$$
as $\lambda\to\infty$ in the sector
(4.6)
$$-\pi+\delta\leq\arg(\lambda)\leq\pi-\delta.$$
Next, we provide an asymptotic expansion of $\mathcal{W}_{-1,1}(a,\lambda)$ as $\lambda\to\infty$ along the rays
in the upper- and lower- half planes.
Corollary 4.4.
Let $m\geq 4$, $a\in\mathbb{C}^{m}$ and $0<\delta<\frac{\pi}{m+2}$ be fixed. Then
$$\mathcal{W}_{-1,1}(a,\lambda)=[2\omega^{\frac{1}{2}+\mu(a)}+O\left(\lambda^{-%
\rho}\right)]\exp\left[L(G^{-1}(a),\omega^{-2}\lambda)-L(a,\lambda)\right],$$
as $\lambda\to\infty$ in the sector $\delta\leq\arg(\lambda)\leq\pi-\delta$.
Also,
$$\mathcal{W}_{-1,1}(a,\lambda)=[2\omega^{\frac{1}{2}+\mu(a)+2\nu(a)}+O\left(%
\lambda^{-\rho}\right)]\exp\left[L(G(a),\omega^{2}\lambda)-L(a,\lambda)\right],$$
as $\lambda\to\infty$ in the sector $-\pi+\delta\leq\arg(\lambda)\leq-\delta$.
Proof.
We will determine which term in (4.5) dominates in the upper and lower half planes.
Since, by (3.8),
$${\rm Re}\,(L(a,\lambda))=K_{m}\cos\left(\frac{m+2}{2m}\arg(\lambda)\right)|%
\lambda|^{\frac{1}{2}+\frac{1}{m}}(1+o(1)),$$
we have
$$\displaystyle\left[{\rm Re}\,(L(G^{-1}(a),\omega^{-2}\lambda))-{\rm Re}\,(L(a,%
\lambda))\right]-\left[{\rm Re}\,(L(G(a),\omega^{2}\lambda))-{\rm Re}\,(L(a,%
\lambda))\right]$$
$$\displaystyle=K_{m}\left[\cos\left(-\frac{2\pi}{m}+\frac{m+2}{2m}\arg(\lambda)%
\right)-\cos\left(\frac{2\pi}{m}+\frac{m+2}{2m}\arg(\lambda)\right)\right]|%
\lambda|^{\frac{1}{2}+\frac{1}{m}}(1+o(1))$$
$$\displaystyle=2K_{m}\sin\left(\frac{2\pi}{m}\right)\sin\left(\frac{m+2}{2m}%
\arg(\lambda)\right)|\lambda|^{\frac{1}{2}+\frac{1}{m}}(1+o(1)).$$
Thus, the first term in (4.5) dominates as $\lambda\to\infty$ along the rays in the upper half plane, and the second term dominates in the lower half plane. This completes the proof.
∎
Proof of Theorem 4.3.
In [24], $C(a,\lambda)$ is used for $\frac{\mathcal{W}_{-1,1}(a,\lambda)}{\mathcal{W}_{0,1}(a,\lambda)}$
and asymptotics of $C(a,\lambda)$ are provided. Notice that
$\mathcal{W}_{-1,1}(a,\lambda)=2\omega^{\mu(a)}C(a,\lambda)$.
Theorem 13 in [24] implies
(4.5) for the sector
(4.7)
$$\pi-\frac{4\lfloor\frac{m}{2}\rfloor\pi}{m+2}+\delta\leq\arg(\lambda)\leq\pi-%
\frac{4\pi}{m+2}-\delta.$$
Theorem 14 in [24] implies that
$$\displaystyle\mathcal{W}_{-1,1}(a,\lambda)=$$
$$\displaystyle[2\omega^{\frac{1}{2}+\mu(a)}+O\left(\lambda^{-\rho}\right)]\exp%
\left[L(G^{-1}(a),\omega^{-2}\lambda)-L(a,\lambda)\right]$$
$$\displaystyle+[2\omega^{1+2\mu(a)+4\nu(a)}+O\left(\lambda^{-\rho}\right)]\exp%
\left[-L(G^{2}(a),\omega^{2-m}\lambda)-L(a,\lambda)\right],$$
as $\lambda\to\infty$ in the sector $\pi-\frac{8\pi}{m+2}+\delta\leq\arg(\lambda)\leq\pi-\delta$. One can check that the first term dominates in this sector, by using an argument similar to that in the proof of Corollary 4.4.
Also, Theorem 15 in [24] implies that
$$\displaystyle\mathcal{W}_{-1,1}(a,\lambda)=$$
$$\displaystyle[2\omega^{1+2\mu(a)}+O\left(\lambda^{-\rho}\right)]\exp\left[-L(a%
,\omega^{-m-2}\lambda)-L(^{-2}(a),\omega^{-4}\lambda)\right]$$
$$\displaystyle+[2\omega^{\frac{1}{2}+\mu(a)+2\nu(a)}+O\left(\lambda^{-\rho}%
\right)]\exp\left[L(G(a),\omega^{-m}\lambda)-L(a,\omega^{-m-2}\lambda)\right],$$
as $\lambda\to\infty$ in the sector $\pi+\delta\leq\arg(\lambda)\leq\pi+\frac{8\pi}{m+2}-\delta$. One can check that the second term dominates in this sector. Then we replace $\lambda$ by $\omega^{m+2}\lambda$ to convert the sector here to $-\pi+\delta\leq\arg(\lambda)\leq-\pi+\frac{8\pi}{m+2}-\delta$. This completes the proof.
∎
Theorem 4.5.
Let $m=3$ and let $a\in\mathbb{C}^{m}$ and $0<\delta<\frac{\pi}{m+2}$ be fixed. Then
$$\displaystyle\mathcal{W}_{-1,1}(a,\lambda)=$$
$$\displaystyle[-2\omega^{-\frac{5}{4}}+O\left(\lambda^{-\rho}\right)]\exp\left[%
L(G^{4}(a),\omega^{-2}\lambda)-L(a,\lambda)\right]$$
$$\displaystyle-[2i\omega^{\frac{5}{2}}+O\left(\lambda^{-\rho}\right)]\exp\left[%
-L(G^{2}(a),\omega^{-1}\lambda)-L(a,\lambda)\right],$$
as $\lambda\to\infty$ in the sector
$-\delta\leq\arg(\lambda)\leq\pi-\delta$.
Also,
$$\displaystyle\mathcal{W}_{-1,1}(a,\lambda)=$$
$$\displaystyle[-2i\omega^{\frac{5}{2}}+O\left(\lambda^{-\rho}\right)]\exp\left[%
-L(a,\omega^{-5}\lambda)-L(G^{-2}(a),\omega^{-4}\lambda)\right]$$
$$\displaystyle+[2\omega^{\frac{15}{4}}+O\left(\lambda^{-\rho}\right)]\exp\left[%
L(G(a),\omega^{-3}\lambda)-L(a,\omega^{-5}\lambda)\right],$$
as $\lambda\to\infty$ in the sector $-\pi+\delta\leq\arg(\lambda)\leq\delta$.
Proof.
See Theorems 14 and 15 in [24] for a proof.
∎
5. Asymptotics of $\mathcal{W}_{-1,n}(a,\lambda)$
In this section, we will provide asymptotic expansions of $\mathcal{W}_{-1,n}(a,\cdot)$, zeros of which will be
closely related with the eigenvalues of $H_{n}$.
First, we treat the cases when $1\leq n<\lfloor\frac{m}{2}\rfloor$.
Theorem 5.1.
Let $1\leq n<\lfloor\frac{m}{2}\rfloor$ be an integer. Then $\mathcal{W}_{-1,n}(a,\cdot)$ admits the following
asymptotic expansion
(5.1)
$$\mathcal{W}_{-1,n}(a,\lambda)=[2\omega^{\frac{2-n}{2}+\mu(G^{n-1}(a))}+O\left(%
\lambda^{-\rho}\right)]\exp\left[L(G^{-1}(a),\omega^{-2}\lambda)-L(G^{n-1}(a),%
\omega^{2(n-1)}\lambda)\right],$$
as $\lambda\to\infty$ in the sector
(5.2)
$$-\frac{2(n-1)\pi}{m+2}+\delta\leq\arg(\lambda)\leq\pi-\frac{4n\pi}{m+2}+\delta.$$
Also,
$$\displaystyle\mathcal{W}_{-1,n}(a,\lambda)$$
$$\displaystyle=[2\omega^{\frac{2-n}{2}+\mu(G^{n-1}(a))}+O\left(\lambda^{-\rho}%
\right)]\exp\left[L(G^{-1}(a),\omega^{-2}\lambda)-L(G^{n-1}(a),\omega^{2(n-1)}%
\lambda)\right]$$
(5.3)
$$\displaystyle+[2\omega^{\frac{2-n}{2}+\mu(G^{-1}(a))}+O\left(\lambda^{-\rho}%
\right)]\exp\left[L(G^{n}(a),\omega^{2n}\lambda)-L(a,\lambda)\right],$$
as $\lambda\to\infty$ in the sector
(5.4)
$$-\frac{2(n-1)\pi}{m+2}-\delta\leq\arg(\lambda)\leq-\frac{2(n-1)\pi}{m+2}+\delta.$$
Proof.
First we will prove (5.1) for the sector
(5.5)
$$-\frac{2(n-1)\pi}{m+2}+\delta\leq\arg(\lambda)\leq\pi-\frac{4n\pi}{m+2}-\delta$$
and the second part of the theorem by induction on $n$.
The case when $n=1$ is trivially satisfied by Theorem 4.3 and Corollary 4.4 since $\mu(a)+2\nu(a)=\mu(G^{-1}(a))$.
Suppose that (5.1) holds in the sector (5.5) for $n-1$. From this induction hypothesis we have
$$\displaystyle\mathcal{W}_{0,n}(a,\lambda)$$
$$\displaystyle=\omega^{-1}\mathcal{W}_{-1,n-1}(G(a),\omega^{2}\lambda)$$
(5.6)
$$\displaystyle=[2\omega^{-\frac{n-1}{2}+\mu(G^{n-1}(a))}+O\left(\lambda^{-\rho}%
\right)]\exp\left[L(a,\lambda)-L(G^{n-1}(a),\omega^{2(n-1)}\lambda)\right],$$
as $\lambda\to\infty$ in the sector
$$-\frac{2(n-2)\pi}{m+2}+\delta\leq\arg(\omega^{2}\lambda)\leq\pi-\frac{4(n-1)%
\pi}{m+2}-\delta,$$
that is,
(5.7)
$$-\frac{2n\pi}{m+2}+\delta\leq\arg(\lambda)\leq\pi-\frac{4n\pi}{m+2}-\delta.$$
Also, from Lemma 4.1 if $1\leq j\leq\frac{m}{2}+1$, then we have
(5.8)
$$\mathcal{W}_{0,j}(a,\lambda)=[2i\omega^{-\frac{j}{2}}+O\left(\lambda^{-\rho}%
\right)]\exp\left[L(G^{j}(a),\omega^{2j-m-2}\lambda)+L(a,\lambda)\right],$$
as $\lambda\to\infty$ in the sector
$$\pi-\frac{4j\pi}{m+2}+\delta\leq\arg(\lambda)\leq\pi-\delta.$$
We solve (3.12) for $\mathcal{W}_{\ell,-1}(a,\lambda)$ and set $\ell=n$ to get
(5.9)
$$\mathcal{W}_{-1,n}(a,\lambda)=\frac{\mathcal{W}_{-1,0}(a,\lambda)\mathcal{W}_{%
n,k}(a,\lambda)}{\mathcal{W}_{0,k}(a,\lambda)}+\frac{\mathcal{W}_{0,n}(a,%
\lambda)\mathcal{W}_{-1,k}(a,\lambda)}{\mathcal{W}_{0,k}(a,\lambda)}$$
Set $k=\lfloor\frac{m}{2}\rfloor$. Then since $1\leq k-n<k=\lfloor\frac{m}{2}\rfloor$, using (3.13),
$$\displaystyle\mathcal{W}_{-1,n}(a,\lambda)$$
$$\displaystyle=\frac{2\omega^{\mu(G^{-1}(a))}\mathcal{W}_{0,k-n}(G^{n}(a),%
\omega^{2n}\lambda)}{\omega^{n-1}\mathcal{W}_{0,k}(a,\lambda)}+\frac{\mathcal{%
W}_{0,n}(a,\lambda)\mathcal{W}_{0,k+1}(G^{-1}(a),\omega^{-2}\lambda)}{\omega^{%
-1}\mathcal{W}_{0,k}(a,\lambda)}$$
$$\displaystyle=\frac{2\omega^{\mu(G^{-1}(a))}[2i\omega^{-\frac{k-n}{2}}+O\left(%
\lambda^{-\rho}\right)]\exp\left[L(G^{n}(a),\omega^{2k-m-2}\lambda)+L(G^{n}(a)%
,\omega^{2n}\lambda)\right]}{\omega^{n-1}[2i\omega^{-\frac{k}{2}}+O\left(%
\lambda^{-\rho}\right)]\exp\left[L(G^{k}(a),\omega^{2k-m-2}\lambda)+L(a,%
\lambda)\right]}$$
$$\displaystyle+\frac{[2\omega^{-\frac{n-1}{2}+\mu(G^{n-1}(a))}+O\left(\lambda^{%
-\rho}\right)]\exp\left[L(a,\lambda)-L(G^{n-1}(a),\omega^{2(n-1)}\lambda)%
\right]}{\omega^{-1}[2i\omega^{-\frac{k}{2}}+O\left(\lambda^{-\rho}\right)]%
\exp\left[L(G^{k}(a),\omega^{2k-m-2}\lambda)+L(a,\lambda)\right]}$$
$$\displaystyle\times[2i\omega^{-\frac{k+1}{2}}+O\left(\lambda^{-\rho}\right)]%
\exp\left[L(G^{k}(a),\omega^{2k-m-2}\lambda)+L(G^{-1}(a),\omega^{-2}\lambda)\right]$$
$$\displaystyle=[2\omega^{\frac{2-n}{2}+\mu(G^{-1}(a))}+O\left(\lambda^{-\rho}%
\right)]\exp\left[L(G^{n}(a),\omega^{2n}\lambda)-L(a,\lambda)\right]$$
(5.10)
$$\displaystyle+[2\omega^{\frac{2-n}{2}+\mu(G^{n-1}(a))}+O\left(\lambda^{-\rho}%
\right)]\exp\left[L(G^{-1}(a),\omega^{-2}\lambda)-L(G^{n-1}(a),\omega^{2(n-1)}%
\lambda)\right],$$
where we used (5.6) for $\mathcal{W}_{0,n}(a,\lambda)$ and (5.8) for everything else, provided
that $\lambda$ lies in (5.7) and that
$$\displaystyle\pi-\frac{4\left(\lfloor\frac{m}{2}\rfloor-n\right)\pi}{m+2}+\delta\leq$$
$$\displaystyle\arg(\omega^{2n}\lambda)\leq\pi-\delta$$
$$\displaystyle\pi-\frac{4\lfloor\frac{m}{2}\rfloor\pi}{m+2}+\delta\leq$$
$$\displaystyle\arg(\lambda)\leq\pi-\delta$$
$$\displaystyle\pi-\frac{4\left(\lfloor\frac{m}{2}\rfloor+1\right)\pi}{m+2}+\delta\leq$$
$$\displaystyle\arg(\omega^{-2}\lambda)\leq\pi-\delta,$$
that is,
$$-\frac{2n\pi}{m+2}+\delta\leq\arg(\lambda)\leq\pi-\frac{4n\pi}{m+2}-\delta.$$
Thus, the second part of the theorem is proved by induction.
Next in order to prove the first part of the theorem for the sector (5.5), we will determine which term in (5.10) dominates as $\lambda\to\infty$. To do that, we look at
$$\displaystyle{\rm Re}\,\left(L(G^{-1}(a),\omega^{-2}\lambda)-L(G^{n-1}(a),%
\omega^{2(n-1)}\lambda)\right)-{\rm Re}\,\left(L(G^{n}(a),\omega^{2n}\lambda)-%
L(a,\lambda)\right)$$
$$\displaystyle=K_{m}\left[\cos\left(-\frac{2\pi}{m}+\frac{m+2}{2m}\arg(\lambda)%
\right)-\cos\left(\frac{2(n-1)\pi}{m}+\frac{m+2}{2m}\arg(\lambda)\right)\right.$$
$$\displaystyle-\left.\left(\cos\left(-\frac{2n\pi}{m}+\frac{m+2}{2m}\arg(%
\lambda)\right)-\cos\left(\frac{m+2}{2m}\arg(\lambda)\right)\right)\right]|%
\lambda|^{\frac{1}{2}+\frac{1}{m}}(1+o(1))$$
$$\displaystyle=2K_{m}\sin\left(\frac{n\pi}{m}\right)\left[\sin\left(\frac{(n-2)%
\pi}{m}+\frac{m+2}{2m}\arg(\lambda)\right)\right.$$
$$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\left.+\sin\left(\frac{n\pi}{%
m}+\frac{m+2}{2m}\arg(\lambda)\right)\right]|\lambda|^{\frac{1}{2}+\frac{1}{m}%
}(1+o(1))$$
(5.11)
$$\displaystyle=4K_{m}\sin\left(\frac{n\pi}{m}\right)\cos\left(\frac{\pi}{m}%
\right)\sin\left(\frac{(n-1)\pi}{m}+\frac{m+2}{2m}\arg(\lambda)\right)|\lambda%
|^{\frac{1}{2}+\frac{1}{m}}(1+o(1)),$$
that tends to positive infinity as $\lambda\to\infty$ (and hence the second term in (5.10) dominates) if $-\frac{2(n-1)\pi}{m+2}+\delta\leq\arg(\lambda)\leq\pi-\frac{4n\pi}{m+2}-\delta$.
We still need to prove (5.1) for the sector
(5.12)
$$\pi-\frac{4n\pi}{m+2}-\delta\leq\arg(\lambda)\leq\pi-\frac{4n\pi}{m+2}+\delta,$$
for which we use induction on $n$ again.
When $n=1$, (5.1) holds by Lemma 4.4.
Suppose that (5.1) in the sector (5.12) for $n-1$ with $2\leq n<\lfloor\frac{m}{2}\rfloor$.
Then (3.13) and (5.9) with $k=n+1$ yield
$$\mathcal{W}_{-1,n}(a,\lambda)=\frac{2\omega^{\mu(G^{-1}(a))}\mathcal{W}_{0,1}(%
G^{n}(a),\omega^{2n}\lambda)}{\omega^{n-1}\mathcal{W}_{0,n+1}(a,\lambda)}+%
\frac{\mathcal{W}_{-1,n-1}(G(a),\omega^{2}\lambda)\mathcal{W}_{0,n+2}(G^{-1}(a%
),\omega^{-2}\lambda)}{\mathcal{W}_{0,n+1}(a,\lambda)}.$$
If $\pi-\frac{4n\pi}{m+2}-\delta\leq\arg(\lambda)\leq\pi-\frac{4n\pi}{m+2}+\delta$, then $\pi-\frac{4(n-1)\pi}{m+2}-\delta\leq\arg(\omega^{2}\lambda)\leq\pi-\frac{4(n-1%
)\pi}{m+2}+\delta$. So
(5.13)
$$\displaystyle\mathcal{W}_{-1,n}(a,\lambda)$$
$$\displaystyle=\frac{4\omega^{\mu(G^{-1}(a))+\mu(G^{n}(a))}}{\omega^{n-1}%
\mathcal{W}_{0,n+1}(a,\lambda)}+\frac{\mathcal{W}_{-1,n-1}(G(a),\omega^{2}%
\lambda)\mathcal{W}_{0,n+2}(G^{-1}(a),\omega^{-2}\lambda)}{\mathcal{W}_{0,n+1}%
(a,\lambda)}$$
$$\displaystyle=\frac{4\omega^{\mu(G^{-1}(a))+\mu(G^{n}(a))}}{\omega^{n-1}[2i%
\omega^{-\frac{n+1}{2}}+O\left(\lambda^{-\rho}\right)]\exp\left[L(G^{n+1}(a),%
\omega^{2n-m}\lambda)+L(a,\lambda)\right]}$$
$$\displaystyle+\frac{[2\omega^{\frac{3-n}{2}+\mu(G^{n-1}(a))}+O\left(\lambda^{-%
\rho}\right)]\exp\left[L(a,\lambda)-L(G^{n-1}(a),\omega^{2(n-1)}\lambda)\right%
]}{[2i\omega^{-\frac{n+1}{2}}+O\left(\lambda^{-\rho}\right)]\exp\left[L(G^{n+1%
}(a),\omega^{2n-m}\lambda)+L(a,\lambda)\right]}$$
$$\displaystyle\times[2i\omega^{-\frac{n+2}{2}}+O\left(\lambda^{-\rho}\right)]%
\exp\left[L(G^{n+1}(a),\omega^{2n-m}\lambda)+L(G^{-1}(a),\omega^{-2}\lambda)\right]$$
$$\displaystyle=[-2i\omega^{\frac{3-n}{2}+\mu(G^{-1}(a))+\mu(G^{n}(a))}+O\left(%
\lambda^{-\rho}\right)]\exp\left[-L(G^{n+1}(a),\omega^{2n-m}\lambda)-L(a,%
\lambda)\right]$$
$$\displaystyle+[2\omega^{\frac{2-n}{2}+\mu(G^{n-1}(a))}+O\left(\lambda^{-\rho}%
\right)]\exp\left[L(G^{-1}(a),\omega^{-2}\lambda)-L(G^{n-1}(a),\omega^{2(n-1)}%
\lambda)\right],$$
where we use the induction hypothesis for $\mathcal{W}_{-1,n-1}(G(a),\omega^{2}\lambda)$ and use (4.1) for
$\mathcal{W}_{0,n+1}(a,\lambda)$ and $\mathcal{W}_{0,n+2}(G^{-1}(a),\omega^{-2}\lambda)$. Next, we use an argument
similar (5.11) to complete the induction step. Thus, the theorem is proved.
∎
Next we investigate $\mathcal{W}_{0,\lfloor\frac{m}{2}\rfloor}(a,\lambda)$.
Theorem 5.2.
If $m\geq 4$ is an even integer, then
$$\displaystyle\mathcal{W}_{-1,\lfloor\frac{m}{2}\rfloor}(a,\lambda)$$
(5.14)
$$\displaystyle=-[2\omega^{2+\mu(G^{-1}(a))+\mu(G^{\frac{m}{2}}(a))}+O\left(%
\lambda^{-\rho}\right)]\exp\left[-L(G^{{\frac{m}{2}+1}}(a),\lambda)-L(a,%
\lambda)\right]$$
$$\displaystyle-[2\omega^{2+\mu(a)+\mu(G^{\frac{m-2}{2}}(a))}+O\left(\lambda^{-%
\rho}\right)]\exp\left[-L(G^{\frac{m-2}{2}}(a),\omega^{m-2}\lambda)-L(G^{m}(a)%
,\omega^{m-2}\lambda)\right].$$
as $\lambda\to\infty$ in the sector
(5.15)
$$-\pi+\frac{4\pi}{m+2}-\delta\leq\arg(\lambda)\leq-\pi+\frac{4\pi}{m+2}+\delta.$$
If $m\geq 4$ is an odd integer, then
$$\displaystyle\mathcal{W}_{-1,\lfloor\frac{m}{2}\rfloor}(a,\lambda)$$
$$\displaystyle=[2\omega^{\frac{5}{4}}+O\left(\lambda^{-\rho}\right)]\exp\left[-%
L(G^{\frac{m+1}{2}}(a),\omega^{-1}\lambda)-L(a,\lambda)\right]$$
(5.16)
$$\displaystyle+[2\omega^{\frac{5}{4}}+O\left(\lambda^{-\rho}\right)]\exp\left[L%
(G^{m+1}(a),\omega^{-2}\lambda)-L(G^{\frac{m-3}{2}}(a),\omega^{m-3}\lambda)%
\right].$$
as $\lambda\to\infty$ in the sector
$$-\pi+\frac{4\pi}{m+2}+\delta\leq\arg(\lambda)\leq-\pi+\frac{6\pi}{m+2}+\delta.$$
Proof.
We will use (5.13) with $n=\lfloor\frac{m}{2}\rfloor$, that is,
$$\mathcal{W}_{-1,\lfloor\frac{m}{2}\rfloor}(a,\lambda)=\frac{4\omega^{\mu(G^{-1%
}(a))+\mu(G^{\lfloor\frac{m}{2}\rfloor}(a))}}{\omega^{\lfloor\frac{m}{2}%
\rfloor-1}\mathcal{W}_{0,\lfloor\frac{m}{2}\rfloor+1}(a,\lambda)}+\frac{%
\mathcal{W}_{-1,\lfloor\frac{m}{2}\rfloor-1}(G(a),\omega^{2}\lambda)\mathcal{W%
}_{0,\lfloor\frac{m}{2}\rfloor+2}(G^{-1}(a),\omega^{-2}\lambda)}{\mathcal{W}_{%
0,\lfloor\frac{m}{2}\rfloor+1}(a,\lambda)}.$$
When $m$ is even, say $m=2k$,
$$\displaystyle\mathcal{W}_{0,k+2}(G^{-1}(a),\omega^{-2}\lambda)$$
$$\displaystyle=\mathcal{W}_{m+2,k+2}(G^{-1}(a),\omega^{-2}\lambda)$$
$$\displaystyle=-\omega^{-k-3}\mathcal{W}_{-1,k-1}(G^{k+2}(a),\omega^{2k+4}\lambda)$$
$$\displaystyle=\omega^{-2}\mathcal{W}_{-1,k-1}(G^{k+2}(a),\omega^{2}\lambda).$$
So
$$\displaystyle\mathcal{W}_{-1,k}(a,\lambda)$$
$$\displaystyle=\frac{4\omega^{\mu(G^{-1}(a))+\mu(G^{k}(a))}}{\omega^{k-1}%
\mathcal{W}_{0,k+1}(a,\lambda)}+\frac{\mathcal{W}_{-1,k-1}(G(a),\omega^{2}%
\lambda)\mathcal{W}_{0,k+2}(G^{-1}(a),\omega^{-2}\lambda)}{\mathcal{W}_{0,k+1}%
(a,\lambda)}$$
$$\displaystyle=\frac{4\omega^{\mu(G^{-1}(a))+\mu(G^{k}(a))}}{\omega^{k-1}%
\mathcal{W}_{0,k+1}(a,\lambda)}+\frac{\mathcal{W}_{-1,k-1}(G(a),\omega^{2}%
\lambda)\mathcal{W}_{-1,k-1}(G^{k+2}(a),\omega^{2}\lambda)}{\omega^{2}\mathcal%
{W}_{0,k+1}(a,\lambda)}$$
Since $\lambda$ lies in (5.15),
$$-\frac{2\left(\lfloor\frac{m}{2}\rfloor-2\right)\pi}{m+2}+\delta\leq-\pi+\frac%
{8\pi}{m+2}-\delta\leq\arg(\omega^{2}\lambda)\leq\pi-\frac{4\left(\lfloor\frac%
{m}{2}\rfloor-1\right)\pi}{m+2}+\delta.$$
$$\displaystyle\mathcal{W}_{-1,k}(a,\lambda)$$
$$\displaystyle=\frac{4\omega^{\mu(G^{-1}(a))+\mu(G^{k}(a))}}{\omega^{k-1}%
\mathcal{W}_{0,k+1}(a,\lambda)}+\frac{\mathcal{W}_{-1,k-1}(G(a),\omega^{2}%
\lambda)\mathcal{W}_{0,k+2}(G^{-1}(a),\omega^{-2}\lambda)}{\mathcal{W}_{0,k+1}%
(a,\lambda)}$$
$$\displaystyle=\frac{4\omega^{\mu(G^{-1}(a))+\mu(G^{k}(a))}}{\omega^{k-1}%
\mathcal{W}_{0,k+1}(a,\lambda)}+\frac{\mathcal{W}_{-1,k-1}(G(a),\omega^{2}%
\lambda)\mathcal{W}_{-1,k-1}(G^{k+2}(a),\omega^{2}\lambda)}{\omega^{2}\mathcal%
{W}_{0,k+1}(a,\lambda)}$$
$$\displaystyle=\frac{4\omega^{\mu(G^{-1}(a))+\mu(G^{k}(a))}}{\omega^{k-1}[2i%
\omega^{-\frac{k+1}{2}}+O\left(\lambda^{-\rho}\right)]\exp\left[L(G^{k+1}(a),%
\omega^{2k-m}\lambda)+L(a,\lambda)\right]}$$
$$\displaystyle+\frac{[2\omega^{\frac{3-k}{2}+\mu(G^{k-1}(a))}+O\left(\lambda^{-%
\rho}\right)]\exp\left[L(a,\lambda)-L(G^{k-1}(a),\omega^{2(k-1)}\lambda)\right%
]}{\omega^{2}[2i\omega^{-\frac{k+1}{2}}+O\left(\lambda^{-\rho}\right)]\exp%
\left[L(G^{k+1}(a),\omega^{2k-m}\lambda)+L(a,\lambda)\right]}$$
$$\displaystyle\times[2\omega^{\frac{3-k}{2}+\mu(G^{2k}(a))}+O\left(\lambda^{-%
\rho}\right)]\exp\left[L(G^{k+1}(a),\lambda)-L(G^{2k}(a),\omega^{2(k-1)}%
\lambda)\right]$$
$$\displaystyle=[-2i\omega^{\frac{3-k}{2}+\mu(G^{-1}(a))+\mu(G^{k}(a))}+O\left(%
\lambda^{-\rho}\right)]\exp\left[-L(G^{k+1}(a),\lambda)-L(a,\lambda)\right]$$
$$\displaystyle-[2i\omega^{\frac{3-k}{2}+\mu(G^{k-1}(a))+\mu(G^{2k}(a))}+O\left(%
\lambda^{-\rho}\right)]\exp\left[-L(G^{k-1}(a),\omega^{2(k-1)}\lambda)-L(G^{2k%
}(a),\omega^{2(k-1)}\lambda)\right]$$
$$\displaystyle=-[2i\omega^{\frac{3-k}{2}+\mu(G^{-1}(a))+\mu(G^{k}(a))}+O\left(%
\lambda^{-\rho}\right)]\exp\left[-L(G^{k+1}(a),\lambda)-L(a,\lambda)\right]$$
$$\displaystyle-[2i\omega^{\frac{3-k}{2}+\mu(a)+\mu(G^{k-1}(a))}+O\left(\lambda^%
{-\rho}\right)]\exp\left[-L(G^{k-1}(a),\omega^{2(k-1)}\lambda)-L(G^{2k}(a),%
\omega^{2(k-1)}\lambda)\right],$$
where we used (5.1) for $\mathcal{W}_{-1,k-1}(G(a),\cdot)$ and $\mathcal{W}_{-1,k-1}(G^{k+2}(a),\cdot)$,
and (4.1) for $\mathcal{W}_{0,k+1}(a,\cdot)$. Finally, we use $\omega^{-\frac{m+2}{4}}=-i$, to get the
desired asymptotic expansion of $\mathcal{W}_{-1,\lfloor\frac{m}{2}\rfloor}(a,\lambda)$.
Next we investigate the case when $m$ is odd, say $m=2k+1$ (so $\lfloor\frac{m}{2}\rfloor=k$).
$$\mathcal{W}_{0,\lfloor\frac{m}{2}\rfloor+1}(a,\lambda)=\omega^{-1}\mathcal{W}_%
{-1,\lfloor\frac{m}{2}\rfloor}(G(a),\omega^{2}\lambda)$$
and
$$\displaystyle\mathcal{W}_{0,k+2}(G^{-1}(a),\omega^{-2}\lambda)$$
$$\displaystyle=\mathcal{W}_{m+2,k+2}(G^{-1}(a),\omega^{-2}\lambda)$$
$$\displaystyle=-\omega^{-k-2}\mathcal{W}_{0,k+1}(G^{k+1}(a),\omega^{2k+2}\lambda)$$
$$\displaystyle=\omega^{-\frac{1}{2}}\mathcal{W}_{0,k+1}(G^{k+1}(a),\omega^{-1}%
\lambda).$$
Similarly to the proof of the theorem for $m$ even,
$$\displaystyle\mathcal{W}_{-1,k}(a,\lambda)$$
$$\displaystyle=\frac{4\omega^{\mu(G^{-1}(a))+\mu(G^{k}(a))}}{\omega^{k-1}%
\mathcal{W}_{0,k+1}(a,\lambda)}+\frac{\mathcal{W}_{-1,k-1}(G(a),\omega^{2}%
\lambda)\mathcal{W}_{0,k+2}(G^{-1}(a),\omega^{-2}\lambda)}{\mathcal{W}_{0,k+1}%
(a,\lambda)}$$
$$\displaystyle=\frac{4\omega^{\mu(G^{-1}(a))+\mu(G^{k}(a))}}{\omega^{k-1}%
\mathcal{W}_{0,k+1}(a,\lambda)}+\frac{\mathcal{W}_{-1,k-1}(G(a),\omega^{2}%
\lambda)\mathcal{W}_{0,k+1}(G^{k+1}(a),\omega^{-1}\lambda)}{\omega^{\frac{1}{2%
}}\mathcal{W}_{0,k+1}(a,\lambda)}$$
$$\displaystyle=\frac{4\omega^{\mu(G^{-1}(a))+\mu(G^{k}(a))}}{\omega^{k-1}[2i%
\omega^{-\frac{k+1}{2}}+O\left(\lambda^{-\rho}\right)]\exp\left[L(G^{k+1}(a),%
\omega^{2k-m}\lambda)+L(a,\lambda)\right]}$$
$$\displaystyle+\frac{[2\omega^{\frac{3-k}{2}+\mu(G^{k-1}(a))}+O\left(\lambda^{-%
\rho}\right)]\exp\left[L(a,\lambda)-L(G^{k-1}(a),\omega^{2(k-1)}\lambda)\right%
]}{\omega^{\frac{1}{2}}[2i\omega^{-\frac{k+1}{2}}+O\left(\lambda^{-\rho}\right%
)]\exp\left[L(G^{k+1}(a),\omega^{2k-m}\lambda)+L(a,\lambda)\right]}$$
$$\displaystyle\times[2i\omega^{-\frac{k+1}{2}}+O\left(\lambda^{-\rho}\right)]%
\exp\left[L(G^{2k+2}(a),\omega^{2k-m-1}\lambda)+L(G^{k+1}(a),\omega^{-1}%
\lambda)\right]$$
$$\displaystyle=[-2i\omega^{\frac{3-k}{2}+\mu(G^{-1}(a))+\mu(G^{k}(a))}+O\left(%
\lambda^{-\rho}\right)]\exp\left[-L(G^{k+1}(a),\omega^{-1}\lambda)-L(a,\lambda%
)\right]$$
$$\displaystyle+[2\omega^{\frac{2-k}{2}+\mu(G^{k-1}(a))}+O\left(\lambda^{-\rho}%
\right)]\exp\left[L(G^{2k+2}(a),\omega^{-2}\lambda)-L(G^{k-1}(a),\omega^{2(k-1%
)}\lambda)\right]$$
$$\displaystyle=-[2i\omega^{\frac{3-k}{2}+\frac{m}{2}}+O\left(\lambda^{-\rho}%
\right)]\exp\left[-L(G^{k+1}(a),\omega^{-1}\lambda)-L(a,\lambda)\right]$$
$$\displaystyle+[2\omega^{\frac{2-k}{2}+\frac{m}{4}}+O\left(\lambda^{-\rho}%
\right)]\exp\left[L(G^{2k+2}(a),\omega^{-2}\lambda)-L(G^{k-1}(a),\omega^{2(k-1%
)}\lambda)\right],$$
where we use (5.1) for $\mathcal{W}_{-1,k-1}(G(a),\cdot)$, and use (4.1) for
$\mathcal{W}_{0,k+1}(a,\cdot)$ and $\mathcal{W}_{0,k+1}(G^{k+1}(a),\cdot)$. Finally, we use
$\omega^{\frac{m+2}{4}}=i$, to get the asymptotic expansion of
$\mathcal{W}_{-1,\lfloor\frac{m}{2}\rfloor}(a,\lambda)$. This completes the proof.
∎
The order of an entire function $g$ is defined by
$$\limsup_{r\rightarrow\infty}\frac{\log\log M(r,g)}{\log r},$$
where $M(r,g)=\max\{|g(re^{i\theta})|:0\leq\theta\leq 2\pi\}$ for $r>0$.
If for some positive real numbers $\sigma,\,c_{1},\,c_{2}$, we have $\exp[c_{1}r^{\sigma}]\leq M(r,g)\leq\exp[c_{2}r^{\sigma}]$ for all large $r$, then the order of $g$ is $\sigma$.
Corollary 5.3.
Let $1\leq n\leq\frac{m}{2}$. Then
the entire functions $\mathcal{W}_{-1,n}(a,\cdot)$ are of order $\frac{1}{2}+\frac{1}{m}$, and hence they have infinitely many zeros in the complex plane. Moreover, $\mathcal{W}_{-1,n}(a,\cdot)$ have at most finitely many zeros in the sector
$$-\frac{2(n-1)\pi}{m+2}+\delta\leq\arg(\lambda)\leq\pi+\frac{4\pi}{m+2}-\delta.$$
6. Relation between eigenvalues of $H_{\ell}$ and zeros of $\mathcal{W}_{-1,n}(a,\cdot)$
In this section, we will relate the eigenvalues of $H_{\ell}$ with zeros of some entire function
$\mathcal{W}_{-1,n}(a,\cdot)$.
Suppose that $\ell=2s-1$ is odd with $1\leq\ell\leq m-1$. Then (1.1) becomes (3.2) by the
scaling $v(z)=u(-iz)$, and $v$ decays in the Stokes sectors $S_{-s}$ and $S_{s}$. Since $f_{s-1}$ and $f_{s}$ are
linearly independent, for some $D_{s}$ and $\widetilde{D}_{s}$ one can write
$$f_{-s}(z,a,\lambda)=D_{s}(a,\lambda)f_{s-1}(z,a,\lambda)+\widetilde{D}_{s}(a,%
\lambda)f_{s}(z,a,\lambda).$$
Then one finds
$$D_{s}(a,\lambda)=\frac{\mathcal{W}_{-s,s}(a,\lambda)}{\mathcal{W}_{s-1,s}(a,%
\lambda)}\quad\text{and}\quad\widetilde{D}_{s}(a,\lambda)=\frac{\mathcal{W}_{-%
s,s-1}(a,\lambda)}{\mathcal{W}_{s,s-1}(a,\lambda)}.$$
Also it is easy to see that $\lambda$ is an eigenvalue of $H_{\ell}$ if and only if $D_{s}(a,\lambda)=0$ if and only
if $\mathcal{W}_{-s,s}(a,\lambda)=0$. Since
$\mathcal{W}_{-s,s}(a,\lambda)=\omega^{s-1}\mathcal{W}_{-1,2s-1}(G^{-s+1}(a),%
\omega^{-2s+2}\lambda)$, by Corollary
5.3 $\mathcal{W}_{-s,s}(a,\lambda)$ has at most finitely many zeros in the sector $-\frac{2(2s-2)\pi}{m+2}+\delta\leq\arg(\omega^{-2s+2}\lambda)\leq\pi+\frac{4%
\pi}{m+2}-\delta,$ that is,
$$\delta\leq\arg(\lambda)\leq\pi+\frac{2(2s+1)\pi}{m+2}-\delta.$$
Next, by symmetry one can show that $\mathcal{W}_{-s,s}(a,\lambda)$ has at most finitely many zeros in the sector
$\pi\leq\arg(\lambda)\leq 2\pi-\delta$. For that, we examine $H_{\ell}$ with $P(z)$ replaced by
$\overline{P(\overline{z})}$ whose coefficient vector is
$\overline{a}:=(\overline{a}_{1},\,\overline{a}_{2},\dots,\overline{a}_{m})$. Then one sees that
$\mathcal{W}_{-s,s}(a,\lambda)=0$ if and only if $\mathcal{W}_{-s,s}(\overline{a},\overline{\lambda})=0$.
$\mathcal{W}_{-s,s}(\overline{a},\overline{\lambda})$ has at most finitely many zeros in the sector
$\delta\leq\arg(\overline{\lambda})\leq\pi$ by the arguments above. Thus, $\mathcal{W}_{-s,s}(a,\lambda)$ has at
most finitely many zeros in the sector $\pi\leq\arg(\lambda)\leq 2\pi-\delta$, and has infinitely many zeros in
the sector $|\arg(\lambda)|\leq\delta$ since it is an entire function of order $\frac{1}{2}+\frac{1}{m}\in(0,1)$.
Suppose that $\ell=2s$ is even with $1\leq\ell\leq m-1$. Then (1.1) becomes (3.4) by the
scaling $y(z)=u(-i\omega^{-\frac{1}{2}}z)$, and $y$ decays in the Stokes sectors $S_{-s}$ and $S_{s+1}$. We then
see that the coefficient vector $\widetilde{a}$ of the polynomial $\omega^{-1}P(\omega^{-\frac{1}{2}}z)$ becomes
$$\widetilde{a}=G^{-\frac{1}{2}}(a).$$
Now one can express $f_{-s}$ as a linear combination of
$f_{s}$ and $f_{s+1}$ as follows.
$$f_{-s}(z,\widetilde{a},\omega^{-1}\lambda)=\frac{\mathcal{W}_{-s,s+1}(%
\widetilde{a},\omega^{-1}\lambda)}{\mathcal{W}_{s,s+1}(\widetilde{a},\omega^{-%
1}\lambda)}f_{s}(z,\widetilde{a},\omega^{-1}\lambda)+\frac{\mathcal{W}_{-s,s}(%
\widetilde{a},\omega^{-1}\lambda)}{\mathcal{W}_{s+1,s}(\widetilde{a},\omega^{-%
1}\lambda)}f_{s+1}(z,\widetilde{a},\omega^{-1}\lambda).$$
Thus, $\lambda$ is an eigenvalue of $H_{\ell}$ if and only if
$\mathcal{W}_{-s,s+1}(\widetilde{a},\omega^{-1}\lambda)=0$. Since
$$\mathcal{W}_{-s,s+1}(\widetilde{a},\omega^{-1}\lambda)=\omega^{s-1}\mathcal{W}%
_{-1,2s}(G^{-s+1}(\widetilde{a}),\omega^{-2s+1}\lambda),$$
by Corollary 5.3,
$\mathcal{W}_{-s,s+1}(\widetilde{a},\omega^{-1}\lambda)$ has at most finitely many zeros in the sector
$-\frac{2(2s-1)\pi}{m+2}+\delta\leq\arg(\omega^{-2s+1}\lambda)\leq\pi+\frac{4%
\pi}{m+2}-\delta,$ that is,
$$\delta\leq\arg(\lambda)\leq\pi+\frac{4s\pi}{m+2}-\delta.$$
This is true for each $a\in\mathbb{C}^{m}$. So one can show that $\mathcal{W}_{-s,s+1}(\widetilde{a},\omega^{-1}\lambda)$
has at most finitely many zeros in the sector $\pi\leq\arg(\lambda)\leq 2\pi-\delta$ by symmetry, similar to the
case when $\ell$ is odd. Thus, $\mathcal{W}_{-s,s+1}(a,\lambda)$ has infinitely many zeros in the sector
$|\arg(\lambda)|\leq\delta$.
7. Proof of Theorem 1.1 when $1\leq\ell<\lfloor\frac{m}{2}\rfloor$
In this section, we prove Theorem 1.1 for
$1\leq\ell<\lfloor\frac{m}{2}\rfloor$ and in doing so, we will use
the following proposition on univalent functions.
Proposition 7.1 ([17, p. 216]).
Let $A(\mu)$ be analytic in the region $S=\{\mu\in\mathbb{C}:\alpha\leq\arg(\mu)\leq\beta,|\mu|\geq M\}$ for some
$\alpha,\beta\in\mathbb{R}$ with $\beta-\alpha<\pi$ and for some $M>0$.
Suppose that $A(\mu)\to 0$ as $\mu\to\infty$ in $S$. Then there
exist $\widetilde{\alpha},\widetilde{\beta},\widetilde{M}$ such
that $\mu(1+A(\mu))$ is univalent in $\{\mu\in\mathbb{C}:\widetilde{\alpha}\leq\arg(\mu)\leq\widetilde{\beta},|\mu|%
\geq\widetilde{M}\}\subset S$.
Proof.
See [17, p. 216] or [27, Thm. 3.4] for a proof.
∎
We will consider two cases; when $\ell$ is odd and when $\ell$ is
even.
Proof of Theorem 1.1 when $1\leq\ell<\lfloor\frac{m}{2}\rfloor$ is odd.
We will closely follow his proof of Theorem 29.1 in [27]
where Sibuya computed the leading term in the asymptotics
(1.3) for $\ell=1$.
Suppose that $1\leq\ell=2s-1<\lfloor\frac{m}{2}\rfloor$ is odd.
Recall that when $\ell$ is odd, $\lambda$ is an eigenvalue of
$H_{\ell}$ if and only if
$\mathcal{W}_{-s,s}(a,\lambda)=0$. Also, in Section 6 we showed that $\mathcal{W}_{-s,s}(a,\lambda)$ has all zeros except finitely many in the sector
$|\arg(\lambda)|\leq\delta$ and hence,
all the eigenvalues $\lambda$ of $H_{\ell}$ lie in the sector $|\arg(\lambda)|\leq\delta$ if $|\lambda|$ is large
enough.
Since
$$\mathcal{W}_{-s,s}(a,\lambda)=\omega^{s-1}\mathcal{W}_{-1,2s-1}(G^{-s+1}(a),%
\omega^{-2s+2}\lambda),$$
we will use (5.3) to investigate asymptotics of large eigenvalues. Suppose that $\mathcal{W}_{-s,s}(a,\lambda)=0$ and $|\lambda|$ is large enough.
Then from (5.3) with $n=2s-1$, and with $a$ and $\lambda$ replaced by $G^{-s+1}(a)$ and $\omega^{-2s+2}\lambda$, respectively, we have
$$\displaystyle\left[1+O\left(\lambda^{-\rho}\right)\right]$$
$$\displaystyle\exp\left[L(G^{s}(a),\omega^{2s}\lambda)-L(G^{-s}(a),\omega^{-2s}%
\lambda)\right]$$
$$\displaystyle\times\exp\left[L(G^{s-1}(a),\omega^{2s-2}\lambda)-L(G^{-s+1}(a),%
\omega^{-2s+2}\lambda)\right]=-\omega^{2\nu(G^{s}(a))}.$$
Also, since $\left[1+O\left(\lambda^{-\rho}\right)\right]=\exp\left[O\left(\lambda^{-\rho}%
\right)\right]$ and
$\omega^{2\nu(G^{s}(a))}=\exp\left[\frac{4\pi\nu(G^{s}(a))}{m+2}\right]$,
$$\displaystyle\exp\left[L(G^{s}(a),\omega^{2s}\lambda)-L(G^{-s}(a),\omega^{-2s}%
\lambda)\right]$$
(7.1)
$$\displaystyle\times\exp\left[L(G^{s-1}(a),\omega^{2s-2}\lambda)-L(G^{-s+1}(a),%
\omega^{-2s+2}\lambda)-\frac{4\pi\nu(G^{s}(a))}{m+2}+O\left(\lambda^{-\rho}%
\right)\right]=-1.$$
For each odd integer $\ell=2s-1$ in $1\leq\ell<\lfloor\frac{m}{2}\rfloor$, we define
$$\displaystyle h_{m,\ell}(\lambda)=$$
$$\displaystyle L(G^{s}(a),\omega^{2s}\lambda)-L(G^{-s}(a),\omega^{-2s}\lambda)$$
$$\displaystyle+L(G^{s-1}(a),\omega^{2s-2}\lambda)-L(G^{-s+1}(a),\omega^{-2s+2}%
\lambda)-\frac{4\pi\nu(G^{s}(a))}{m+2}+O\left(\lambda^{-\rho}\right).$$
Then by Corollary 3.4,
$$\displaystyle h_{m,\ell}(\lambda)$$
$$\displaystyle=K_{m}\left(e^{\frac{2s\pi}{m}i}-e^{-\frac{2s\pi}{m}i}+e^{\frac{2%
(s-1)\pi}{m}i}-e^{-\frac{2(s-1)\pi}{m}i}\right)\lambda^{\frac{m+2}{2m}}(1+o(1))$$
$$\displaystyle=2iK_{m}\left(\sin\left(\frac{2s\pi}{m}\right)+\sin\left(\frac{2s%
\pi}{m}-\frac{2\pi}{m}\right)\right)\lambda^{\frac{m+2}{2m}}(1+o(1))$$
(7.2)
$$\displaystyle=4iK_{m}\cos\left(\frac{\pi}{m}\right)\sin\left(\frac{(2s-1)\pi}{%
m}\right)\lambda^{\frac{m+2}{2m}}(1+o(1))\quad\text{as}\quad\lambda\to\infty,$$
in the sector $|\arg(\lambda)|\leq\delta$. Since $K_{m}>0$ and $0<\frac{(2s-1)\pi}{m}<\pi$, the function
$h_{m,\ell}(\cdot)$ maps the region $|\lambda|\geq M_{1}$ for some large $M_{1}$ and $|\arg(\lambda)|\leq\delta$ into
a region containing $|\lambda|\geq M_{2}$ for some large $M_{2}$ and $|\arg(\lambda)-\frac{\pi}{2}|\leq\varepsilon_{1}$
for some $\varepsilon_{1}>0$. Also, $h_{m,\ell}(\cdot)$ is analytic in this region due to the analyticity of
$f(0,a,\lambda)$ in Theorem 3.2. Following Sibuya, we will show that for every large positive integer $n$
there exists $\lambda_{n}$ such that
(7.3)
$$h_{m,\ell}(\lambda_{n})=\left(2n+1\right)\pi i.$$
Next, Proposition 7.1 with $\mu=\lambda^{\frac{m+2}{2m}}$ implies that there exist $M_{1}^{\prime}>0$ and
$0<\delta^{\prime}<\frac{\pi}{6}$ such that $\mu(1+A(\mu))$ is univalent in the sector $|\arg(\mu)|\leq\delta^{\prime}$ and
$|\mu|\geq M_{1}^{\prime}.$ Thus, for each $n\in\mathbb{Z}$, there is at most one $\mu_{n}$ in the sector $|\arg(\mu)|\leq\delta^{\prime}$
and $|\mu|\geq M_{1}^{\prime}$ such that
(7.4)
$$\mu_{n}(1+A(\mu_{n}))=B_{n},$$
where $A$ is the error term in (7.2) and
$$B_{n}=\frac{\left(2n+1\right)\pi}{4K_{m}\cos\left(\frac{\pi}{m}\right)\sin%
\left(\frac{(2s-1)\pi}{m}\right)}.$$
Then there exists $M_{1}^{\prime\prime}\geq M_{1}^{\prime}$ such that in the sector $|\arg(\mu)|\leq\delta^{\prime}$ and $|\mu|\geq M_{1}^{\prime\prime}$,
$$|A(\mu)|<\frac{1}{5}.$$
Also we can take $n$ large enough so that $\frac{1}{2}B_{n}\geq M_{1}^{\prime\prime}$. Next, we define
$$A_{n}:=\{|A(\mu)|:|\arg(\mu)|\leq\delta^{\prime},|\mu|\geq B_{n}\}.$$
Then $\lim_{n\to\infty}A_{n}=0$
Suppose that $n$ is so large that $A_{n}\leq\frac{1}{2}\sin(\delta^{\prime})<\frac{1}{4}$. Then the disk defined by
$|\mu-B_{n}|\leq\frac{A_{n}}{1-2A_{n}}B_{n}$ is contained in the sector $|\arg(\mu)|\leq\delta^{\prime}$ and
$|\mu|\geq\frac{1}{2}B_{n}$. Moreover, on the circle $|\mu-B_{n}|=\frac{A_{n}}{1-2A_{n}}B_{n}$,
$$\displaystyle|\mu-B_{n}|-|\mu||A(\mu)|$$
$$\displaystyle\geq|\mu-B_{n}|-|\mu|A_{n}$$
$$\displaystyle\geq|\mu-B_{n}|(1-A_{n})-A_{n}B_{n}$$
$$\displaystyle\geq\frac{1-A_{n}}{1-2A_{n}}A_{n}B_{n}-A_{n}B_{n}>0.$$
Thus, by the Rouché’s theorem in complex analysis (see, e. g., [7, p. 125]), $\mu-B_{n}$ and
$\mu(1+A(\mu))-B_{n}$ have the same number of zeros in the disk and hence, (7.4) has exactly one $\mu_{n}$ in
the sector. Therefore, there exists $N=N(m,a)>0$ such that (7.3) has exactly one solution $\lambda_{n}$ for
all integers $n\geq N$.
Next, since
$$\displaystyle-\frac{\nu(G^{s}(a))}{m}\ln(\omega^{2s}\lambda)+\frac{\nu(G^{-s}(%
a))}{m}\ln(\omega^{-2s}\lambda)$$
$$\displaystyle-\frac{\nu(G^{s-1}(a))}{m}\ln(\omega^{2s-2}\lambda)+\frac{\nu(G^{%
-s+1}(a))}{m}\ln(\omega^{-2s+2}\lambda)-\frac{4\nu(G^{s}(a))}{m+2}\pi i$$
$$\displaystyle=$$
$$\displaystyle\frac{\nu(G^{s}(a))}{m}\left(-\ln(\omega^{2s}\lambda)+\ln(\omega^%
{-2s}\lambda)+\ln(\omega^{2s-2}\lambda)-\ln(\omega^{-2s+2}\lambda)\right)-%
\frac{4\nu(G^{s}(a))}{m+2}\pi i$$
$$\displaystyle=$$
$$\displaystyle-\frac{4}{m}\nu(G^{s}(a))\pi i,$$
using Corollary 3.3, (7.3) becomes
(7.5)
$$\left(2n+1\right)\pi i=\sum_{j=0}^{m+1}e_{\ell,j}(a)\lambda_{n}^{\frac{1}{2}+%
\frac{1-j}{m}}-\frac{4}{m}\nu(G^{s}(a))\pi i+O\left(\lambda^{-\rho}\right),$$
where
$$\displaystyle e_{\ell,j}(a)=$$
$$\displaystyle K_{m,j}(G^{s}(a))\omega^{2s\left(\frac{1}{2}+\frac{1-j}{m}\right%
)}-K_{m,j}(G^{-s}(a))\omega^{-2s\left(\frac{1}{2}+\frac{1-j}{m}\right)}$$
$$\displaystyle+K_{m,j}(G^{s-1}(a))\omega^{2(s-1)\left(\frac{1}{2}+\frac{1-j}{m}%
\right)}-K_{m,j}(G^{-s+1}(a))\omega^{-2(s-1)\left(\frac{1}{2}+\frac{1-j}{m}%
\right)}$$
$$\displaystyle=$$
$$\displaystyle K_{m,j}(a)\left(\omega^{-sj+2s\left(\frac{1}{2}+\frac{1-j}{m}%
\right)}-\omega^{sj-2s\left(\frac{1}{2}+\frac{1-j}{m}\right)}\right)$$
$$\displaystyle+K_{m,j}(a)\left(\omega^{-(s-1)j+2(s-1)\left(\frac{1}{2}+\frac{1-%
j}{m}\right)}-\omega^{(s-1)j-2(s-1)\left(\frac{1}{2}+\frac{1-j}{m}\right)}\right)$$
$$\displaystyle=$$
$$\displaystyle 4iK_{m,j}(a)\sin\left(\frac{(1-j)\ell\pi}{m}\right)\cos\left(%
\frac{(1-j)\pi}{m}\right),$$
where we used Lemma 3.5.
In summary, we have showed that for each $a\in\mathbb{C}^{m}$, there exists
$N_{0}=N_{0}(m,a)\leq N$ such that $\{\lambda_{n}\}_{n\geq N_{0}}$ is the
set of all eigenvalues that satisfy (7.5), since
there are only at most finitely many eigenvalues outside
$|\arg(\lambda)|\leq\varepsilon$ or inside $|\lambda|\leq M$ for
any $\varepsilon>0,\,M>0$. In order to complete proof, we need to
show that $N_{0}$ does not depend on $a\in\mathbb{C}^{m}$.
We will prove that for each $R>0$, if $|a|\leq R$, then $N_{0}(m,a)=N_{0}(m,0)$.
The error terms in (3.5) and (3.6) are uniform on the closed ball $|a|\leq R$ and hence, so is $A(\mu)$ as $\mu\to\infty$
in the sector $|\arg(\mu)|\leq\delta^{\prime}$. Thus, we can choose $N$ independent of $|a|\leq R$. Since $\mu(1+A(\mu))$ is univalent
in the sector $|\arg(\mu)|\leq\delta^{\prime}$ and $|\mu|\geq M_{1}^{\prime\prime}\geq M_{1}^{\prime}$, by the implicit function theorem and (7.4) the function $a\mapsto\mu_{n}(a)$ is
continuous on $|a|\leq R$ for each $n\geq N$ and hence, so is $a\mapsto\lambda_{n}(a)$ for $n\geq N$.
Next, we claim that for all $|a|\leq R$, there are exactly the same number of
eigenvalues that are not in $\{\lambda_{n}(a)\}_{n\geq N}$. To prove this, we will use the Hurwitz’s theorem (see, e. g., [7, p. 152]) in complex analysis. That is,
if a sequence of analytic functions converges uniformly to an
analytic function on any compact sets, then eventually functions in the sequence and the limit function have the
same number of zeros in any open set whose boundary does not contain any zeros of the limit function. The
Hurwitz’s theorem implies that since the eigenvalues are the zeros of the entire function
$\mathcal{W}_{-s,s}(a,\lambda)$, they vary continuously as $a$ and hence, there is no sudden appearance or
disappearance of eigenvalues. Also, none of the eigenvalues that are not in $\{\lambda_{n}(a)\}_{n\geq N}$ can be
continuously mapped to $\lambda_{n}(a)$ for some $n\geq N$ as $a$ varies and hence, the claim is proved. This
completes the proof.
∎
Next we prove Theorem 1.1 for $1\leq\ell<\lfloor\frac{m}{2}\rfloor$ is even.
Proof of Theorem 1.1 when $1\leq\ell<\lfloor\frac{m}{2}\rfloor$ is even.
Proof is very similar to the case when $1\leq\ell<\lfloor\frac{m}{2}\rfloor$ is odd. Let $\ell=2s$ for some
$s\in\mathbb{N}$.
Recall that $\lambda$ is an eigenvalue of $H_{\ell}$ if and only if
$\mathcal{W}_{-1,2s}(G^{-s+1}(\widetilde{a}),\omega^{-2s+1}\lambda)=0$. Then from (5.3), we have
$$\displaystyle\exp\left[L(G^{s+1}(\widetilde{a}),\omega^{2s+1}\lambda)-L(G^{-s}%
(\widetilde{a}),\omega^{-2s-1}\lambda)\right]$$
(7.6)
$$\displaystyle\times\exp\left[L(G^{s}(\widetilde{a}),\omega^{2s-1}\lambda)-L(G^%
{-s+1}(\widetilde{a}),\omega^{-2s+1}\lambda)+O\left(\lambda^{-\rho}\right)%
\right]=-1,$$
where $\widetilde{a}=G^{-\frac{1}{2}}(a)$ and where we used
$\left[1+O\left(\lambda^{-\rho}\right)\right]=\exp[O\left(\lambda^{-\rho}\right)]$
again.
Like in the case when $1\leq\ell<\lfloor\frac{m}{2}\rfloor$ is
odd, from Lemma 3.3, we have
(7.7)
$$\left(2n+1\right)\pi i=\sum_{j=0}^{m+1}e_{\ell,j}(a)\lambda_{n}^{\frac{1}{2}+%
\frac{1-j}{m}}+O\left(\lambda^{-\rho}\right),$$
where for $0\leq j\leq m+1$,
$$\displaystyle e_{\ell,j}(a)=$$
$$\displaystyle K_{m,j}(G^{s+\frac{1}{2}}(a))\omega^{(2s+1)\left(\frac{1}{2}+%
\frac{1-j}{m}\right)}-K_{m,j}(G^{-s-\frac{1}{2}}(a))\omega^{-(2s+1)\left(\frac%
{1}{2}+\frac{1-j}{m}\right)}$$
$$\displaystyle+K_{m,j}(G^{s-\frac{1}{2}}(a))\omega^{(2s-1)\left(\frac{1}{2}+%
\frac{1-j}{m}\right)}-K_{m,j}(G^{-s+\frac{1}{2}}(a))\omega^{-(2s-1)\left(\frac%
{1}{2}+\frac{1-j}{m}\right)}$$
$$\displaystyle=$$
$$\displaystyle\sum_{k=0}^{j}(-1)^{k}K_{m,j,k}b_{j,k}(a)\left(\omega^{-j\frac{%
\ell+1}{2}+(\ell+1)\left(\frac{1}{2}+\frac{1-j}{m}\right)}-\omega^{j\frac{\ell%
+1}{2}-(\ell+1)\left(\frac{1}{2}+\frac{1-j}{m}\right)}\right.$$
$$\displaystyle\qquad\qquad\qquad\qquad\qquad\left.+\omega^{-j\frac{\ell-1}{2}+(%
\ell-1)\left(\frac{1}{2}+\frac{1-j}{m}\right)}-\omega^{j\frac{\ell-1}{2}-(\ell%
-1)\left(\frac{1}{2}+\frac{1-j}{m}\right)}\right)$$
$$\displaystyle=$$
$$\displaystyle 4i\sum_{k=0}^{j}(-1)^{k}K_{m,j,k}b_{j,k}(a)\sin\left(\frac{(1-j)%
\ell\pi}{m}\right)\cos\left(\frac{(1-j)\pi}{m}\right),$$
where we used Lemma 3.5 and $\ell=2s$. Then we use the
arguments in the case when $\ell$ is odd to show that there exists
$N_{0}=N_{0}(m)$ such that the set $\{\lambda_{n}\}_{n\geq N_{0}}$ of all
eigenvalues satisfy (7.7) and hence, the proof is
completed.
∎
8. Proof of Theorem 1.1 when $\ell=\lfloor\frac{m}{2}\rfloor$ and when $\frac{m}{2}<\ell\leq m-1$.
In this section, we prove Theorem 1.1 for
$\ell=\lfloor\frac{m}{2}\rfloor$. We first prove the theorem when
$m$ is even, and later, we will treat the cases when $m$ is odd.
Then at the end of the section, we will prove the theorem when
$\frac{m}{2}<\ell\leq m-1$, by scaling. Proof of existence of
$N_{0}=N_{0}(m)$ is the same as in Section 7, so below we
will omit this part of proof.
8.1. When $m$ is even
We further divide the case into when $\ell$ is odd and when $\ell$ is even.
Proof of Theorem 1.1 when $m$ is even and $\ell=\lfloor\frac{m}{2}\rfloor$ is odd.
Suppose that $m$ is even and $\ell=\frac{m}{2}=2s-1$ for some $s\in\mathbb{N}$. Recall that $\lambda$ is an eigenvalue of
$H_{\ell}$ if and only if $\mathcal{W}_{-s,s}(a,\lambda)=0$ if and only if
$\mathcal{W}_{-1,2s-1}(G^{-s+1}(a),\omega^{-2s+2}\lambda)=0$.
So if $\mathcal{W}_{-1,2s-1}(G^{-s+1}(a),\omega^{-2s+2}\lambda)=0$ and if $|\lambda|$ is large enough, then from
(5.14),
$$\displaystyle\exp\left[L(G^{3s-1}(a),\omega^{2s-2}\lambda)-L(G^{s+1}(a),\omega%
^{-2s+2}\lambda)\right]$$
$$\displaystyle\times\exp\left[L(G^{s-1}(a),\omega^{2s-2}\lambda)-L(G^{-s+1}(a),%
\omega^{-2s+2}\lambda)+O\left(\lambda^{-\rho}\right)\right]=-\omega^{4\nu(G^{s%
}(a))}.$$
Then by Lemma 3.3,
$$\left(\frac{8\nu(G^{s}(a))}{m+2}+2n+1\right)\pi i=\sum_{j=0}^{m+1}e_{\ell,j}(a%
)\lambda_{n}^{\frac{1}{2}+\frac{1-j}{m}}+\frac{16(s-1)\nu(G^{s}(a))}{m(m+2)}%
\pi i+O\left(\lambda^{-\rho}\right),$$
where for $0\leq j\leq m+1$, the coefficients $e_{\ell,j}(a)$ are given by
$$\displaystyle e_{\ell,j}(a)=$$
$$\displaystyle K_{m,j}(G^{3s-1}(a))\omega^{2(s-1)\left(\frac{1}{2}+\frac{1-j}{m%
}\right)}-K_{m,j}(G^{s+1}(a))\omega^{-2(s-1)\left(\frac{1}{2}+\frac{1-j}{m}%
\right)}$$
$$\displaystyle+K_{m,j}(G^{s-1}(a))\omega^{2(s-1)\left(\frac{1}{2}+\frac{1-j}{m}%
\right)}-K_{m,j}(G^{-s+1}(a))\omega^{-2(s-1)\left(\frac{1}{2}+\frac{1-j}{m}%
\right)}$$
$$\displaystyle=$$
$$\displaystyle K_{m,j}(a)\left(\omega^{(s+1)j+2(s-1)\left(\frac{1}{2}+\frac{1-j%
}{m}\right)}-\omega^{-(s+1)j-2(s-1)\left(\frac{1}{2}+\frac{1-j}{m}\right)}\right.$$
$$\displaystyle\qquad\qquad\qquad\qquad\left.+\omega^{-(s-1)j+2(s-1)\left(\frac{%
1}{2}+\frac{1-j}{m}\right)}-\omega^{(s-1)j-2(s-1)\left(\frac{1}{2}+\frac{1-j}{%
m}\right)}\right)$$
$$\displaystyle=$$
$$\displaystyle 2iK_{m,j}(a)\left(1+(-1)^{j}\right)\sin\left(\frac{(1-j)(2s-2)%
\pi}{m}\right)$$
$$\displaystyle=$$
$$\displaystyle 4iK_{m,j}(a)\sin\left(\frac{(1-j)\pi}{2}\right)\cos\left(\frac{(%
1-j)\pi}{m}\right),$$
where we used Lemma 3.5.
∎
Proof of Theorem 1.1 when $m$ is even and $\ell=\lfloor\frac{m}{2}\rfloor$ is even.
Let $\ell=\frac{m}{2}=2s$ for some $s\in\mathbb{N}$. Recall that $\lambda$ is an eigenvalue of $H_{\ell}$ if and only if
$\mathcal{W}_{-1,2s}(G^{-s+1}(\widetilde{a}),\omega^{-2s+1}\lambda)=0$.
Suppose that
$\mathcal{W}_{-1,2s}(G^{-s+1}(\widetilde{a}),\omega^{-2s+1}\lambda)=\mathcal{W}%
_{-1,2s}(G^{-s+\frac{1}{2}}(a),\omega^{-2s+1}\lambda)=0$.
Then by (5.14),
$$\displaystyle\exp\left[L(G^{3s+\frac{1}{2}}(a),\omega^{2s-1}\lambda)-L((G^{s+%
\frac{3}{2}}(a),\omega^{-2s+1}\lambda)\right]$$
$$\displaystyle\times\exp\left[L(G^{s-\frac{1}{2}}(a),\omega^{2s-1}\lambda)-L(G^%
{-s+\frac{1}{2}}(a),\omega^{-2s+1}\lambda)+O\left(\lambda^{-\rho}\right)\right%
]=-1.$$
Then like before,
$$\left(2n+1\right)\pi i=\sum_{j=0}^{m+1}e_{\ell,j}(a)\lambda_{n}^{\frac{1}{2}+%
\frac{1-j}{m}}+O\left(\lambda^{-\rho}\right),$$
where for $0\leq j\leq m+1$,
$$\displaystyle e_{\ell,j}(a)=$$
$$\displaystyle K_{m,j}(G^{3s+\frac{1}{2}}(a))\omega^{(2s-1)\left(\frac{1}{2}+%
\frac{1-j}{m}\right)}-K_{m,j}((G^{s+\frac{3}{2}}(a))\omega^{-(2s-1)\left(\frac%
{1}{2}+\frac{1-j}{m}\right)}$$
$$\displaystyle+K_{m,j}(G^{s-\frac{1}{2}}(a))\omega^{(2s-1)\left(\frac{1}{2}+%
\frac{1-j}{m}\right)}-K_{m,j}(G^{-s+\frac{1}{2}}(a))\omega^{-(2s-1)\left(\frac%
{1}{2}+\frac{1-j}{m}\right)}$$
$$\displaystyle=$$
$$\displaystyle 2i\sum_{k=0}^{j}(-1)^{k}K_{m,j,k}b_{j,k}(a)\left(1+(-1)^{j}%
\right)\sin\left(\frac{(1-j)(2s-1)\pi}{m}\right)$$
$$\displaystyle=$$
$$\displaystyle 4i\sum_{k=0}^{j}(-1)^{k}K_{m,j,k}b_{j,k}(a)\sin\left(\frac{(1-j)%
\pi}{2}\right)\cos\left(\frac{(1-j)\pi}{m}\right).$$
∎
8.2. When $m$ is odd
We divide the case into when $\ell$ is odd and when $\ell$ is even.
Proof of Theorem 1.1 when $m$ and $\ell=\lfloor\frac{m}{2}\rfloor$ are odd.
Let $m$ and $\ell=\frac{m-1}{2}=2s-1$ be odd. Suppose that $\lambda$ is an eigenvalue of $H_{\ell}$. Since
$\lambda$ is an eigenvalue of $H_{\ell}$ if and only if
$\mathcal{W}_{-1,2s-1}(G^{-s+1}(a),\omega^{-2s+2}\lambda)=0$,
if $|\lambda|$ is large enough, then from (5.16),
$$\displaystyle\exp\left[L(G^{s-1}(a),\omega^{2s-2}\lambda)-L(G^{-s+1}(a),\omega%
^{-2s+2}\lambda)\right]$$
$$\displaystyle\times\exp\left[-L(G^{3s+1}(a),\omega^{-2s}\lambda)-L(G^{s+1}(a),%
\omega^{-2s+1}\lambda)+O\left(\lambda^{-\rho}\right)\right]=-1.$$
Then, from Lemma 3.3,
$$\left(2n+1\right)\pi i=\sum_{j=0}^{m+1}e_{\ell,j}(a)\lambda_{n}^{\frac{1}{2}+%
\frac{1-j}{m}}+O\left(\lambda^{-\rho}\right),$$
where for $0\leq j\leq m+1$,
$$\displaystyle e_{\ell,j}(a)=$$
$$\displaystyle K_{m,j}(G^{s-1}(a))\omega^{(2s-2)\left(\frac{1}{2}+\frac{1-j}{m}%
\right)}-K_{m,j}(G^{-s+1}(a))\omega^{(-2s+2)\left(\frac{1}{2}+\frac{1-j}{m}%
\right)}$$
$$\displaystyle-K_{m,j}(G^{3s+1}(a))\omega^{-2s\left(\frac{1}{2}+\frac{1-j}{m}%
\right)}-K_{m,j}(G^{s+1}(a))\omega^{(-2s+1)\left(\frac{1}{2}+\frac{1-j}{m}%
\right)}$$
$$\displaystyle=$$
$$\displaystyle K_{m,j}(a)\left(\omega^{-j(s-1)+(2s-2)\left(\frac{1}{2}+\frac{1-%
j}{m}\right)}-\omega^{j(s-1)-(2s-2)\left(\frac{1}{2}+\frac{1-j}{m}\right)}\right.$$
$$\displaystyle\qquad\qquad\qquad\left.-\omega^{-j(3s+1)-2s\left(\frac{1}{2}+%
\frac{1-j}{m}\right)}-\omega^{-j(s+1)-(2s-1)\left(\frac{1}{2}+\frac{1-j}{m}%
\right)}\right)$$
$$\displaystyle=$$
$$\displaystyle K_{m,j}(a)\left(\omega^{(1-j)(2s-2)\left(\frac{1}{2}+\frac{1}{m}%
\right)}-\omega^{-(1-j)(2s-2)\left(\frac{1}{2}+\frac{1}{m}\right)}\right.$$
$$\displaystyle\qquad\qquad\qquad\left.-\omega^{js-2s\left(\frac{1}{2}+\frac{1-j%
}{m}\right)}+\omega^{-js+2s\left(\frac{1}{2}+\frac{1-j}{m}\right)}\right)$$
$$\displaystyle=$$
$$\displaystyle 2iK_{m,j}(a)\left(\sin\left(\frac{(m-3)(1-j)\pi}{2m}\right)+\sin%
\left(\frac{(m+1)(1-j)\pi}{2m}\right)\right)$$
$$\displaystyle=$$
$$\displaystyle 4iK_{m,j}(a)\sin\left(\frac{(1-j)\ell\pi}{m}\right)\cos\left(%
\frac{(1-j)\pi}{m}\right).$$
∎
Proof of Theorem 1.1 when $m$ is odd and $\ell=\lfloor\frac{m}{2}\rfloor$ is even.
Let $\ell=\frac{m-1}{2}=2s$ for some $s\in\mathbb{N}$. Then $\lambda$ is an eigenvalue of $H_{\ell}$ if and only if
$\mathcal{W}_{-1,2s}(G^{-s+1}(\widetilde{a}),\omega^{-2s+1}\lambda)=0$.
If $\mathcal{W}_{-1,2s}(G^{-s+1}(\widetilde{a}),\omega^{-2s+1}\lambda)=0$ and if $|\lambda|$ is large enough, then
from (5.16),
$$\displaystyle\exp\left[L(G^{s}(\widetilde{a}),\omega^{2s-1}\lambda)-L(G^{-s+1}%
(\widetilde{a}),\omega^{-2s+1}\lambda)\right]$$
$$\displaystyle\times\exp\left[-L(G^{3s+3}(\widetilde{a}),\omega^{-2s-1}\lambda)%
-L(G^{s+2}(\widetilde{a}),\omega^{-2s}\lambda)+O\left(\lambda^{-\rho}\right)%
\right]=-1.$$
Then,
from Lemma 3.3,
$$\left(2n+1\right)\pi i=\sum_{j=0}^{m+1}e_{\ell,j}(a)\lambda_{n}^{\frac{1}{2}+%
\frac{1-j}{m}}+O\left(\lambda^{-\rho}\right),$$
where for $0\leq j\leq m+1$,
$$\displaystyle e_{\ell,j}(a)=$$
$$\displaystyle K_{m,j}(G^{s-\frac{1}{2}}(a))\omega^{(2s-1)\left(\frac{1}{2}+%
\frac{1-j}{m}\right)}-K_{m,j}((G^{-s+\frac{1}{2}}(a))\omega^{-(2s-1)\left(%
\frac{1}{2}+\frac{1-j}{m}\right)}$$
$$\displaystyle-K_{m,j}(G^{3s+\frac{5}{2}}(a))\omega^{-(2s+1)\left(\frac{1}{2}+%
\frac{1-j}{m}\right)}-K_{m,j}(G^{s+\frac{3}{2}}(a))\omega^{-2s\left(\frac{1}{2%
}+\frac{1-j}{m}\right)}$$
$$\displaystyle=$$
$$\displaystyle 2i\sum_{k=0}^{j}(-1)^{k}K_{m,j,k}b_{j,k}(a)\left[\sin\left(\frac%
{(1-j)(2s-1)\pi}{m}\right)+\sin\left(\frac{(1-j)(2s+1)\pi}{m}\right)\right]$$
$$\displaystyle=$$
$$\displaystyle 4i\sum_{k=0}^{j}(-1)^{k}K_{m,j,k}b_{j,k}(a)\sin\left(\frac{(1-j)%
\ell\pi}{m}\right)\cos\left(\frac{(1-j)\pi}{m}\right).$$
∎
Theorem 1.1 for $1\leq\ell\leq\frac{m}{2}$ has been proved. Next we prove Theorem 1.1 for
$\frac{m}{2}<\ell\leq m-1$, by the change of the variables $z\mapsto-z$.
8.3. When $\frac{m}{2}<\ell\leq m-1$
Proof of Theorem 1.1 when $\frac{m}{2}<\ell\leq m-1$.
Suppose that $\ell>\frac{m}{2}$. If $u$ is an eigenfunction of $H_{\ell}$, then $v(z)=u(-z,\lambda)$ solves
$$-v^{\prime\prime}(z)+\left[(-1)^{-\ell}(-iz)^{m}-P(-iz)\right]v(z)=\lambda v(z),$$
and
$$\text{$v(z)\rightarrow 0$ exponentially, as $z\rightarrow\infty$ along the two%
rays}\quad\arg z=-\frac{\pi}{2}\pm\frac{((m-\ell)+1)\pi}{m+2}.$$
The coefficient vector of $P(-z)$ is $((-1)^{m-1}a_{1},(-1)^{m-2}a_{2},\dots,(-1)^{1}a_{m-1},a_{m}).$ Certainly,
$$\sin\left(\frac{(1-j)(m-\ell)\pi}{m}\right)=(-1)^{j}\sin\left(\frac{(1-j)\ell%
\pi}{m}\right).$$
Also, one can find from Lemma 3.5 that for $0\leq k\leq j$,
$$b_{j,k}((-1)^{m-1}a_{1},(-1)^{m-2}a_{2},\dots,-a_{m-1},a_{m})=(-1)^{mk-j}b_{j,%
k}(a_{1},a_{2},\dots,a_{m-1},a_{m}).$$
Moreover, $c_{m-\ell,j}((-1)^{m-1}a_{1},(-1)^{m-2}a_{2},\dots,-a_{m-1},a_{m})=c_{\ell,j}(%
a_{1},a_{2},\dots,a_{m-1},a_{m}).$
This completes proof of Theorem 1.1.
∎
Acknowledgments
The author thanks Mark Ashbaugh, Fritz Gesztesy, Richard Laugesen, Boris Mityagin, Grigori Rozenblioum, and Alexander Turbiner for helpful discussions and references.
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email contact: [email protected] |
Quasiparticle velocities in 2D electron/hole liquids with spin-orbit coupling
D. Aasen
Stefano Chesi
W. A. Coish
Department of Physics, McGill University, Montréal, Québec H3A 2T8, Canada
(December 5, 2020)
Abstract
We study the influence of spin-orbit interactions on quasiparticle dispersions in two-dimensional electron and heavy-hole liquids in III-V semiconductors. To obtain closed-form analytical results, we restrict ourselves to spin-orbit interactions with isotropic spectrum and work within the screened Hartree-Fock approximation, valid in the high-density limit. For electrons having a linear-in-momentum Rashba (or, equivalently, Dresselhaus) spin-orbit interaction, we show that the screened Hartree-Fock approximation recovers known results based on the random-phase approximation and we extend those results to higher order in the spin-orbit coupling. While the well-studied case of electrons leads only to a weak modification of quasiparticle properties in the presence of the linear-in-momentum spin-orbit interaction, we find two important distinctions for hole systems (with a leading nonlinear-in-momentum spin-orbit interaction). First, the group velocities associated with the two hole-spin branches acquire a significant difference in the presence of spin-orbit interactions, allowing for the creation of spin-polarized wavepackets in zero magnetic field. Second, we find that the interplay of Coulomb and spin-orbit interactions is significantly more important for holes than for electrons and can be probed through the quasiparticle group velocities. These effects should be directly observable in magnetotransport, Raman scattering, and femtosecond-resolved Faraday rotation measurements. Our results are in agreement with a general argument on the velocities, which we formulate for an arbitrary choice of the spin-orbit coupling.
pacs: 71.10.-w, 71.70.Ej, 71.45.Gm, 73.61.Ey
I Introduction
Semiconductor heterostructures offer the possibility of forming two-dimensional liquids with tunable sheet density $n_{s}$. In an idealized model where the carriers have parabolic dispersion with band mass $m$ and interact with Coulomb forces,TheBook the only relevant quantity is the dimensionless Wigner-Seitz radius $r_{s}=1/\sqrt{\pi a_{B}^{2}n_{s}}$, where $a_{B}=\hbar^{2}\epsilon_{r}/me^{2}$ is the effective Bohr radius ($\epsilon_{r}$ is the dielectric constant). Since $r_{s}$ serves as the interaction parameter, changing $n_{s}$ allows for a systematic study of the effects of the Coulomb interaction. In particular, properties of quasiparticle excitations such as their dispersion and lifetime are significantly modified due to electron-electron interactions.TheBook
Great attention has been paid in recent years to band-structure effects involving the spin degree of freedom.WinklerBook The strength and form of spin-orbit interaction (SOI) can be controlled in two-dimensional liquids through the choice of materials, the type of carriers (electrons/holes), and details of the confinement potential. For example, it is possible to change the coupling constant with external gates.Datta1989 ; Nitta1997 ; Johnson2009 In addition to detailed studies of single-particle properties, the problem of understanding the effects of SOI in the presence of Coulomb interactions is a topic of ongoing investigations.
Quasiparticle properties in the presence of SOIChen1999 ; Saraga2005 ; Nechaev2009 ; Nechaev2010 ; Agarwal2011 have been examined primarily accounting for RashbaBychov1984a ; Bychov1984b and/or Dresselhaus SOI,Dresselhaus1955 ; Dyakonov1986 which are dominant in electronic systems. The SOI results in two distinct spin subbands, with two associated Fermi surfaces. The effects on quasiparticles are usually very small; at each of the two Fermi surfaces the quasiparticle dispersionChen1999 ; Saraga2005 ; Agarwal2011 and lifetimeSaraga2005 ; Nechaev2009 ; Nechaev2010 are almost unaffected by SOI, except in the case of very large SOI coupling.Nechaev2010 ; Berman2010 In fact, it was found with Rashba SOI that the corrections to these quantities linear in the SOI coupling are absent.Saraga2005 Although explicit calculations are performed within perturbative approximation schemes, notably the random-phase approximation (RPA),Saraga2005 ; Nechaev2009 ; Nechaev2010 the SOI leading-order cancellation is valid non-perturbatively (to all orders in $r_{s}$).ChesiExact Similar arguments hold for other physical quantities.ChesiExact ; Abedinpour2010 For example, values of the ground-state energy obtained with Monte Carlo simulationsAmbrosetti2009 for up to $r_{s}=20$ could be reproduced with excellent accuracy by simply neglecting SOI corrections to the exchange-correlation energy.ChesiHighDens Noticeable exceptions exhibiting larger SOI effects are spin-textured broken symmetry phases,ChesiHFstates ; Juri2008 non-analytic corrections to the spin susceptibility,Zak2010 ; Zak2011 and the plasmon dispersion.Agarwal2011 All these examples involve the presence of spin polarization (either directlyChesiHFstates ; Juri2008 ; Zak2010 or indirectlyAgarwal2011 ), in which case the arguments of Ref. ChesiExact, do not apply.
Another interesting situation occurs when the SOI has a nonlinear dependence on momentum, thus cannot be written as a spin-dependent gauge potential.Aleiner2001 ; Coish2006 ; Abedinpour2010 Then, the approximate cancellations mentioned above are not expected. Winkler has shown that the dominant SOI induced by heterostructure asymmetry is cubic in momentum for heavy holes in III-V semiconductors.Winkler2000 This theoretical analysis was later shown to be in good agreement with magnetoresistance experiments.Winkler2002 Recently, the relevance of this cubic-in-momentum model was supported by the anomalous sign and magnetic-field dependence of spin polarization in quantum point contacts.Rokhinson2004 ; ChesiQPC SOI quadratic in momentum can also be induced for heavy holes by an in-plane magnetic field.Bulaev2007 ; Chesi07April
That these band-structure effects can substantially modify standard many-body results is confirmed by recent Shubnikov-de Haas oscillation measurements in low-density hole systems.Winkler2005 ; Chiu2011 For example, a surprisingly small Coulomb enhancement of the $g$-factor has been reported Winkler2005 and puzzling results have also been obtained for the effective masses $m_{\pm}$ of the two hole-spin sub-bands ($\sigma=\pm$).Chiu2011 A mechanism for the small $g$-factor enhancement was suggested in Ref. Chesi07April, : if SOI strongly distorts the ground state spin structure, the exchange energy becomes ineffective in promoting full polarization of the hole system.
In this paper, we focus on the effective masses $m_{\pm}$. We note that $m_{\pm}$ are directly related to the quasiparticle group velocities $v_{\pm}$ at the Fermi surfaces. With this in mind, we find it more transparent to discuss the effects of SOI and electron-electron interactions on $m_{\pm}$ in terms of wavepacket motion, as illustrated in Fig. 1. If an unpolarized wavepacket is injected at the Fermi energy (with average momentum along a given direction), the subsequent motion is very different depending on whether the SOI is linear ($n=1$) or non-linear ($n=2,3$). In the former case ($n=1$), corresponding to electrons, we have $v_{+}\simeq v_{-}$ and the motion is essentially equivalent to the case without SOI. In contrast, for holes with strong SOI, $v_{+}\neq v_{-}$ and the two spin components become spatially separated. The separation between the two spin components can become quite sizable, and it should be possible to observe such an effect with, e.g., Faraday-rotation imaging techniques.Crooker1996 ; Kikkawa1997 Electron-electron interactions, in addition to modifying the average velocity $v=(v_{+}+v_{-})/2$ (an effect which is well-known without SOITheBook ; Tan2005 ; Drummond2009 ), are also reflected on the velocity difference $(v_{+}-v_{-})$ between the two spin branches.
With these motivations in mind, we pursue a study of the quasiparticle group velocity in the presence of linear Rashba and non-linear SOI. An accurate analytical treatment of the electron-electron interactions can be carried out at high density, and we restrict ourselves to this interesting limit. The regime of strong electron-electron interactions is much more difficult to treat (see Ref. Drummond2009, for a Monte Carlo study without SOI) but it represents a relevant topic for future investigations ($r_{s}\sim 6-12$ in Refs. Winkler2005, ; Chiu2011, ).
This paper is organized as follows: In Sec. II we introduce a model Hamiltonian including a generalized SOIChesi07April and we briefly review its non-interacting properties, demonstrating that injected holes will separate into spin-polarized wavepackets. The Coulomb interaction is treated in Sec. III by extending the classic treatment of Ref. Janak1969, . We describe several results of this screened Hartree-Fock approach in detail, focusing on the quasiparticle group velocities and the interplay of spin-orbit and Coulomb interaction effects. A discussion of these results is given in Sec. IV. In Appendix A, we give a general argument showing that corrections to the velocity from any linear-in-momentum SOI can always be neglected to lowest order. Finally, a number of technical details are provided in Appendices B and C.
II Noninteracting problem
In the high-density limit, $r_{s}$ is small and the system is well-described by a non-interacting single-particle Hamiltonian. We consider here the following model,Chesi07April including a generalized SOI with a linear-, quadratic-, and cubic-in-momentum dependence for $n=1,2,3$:
$$H_{0}=\frac{p^{2}}{2m}+i\gamma\frac{p_{-}^{n}\sigma_{+}-p_{+}^{n}\sigma_{-}}{2},$$
(1)
where ${\bf p}$ is the momentum operator, $m$ the band mass, $p_{\pm}=p_{x}\pm ip_{y}$, $\sigma_{\pm}=\sigma_{x}\pm i\sigma_{y}$, and $\gamma$ the generalized spin-orbit coupling, with $\boldsymbol{\sigma}$ the vector of Pauli matrices. The physical justification of this model has been given in Ref. Chesi07April, : the $n=1$ Hamiltonian contains the Rashba SOI present in electronic systems,Bychov1984a ; Bychov1984b while $n=3$ includes the analogous term generated by an asymmetric confinement potential for holes.Winkler2000 ; Winkler2002 ; ChesiQPC Finally, the $n=2$ case is also relevant for holes, in the presence of an in-plane magnetic field.Chesi07April ; Bulaev2007 While SOI terms of different form generally coexist (see Appendix A), we assume here that one $n$ value is dominant. This greatly simplifies the problem by preserving the isotropy of the electron liquid in the x-y plane and is a good approximation for several relevant situations. For example, it was found for holesWinkler2000 ; Winkler2002 ; ChesiQPC that the $n=3$ term can be much larger than corrections to the SOI due to bulk inversion asymmetry.WinklerBook ; Rashba1988 Diagonalizing $H_{0}$ yields the energy spectrum
$$E_{\sigma}^{0}(k)=\frac{\hbar^{2}k^{2}}{2m}+\sigma\gamma\hbar^{n}k^{n},$$
(2)
with $\sigma=\pm$ labeling the two chiral spin branches. The corresponding eigenfunctions areChesi07April
$$\varphi_{\mathbf{k}\sigma}(\mathbf{r})=\frac{e^{i\mathbf{k}\cdot\mathbf{r}}}{%
\sqrt{2L^{2}}}\begin{pmatrix}1\\
i\sigma e^{in\theta_{\mathbf{k}}}\end{pmatrix},$$
(3)
where $\mathbf{k}$ is a wavevector in the x-y plane, $\theta_{\bf k}$ is the angle $\mathbf{k}$ makes with the x-axis, and $L$ is the linear size of the system.
It is useful at this point to introduce a dimensionless quantity $g$ characterizing the strength of the spin-orbit coupling:Chesi07April
$$g=\frac{\gamma\hbar^{n}k_{F}^{n}}{E_{F}},$$
(4)
where $E_{F}=\hbar^{2}k_{F}^{2}/2m$ is the Fermi energy without SOI, written in terms of the Fermi wavevector $k_{F}=\sqrt{2\pi n_{s}}=\sqrt{2}/(a_{\mathrm{B}}r_{s})$. While $\gamma$ has different physical dimensions for each value, $n=1,2,3$, the dimensionless coupling $g$ always gives the ratio of the spin-orbit energy to the kinetic energy. The coupling $g$ thus plays a role analogous to that of $r_{s}$ for the Coulomb interaction. Taking $\gamma$ to be independent of the density (this is not always the caseWinklerBook ), then from Eq. (4) we have $g\propto k_{F}^{n-2}\propto r_{s}^{2-n}$ (since $E_{F}\propto k_{F}^{2}$). This suggests that in the high-density limit ($r_{s}\to 0$), the effects of SOI are suppressed for electrons ($n=1$), but remain constant ($n=2$) or are enhanced ($n=3$) for holes. This simple estimate already indicates a qualitative difference for holes relative to electrons. We will see that this difference is indeed significant in the following sections.
In the presence of SOI the two spin bands $(\sigma=\pm)$ have different densities $n_{\pm}$, giving the total 2D sheet density $n_{s}=n_{+}+n_{-}$. Keeping $n_{s}$ (thus $k_{F}$) fixed gives a constraint on the Fermi wavevectors $k_{\pm}$ for the $\sigma=\pm$ bands, $k_{+}^{2}+k_{-}^{2}=2k_{F}^{2}$. We can then characterize the solution to this equation with a single parameter $\chi$:
$$k_{\pm}=k_{F}\sqrt{1\mp\chi}.$$
(5)
The parameter $\chi=(n_{-}-n_{+})/n_{s}$ gives the chirality and is determined by both the SOI and electron-electron interaction.comment_gen_chi We will assume for definiteness that $\gamma\geq 0$ such that $\chi\geq 0$ and $k_{+}\leq k_{-}$. For a fixed generalized spin-orbit coupling $\gamma$, the non-interacting value of $\chi$ can be determined by setting the Fermi energies of the two bands equal, i.e., $E^{0}_{+}(k_{+})=E^{0}_{-}(k_{-})$. This equation immediately gives a relationship between $g$ and $\chi$,
$$g=\frac{2\chi}{(1+\chi)^{n/2}+(1-\chi)^{n/2}}.$$
(6)
We denote the solution of Eq. (6) by $\chi_{0}(g)$, which gives the non-interacting Fermi wavevectors $k_{\pm}^{0}=k_{F}\sqrt{1\mp\chi_{0}}$. Explicit expressions for $\chi_{0}(g)$ are given in Ref. Chesi07April, . We only cite here the small-$g$ behavior, which is easily found from Eq. (6):
$$\chi_{0}(g)\simeq g.$$
(7)
We are mainly interested here in the properties of the quasiparticles and, in particular, their group velocities $v_{\pm}$. The fact that the dispersion relation (2) is a function only of the magnitude $k$ is a consequence of the model being isotropic with respect to $\mathbf{k}$, which allows us to discuss the magnitude of the group velocity on the Fermi surfaces,
$$v_{\pm}^{0}=\frac{1}{\hbar}\frac{\partial E_{\pm}^{0}(k)}{\partial k}\Big{|}_{%
k=k^{0}_{\pm}}=\frac{\hbar k^{0}_{\pm}}{m}\pm n\gamma(\hbar k^{0}_{\pm})^{n-1}.$$
(8)
The above expression can be evaluated explicitly in terms of the non-interacting chirality $\chi_{0}(g)$:
$$\displaystyle\frac{v_{\pm}^{0}}{v_{F}}=$$
$$\displaystyle\sqrt{1\mp\chi_{0}(g)}\pm\frac{n}{2}g[1\mp\chi_{0}(g)]^{(n-1)/2}$$
(9)
$$\displaystyle\simeq 1\pm\frac{g}{2}(n-1),$$
where $v_{F}=\hbar k_{F}/m$ is the Fermi velocity in the absence of SOI and in the second line we have expanded the result to lowest order in $g$, by making use of Eq. (7). Equation (9) shows that there is no relative difference in group velocity for the $\sigma=\pm$ bands when $n=1$. In contrast, when $n=2,3$ a sizable correction linear in $g$ is present. This difference reflects itself on the evolution of an initially unpolarized wavepacket injected at the Fermi surface, schematically illustrated in Fig. 1. While for electrons the wavepacket remains unpolarized, for holes the two spin components spatially separate with time. Electron-electron interactions modify the non-interacting velocities $v^{0}_{\pm}$, but the qualitative difference introduced by SOI between electrons ($n=1$) and holes ($n=2,3$) remains essentially unchanged. A similar behavior holds for other properties of the electron liquid as well:ChesiExact vanishing corrections to lowest order in $g$ were found for the quasiparticle lifetime,Saraga2005 ; Nechaev2010 the occupation,Chesi07April and the exchange-correlation energy,ChesiHighDens if only Rashba SOI ($n=1$) is included.
Finally, another relevant quasiparticle observable is the effective mass. This has recently been measured in hole systems through experiments on quantum oscillations.Chiu2011 This physical quantity is simply given by $m_{\pm}=\hbar k_{\pm}/v_{\pm}$ and is thus essentially equivalent to $v_{\pm}$.
III Screened Hartree-Fock approximation
Realistically, charged particles interact through the Coulomb potential so it is interesting to understand how the presence of SOI modifies the behavior of the quasiparticles. The fully interacting Hamiltonian is given by:
$$H=\sum_{i}H_{0}^{(i)}+\frac{1}{2}\sum_{i\neq j}\frac{e^{2}}{\epsilon_{r}|{\bf r%
}_{i}-{\bf r}_{j}|},$$
(10)
where $H_{0}^{(i)}$ (for electron $i$) is as in Eq. (1) and the presence in (10) of a uniform neutralizing background is understood. Although many sophisticated techniques exist to approach this problem,TheBook ; Drummond2009 the simplest approximation to the quasiparticle self-energy is obtained by only including the exchange contribution:
$$E_{\sigma}(k)=E_{\sigma}^{0}(k)+\Sigma^{x}_{\sigma}(k),$$
(11)
where
$$\Sigma^{x}_{\sigma}(k)=-\sum_{\mathbf{k}^{\prime}\sigma^{\prime}}\frac{(1+%
\sigma\sigma^{\prime}\cos{n\theta^{\prime}})}{2L^{2}}n_{\mathbf{k}^{\prime}%
\sigma^{\prime}}V(|\mathbf{k}-\mathbf{k}^{\prime}|).$$
(12)
Here, $n_{\mathbf{k^{\prime}}\pm}=\Theta(k_{\pm}-k^{\prime})$ is the occupation at $T=0$ for the $\sigma=\pm$ band, respectively, with $\Theta(x)$ the Heaviside step function. The first factor in the summation of Eq. (12), involving the angle $\theta^{\prime}$ between $\mathbf{k^{\prime}}$ and $\mathbf{k}$, arises from the scalar product of the non-interacting spinors [Eq. (3)], and takes into account the specific nature of the spin-orbit interaction ($n=1,2,3$).
To lowest order, $V(q)$ is the Fourier transform of the bare Coulomb potential, $2\pi e^{2}/(\epsilon_{r}q)$. As is well known,TheBook this form of the Coulomb interaction leads to an unphysical divergence in the quasiparticle velocity. By considering an infinite resummation in perturbation theory, the screening of the Coulomb interaction removes the divergence, e.g., in the RPA approximation. Finally, by approximating the dielectric function in the effective interaction by its zero-frequency long-wavelength limit, the RPA self-energy gives Eq. (12) with
$$V(q)=\frac{2\pi e^{2}/\epsilon_{r}}{q+\sqrt{2}r_{s}k_{F}}.$$
(13)
This screened Hartree-Fock approximation with SOI,Chen1999 ; Agarwal2011 notwithstanding its simplicity, becomes accurate in the high-density limit.
III.1 Renormalized occupation
By including the SOI, we can verify that Eq. (12) gives the correct high-density behavior for the Fermi wavevectors $k_{\pm}$. These are modified by electron-electron interactions from their non-interacting values $k_{\pm}^{0}$.Schliemann2006 ; Chesi07April From Eq. (12), $k_{\pm}$ can be obtained by equating the chemical potentials in the two spin branches:
$$E_{+}(k_{+})=E_{-}(k_{-}).$$
(14)
After taking the continuum limit, this equation is rewritten in dimensionless form as follows:
$$\displaystyle(y_{+}^{n}+y_{-}^{n})g=$$
$$\displaystyle 2\chi+\frac{r_{s}}{\sqrt{2}}\sum_{\sigma\sigma^{\prime}}\int_{0}%
^{2\pi}\!\frac{d\theta}{2\pi}\!\int_{0}^{y_{\sigma^{\prime}}}\!dy$$
(15)
$$\displaystyle\times\frac{y(\sigma+\sigma^{\prime}\cos{n\theta})}{\sqrt{y^{2}+y%
_{\sigma}^{2}-2yy_{\sigma}\cos{\theta}}+\sqrt{2}r_{s}},$$
where we have rescaled the wavevectors $k=k_{F}y$ and defined $y_{\pm}=\sqrt{1\mp\chi}$. The integral on the right-hand side is the correction from the exchange term and we have verified that Eq. (15) gives Eq. (6) for $r_{s}=0$. We note that, for a given value of the SOI, $\chi(r_{s},g)$ enters in a rather complicated way in Eq. (15), being involved in the integration limits of the exchange term as well as the integrand. In practice, instead of solving Eq. (15) for $\chi$, it is convenient to evaluate $g$ for a given value of $\chi$ and numerically invert the function $g(r_{s},\chi)$.
The values of $k_{\pm}=k_{F}y_{\pm}$ were obtained in Ref. Chesi07April, through a different procedure, i.e., by minimizing the total energy (including the exchange contribution) of non-interacting states.comment_variational Although both methods are unreliable at $r_{s}>1$, they both become accurate in the high-density limit, $r_{s}<1$. In fact, the only difference in the two approaches is due to the presence of the Thomas-Fermi screening wavevector in Eq. (15) and, by neglecting $\sqrt{2}r_{s}$ in the denominator of the second line, the equation from the variational treatment is recovered. In particular, expanding Eq. (15) at small $r_{s}$ and $g$ gives the same result found in Ref. Chesi07April, :
$$\chi(r_{s},g)\simeq g\left(1-\frac{\sqrt{2}r_{s}}{\pi}\sum_{j=0}^{n}\frac{1}{2%
j-1}\right).$$
(16)
Details of the derivation of Eq. (16) are given in Appendix B. Two salient features of Eq. (16) are:Chesi07April (i) For $n=1$ there is no correction to the noninteracting result $\chi(r_{s},g)\simeq g$. On the other hand, the linear dependence on $g$ is actually modified by electron-electron interactions at $n=2,3$. (ii) The effect of the electron-electron interactions is a reduction of $\chi(r_{s},g)$ from the non-interacting value. This result could be rather surprising, having in mind the well-known enhancement of spin polarization caused by the exchange energyTheBook (when the spin-splitting is generated by a magnetic field). However, $\chi$ does not correspond here to a real spin-polarization, which is zero. Instead, $\chi$ is simply related to the population difference of the two chiral spin subbands.
III.2 Quasiparticle velocity
In the screened Hartree-Fock approximation, the group velocities at the Fermi surfaces are given by:
$$\displaystyle v_{\pm}=$$
$$\displaystyle\,\frac{1}{\hbar}\frac{\partial E_{k\pm}}{\partial k}\Big{|}_{k=k%
_{\pm}}$$
$$\displaystyle=$$
$$\displaystyle\,\frac{\hbar k_{\pm}}{m}\pm n\gamma(\hbar k_{\pm})^{n-1}$$
(17)
$$\displaystyle-\sum_{\mathbf{k}^{\prime}\sigma^{\prime}}\frac{(1\pm\sigma^{%
\prime}\cos{n\theta^{\prime}})}{2L^{2}}n_{\mathbf{k}^{\prime}\sigma^{\prime}}%
\left[\frac{\partial}{\partial k}V(|\mathbf{k}\!-\!\mathbf{k^{\prime}}|)\right%
]_{k=k_{\pm}}.$$
In general, we can discuss all corrections to $v_{\pm}$ by introducing the following notation:
$$\frac{v_{\pm}}{v_{F}}=1+\delta v(r_{s})+\delta v_{\pm}^{0}(g)+\delta v_{\pm}(r%
_{s},g),$$
(18)
where $\delta v(r_{s})$ is the (spin-independent) correction due to electron-electron interactions at $g=0$, which has been the subject of many theoretical and experimental studies (see, e.g., Ref. TheBook, , Chiu2011, ; Janak1969, ; Tan2005, ; Drummond2009, , and references therein). In the approximation (III.2), it is given byJanak1969
$$\delta v(r_{s})=-\frac{\sqrt{2}r_{s}}{\pi}+\frac{r_{s}^{2}}{2}+\frac{r_{s}(1-r%
_{s}^{2})}{\sqrt{2}\pi}\frac{\cosh^{-1}(\sqrt{2}/r_{s})}{\sqrt{1-r_{s}^{2}/2}}.$$
(19)
As is known,TheBook this approximation gives the correct leading behavior at small $r_{s}$: $\delta v\simeq-(r_{s}\ln r_{s})/(\sqrt{2}\pi)$. The second nontrivial term in Eq. (18) is the non-interacting correction purely due to SOI
$$\delta v^{0}_{\pm}(g)=\frac{v^{0}_{\pm}(g)}{v_{F}}-1,$$
(20)
which only depends on $g$ [see Eq. (9)]. Finally, $\delta v_{\pm}(r_{s},g)$ collects all remaining corrections.
A pictorial representation of the physical meaning of the three terms is shown in Fig. 1 for an unpolarized wavepacket injected at the Fermi energy. At high density, as seen in Eq. (19) and illustrated in Fig. 1(a), the group velocity is larger than without electron-electron interactions. In Fig. 1(b) we depict the generic situation with SOI. In the presence of SOI, the two spin branches have different group velocities. An initially unpolarized wavepacket then splits into its two spin components. Both the SOI and electron-electron interactions influence the relative velocity $(v_{+}-v_{-})$. The separation after a time $\tau$ is $\delta=|v^{0}_{+}-v_{-}^{0}|\tau$ for the non-interacting case and is modified by $\delta v_{\pm}$ with electron-electron interactions: $\Delta=|v_{+}-v_{-}|\tau$ [see Fig. 1(b)].
The presence of ‘interference’ terms, $\delta v_{\pm}(r_{s},g)$, becomes clear from Eq. (III.2). These terms are due to the interplay of many-body interactions with SOI. A first contribution to $\delta v_{\pm}(r_{s},g)$, which we refer to as the ‘self-energy contribution’, comes directly from the exchange integral [third line of Eq. (III.2)]: due to the presence of two distinct Fermi wavevectors $k_{\pm}$, the result obviously contains corrections to Eq. (19) which depend on $g$ (in addition to $r_{s}$). A second contribution to $\delta v_{\pm}(r_{s},g)$ comes indirectly from the non-interacting part and we refer to it as the ‘repopulation contribution’. Since the Fermi wavevectors $k_{\pm}$ are modified from the non-interacting values $k_{\pm}^{0}$, the second line of Eq. (III.2) gives a result distinct from $v^{0}_{\pm}$. The sign of this repopulation contribution is easily found by noting that, as discussed in Sec. III.1, the exchange energy reduces the value of $\chi$ (for $n=2,3$). This corresponds to an increase (decrease) of $k_{+}$ ($k_{-}$), i.e., a positive (negative) correction to $v_{+}$ ($v_{-}$). The effect would be to enhance the difference in velocity between the two branches, as illustrated in Fig. 1(b) ($\Delta>\delta$). However, to establish the ultimate form and sign of $\delta v_{\pm}(r_{s},g)$ requires a detailed calculation of both self-energy and repopulation contributions, which is presented below for some interesting cases. The total spin-dependent part of the velocity, $\delta v_{\pm}^{0}+\delta v_{\pm}$, can then be compared to the simple non-interacting effect, $\delta v_{\pm}^{0}$.
III.3 $n=1$: higher-order corrections in $g$
We begin the analysis of Eq. (III.2) by rewriting it in a more explicit way. To this end, we use
$$\frac{\partial}{\partial k}V(|\mathbf{k}-\!\mathbf{k}^{\prime}|)=-\hat{\mathbf%
{k}}\cdot\frac{\partial}{\partial{\mathbf{k}^{\prime}}}V(|\mathbf{k}-\mathbf{k%
}^{\prime}|),$$
(21)
where $\hat{\bf k}={\bf k}/k$. This allows us to integrate Eq. (III.2) by parts, which leads to:
$$\displaystyle\frac{v_{\pm}}{v_{F}}=y_{\pm}\pm\frac{g}{2}ny_{\pm}^{n-1}$$
$$\displaystyle+r_{s}\sum_{\sigma}\int_{0}^{2\pi}\frac{d\theta^{\prime}}{8\pi}%
\frac{\sqrt{2}\cos{\theta^{\prime}}(1\pm\sigma\cos{n\theta^{\prime}})y_{\sigma%
}}{\sqrt{y_{\pm}^{2}+y_{\sigma}^{2}-2y_{\pm}y_{\sigma}\cos{\theta^{\prime}}}+r%
_{s}\sqrt{2}}$$
$$\displaystyle\pm r_{s}\int_{0}^{2\pi}\frac{d\theta^{\prime}}{8\pi}\int_{y_{+}}%
^{y_{-}}\frac{\sqrt{2}n\sin{n\theta^{\prime}}\sin{\theta^{\prime}}\,dy^{\prime%
}}{\sqrt{y_{\pm}^{2}+{y^{\prime}}^{2}-2y^{\prime}y_{\pm}\cos{\theta^{\prime}}}%
+r_{s}\sqrt{2}}$$
$$\displaystyle=L_{1}+L_{2}+L_{3}.$$
(22)
Notice that, in the integration by parts of Eq. (III.2) in the continuum limit, two types of terms enter: those corresponding to the second line of Eq. (III.3) ($L_{2}$) involve the derivative of $n_{\mathbf{k^{\prime}}\pm}$. This results in a delta function in the $dk^{\prime}$ integral, which can then be easily evaluated. Thus, only the integral in $d\theta^{\prime}$ is left. The second type of term involves the derivative of $\cos{n\theta^{\prime}}$:
$$\hat{\mathbf{k}}\cdot\frac{\partial}{\partial\mathbf{k^{\prime}}}\cos{n\theta^%
{\prime}}=\frac{n}{k^{\prime}}\sin{n\theta^{\prime}}\sin{\theta^{\prime}},$$
(23)
and indeed this angular factor appears in the third line of Eq. (III.3) ($L_{3}$).
Eq. (III.3) can always be evaluated numerically, after obtaining the values of $\chi$ (thus $y_{\pm}=\sqrt{1\mp\chi}$) from Eq. (15). By specializing to the small $g,r_{s}$ limit for $n=1$ SOI, it is known that the linear-in-$g$ correction to $v_{\pm}$ vanishes.Saraga2005 ; ChesiExact Thus, an expansion to second order in $g$ has to be performed, which has been done in Ref. Saraga2005, in the context of the RPA treatment of the quasiparticle properties. To verify the validity of the simpler screened Hartree-Fock procedure, it is interesting to perform the same expansion for our Eq. (III.3).
In fact, the two calculations bear some similarities since $L_{2}$ is the same as the boundary term $B_{\rm boundary}^{(u\to 0_{+})}$ of the RPA treatment, see Eq. (69) of Ref. Saraga2005, . Thus, we can borrow the expansion in small $g,r_{s}$:
$$L_{2}\simeq-\frac{r_{s}}{\sqrt{2}\pi}\left(\ln\frac{r_{s}}{2\sqrt{2}}+2\pm%
\frac{2}{3}\,g-\frac{g^{2}}{8}\ln g\right);\quad(n=1),$$
(24)
where terms of order $O(r_{s}^{2},r_{s}g^{2})$ have been omitted. To the same order of approximation, we have for $L_{1}$:
$$L_{1}\simeq 1-\frac{g^{2}}{8};\quad(n=1),$$
(25)
which can also be obtained by expanding Eq. (9) (with $n=1$). Since $\chi$ only receives $O(r_{s}g^{3}\ln g)$ corrections from electron-electron interactions,ChesiHighDens it is sufficient to use the non-interacting value, $\chi_{0}(g)$, to this order of approximation. Thus, the repopulation induced by electron-electron interactions has a negligible effect on $v_{\pm}$ in this case. The situation will be different for $n=2,3$. Expanding $L_{3}$ for small $r_{s},g$ yields
$$L_{3}\simeq\pm\frac{\sqrt{2}r_{s}}{3\pi}g;\quad(n=1),$$
(26)
which cancels the linear-in-$g$ term of Eq. (24), as expected for $n=1$.
We note that in the screened Hartree-Fock approximation we are able to obtain an analytic result for the leading term of $L_{3}$, and some details of the derivation can be found in Appendix C. In contrast, in the RPA treatment of Ref. Saraga2005, it was not possible to expand the more involved corresponding term, $B_{\rm int}$, in a fully analytic fashion. The cancellation of the linear-in-$g$ contribution was indicated by a general argumentSaraga2005 ; ChesiExact and confirmed through numerical study. Additionally, the absence of higher-order terms which modify Eq. (24) was inferred numerically in Ref. Saraga2005, . Although the final results of both approaches (RPA and screened Hartree-Fock) agree, the situation is clearly more satisfactory within the framework of Eq. (III.3), since cancellation of the linear term in Eq. (24) can be checked exactly and the expansion can be carried out to higher order systematically. The final result, computed to higher order in $g$, reads (for $n=1$):
$$\displaystyle\frac{v_{\pm}}{v_{F}}$$
$$\displaystyle\simeq 1+\delta v(r_{s})+\delta v^{0}_{\pm}(g)$$
(27)
$$\displaystyle+\frac{\sqrt{2}r_{s}}{16\pi}\left[g^{2}\left(\ln\frac{g}{8}+\frac%
{3}{2}\right)\pm\frac{g^{3}}{6}\left(\ln\frac{g^{7}}{8^{5}}+\frac{319}{20}%
\right)\right].$$
The second line of Eq. (27) represents the expansion of $\delta v_{\pm}(r_{s},g)$ including all terms up to $O(r_{s}g^{3}$). For $n=1$, the non-interacting result gives the same velocity for both spin branches [$\delta v^{0}_{\pm}(g)$ is actually independent of $\pm$ for this particular model]. The $O(r_{s}g^{3})$ term in Eq. (27) is therefore quite interesting. It shows that a small difference in velocity exists. This is a genuine effect of electron-electron interactions.
The accuracy of Eq. (27) can be seen in Fig. 2: it becomes accurate at very small values of $r_{s}$ (as shown in the inset) while at larger more realistic values of $r_{s}$ it still gives the correct magnitude of the effect. The numerical example of Fig. 2 also shows that $\delta v_{\pm}(r_{s},g)$ is very small. In fact, it is generally much smaller than the non-interacting correction: since $\delta v^{0}_{\pm}(g)\simeq-g^{2}/8$ [see Eq. (25)] the $~{}r_{s}g^{2}\ln g$ term becomes larger only if $r_{s}\ln g\gg 1$. This condition is only satisfied at extremely small values of $g$ (if $r_{s}$ is small as well) at which SOI effects are hardly of any relevance. Thus, Eq. (27) implies that the effects of SOI and electron-electron interactions are essentially decoupled for $n=1$. This picture changes substantially for hole systems ($n=2,3$), as we show in the next section.
III.4 Corrections to the velocity for $n=2,3$
We now apply the discussion of the previous section to the SOI more appropriate for holes and point out some important differences. Expansions for small $r_{s}$ and $g$ are given in Appendix C. From Eqs. (56) and (57) and using $\chi=g+O(r_{s}g,g^{3})$ [Eq. (16)], we find the following expressions for the self-energy contribution. For $n=2$:
$$L_{2}+L_{3}\simeq\,\delta v(r_{s})+\frac{\sqrt{2}r_{s}}{4\pi}\left[\pm\frac{g}%
{3}+g^{2}\left(\ln\frac{g}{8}+2\right)\right],$$
(28)
and for $n=3$:
$$L_{2}+L_{3}\simeq\,\delta v(r_{s})+\frac{\sqrt{2}r_{s}}{4\pi}\left[\pm\frac{8}%
{15}g+\frac{g^{2}}{4}\left(9\ln\frac{g}{8}+\frac{613}{30}\right)\right].$$
(29)
At variance with the case of $n=1$, the linear-in-$g$ term does not vanish here. Thus, we find an appreciable correction to the velocity. This correction has opposite sign in the two branches and is positive for the $+$ (higher-energy) branch.
A second contribution to the $g$-linear correction comes from $L_{1}$. Expanding $L_{1}$ in terms of $\chi$ and using Eq. (16) gives, for $n=2$,
$$L_{1}\simeq 1+\delta v^{0}_{\pm}(g)+\frac{\sqrt{2}r_{s}}{4\pi}\left(\pm\frac{2%
}{3}g+g^{2}\right);\quad(n=2),$$
(30)
and for $n=3$:
$$L_{1}\simeq 1+\delta v^{0}_{\pm}(g)+\frac{\sqrt{2}r_{s}}{4\pi}\left(\pm\frac{1%
6}{15}g+\frac{32}{15}g^{2}\right);\quad(n=3).$$
(31)
Thus, the repopulation contribution is present in this case and has the sign discussed at the end of Sec. III.2 (it is positive for the $+$ branch).
As it turns out, the self-energy, repopulation, and non-interacting contributions to the velocity have the same sign. The three contributions therefore have a cooperative effect in enhancing the difference in velocity between the two spin branches. Of course, based on the high-density theory presented here, we cannot tell if this conclusion holds at all densities. We also note that the $g$-linear term of the self-energy correction [Eq. (28) or (29)] is always half of the corresponding repopulation correction [Eq. (30) or (31)]. Again, we have not investigated if this curious relation only occurs within this approximation scheme or if it is more general.
Finally, we give the complete result for $n=2$
$$\displaystyle\frac{v_{\pm}}{v_{F}}$$
$$\displaystyle\simeq 1+\delta v(r_{s})+\delta v^{0}_{\pm}(g)$$
(32)
$$\displaystyle+\frac{\sqrt{2}r_{s}}{4\pi}\left[\pm g+g^{2}\left(\ln\frac{g}{8}+%
3\right)\right];\quad(n=2),$$
and for $n=3$
$$\displaystyle\frac{v_{\pm}}{v_{F}}$$
$$\displaystyle\simeq 1+\delta v(r_{s})+\delta v^{0}_{\pm}(g)$$
(33)
$$\displaystyle+\frac{\sqrt{2}r_{s}}{4\pi}\left[\pm\frac{8}{5}g+g^{2}\left(9\ln%
\frac{g}{8}+\frac{869}{120}\right)\right];\quad(n=3),$$
and show two numerical examples in Figs. 3 and 4.
Fig. 3 is a comparison of $\delta v_{\pm}(r_{s},g)$ for $n=1$ and $n=3$: it is clear that the dependence on $g$ is very weak for the electron case ($n=1$) and the magnitude of $\delta v_{\pm}(r_{s},g)$ is much larger for holes ($n=3$) with SOI of comparable strength. Fig. 4 shows the separation $\Delta$ between the two spin components of an initially unpolarized wave packet for a fixed travel distance of $1~{}\mu$m:
$$\Delta=2\,\frac{|v_{+}-v_{-}|}{v_{+}+v_{-}}\times(1~{}\mu{\rm m}).$$
(34)
In addition to including both $n=1$ and $n=3$, in Fig. 4 we plot the non-interacting value $\Delta_{0}$, obtained by substituting $v_{\pm}\to v^{0}_{\pm}$ in Eq. (34). For $n=1$ the non-interacting velocities $v_{\pm}^{0}$ are the same ($\Delta_{0}=0$). The effect of electron-electron interactions is not visible. Thus, the wavepacket remains essentially unsplit and unpolarized ($\Delta\simeq 0$) and the only significant influence is on the average velocity $(v_{+}+v_{-})/2$, from Eq. (19) [see Fig. 1(a)]. In contrast, a large splitting is found for $n=3$ where $\Delta,\Delta_{0}$ can reach a large fraction of the traveling distance. Since spin-polarized ballistic transport is observed in two-dimensional hole systems on $\mu$m-scales (e.g., in spin-focusing experimentsRokhinson2004 ; ChesiQPC ) and the typical wavepacket traveling time is $1-50$ ps (depending on the density), Fig. 4 suggests that a direct optical imaging of the wavepacket separation should be within reach of femtosecond-resolved Faraday rotation measurements.Crooker1996 ; Kikkawa1997
As for electron-electron interaction effects, we see in Fig. 4 that a visible difference between $\Delta$ and $\Delta_{0}$ exists. The difference is quite small, due to the fact that we are considering here the weak-coupling limit, and all the interaction corrections are proportional to $r_{s}<1$. $\Delta_{0}$ is modified here by $\sim 2-3\%$ and we collect some representative values in Table 1. It can be seen in Table 1 that, while $\Delta_{0}$ does not change with $r_{s}$, the interaction correction $\Delta-\Delta_{0}$ grows at lower densities (see Table 1). Since experiments on hole systems can reach values as large as $r_{s}=6-12$,Winkler2005 ; Chiu2011 it is reasonable to expect significant effects from $\delta v_{\pm}$ in this low-density regime. As a reference, in electron systems, $\delta v$ changes from $\sim 5\%$ to $-30\%$ for $r_{s}\sim 1$ to $6$.Tan2005 ; Drummond2009
It might be surprising to see $\Delta<\Delta_{0}$ in Fig. 4. This is due to the interplay of two competing interaction effects. It is true that, as discussed already, the difference between $v_{\pm}$ is enhanced by $\delta v_{\pm}$. The two spin components therefore split faster in the interacting case. However, the situation shown in Fig. 4 is distinct from that shown in Fig. 1 since a constant traveling distance (and not time $\tau$) is assumed. At high density the mean velocity $(v_{+}+v_{-})/2$ is greater for the interacting gas ($\delta v>0$), which allows the wavepackets less time to separate. As it turns out, the latter effect is dominant in Fig. 4.
Interestingly, the sign of $\delta v$ changes at low density.TheBook This would imply a cooperative effect of $\delta v_{\pm}$ and $\delta v$ on $\Delta$ if $\delta v_{\pm}$ does not change sign. Unfortunately, to infer the behavior of $\delta v_{\pm}$ at low density requires a much more sophisticated approach.
IV Conclusion
We have presented a theory of the quasiparticle group velocity at high density, in the presence of SOI of different types. Contrasting the behavior of electron and hole systems, we find several intriguing differences. We have shown explicitly that the lowest-order cancellations of SOI effects occur only for the electronic case, when the SOI is approximately linear in momentum (e.g., a strong Rashba or Dresslhaus SOI is present). On the other hand, SOI terms non-linear in ${\bf p}$ are often dominant in hole systems.Winkler2000 ; Winkler2002 ; ChesiQPC Thus, larger effects of the SOI and a non-trivial interplay with electron-electron interactions are expected for holes on general grounds.
As an important motivation for future theoretical studies, hole liquids can be realized in the laboratory with strong SOI and large values of $r_{s}$. For example, the spin-subband population difference at zero field is of order $15-20\%$ in Ref. Winkler2002, , with $n_{s}\simeq 2-4\times 10^{10}~{}{\rm cm}^{-2}$. With a hole effective mass $m\simeq 0.2m_{0}$ in GaAs these densities correspond to $r_{s}\simeq 9-12$. For electrons, materials with strong SOI typically have small effective masses, which results in much lower values of $r_{s}$. A diluted electron liquid with $n_{s}\simeq 2\times 10^{10}~{}{\rm cm}^{-2}$ gives $r_{s}\simeq 1.2$, using InAs parameters ($m=0.023m_{0}$).
Discussing the large-$r_{s}$ regime of holes would require extending many-body perturbation theorySaraga2005 ; Nechaev2009 ; Nechaev2010 or Monte CarloAmbrosetti2009 approaches, so far only applied to linear SOI. The high-density regime studied here would represent a well-controlled limit of these theories for the quasiparticle dispersion. In addition to being relevant for transport measurements of the effective mass, the significant difference in group velocity at the Fermi surface of the two spin branches could also be addressed by Raman scattering experiments, demonstrated for electron systems in Ref. Jusserand1992, , or via time-resolved Faraday-rotation detection of the spin-polarization.Crooker1996 ; Kikkawa1997 A similar discussion should hold with $n=2,3$ for other physical observables and many-body effects. For example, studying the compressibilityTheBook ; Eisenstein1992 in the presence of SOI and extending the $n=1$ discussion of the quasiparticle lifetimeSaraga2005 ; Nechaev2009 ; Nechaev2010 would also be topics of interest. Finally, the problem of including in our framework a more general form of SOI than Eq. (1) is clearly of practical relevance. However, as discussed in Appendix A, we expect that our qualitative picture on the different role of linear and non-linear SOI remains valid also in this more general situation.
Acknowledgements.We thank J. P. Eisenstein and D. L. Maslov for useful discussions and acknowledge financial support from CIFAR, NSERC, and FQRNT.
Appendix A General spin-orbit coupling
A general SOI contains terms with linear and non-linear dependence on momentum, and does not necessarily have the isotropic form assumed in Eq. (1). For definiteness, we suppose that there is no magnetic field, so that quadratic terms are not present:
$$H=\sum_{i}\frac{p_{i}^{2}}{2m}+H^{\rm so}_{1}+H^{\rm so}_{3}+H_{\rm el-el},$$
(35)
where $H^{\rm so}_{1}$, $H^{\rm so}_{3}$, and $H_{\rm el-el}$ give, respectively, the linear-in-momentum spin-orbit, the cubic-in-momentum spin-orbit, and electron-electron interactions. To show that $H^{\rm so}_{1}$ always has a small effect, we consider the unitary transformation $H^{\prime}=e^{-S}He^{S}$ with $S$ defined by $[S,\sum_{i}p_{i}^{2}/2m]=H^{\rm so}_{1}$, giving
$$H^{\prime}\simeq\frac{p^{2}}{2m}+H^{\rm so}_{3}+H_{\rm el-el}-[S,H^{\rm so}_{1%
}+H^{\rm so}_{3}].$$
(36)
In the same transformed frame, the velocity operator of electron $i$ is given by ${\bf v}^{\prime}_{i}=e^{-S}(\partial H/\partial{\bf p}_{i})e^{S}$, which yields
$${\bf v}^{\prime}_{i}\simeq\frac{{\bf p}_{i}}{m}+\frac{\partial{H^{\rm so}_{3}}%
}{\partial{\bf p}_{i}}-[S,\frac{\partial{H^{\rm so}_{1}}}{\partial{\bf p}_{i}}%
+\frac{\partial{H^{\rm so}_{3}}}{\partial{\bf p}_{i}}].$$
(37)
In deriving Eq. (36) we have used the fact that $[S,H_{\rm el-el}]=0$. For example, for an isotropic SOI as in Eq. (1),
$$S=i\frac{m\gamma}{\hbar}\sum_{i}(x_{i}\sigma_{y,i}-y_{i}\sigma_{x,i}).$$
(38)
The property $[S,H_{\rm el-el}]=0$ is valid for a general linear-in-momentum SOI, including a combination of Rashba and Dresselhaus SOI.Aleiner2001 However, the same identity $[S,H_{\rm el-el}]=0$ does not hold for a transformation with $[S,\sum_{i}p_{i}^{2}/2m]=H^{\rm so}_{3}$, i.e., a transformation that is designed to remove the non-linear component from the non-interacting Hamiltonian. In writing Eq. (37), we have used $[S,{\bf p}_{i}/m]=\partial{H^{\rm so}_{1}}/\partial{\bf p}_{i}$, which implies a cancellation of the $H^{\rm so}_{1}$ contribution to ${\bf v}^{\prime}_{i}$ to lowest order. Again, this cancellation is only valid for linear-in-momentum SOI.
By introducing dimensionless couplings $g_{1,3}$ associated with $H^{\rm so}_{1,3}$, in direct analogy with Eq. (4), we see that the commutators in Eqs. (36) and (37) are of quadratic or bilinear order in the couplings ($\sim g_{1}^{2}$ and $\sim g_{1}g_{3}$). This indicates on general grounds that $H^{\rm so}_{3}$ has the largest effect on the quasiparticle velocity if $g_{1}\lesssim g_{3}\ll 1$. In this case, if we are interested in lowest-order effects, we can neglect both anticommutators in Eqs. (36) and (37), which is equivalent to neglecting $H^{\rm so}_{1}$ in the original Hamiltonian (35). Thus, to leading order all results we report for the quasiparticle velocities due to a pure cubic-in-momentum spin-orbit interaction also apply in the case of a mixed linear-plus-cubic spin-orbit interaction (with the caveat that we consider only the isotropic form of cubic SOI).
Appendix B Derivation of Eq. (16)
As discussed in the text, at high density we can neglect the $\sqrt{2}r_{s}$ in the integrand of Eq. (15) (second line). For small $g$, the value of $\chi$ is also small and we can perform an expansion of the exchange integral. First notice that the constant term at $\chi=0$ is missing, because the integration limits simply become $y_{\pm}=1$ and the integrand vanishes upon the summation on $\sigma,\sigma^{\prime}$. Therefore we only need to compute the linear term in $\chi$:
$$\displaystyle\left[\frac{\partial}{\partial\chi}\sum_{\sigma\sigma^{\prime}}%
\int_{0}^{2\pi}\frac{d\theta}{2\pi}\int_{0}^{y_{\sigma^{\prime}}}\frac{(\sigma%
+\sigma^{\prime}\cos{n\theta})ydy}{\sqrt{y^{2}+y_{\sigma}^{2}-2yy_{\sigma}\cos%
{\theta}}}\right]_{\chi=0}$$
$$\displaystyle=$$
$$\displaystyle\int_{0}^{2\pi}\frac{d\theta}{2\pi}\left[\int_{0}^{1}\frac{2(1-y%
\cos{\theta})ydy}{(1+y^{2}-2y\cos{\theta})^{\frac{3}{2}}}-\frac{\cos{n\theta}}%
{\sin{\frac{\theta}{2}}}\right],$$
(39)
and after evaluating the $dy$ integral in the square parenthesis, Eq. (B) gives
$$\displaystyle\int_{0}^{2\pi}$$
$$\displaystyle\frac{d\theta}{2\pi}\Bigg{[}2+\frac{2\cos{\theta}-\cos n\theta-1}%
{\sin{\frac{\theta}{2}}}$$
$$\displaystyle-2$$
$$\displaystyle\,\ln\left(1+\frac{1}{\sin{\frac{\theta}{2}}}\right)\cos{\theta}%
\Bigg{]}=\frac{4}{\pi}\sum_{j=0}^{n}\frac{1}{2j-1}.$$
(40)
Equation (16) is then easily obtained from Eq. (15) by neglecting all the cubic terms in the small parameters $g,\chi,r_{s}$ [e.g., the left side of Eq. (15) is $(y_{+}^{2}+y_{-}^{2})g=2g+O(g\chi^{2})$].
Appendix C Small $r_{s},g$ expansions
We give in this appendix some details on the expansions of Eq. (III.3):
$$\frac{v_{F\pm}}{v_{F}}=y_{\pm}\pm\frac{g}{2}ny_{\pm}^{n-1}+\frac{\sqrt{2}r_{s}%
}{16\pi}(I_{1}+I_{2}+I_{3}),$$
(41)
where we have split $L_{2}$ in its two contributions ($I_{1,2}$ refer to $\sigma=\pm$) and $I_{3}$ corresponds to $L_{3}$:
$$\displaystyle I_{1}$$
$$\displaystyle=$$
$$\displaystyle\int_{0}^{2\pi}d\theta\frac{\sqrt{2}y_{\pm}\cos{\theta}(1+\cos{n%
\theta})}{\sqrt{2}y_{\pm}\sin{\theta/2}+r_{s}},$$
(42)
$$\displaystyle I_{2}$$
$$\displaystyle=$$
$$\displaystyle\int_{0}^{2\pi}d\theta\frac{\sqrt{2}y_{\mp}\cos{\theta}(1-\cos{n%
\theta})}{\sqrt{1-y_{+}y_{-}\cos{\theta}}+r_{s}},$$
(43)
$$\displaystyle I_{3}$$
$$\displaystyle=$$
$$\displaystyle\pm\int_{0}^{2\pi}d\theta\int_{y_{+}}^{y_{-}}dy\frac{2n\sin{n%
\theta}\sin{\theta}}{\sqrt{y_{\pm}^{2}+y^{2}-2yy_{\pm}\cos{\theta}}+r_{s}\sqrt%
{2}},$$
(44)
Notice that these integrals have an explicit dependence on $r_{s}$ and $y_{\pm}=\sqrt{1\mp\chi}$. So, it is easier to perform first the expansion in the two small parameters $r_{s},\chi$. The final results in the main text are given in terms of the physical couplings of the hamiltonian: $r_{s}$ and $g$. Those final expression are easily obtained by substituting the value of $\chi$ in terms of $r_{s}$ and $g$ ($\chi\simeq g$ in first approximation).
The first integral, Eq. (42), can be evaluated exactly. In particular for $n=1$ we obtain
$$\displaystyle I_{1}=$$
$$\displaystyle-\frac{40}{3}+8\pi\delta(1-\delta^{2})+16\delta^{2}$$
(45)
$$\displaystyle+8(1-3\delta^{2}+2\delta^{4})\frac{\tanh^{-1}\sqrt{1-\delta^{2}}}%
{\sqrt{1-\delta^{2}}}$$
where $\delta=r_{s}/\sqrt{2}y_{\pm}$. This expression can then be easily expanded in $r_{s},\chi$ and an analogous procedure is followed for $n=2,3$. To lowest-order in $r_{s}$, we can set $r_{s}=0$ in $I_{2}$ and $I_{3}$. Similarly to the $I_{1}$ angular integral above, the $d\theta$ integrals of $I_{2}$ and $I_{3}$ at $r_{s}=0$ can be computed analytically for $n=1,2,3$. For $I_{2}$ this yields directly the desired function of $\chi$. For $I_{3}$ we still need to perform a last integration in $dy$. Since the integration region is of size $\sim\chi$ around $y=1$, we can expand the integrand in the small parameter $(y-1)$ and perform the integration in $dy$ order-by-order, which allows us to extract the leading terms of the expansion in $\chi$. For $n=1$ all this gives
$$\displaystyle\delta I_{1}$$
$$\displaystyle\simeq$$
$$\displaystyle\mp 4\chi-2\chi^{2}\mp\frac{4}{3}\chi^{3},$$
(46)
$$\displaystyle\delta I_{2}$$
$$\displaystyle\simeq$$
$$\displaystyle\mp\frac{4}{3}\chi+\chi^{2}\left(\ln\frac{\chi}{8}+\frac{13}{6}%
\right)\pm\frac{\chi^{3}}{2}\left(\ln\frac{\chi}{8}+\frac{3}{2}\right),$$
(47)
$$\displaystyle\delta I_{3}$$
$$\displaystyle\simeq$$
$$\displaystyle\pm\frac{16}{3}\chi+\frac{4}{3}\chi^{2}\pm\chi^{3}\left(\frac{2}{%
3}\ln\frac{\chi}{16}+\frac{389}{120}\right),$$
(48)
where only the corrections $\delta I_{\alpha}=I_{\alpha}(\chi)-I_{\alpha}(\chi=0)$ are listed, since terms independent on $\chi$ simply give the small $r_{s}$ expansion of the well known Eq. (19). Here, terms of order $O(r_{s}\chi,\chi^{4})$ are omitted, while it is interesting to keep the $O(\chi^{3})$ terms, since they give the leading spin splitting. Indeed, it is easily checked that the linear terms cancel
$$\sum_{\alpha=1}^{3}\delta I_{\alpha}\simeq\,\chi^{2}\left(\frac{3}{2}+\ln\frac%
{\chi}{8}\right)\pm\frac{\chi^{3}}{6}\left(\ln\frac{\chi^{7}}{8^{5}}+\frac{319%
}{20}\right),$$
(49)
which immediately leads to Eq. (27).
For $n=2,3$ we can proceed in a similar way. The spin splitting appears now already to linear order in $\chi$. By keeping the first subleading correction in $\chi$ we have for $n=2$:
$$\displaystyle\delta I_{1}$$
$$\displaystyle\simeq$$
$$\displaystyle\mp 4\chi-2\chi^{2},$$
(50)
$$\displaystyle\delta I_{2}$$
$$\displaystyle\simeq$$
$$\displaystyle\pm\frac{16}{15}\chi+\chi^{2}\left(4\ln{\frac{\chi}{8}}+\frac{134%
}{15}\right),$$
(51)
$$\displaystyle\delta I_{3}$$
$$\displaystyle\simeq$$
$$\displaystyle\pm\frac{64}{15}\chi+\frac{16}{15}\chi^{2},$$
(52)
and for $n=3$:
$$\displaystyle\delta I_{1}$$
$$\displaystyle\simeq$$
$$\displaystyle\mp 4\chi-2\chi^{2},$$
(53)
$$\displaystyle\delta I_{2}$$
$$\displaystyle\simeq$$
$$\displaystyle\pm\frac{212}{105}\chi+\chi^{2}\left(9\ln{\frac{\chi}{8}}+\frac{8%
99}{42}\right),$$
(54)
$$\displaystyle\delta I_{3}$$
$$\displaystyle\simeq$$
$$\displaystyle\pm\frac{144}{35}\chi+\frac{36}{35}\chi^{2}.$$
(55)
The final results are for $n=2$:
$$\sum_{\alpha=1}^{3}\delta I_{\alpha}\simeq\,\pm\frac{4}{3}\chi+4\chi^{2}\left(%
\ln\frac{\chi}{8}+2\right),$$
(56)
and for $n=3$:
$$\sum_{\alpha=1}^{3}\delta I_{\alpha}\simeq\,\pm\frac{32}{15}\chi+\chi^{2}\left%
(9\ln\frac{\chi}{8}+\frac{613}{30}\right).$$
(57)
From Eqs. (56) and (57) we immediately obtain Eqs. (28) and (29).
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Coupled-Channel Effects for the Bottomonium with Realistic Wave Functions
Yu Lu${}^{1,3}$
111Email address:[email protected] ,
Muhammad Naeem Anwar ${}^{2,3}$
222Email address:[email protected] ,
Bing-Song Zou${}^{2,3}$
333Email address:[email protected]
${}^{1}$Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China,
${}^{2}$CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,
Chinese Academy of Sciences, Beijing 100190, China
${}^{3}$University of Chinese Academy of Sciences, Beijing 100049, China
(November 19, 2020)
Abstract
With Gaussian expansion method (GEM),
realistic wave functions are used to calculate coupled-channel effects for the bottomonium under the framework of ${}^{3}P_{0}$ model.
The simplicity and accuracy of GEM are explained.
We calculate the mass shifts, probabilities of the $B$ meson continuum, $S-D$ mixing angles,
strong and dielectric decay widths.
Our calculation shows that both $S-D$ mixing and the $B$ meson continuum can contribute to
the suppression of the vector meson’s dielectric decay width.
We suggest more precise measurements on the radiative decays of $\Upsilon(10580)$ and $\Upsilon(11020)$ to distinguish these two effects.
The above quantities are also calculated with simple harmonic oscillator (SHO) wave function approximation for comparison.
The deviation between GEM and SHO indicates that it is essential to treat the wave functions accurately for near threshold states.
1 Introduction
Heavy quarkonium is a multi-scale system covering all regimes of quantum chromodynamics (QCD) which make it an ideal place to study strong interactions [1].
Despite the success of QCD in high energy region, due to asymptotic freedom,
non-perturbative effect dominates at low energies and brings problems to perturbative calculation.
One tool to study this non-perturbative effect is lattice QCD.
However, due to its huge calculation work, it is still unable to calculate all the physical quantities with the current computation power.
Another important approach is to develop various phenomenological models.
Among these phenomenological models, quark model is a prominent one.
Under quark model framework,
various types of interactions have been suggested by various groups,
and they have achieved many impressive successes (see e.g. Refs. [2, 3, 4, 5, 6]).
However, these potential models cannot be the whole story.
One important missing ingredient is the mechanism to generate quark-antiquark pairs,
which enlarge the Fock space of the initial state,
i.e. the initial state contains multi-quark components.
These multi-quark components will change the Hamiltonian of the potential model,
causing mass shift and mixing between states with same quantum numbers
or directly contributing to open channel strong decay if the initial state is above threshold.
These consequences can be summarized as unquenched effects or coupled-channel effects.
Coupled-channel effects have been considered at least 30 years ago by
Törnqvist et al. in Refs. [7, 8, 9, 10, 11];
they extended the quark model to be unquenched quark model.
Despite the fact that the underlying quark pair creation mechanism is not fully understood up to now,
still there are different phenomenological models to decode the mystery,
such as ${}^{3}P_{0}$ model [12, 13, 14],
flux-tube breaking model [15, 16],
microscopic decay models [4, 17, 18].
Among these, the most simple and successful one is the ${}^{3}P_{0}$ model,
where the generated light quark pair share the same quantum number as vacuum.
Even though ${}^{3}P_{0}$ model is extensively studied by many people,
however, almost all the calculations are using SHO wave function approximation to simplify the calculation
(see e.g. [19, 18, 20, 21, 22, 23, 24, 25, 26]).
A simple yet powerful method to handel the wave function precisely is still not widely known.
We propose using the Gaussian expansion method (GEM) [27] to accurately evaluate the wave function convolution.
There have been already some works related to GEM.
In Refs. [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41],
GEM is adopted to calculate the wave functions under variational method approach.
Coupled-channel effects with GEM are only studied for some specific cases,
such as $X(3872)$ and $P$ wave $D_{s}$ mesons [42, 43].
In Refs. [44, 45, 46],
the authors also use GEM to calculate the spectrum and open channel strong decays of light mesons and some specific charmonia,
where the coupled channel induced mass shift is not considered.
Even though the mass shifts can be partly absorbed by redefining the potential,
the potential model cannot describe near-threshold effects [47].
We want to emphasize that the mass shifts and open channel strong decays are directly correlated by coupled-channel effects,
so it is essential to evaluate them under a same framework and calculate them precisely.
So far, a precise evaluation and a thorough discussion of the coupled-channel effects are still missing,
and the validity of SHO approximation is yet to be clarified.
In this paper, we fill these gaps by a thorough discussion of coupled-channel effects for the bottomonium
and we also predict some important results on the dielectric and radiative decays of vector mesons
which are going to be tested by experiment.
The paper is organized as follows.
In Sec. 2, we explain the details of Cornell potential and ${}^{3}P_{0}$ model,
where we deduce the formula of mass shift, open channel strong decay width and $S-D$ mixing.
In Sec. 3, we focus on the calculation details and GEM,
where the advantages of GEM are elucidated and the procedure to fit the wave function is explained.
Sec. 4 is devoted to discussing the possible impacts of coupled-channel effects for the bottomonium
on the spectrum, open channel strong decays, probabilities of the $B$ meson continuum,
the $S-D$ mixing and the vector meson’s dielectric and radiative decays.
We also explicitly show the deviation between GEM and SHO approximation.
Finally, we give a short summary of this work in Sec. 5.
2 Theoretical Framework
2.1 Cornell Potential Model
As the quenched limit,
the wave functions for the heavy quarkonium are obtained by solving the Schrödinger equation
with the well-known Cornell potential [4, 17]
$$V(r)=-\frac{4}{3}\frac{\alpha}{r}+\lambda r+c,$$
(1)
where $\alpha,\lambda$ and $c$ stand for the strength of color Coulomb potential,
the strength of linear confinement and mass renormalization, respectively.
To restore the hyperfine or fine structures of the bottomonium,
we use the following form of the spin dependent interactions
$$V_{s}(r)=\left(\frac{2\alpha}{m^{2}_{b}r^{3}}-\frac{\lambda}{2m^{2}_{b}r}%
\right)\vec{L}\cdot\vec{S}+\frac{32\pi\alpha}{9m_{b}^{2}}\tilde{\delta}(r)\vec%
{S}_{b}\cdot\vec{S}_{\bar{b}}+\frac{4\alpha}{m^{2}_{b}r^{3}}\left(\frac{\vec{S%
}_{b}\cdot\vec{S}_{\bar{b}}}{3}+\frac{(\vec{S}_{b}\cdot\vec{r})(\vec{S}_{\bar{%
b}}\cdot\vec{r})}{r^{2}}\right),$$
(2)
where $\vec{L}$ denotes relative orbital angular momentum,
$\vec{S}=\vec{S}_{b}+\vec{S}_{\bar{b}}$ is the total spin of the $b$ quark pairs
and $m_{b}$ is the $b$ quark mass.
Since the non-relativistic expansion will fail if two composite quarks are very close to each other,
instead of Dirac $\delta$ function in the second term,
we use the smeared delta function,
which can be written as
$\tilde{\delta}(r)=(\sigma/\sqrt{\pi})^{3}e^{-\sigma^{2}r^{2}}$ [48, 47].
The Hamiltonian of the Schrödinger equation in quenched limit is represented as
$$H_{0}=2m_{b}+\frac{p^{2}}{m_{b}}+V(r)+V_{s}(r).$$
(3)
We treat the spin dependent term as a perturbation and
the spatial wave functions are obtained by solving Schrödinger equation numerically using Numerov’s method [49].
2.2 ${}^{3}P_{0}$ Model and Coupled-Channel Effects
For the coupled channel calculation,
we adopt the widely used ${}^{3}P_{0}$ model or quark pair creation model,
which is first proposed by L. Micu [12] in 1969 and then extended by A. Le Yaouanc et al. in 1970s [13, 14].
In this model, the generated quark pairs have vacuum quantum number $J^{PC}=0^{++}$.
After simple arithmetic, one can conclude that the relative orbital angular momentum and total spin are both equal to 1
and total angular momentum is 0.
In the notation of ${}^{2S+1}L_{J}$, one should write it as ${}^{3}P_{0}$ which explains the model’s name.
The interaction Hamiltonian can be expressed as
$$H_{I}=2m_{q}\gamma\int d^{3}x\bar{\psi}_{q}\psi_{q},$$
(4)
where $m_{q}$ is the produced quark mass,
$\gamma$ is the dimensionless coupling constant.
Since the probability to generate heavier quarks is suppressed,
we use the effective strength $\gamma_{s}=\frac{m_{q}}{m_{s}}\gamma$ in the following calculation,
where $m_{q}=m_{u}=m_{d}$ is the constituent quark mass of up (or down) quark and $m_{s}$ is strange quark mass.
The ${}^{3}P_{0}$ Hamiltonian induces not only open-flavor strong decays of the heavy quarkonium above threshold,
but also coupled-channel effects.
As sketched by Fig. 1, the experimentally observed state should be a mixture of pure quarkonium state (bare state) and $B$ meson continuum.
Put it in to formula, the physical or experimentally observed state $|{A}\rangle$ should be expressed as
$$|{A}\rangle=c_{0}|{\psi_{0}}\rangle+\sum_{BC}\int d^{3}p\,c_{BC}(p)|{BC;p}\rangle,$$
(5)
where $c_{0}$ and $c_{BC}$ stand for the normalization constants of the bare state and $B$ meson continuum, respectively.
$|{\psi_{0}}\rangle$ is normalized to 1 and $|{A}\rangle$ is also normalized to 1 if it lies below $B\bar{B}$ threshold.
$|{BC;p}\rangle$ is normalized as $\langle{BC;p_{1}}|{B^{\prime}C^{\prime};p_{2}}\rangle=\delta^{3}(p_{1}-p_{2})%
\delta_{BB^{\prime}}\delta_{CC^{\prime}}$,
where $p$ is the momentum of $B$ meson in $|{A}\rangle$’s rest frame.
Combine the Cornell potential and the dynamics of quark pair generation,
we get the full Hamiltonian,
$$H=H_{0}+H_{BC}+H_{I},$$
(6)
with the following relations
$$\displaystyle H_{0}|{\psi_{0}}\rangle$$
$$\displaystyle=$$
$$\displaystyle M_{0}|{\psi_{0}}\rangle$$
(7)
$$\displaystyle H_{0}|{BC;p}\rangle$$
$$\displaystyle=$$
$$\displaystyle 0$$
(8)
$$\displaystyle H_{BC}|{\psi_{0}}\rangle$$
$$\displaystyle=$$
$$\displaystyle 0$$
(9)
$$\displaystyle H_{BC}|{BC;p}\rangle$$
$$\displaystyle=$$
$$\displaystyle E_{BC}|{BC;p}\rangle$$
(10)
$$\displaystyle H|{A}\rangle$$
$$\displaystyle=$$
$$\displaystyle M|{A}\rangle,$$
(11)
where $M_{0}$ is the bare mass of the charmonium state and can be solved directly from Schrödinger equation.
The interaction between $B$ mesons is neglected,
so the energy of meson continuum can be expressed as
$E_{BC}=\sqrt{m_{B}^{2}+p^{2}}+\sqrt{m_{C}^{2}+p^{2}}.$
When Eq. (11) is projected onto each component, we immediately get
$$\displaystyle\langle{\psi_{0}}|H|{\psi}\rangle=c_{0}M=c_{0}M_{0}+\int d^{3}p\,%
c_{BC}(p)\langle{\psi_{0}}|H_{I}|{BC;p}\rangle,$$
(12)
$$\displaystyle\langle{BC;p}|H|{\psi}\rangle=c_{BC}(p)M=c_{BC}(p)E_{BC}+c_{0}%
\langle{BC;p}|H_{I}|{\psi_{0}}\rangle.$$
(13)
Solve $c_{BC}$ from Eq. (13), substitute back to Eq. (12) and eliminate the $c_{0}$ on both sides,
we get a integral equation
$$M=M_{0}+\Delta M,$$
(14)
where
$$\Delta M=\sum_{BC}\int d^{3}p\,\frac{|\langle{BC;p}|H_{I}|{\psi_{0}}\rangle|^{%
2}}{M-E_{BC}-i\epsilon}.$$
(15)
The sum of $BC$ is restricted to the ground state $B_{(s)}$ mesons, i.e. $B\bar{B},B\bar{B}^{*}+h.c.,B^{*}\bar{B}^{*},B_{s}\bar{B}_{s},B_{s}\bar{B}^{*}_%
{s}+h.c.,B^{*}_{s}\bar{B}^{*}_{s}$.
Note that the $i\epsilon$ term is added to handle the situation when $m_{A}>m_{B}+m_{C}$.
In this case, $\Delta M$ will pick up an imaginary part
$$\mathrm{Im}(\Delta M)=\sum_{BC}\pi P_{B}\frac{E_{B}E_{C}}{m_{A}}|\langle{BC;P_%
{B}}|H_{I}|{\psi_{0}}\rangle|^{2},$$
(16)
which is equal to one half of the the decay width.
$P_{B}$ and $E_{B}$ respectively denote the momentum and energy of $B$ meson.
The wave function overlap integration lies in the term
$$\langle{BC;P_{B}}|H_{I}|{\psi_{0}}\rangle=\sum_{\text{polarization}}\int d^{3}%
k\phi_{0}(\vec{k}+\vec{P}_{B})\phi_{B}^{*}(\vec{k}+x_{B}\vec{P}_{B})\phi_{C}^{%
*}(\vec{k}+x_{C}\vec{P}_{B})|\vec{k}|Y_{1}^{m}(\theta_{\vec{k}},\phi_{\vec{k}}),$$
(17)
where $x_{B}=m_{4}/(m_{1}+m_{4}),x_{C}=m_{3}/(m_{2}+m_{3})$ and
$m_{1}=m_{2}=m_{Q},m_{3}=m_{4}$ respectively denote the $b$ quark and the light quark mass.
Once $M$ is solved, the coefficient of different components can be worked out either.
For states below threshold, the normalization condition $|{A}\rangle$ can be rewritten as
$$|c_{0}|^{2}+\int d^{3}p|c_{BC}|^{2}=1$$
(18)
after the substitution of $c_{BC}$, we get the probability of the $b\bar{b}$ component
$$P_{b\bar{b}}:=|c_{0}|^{2}=1/\left(1+\sum_{BCLS}\int_{0}^{\infty}dp\frac{p^{2}|%
\mathcal{M}^{LS}|^{2}}{(M-E_{BC})^{2}}\right),$$
(19)
where $|\mathcal{M}^{LS}|^{2}$ is represented as
$$|\mathcal{M}^{LS}|^{2}=\int d\Omega_{B}\,|\langle{BC;P_{B}}|H_{I}|{\psi_{0}}%
\rangle|^{2}.$$
(20)
2.3 Coupled Channel Induced S-D Mixing
From the quark model’s perspective,
the spatial wave functions of $J^{PC}=1^{--}$ family can be both $S$ and $D$ wave.
It is natural to expect that the experimentally observed vector states are mixing of $S$ and $D$ waves.
As in the case of conventional meson coupling with $B\bar{B}$ continuum,
we rewrite it into a matrix form
$$\left(\begin{array}[]{cc}M_{0}&\int d^{3}p\langle{\psi_{0}}|H_{I}|{BC}\rangle%
\\
\langle{BC}|H_{I}|{\psi_{0}}\rangle&E_{BC}\\
\end{array}\right)\left(\begin{array}[]{c}c_{0}\\
c_{BC}\\
\end{array}\right)=M\left(\begin{array}[]{c}c_{0}\\
c_{BC}\\
\end{array}\right),$$
(21)
where the integration part should be understood as a formal notation,
and one need to insert all the $p$ dependent part into the integral.
For example, in the above case,
one may naively get the following form after diagonalization
$$(M-M_{0})(M-E_{BC})=\int d^{3}p|\langle{\psi_{0}}|H_{I}|{BC}\rangle|^{2}.$$
(22)
However, the correct form should be understand as Eq.(15),
where the $(M-E_{BC})$ term have moved to the integration.
The advantage of the matrix form is that one can easily see its structure and
can be easily generalized to $S-D$ mixing case.
Under the assumption that $(n+1)S$ mix only with $nD$, we have
$$\left(\begin{array}[]{ccc}M_{S}^{0}&H_{T}&\int d^{3}p\langle{\psi_{S}}|H_{I}|{%
BC}\rangle\\
H_{T}&M_{D}^{0}&\int d^{3}p\langle{\psi_{D}}|H_{I}|{BC}\rangle\\
\langle{BC}|H_{I}|{\psi_{S}}\rangle&\langle{BC}|H_{I}|{\psi_{D}}\rangle&E_{BC}%
\\
\end{array}\right)\left(\begin{array}[]{c}c_{S}\\
c_{D}\\
c_{BC}\\
\end{array}\right)=M\left(\begin{array}[]{c}c_{S}\\
c_{D}\\
c_{BC}\\
\end{array}\right).$$
(23)
The $S-D$ mixing induced by tensor part of the potential is so small,
typically around $0.8^{\circ}$ in our calculation (see also the appendix A in Ref. [50]),
so its quite reasonable to set $H_{T}=0$.
After this approximation, one can reexpress $c_{BC}$ in terms of $c_{S},c_{D}$ and easily get
$$\left(\begin{array}[]{cc}M_{S}^{0}+\Delta M_{S}&\Delta M_{SD}\\
\Delta M_{DS}&M_{D}^{0}+\Delta M_{D}\\
\end{array}\right)\left(\begin{array}[]{c}c_{S}\\
c_{D}\\
\end{array}\right)=M\left(\begin{array}[]{c}c_{S}\\
c_{D}\\
\end{array}\right),$$
(24)
where
$$\displaystyle\Delta M_{f}$$
$$\displaystyle=$$
$$\displaystyle\int d^{3}p\frac{|\langle{\psi_{f}}|H_{I}|{BC}\rangle|^{2}}{M-E_{%
BC}-i\epsilon}\qquad(f=S,D),$$
(25)
$$\displaystyle\Delta M_{SD}=\Delta M_{DS}^{*}$$
$$\displaystyle=$$
$$\displaystyle\int d^{3}p\frac{\langle{\psi_{S}}|H_{I}|{BC}\rangle\langle{BC}|H%
_{I}|{\psi_{D}}\rangle}{M-E_{BC}-i\epsilon}.$$
(26)
From the above equation, both the mass and the relative ratio $c_{S}/c_{D}$ can be worked out.
For states below threshold, the probability can be solved once the mass is known,
which is a generalization of Eq. (19)
$$|c_{S}|^{2}+|c_{D}|^{2}+\sum_{BC}\int d^{3}p\frac{1}{(M-E_{BC})^{2}}(|c_{S}|^{%
2}H_{S,BC}^{2}+|c_{D}|^{2}H_{D,BC}^{2}+2\text{Re}[c_{S}c_{D}^{*}H_{S,BC}H_{BC,%
D}])=1,$$
(27)
where $H_{f,i}$ stands for$\langle{f}|H_{I}|{i}\rangle$.
One will get a complex solution of $M=M_{\text{BW}}+i\Gamma/2$ if $M_{\text{BW}}>m_{B}+m_{\bar{B}}$,
where $M_{\text{BW}}$ represents the Breit-Wigner mass of the resonance,
and $\Gamma$ is the decay width after considering $S-D$ mixing.
As a cross check, one can also calculate the decay width directly with the following formula,
$$\Gamma_{SD}=2\Big{(}|c_{S}|^{2}\mathrm{Im}(\Delta M_{S})+|c_{D}|^{2}\mathrm{Im%
}(\Delta M_{D})+2\mathrm{Re}\big{(}c_{S}^{*}c_{D}\mathrm{Im}(\Delta M_{SD})%
\big{)}\Big{)}.$$
(28)
Eq. (24) is much more difficult to solve than the Eq. (15).
The method we use to solve this equation will be discussed in the next section.
3 Parameter Selection and Gaussian Expansion Method
3.1 Parameter Selection
As a first step, we tune the wave functions to be consistent with the dielectric decay widths of $\Upsilon(nS)$ for $n\leq 3$.
Theoretically, dielectric decay widths can be expressed as [51, 52, 53, 54]
$$\Gamma_{ee}=\beta\frac{4\alpha^{2}e^{2}_{b}}{M^{2}_{nS}}|c_{S}\,R_{nS}(0)+c_{D%
}\frac{5}{2\sqrt{2}m_{b}^{2}}R_{nD}^{\prime\prime}(0)|^{2}\\
$$
(29)
where $\beta=(1-16\alpha_{s}/3\pi)$ is the QCD radiative correction,
and $e_{b}=-1/3$ is the $b$ quark charge in the unit of electron charge.
$R_{nS}(0)$ denotes the radical $S$ wave function at the origin,
and $R_{nD}^{\prime\prime}(0)$ is the second derivative of the radical $D$ wave function at the origin.
$c_{S}$ and $c_{D}$ respectively denote the normalization coefficients before $S$ and $D$ wave.
Note that from the perspective of coupled channels,
they are not restricted to be real-valued and $|c_{S}|^{2}+|c_{D}|^{2}\neq 1$.
For below threshold states, the correct normalization is given by Eq. (27).
Nevertheless, if the imaginary part of the $\Delta M$ are neglected in Eq. (24),
the corresponding solutions will be real,
and one can easily get the feel of how big the mixing is by defining
$\tan\theta:=|c_{S}/c_{D}|$ for $S$ wave dominate states and $\tan\theta:=|c_{D}/c_{S}|$ for $D$ wave dominate states.
There is also an argument for the above formula that
the QCD corrections of higher order may be important [50],
thus $\beta$ has to be treat as an effective constant.
So in order to reduce the parameter’s uncertainty,
we tune the wave functions to reproduce $\Gamma_{ee}(nS)/\Gamma_{ee}(1S),(n=2,3)$(see Fig. 6).
3.2 Gaussian Expansion Method
There are at least two ways to solve Eq. (15).
First one is recursion method,
which based on the observation that the mass shift is expected to be small compared with the bare mass.
i.e. set $M=m_{0}$ as the first step and do the integration in Eq. (15) to get the mass shift,
then set $M=m_{0}+\Delta m$ again and so on until the result converges.
One can even make a further approximation and only do the first step recursion.
However, this method only applies to the single channel mass shift formula (15).
In $S-D$ mixing cases, such as Eq. (24),
the mass difference between $M_{S}^{0}+\Delta M_{S},M_{D}^{0}+\Delta M_{D}$ is small,
even a small error in the off-diagonal term in Eq. (24) will ruin the prediction of the $S-D$ mixing angle.
The second way is to solve the equation by brute-force,
i.e. for the energy ranges we are interested in,
work out all of the integrations in Eq. (15) or Eq. (24) at specific energy point.
We use this method despite of its huge calculation work.
The benefit is that we can extract a lot of information about the wave function’s impact on the mass shift.
One can also change the ${}^{3}P_{0}$’s coupling constant $\gamma$ or the mass renormalization constant $c$ to see the possible consequences.
The high precision work will be not convincing if there is no way to precisely evaluate the integration,
which has a key ingredient — the wave function.
One can indeed evaluate the amplitude pure numerically as the authors do in Refs. [55, 47],
however, we still want analytic expressions which is more convenient
if we want to change the parameters and then repeat the calculations.
In order to achieve that,
various groups approximate the wave functions by simple harmonic oscillators (SHOs) approximation (see e.g. [19, 18, 20, 23, 22]).
The oscillator parameters $\beta$s are usually settled down by requiring that
the root mean square radii to be equal to the initial states [56, 57, 58]
or maximizing its overlap with the numerical wave function [18].
To improve the accuracy, people also expand the true wave function in terms of SHOs (see e.g. Refs. [59, 60]).
As a consequence, one will get fairly complicated analytic expression for highly excited states.
For example, expression (2.12) in Ref. [59].
Due to the highly oscillated behavior of the excited SHOs,
one would need a large number of SHOs to achieve an ideal precision.
We think that it is necessary to fully respect the wave function and make a precise calculation of the transition amplitude,
which is shown to be essential for the states near threshold.
In this work, we get both the analytic expression and the high precision
by using the Gaussian expansion method (GEM) proposed by Hiyama et al. [27].
This method has the observation that bound state’s wave function can be expanded in Gaussian bases as the following
$$\psi_{NLM}(r)=\left(\sum_{i=1}^{n}c_{i}\beta_{i}^{L+\frac{3}{2}}e^{-\frac{1}{2%
}\beta_{i}^{2}r^{2}}r^{L}\right)Y_{L}^{M}(\theta,\varphi),$$
(30)
where $\beta_{i}$s and $c_{i}$s denote oscillator parameters and corresponding coefficients, respectively.
$n$ is the number of Gaussian basis.
In this work, $n=5\sim 20$ for initial states and 5 for $B$ mesons,
and $\beta_{i}$s lie in the range of $0.1\sim 5$GeV.
Compared with the SHO basis, the Gaussian basis is no longer orthogonal.
So a little trick is used to speed up the fitting procedure.
As explained in Ref [27],
$\beta_{i}$s are set to be a geometric series.
Instead of increasing the number of SHO basis for fixed $\beta$,
GEM can both increase the bases number and change $\beta$ to improve the fit.
The quality of the fitting by GEM is quite impressive which is shown in Fig. 2.
Because the ${}^{3}P_{0}$ model calculation is easily done in momentum space,
we need to make a Fourier transformation of the position space wave function.
One benefit of the SHO wave function is that it is invariant after Fourier transform
apart from the substitution $\beta\to 1/\beta$.
Since Gaussian basis is ground state SHO wave function, it naturally keeps this property.
That means, after fitting position space wave functions,
we can rebuild the momentum space wave functions by
$$\psi_{NLM}(p)=\left(\sum_{i=1}^{n}c_{i}\beta_{i}^{-(L+\frac{3}{2})}e^{-\frac{p%
^{2}}{2\beta_{i}^{2}}}p^{L}\right)Y_{L}^{M}(\theta,\varphi).$$
(31)
What makes GEM simple is that there are a minimum number of polynomials in the integration
which simplifies the expression from the beginning.
GEM is quite universal and it is not limited to the wave functions obtained by solving the non-relativistic Schrödinger equation.
To work out the analytic expression,
one has to deal with the associated Laguerre polynomials if SHO wave functions are involved
and also the sophisticated angular integration.
Although in Ref. [61], Roberts and B. Silvestre-Brac show us the general method to do the integration,
and there are analytical expressions [59],
however, these expressions are quite lengthy and they only apply to ${}^{3}P_{0}$ model.
The complexity can be bypassed if we transfer the spherical harmonics into Cartesian form [62] from the beginning.
After this transformation, the general form of the integration can be compactly expressed as
$$\iiint dp_{x}dp_{y}dp_{z}\exp(-p^{\mathrm{T}}.A.p-B.p-C)f(p_{x},p_{y},p_{z}),$$
(32)
where $p,A,B$ and $C$ in the exponent denote $(p_{x},p_{y},p_{z})^{\mathrm{T}}$, $3\times 3$ real symmetric matrix,
three-vector and constant, respectively.
$f(p_{x},p_{y},p_{z})$ is nothing but polynomial, so Eq. (32) is a standard form of Gaussian integration
which can be easily done even manually.
After the integration is done, we can easily transform it back to spherical basis by the substitution
$P_{x}=P\sin\theta\cos\varphi,P_{y}=P\sin\theta\sin\varphi,P_{z}=P\cos\theta$.
Another benefit of this transformation is that it can easily handle much more complicated polynomials $f(p_{x},p_{y},p_{z})$
which may show up in other quark pair creation models.
4 Result and Discussion
4.1 Mass Shift and Open Channel Strong Decay
From Tab. 2 and 3,
one can find that the mass shifts are generally same between GEM and SHOs with a few exceptions for near threshold states.
The mass shifts in a same multiplet are also almost equal
simply because that their wave functions are identical and their bare masses are approximately equal.
This conclusion is consistent with the loop theorem in Ref. [63].
For the states below threshold, $\Delta M$s are all negative,
and closer to the threshold gets more deduction of the mass.
With GEM, this conclusion is true even for states slightly above threshold.
This conclusion differs with Refs. [22, 23], where SHO are used to calculate the mass shift.
Take $h_{b}$ family as an example, our mass shift grows with the mass going higher no matter whether we use GEM or SHO,
however, in Ref. [22] and Ref. [23],
the largest mass shift happens to $h_{b}(1P)$ and $h_{b}(2P)$, respectively.
For states above threshold,
the mass shift behavior becomes complicated (see Fig. 4)
and it is not appropriate to draw the conclusion that the mass shift of state above the $B\bar{B}$ threshold is positive.
This conclusion is only true for asymptotically large mass, and in this case, $\Delta M\propto 1/M$.
We should also point out that this mathematical fact does not mean it will definitely happen.
The reason is that when mass becomes bigger, more $B$ meson channels will contribute,
and one cannot tell the sign of $\Delta M$ before summing all possible channels’ contributions in Eq. (15).
In order to study this sensitivity,
we also plot the dependence on the initial state mass of $\Delta M$ and decay width
for the vector meson above threshold in Fig. 4 and Fig. 5, respectively.
As a concrete example, one can see this sensitivity by comparing $\Upsilon(4S)$ with $\Upsilon(6S)$.
Compared with $\Upsilon(4S)$, the wave function of $\Upsilon(6S)$ has more nodes,
however, basing on this fact one cannot conclude that the $\Delta M$’s behavior is more complicated.
The important reason is that the bare mass of $\Upsilon(6S)$ is also farther from threshold,
causing the average of the wave function overlap integration in Eq. (17).
Note also that absolute value of the mass shift of $\Upsilon(6S)$ calculated in SHO is larger than GEM,
however, in $\Upsilon(4S)$ case,
we have the opposite conclusion if we choose the lowest intersection point of $\Delta M$ and $M-M_{0}$.
From Tab. 2,
one can find that the masses predicted in Ref. [22] are generally closer to the experimental data.
However, we want to stress that the spectrum is an important but not the only criterion to judge whose parameters are better.
As shown in Fig. 6,
the dielectric decay ratios $\Gamma_{ee}/\Gamma_{ee}(1S)$ calculated with the parameters given in Ref. [22]
are generally smaller than experimental measurements before coupled-channel effects are taken into account.
As will be discussed in Sec. 4.2,
coupled-channel effects will suppress rather than enhance these ratios,
so their parameters are difficult to explain the dielectric decays of vector mesons despite of their success in the spectrum.
The complicated structure of the mass shift of $\Upsilon(4S)$ needs further discussion.
Even though the curve of GEM and SHO share some common features,
the small difference is sufficient to generate a large discrepancy of the mass shift.
Another interesting feature of this plot is that GEM has three solutions,
implying that more resonances may pop up compared with potential model prediction.
This sensitivity can also be seen by the decay width behavior in Fig. 5.
Over a large energy range $10.58\sim 10.73$ GeV,
the decay width of $\Upsilon(4S)$ calculated by GEM can be around two times as large as SHO.
Our decay width plot of $\Upsilon(4S)$ also share some resemblance with Fig.2 in Ref. [64],
where the prediction of SHO is not calculated.
The deviations of mass shift and decay width tell us that
it is necessary to adopt the realistic wave functions other than SHO approximation in the coupled channel calculation.
Of course, one may argue that, since the bare mass is 140MeV heavier
than the experimental measurement,
if the bare mass is tuned closer to the $B\bar{B}$ threshold,
the difference between GEM and SHO would be small and we will get one solution.
However, we want to stress that the bare mass is directly related to the wave function,
in the case where we have a smaller bare mass, the wave function will also be different,
thus cause different mass shift behavior.
This sensitivity also reminds us that taking only one step approximation
in the recursive method to solve Eq. (15) may cause a large error,
so an accurate treatment of wave function and a precise method to solve Eq. (15)
are essential for near threshold states.
If $\Upsilon(10580),\Upsilon(10860)$ and $\Upsilon(11020)$ are treated to be pure $S$ or $D$ wave,
we get the open channel decay width shown in Tab. 5.
It is worthy to note that this assumption is oversimplified, so the absolute value cannot be treated too seriously.
4.2 S-D Mixing and Dielectric Decay
As explained in Sec. 2 and sketched in Fig. 1,
coupled-channel effects will also induce mixing among states with same $J^{PC}$.
In this paper, we focus on the mixing between $\Upsilon(S)$ and $\Upsilon_{1}(D)$ family.
The Cornell potential model tells us that
the mass splitting between $\Upsilon((n+1)S)$ and $\Upsilon_{1}(nD)$ is smaller than other configurations,
such as $\Upsilon(nS)$ and $\Upsilon((n+1)S)$ or $\Upsilon_{1}(nD)$ and $\Upsilon_{1}((n+1)D)$ states,
so its quite reasonable to assume that the mixing only happens to $\Upsilon((n+1)S)$ and $\Upsilon_{1}(nD)$.
The masses and corresponding mixing angles after considering $S-D$ mixing is listed in Tab. 6.
In Eq. 24, the overall phase before $c_{S}$ and $c_{D}$ is nonphysical,
so we are free to set the phase of $c_{D}$ to be 0, i.e. $c_{D}\geq 0$.
Under this convention and the normalization condition Eq. (27),
the ratio $c_{S}/c_{D}$ is adequate to fix the value of $c_{S}$ and $c_{D}$.
If the imaginary part of $\Delta M_{f}$ and $\Delta M_{SD}$ in Eq. (24) are neglected,
one would get real solutions both for $M$ and $c_{S}/c_{D}$.
After this approximation, one can deduce the mixing angles.
However, the definition of $c_{S}$ and $c_{D}$ doesn’t exist for states above threshold [56, 65].
Despite of this difficulty, we follow Ref. [7],
assuming that these open channels’ contribution are neglected.
Under this assumption, quantities related to $S-D$ mixing are shown in Tab. 6.
From Tab. 6, we found that the masses barely change
after considering $S-D$ mixing for states below threshold,
indicating that the mixing angles are approximately 0.
So it is reasonable to treat $\Upsilon(1S),\Upsilon(2S)$ and $\Upsilon(3S)$ as pure $S$ wave states.
This conclusion also agrees with the loop theorem in Ref. [63].
From Eq. (29),
we also learn that the dielectric decay of $\Upsilon_{1}(D)$ is suppressed by the $b$ quark mass $m_{b}$,
so the small mixing also provide a natural explanation why these $D$ wave vector mesons are difficult to find at $e^{+}e^{-}$ collider.
From Tab. 6, one can also read off the open channel strong decay widths after considering $S-D$ mixing for states above threshold.
However, one cannot compare the imaginary part of $M_{\text{comp}}$ directly with experimental data
because its real part (which is the Breit-Wigner mass) does not equal to the experimental mass,
thus their phase space for $B\bar{B}$ are different from experiment.
A natural and direct consequence of non-negligible $S-D$ mixing
is the suppression of $\Gamma_{ee}(S)$ or enhancement of $\Gamma_{ee}(D)$.
As can be seen from Fig. 6,
the dielectric decay width of $\Upsilon(10580)$ and $\Upsilon(11020)$ is highly suppressed experimentally.
Under the assumption that $\Upsilon(10580)$ and $\Upsilon(11020)$ are $S$ wave dominate states,
one may be tempted to introduce a large $S-D$ mixing angle for these highly excited states (see e.g. Ref. [50]).
The unexpected large central value of $\Gamma_{ee}(\Upsilon(10860))$ seems to favor a small $S-D$ mixing angle,
however, due to its large errors,
a mixing angle as large as $27^{\circ}$ can also reproduce the data which gives the decay width lies at the lower bound [50].
Of course, more precise measurement of $\Upsilon(10860)$’s dielectric decay will tell us
whether the claim of large $S-D$ mixing is correct or not,
if $S-D$ mixing is fully responsible for this suppression.
As can be seen from Tab. 6,
except for $4S-3D$ case, which will be discussed shortly,
we get a rather small mixing angle not only for below threshold states,
but also for highly excited states.
It seems that the coupled channel formalism cannot explain the suppression of $\Gamma_{ee}$.
However, we want to point it out that $S-D$ mixing is not the only way to suppress $\Gamma_{ee}(S)$.
Even though the $S-D$ mixing angles are small,
$\Gamma_{ee}$ can still be suppressed by the $B$ meson continuum.
The suppression due to $B$ meson continuum is not difficult to understand.
If the conventional mesons have non-negligible components of $B$ meson pairs,
these meson pairs have to undergo one more ${}^{3}P_{0}$ vertex before annihilating into $e^{+}e^{-}$ pairs.
Since the Hamiltonian is small compared with Cornell potential,
it is reasonable to discard the contribution of these meson pairs, then $\Gamma_{ee}$ is suppressed.
This suppression is universal both for $S$ and $D$ wave vector m considering esons,
in contrast to $S-D$ mixing, which enhances the $\Gamma_{ee}(D)$.
We take into account both $S-D$ mixing (see Tab. 6) and $B$ meson pair suppression mechanism (see Tab. 4) in this work,
and the results of $\Gamma_{ee}/\Gamma_{ee}(1S)$ are shown in Fig. 6.
From Fig. 6, one can see that the coupled channel results agree well with experiment except for $\Upsilon(11020)$.
The large suppression of $\Gamma_{ee}(\Upsilon(10580))$ deserves more explanation.
From quark model’s perspective, $\Gamma_{ee}(\Upsilon(10580))$ is suggested to be $4S$ or $3D$ state.
If it is $S$ wave dominate, the mixing angle is about $9^{\circ}$,
which is still not big enough to reproduce $\Gamma_{ee}$.
In fact, according to our calculation, the major suppression comes from the $B$ meson pairs.
Even though $\Upsilon(10580)$ is above $B\bar{B}$ threshold,
it is still below and quite close to $B\bar{B}^{*}$ threshold.
As $h_{b}(3P),\chi_{b0}(3P)$ and $\chi_{b0}(3P)$ shown to us in Tab. 4,
closer to the threshold means bigger probabilities of $B$ meson continuum.
Like the authors did in Ref. [7],
we neglect $B\bar{B}$’s probability and work out the probabilities of other channels.
The probabilities of $B\bar{B}^{*}+h.c.,B^{*}\bar{B}^{*},B_{s}\bar{B}_{s},B_{s}\bar{B}_{s}^{*}+h.c.,%
B_{s}^{*}\bar{B}^{*}_{s}$ are
$20.06\%,11.7\%,0.125\%,0.37\%,0.51\%$, respectively, that means $P_{b\bar{b}}=67.2\%$.
So as an estimation, one will get only two thirds of the decay width predicted by the potential model.
For $3D$ dominant states with Breit-Wigner mass $10.731$GeV in Tab. 6,
its large mixing angle may grasp one’s attention.
Because that it is more difficult to generate at $e^{+}e^{-}$ collider compared with $4S$ dominant states,
and its mass is $80$ MeV heavier than $4S$ dominant states,
we do not consider it as $\Upsilon(10580)$.
In the $4S-3D$ mixing case,
because of the oscillation behavior of $\Delta M_{f}$ and $\langle{\psi_{f}}|H_{I}|{\psi_{i}}\rangle$,
there is one more pair of solutions of $M$ in Eq. (24) with GEM.
For $4S$ dominate state, $m_{\text{comp}}=10.673+0.0989i,m_{\text{real}}=10.675,\theta=18.18^{\circ}$,
and for $3D$ dominate state, $m_{\text{comp}}=10.718+0.0441i,m_{\text{real}}=10.7233,\theta=18.4^{\circ}$.
With the same reasons we also do not consider it as $\Upsilon(10580)$.
Another interesting detail of the $B$ meson continuum is
the slightly increased ratio of $\Gamma_{ee}/\Gamma_{ee}(1S)$.
The mixing angles of $5S-4D$ and $6S-5D$ are so small that $\Gamma_{ee}$ barely change,
nevertheless, due to the small $B$ meson continuum component of $\Upsilon(1S)$,
$\Gamma_{ee}(1S)$ will be suppressed about 0.013, as a consequence,
the ratio $\Gamma_{ee}/\Gamma_{ee}(1S)$ becomes slightly larger after considering coupled-channel effects.
For $\Upsilon(10876)$ and $\Upsilon(11020)$,
all the ground state $B$ meson channels are open.
we can no longer deduce the probabilities of these $B$ meson pairs with Eq. (24).
There is no $B$ meson continuum suppression in this work.
This can reproduce the dielectric decay width of $\Upsilon(10876)$ but not $\Upsilon(11020)$.
As shown in Fig. 6,
there is a notable discrepancy between our calculation and experiment on $\Upsilon(11020)$ dielectric decay width.
This issue may come from the two assumptions we use to simplify the calculation.
One is that we only consider the mixing between $6S$ and $5D$.
In fact, with the increase of radial quantum number,
the energy levels of $S$ or $D$ wave will become denser,
so the mixing may exist between several $S$ and $D$ wave states.
Another is the probabilities of excited $B$ meson pairs are neglected.
This may also cause problems.
For example, the $\Upsilon(11020)$ is only $26$ MeV lighter than $m_{B^{*}}+m_{B_{1}}$,
so a large suppression of $P_{b\bar{b}}$ is naturally expected,
causing a large suppression of the dielectric decay width.
It is possible to distinguish $S-D$ mixing and $B$ meson pairs suppression mechanism by the measurement of the radiative decay.
Theoretically, the $E1$ transition can be represented by [67, 54, 68]
$$\Gamma(n^{2S+1}L_{J}\to n^{\prime 2S^{\prime}+1}L^{\prime}_{J^{\prime}}+\gamma%
)=\frac{4}{3}C_{fi}\delta_{SS^{\prime}}e^{2}_{b}\alpha|\langle{f}|r|{i}\rangle%
|^{2}E^{3}_{\gamma},$$
(33)
where $e_{b}=-\frac{1}{3}$. $\alpha$ and $E_{\gamma}$ respectively denote the fine structure constant and the energy of the emitted photon.
$\langle{f}|r|{i}\rangle$ and $C_{fi}$ are represented by
$$\displaystyle\langle{f}|r|{i}\rangle$$
$$\displaystyle=$$
$$\displaystyle\int_{0}^{\infty}R_{f}(r)R_{i}(r)r^{3}dr,$$
(34)
$$\displaystyle C_{fi}$$
$$\displaystyle=$$
$$\displaystyle\max(L,L^{\prime})(2J^{\prime}+1)\left\{\begin{array}[]{ccc}L^{%
\prime}&J^{\prime}&S\\
J&L&1\end{array}\right\}^{2}.$$
(35)
From Eq. (33), we have
$$\displaystyle r_{\gamma}(S)$$
$$\displaystyle:=$$
$$\displaystyle\frac{\Gamma(\Upsilon(S)\to\chi_{b2}(1P)+\gamma)}{\Gamma(\Upsilon%
(S)\to\chi_{b0}(1P)+\gamma)}=5\left(\frac{E_{\gamma 2}}{E_{\gamma 0}}\right)^{%
3},$$
(36)
$$\displaystyle r_{\gamma}(D)$$
$$\displaystyle:=$$
$$\displaystyle\frac{\Gamma(\Upsilon_{1}(D)\to\chi_{b2}(1P)+\gamma)}{\Gamma(%
\Upsilon_{1}(D)\to\chi_{b0}(1P)+\gamma)}=\frac{1}{20}\left(\frac{E_{\gamma 2}}%
{E_{\gamma 0}}\right)^{3},$$
(37)
where $E_{\gamma 2}$ and $E_{\gamma 0}$ respectively represent the photon energy of
$V\to\chi_{b2}+\gamma$ and $V\to\chi_{b0}+\gamma$. ($V$ stands for initial vector state.)
From PDG data [66], we have $r_{\gamma}(2S)=1.91\pm 0.29$ and $r_{\gamma}(3S)=3.82\pm 1.05$,
and the theoretical predictions of Eq. (36) and Eq. (37) are $r_{\gamma}(2S)=1.57$, $r_{\gamma}(3S)=3.6$,
$r_{\gamma}(1D)=0.0157$ and $r_{\gamma}(2D)=0.036$.
So it is reasonable to treat $\Upsilon(2S)$ and $\Upsilon(3S)$ as pure $S$ wave,
and our conclusion of small $S-D$ mixing angle is consistent with experiment for vector bottomonia below threshold.
If the all vector bottomonia observed are $S$ wave dominant,
the small $\Gamma_{ee}$ of $\Upsilon(10580)$ and $\Upsilon(11020)$ naturally requires a large mixing angle
under $S-D$ mixing mechanism, causing a large suppression of $r_{\gamma}(S)$.
On the contrary, $B$ meson continuum suppresses $P_{b\bar{b}}$, leaving the ratio $r_{\gamma}$ unchanged.
Given that the deduced $S-D$ mixing angle is small, we expect a large $r_{\gamma}$.
Unfortunately, the data on the radiative decay widths of
$\Upsilon(10580),\Upsilon(10860)$ and $\Upsilon(11020)$ is still not available so far.
A precise measurement of radiative decay will definitely tell us more about their internal structures.
For states above threshold,
the predicted spectra and decay widths agree not very well with experimental data.
There are two reasons to cause this issue.
As with most work, the meson loops of excited $B$ mesons are ignored,
however, this assumption may be not appropriate for highly excited states.
For example, $\Upsilon(11020)$ is already $20$ MeV heavier than $B_{1}\bar{B}$ threshold.
Second reason comes from the non-relativistic approximation of our bare mass.
In principle, relativistic corrections will be more important with the binding energy goes high,
so both the wave functions and the bare masses will change accordingly.
However, including contributions of excited $B$ mesons
and re-fitting the spectrum and decay widths involves much more work, which lies beyond this work.
It still remains a challenge to reproduce the spectra and dielectric or hadronic decay patterns.
5 Summary
In this paper, we make a thorough and precise calculation of coupled-channel effects in the framework of ${}^{3}P_{0}$ model with GEM
for the bottomonium.
The results of the spectrum, open channel strong decays, probabilities of the $B$ meson continuum,
the $S-D$ mixing and the vector meson’s dielectric decays are explicitly shown.
In order to study the near threshold effects,
we also plot the mass dependence of the mass shift and open channel decay widths for pure $S$ and $D$ wave vector mesons.
For $\Upsilon(4S)$, the decay width of GEM can be two times as large as SHO over a wide energy range,
and the mass shift is around three times as large.
These big deviations indicate that SHO is not a good approximation for near threshold states,
even though the oscillator parameters are carefully selected to reproduce the root mean square radius of the corresponding mesons.
With the consideration of coupled-channel effects, we get small $S-D$ mixing angles except for $\Upsilon(4S)$.
We point it out that, for $S$ wave dominant vector states,
$S-D$ mixing is not the only mechanism to suppress their dielectric decay widths,
$B$ meson continuum can also lead to the suppression.
With this $B\bar{B}$ suppression mechanism at hand,
we still succeed to reproduce the dielectric decays of vector bottomonia except for $\Upsilon(11020)$.
The deviation of the spectrum and decays between our predictions and experimental measurements
may be due to the neglect of excited $B$ meson continuum in coupled-channel effects or
the non-relativistic approximation in the quenched limit.
$S-D$ mixing will cause the suppression of the ratio in Eq. (36) for $S$ wave dominant state,
as a contrary, the $B$ meson continuum does not change this ratio.
We suggest BaBar and Belle to make precise measurements on the radiative decays of
$\Upsilon(10580),\Upsilon(10860)$ and $\Upsilon(11020)$ to distinguish these two effects.
Acknowledgements
The authors are grateful to
Meng Ce, Gui-Jun Ding, David R. Entem, Feng-Kun Guo, Yu. S. Kalashnikova, Bilal Masud, Jia-Lun Ping and E. Santopinto
for useful discussions and suggestions.
This work is supported by the National Natural Science Foundation of China under Grants No. 11261130311 (CRC110 by DFG and NSFC).
M. Naeem Anwar is supported by CAS-TWAS President’s Fellowship for International Ph.D Students.
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What Makes Kevin Spacey Look Like Kevin Spacey
Supasorn Suwajanakorn Ira Kemelmacher-Shlizerman Steven M. Seitz
University of Washington
Abstract
We reconstruct a controllable model of a person from a large photo collection that captures his or her persona, i.e., physical appearance and behavior. The ability to operate on unstructured photo collections enables modeling a huge number of people, including celebrities and other well photographed people without requiring them to be scanned. Moreover, we show the ability to drive or puppeteer the captured person B using any other video of a different person A. In this scenario, B acts out the role of person A, but retains his/her own personality and character. Our system is based on a novel combination of 3D face reconstruction, tracking, alignment, and multi-texture modeling, applied to the puppeteering problem. We demonstrate convincing results on a large variety of celebrities derived from Internet imagery and video.
1 Introduction
Kevin Spacey has appeared in many acting roles over the years. He’s played characters with a wide variety of temperaments and personalities. Yet, we always recognize him as Kevin Spacey.
Why? Is it his shape? His appearance? The way he moves?
Inspired by Doersch et al’s “What Makes Paris Look Like Paris” [10], who sought to capture the essence of a city, we seek to capture an actor’s persona.
But what defines a persona, how can we capture it, and how will we know if we’ve succeeded?
Conceptually, we want to capture how a person appears in all possible scenarios. In the case of famous actors, there’s a wealth of such data available, in the form of photographic and video (film and interview) footage.
If, by using this data, we could somehow synthesize footage of Kevin Spacey in any number and variety of new roles, and they all look just like Kevin Spacey, then we have arguably succeeded in capturing his persona.
Rather than creating new roles from scratch, which presents all sorts of challenges unrelated to computer vision, we will assume that we have video footage of one person (actor A), and we wish to replace him with actor B, performing the same role.
More specifically, we define the following problem:
Input:
1) all available photos and/or videos of actor B, and 2) a photo collection and a single video V of actor A
Output:
a video V${}^{\prime}$ of actor B performing the same role as actor A in V, but with B’s personality and character.
Figure LABEL:fig:teaser presents example results with Kevin Spacey as actor B, and two other celebrities (Daniel Craig and George Bush) as actor A.
The problem of using one face to drive another is a form of puppetry, which has been explored in the graphics literature
e.g., [20, 25, 18]. The term avatar is also used sometimes to denote this concept of a puppet. What makes our work unique is that we derive the puppet (actor B) automatically from large photo collections.
Our answer to the question of what makes Kevin Spacey look like Kevin Spacey is in the form of a demonstration, i.e., a system that is capable of very convincing renderings of one actor believably mimicking the behavior of another. Making this work well is challenging, as we need to determine what aspects are preserved from actor A’s performance and actor B’s personality. For example, if actor A smiles, should actor B smile in the exact same manner? Or use actor B’s own particular brand of smile?
After a great deal of experimentation, we obtained surprisingly convincing results using the following simple recipe: use actor B’s shape, B’s texture, and A’s motion (adjusted for the geometry of B’s face). Both the shape and texture model are derived from large photo collections of B, and A’s motion is estimated using a 3D optical flow technique.
We emphasize that the novelty of our approach is in our application and system design, not in the individual technical ingredients.
Indeed, there is a large literature on face reconstruction [9, 2, 14, 21] and tracking and alignment techniques e.g., [24, 15]. Without a doubt, the quality of our results is due in large part to the strength of these underlying algorithms from prior work. Nevertheless, the system itself, though seemingly simple in retrospect, is the result of many design iterations, and shows results that no other prior art is capable of. Indeed, this is the first system capable of building puppets automatically from large photo collections and driving them from videos of other people.
2 Related Work
Creating a realistic controllable model of a person’s face is an extremely challenging problem due to the high degree of variability in the human face shape and appearance. Moreover, the shape and texture are highly coupled:
when a person smiles, the 3D mouth and eye shape changes, and wrinkles and creases appear and disappear which changes the texture of the face.
Most research on avatars focuses on animated, non-human faces [18, 24]. The canonical example is that a person drives an animated character, e.g., a dog, with his/her face. The driverâs face can be captured by a webcam or structured light device such as Kinect, the facial expressions are then transferred to a blend shape model that connects the driver and the puppet and then coefficients of the blend shape model are applied to the puppet to create a similar facial expression. Recent techniques can operate in real-time, with a number of commercial systems now available, e.g., faceshift.com (based on [24]), and Adobe Project Animal [1].
The blend shape model that is used for non-human avatars typically capture only large scale expression deformations. Capturing fine details remains an open challenge. Some authors have explored alternatives to blend shape models for non-human characters by learning shape transfer functions [29], and dividing the shape transfer to several layers of detail [27, 18].
Creating a model of a real person, however, is an even more challenging task due to the extreme detail that is required for creation of a realistic face. One way of capturing fine details is by having the person participate in a sequence of lab sessions and use multiple synchronized and calibrated lights and camera rigs [2]. For example, light stages were used for creation of the Benjamin Button movie–to create an avatar of Brad Pitt in an older age [9]. For this, Brad Pitt had to participate in numerous sessions of capturing every possible expression his face can make according to the Facial Action coding system [11]. These expressions were later used to create a personalized blend shape model and transferred to an artist created sculpture of an older version of him. This approach produces amazing results, however, requires active participation of the person and takes months to execute.
Automatic methods for expression transfer of people used multilinear models created from 3D scans [22] or structured light data [6], and transfered differences in expressions of the driver’s mesh to the puppet’s mesh through direct deformation transfer, e.g., [20, 25, 18] or through coefficients that represent different face shapes [24, 22], or driven by speech [7]. These approaches either account only for large scale deformations or do not handle texture changes on the puppet.
This paper is about creating expression transfer in 3D with high detail models and accounting for texture changes that occur due to expression change. Change in texture was previously considered by [19] via image based wrinkles transfer using ratio images, where editing of facial expression used only a single photo [28], face swapping [3, 8], reenactment [12], and age progression [17]. These approaches changed a person’s appearance by transferring changes in texture from another person, and typically focus on a small range of expressions. Finally, [13] showed that it is possible to create a puppetry effect by simply comparing two youtube videos (of the driver and puppet) and finding similarly looking (based on metrics of [16]) pairs of photos. However, the results simply recalled the best matching frame at each time instant, and did not synthesize continuous motion.
In this paper, we show that it is possible to leverage a completely unconstrained photo collection of the person (e.g., Internet photos) in a simple but highly effective way to create texture changes, applied in 3D.
3 Overview
Given a photo collection of the driver and the puppet, our system (illustrated in Figure 1) first reconstructs a rigid 3D average model of the driver and the puppet. Next, given a video of the driver, it estimates 3D flow from each video frame to the driver’s average model. This flow is then transfered onto the average model of the puppet creating a sequence of shapes that move like the driver (Section 4). In the next stage, high detail consistent texture is generated for each frame that accounts for changes in facial expressions (Section 5).
4 3D Dynamic Mesh Creation
By searching for “Kevin Spacey” on Google’s image search we get a large collection of photos that are captured under various poses, expressions, and lightings. In this section, we describe how we estimate an average 3D model of the driver and the puppet, and deform it according to a video or a sequence of photos of the driver. Figure 2 illustrates the shape creation process.
3D Average Model Estimation.
We begin by detection of face and fiducial points (corners of eyes, mouth, nose) in each photo using CMU’s face tracker IntraFace [26]. We next align all the faces to a canonical coordinate frame and reconstruct an average rigid 3D shape of Spacey’s face. For 3D average shape reconstruction we follow [14] with the modification of non-rigidly aligning photos prior to 3D reconstruction. We describe the non-rigid alignment step in Section 5. The same reconstruction pipeline is applied on the driver and the puppet photo collections, resulting in two average 3D rigid models.
Dynamic 3D Model.
Next, we create a dynamic model of the puppet that is deformed according to the driver’s non-rigid motions. For the driver, we are given a video or sequence of photos. The first step is to reconstruct the 3D flow that deforms the driver’s 3D average model to the expression of the driver in every single frame of the input video, using the method of [21]. The geometric transformation is given as a 3D translation field $T:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ applied on a driver’s average shape.
Given a reconstructed mesh at frame $i$ of a driver $M_{D}^{i}(u,v):\mathbb{R}^{2}\rightarrow\mathbb{R}^{3}$ parametrized on an image plane $(u,v)$ from a depth map, and the average mesh over the entire frame sequence $\overline{M}_{D}$, the goal is to transfer the translation field $M_{D}^{i}-\overline{M}_{D}$ to the puppet’s base mesh $M_{P}$ to produce $M_{P}^{i}$. To transfer the deformation, we first establish correspondence between $M_{D}$ and $M_{P}$ through a 2D optical flow algorithm between the puppet’s and driver’s 2D averages from their photo collections.
[15] has shown that we can obtain correspondence between two very different people by projecting one average onto the appearance subspace of the other by this matching illumination, and then run an optical flow algorithm between the resulting projections. With this flow, we can apply the deformation of the driver on the same facial features of the puppet. However, the direct deformation from the driver may not be suitable for the puppet, for example, if their eye sizes are different, the deformation needed to blink will be different. We solve this by scaling the magnitude of the deformation to fit each puppet as follows (Figure 2): Let the deformation vector from the driver at vertex $M_{D}(u,v)$, be $\Delta(u,v)$. We first find the nearest vertex to $M_{D}(u,v)+\Delta(u,v)$ in euclidean distance on the driver mesh, denoted by $M_{D}(s,t)$. Through the flow between $M_{D}$ and $M_{P}$ we computed earlier, we can establish a corresponding pair $(M_{P}(u^{\prime},v^{\prime}),M_{P}(s^{\prime},t^{\prime}))$ on the puppet mesh. The magnitude-adjusted deformation at $M_{P}(u^{\prime},v^{\prime})$ is then computed by $\hat{\Delta}(u,v)(\hat{\Delta}(u,v)\cdot\Delta^{\prime})$ where $\hat{\Delta}=\frac{\Delta}{\|\Delta\|}$ and $\Delta^{\prime}=M_{P}(s^{\prime},t^{\prime})-M_{P}(u^{\prime},v^{\prime})$. In addition, since the flow between the driver and puppet can be noisy around ambiguous, untextured regions, we perform the standard denoising on the term $f(u,v)=(\hat{\Delta}(u,v)\cdot\Delta^{\prime})$ to obtain a regularized field $f^{*}(u,v)$. The particular denoising algorithm we use is ROF denoising with the Huber norm and TV regularization. The final puppet’s mesh is constructed as $M_{P}^{i}(u,v)=M_{P}(u,v)+\hat{\Delta}(u,v)f^{*}(u,v)$.
5 High detail Dynamic Texture Map Creation
In the previous section, we have described how to create a dynamic mesh of the puppet. This section will focus on creation of a dynamic texture. The ultimate set of texture maps should be consistent over time (no flickering, or color change), have the facial details of the puppet, and change according to the driver’s expression, i.e., when the driver is laughing, creases around the mouth and eye wrinkles may appear on the face. For the latter it is particularly important to account for the puppet’s identity–some people may have wrinkles while others won’t. Thus, a naive solution of copying the expression detail from the driver’s face will generally not look realistic. Instead, we leverage a large unconstrained photo collection of the puppet’s face. The key intuition is that to create a texture map of a smile, we can find many more smiles of the person in the collection. While these smiles are captured under different pose, lighting, white balance, etc. they have a common high detail that can be transfered to a new texture map.
Our method works as follows. Given the target expression which is either the configuration of fiducials on the driver’s face (that represents e.g., a rough shape of a smile) or by a reference photo if the driver is the same person as the puppet, we first warp all the photos in the puppet’s collection to the given expression. We then create a multi-scale weighted average that perserves a uniform illumination represented by the lower frequency bands and enhances details in the higher frequency bands. Next we explain each of these in more detail.
Non-rigid warping of the photos.
Each photo in the puppet’s photo collection has 49 fiducial points that we detected. Next we frontalize the face by marking the same fiducials on a generic 3D face model and solve a Perspective-n-Point problem to estimate the 3D pose of the face in the photo. The model is then back-projected to produce a frontal-warp version of each photo. Let the rigid pose-corrected fiducials in each photo be $F^{i}\in\mathbb{R}^{2\times 49}$ and the target fiducials be $F^{T}$. Given two sets of fiducials we estimate a smooth mapping $r$ that transforms the $i$-th photo to the target expression using a smooth variant of thin-plate splines [23] which minimizes the following objective:
$$\min_{r}\sum_{i=1}^{n}\|F^{i}-r(F^{T})\|^{2}+\lambda\iint r_{xx}^{2}+2r_{xy}^{%
2}+r_{yy}^{2}\,\textrm{d}x\,\textrm{d}y$$
(1)
The optimal mapping $r$ satisfying this objective can be represented with a radial basis function $\phi(x)=x^{2}\log x$ and efficiently solved with a linear system of equations [23]. Given the optimal $r$, we can then warp each face photo to the target expression by backward warping. However, this warping relies only on a sparse set of fiducials and the resulting warp field can be too coarse to capture shape changes required to match the target expression which results in a blurry average around eyes and mouth (Figure 5 b). To refine the alignment, we perform an additional dense warping step by exploiting appearance subspaces based on [21, 15]. The idea is to warp all photos to their average, which now has the target expression, through optical flow between illumination-matched pairs. Specifically, let the $i^{\text{th}}$ face image after TPS warping be $I^{i}$, its projection onto the rank 4 appearance subspace of the TPS warped photos be $\hat{I}^{i}$. The refined warp field is then simply the flow from $I^{i}$ to $\hat{I}^{i}$. In the case where a reference photo is available, (Figure 4), we can further warp the entire collection to the reference photo by computing an optical flow that warps $\hat{I}^{T}$ to $I^{T}$, denoted by $F_{{\hat{I}^{T}}\rightarrow I^{T}}$, and compute the final warp field by composing $F_{{I^{i}}\rightarrow\hat{I}^{i}}\circ F_{\hat{I}^{i}\rightarrow{\hat{I}^{T}}}%
\circ F_{{\hat{I}^{T}}\rightarrow I^{T}}=F_{{I^{i}}\rightarrow\hat{I}^{i}}%
\circ F_{{\hat{I}^{T}}\rightarrow I^{T}}$ from the fact that $F_{\hat{I}^{i}\rightarrow{\hat{I}^{T}}}$ is an identity mapping.
Adding high-detail.
Given the set of aligned face photos, we compute a weighted average of the aligned photos where the weights measure the expression similarity to the target expression and the confidence of high-frequency details. We measure expression similarity by $L_{2}$ norm of the difference between the source and target fiducial points, and high-frequency details by the response of a Laplacian filter. A spatially-varying weight $W^{i}_{jk}$ for face $i$ at pixel $(j,k)$ is computed as:
$$W^{i}_{jk}=\exp\left(\frac{-\|F^{T}-F^{i}\|^{2}}{2\sigma^{2}}\right)\cdot(L^{i%
}_{jk})^{\alpha}$$
(2)
where $L^{i}_{jk}$ is the response of a Laplacian filter on face image $i$ at pixel $(j,k)$. An average produced with this weighting scheme produces blending artifacts, for example if high-frequency details from many photos with various illuminations are blended together (Figure 6). To avoid this problem, the blending is done in a multi-scale framework, which blends different image frequency separately. In particular, we construct a Laplacian pyramid for every face photo and compute the weighted average of each level from all the pyramids according to the normalized $W^{i}_{jk}$, then collapse the average pyramid to create a final texture.
With real photo collections, it is rarely practical to assume that the collection spans any expression under every illumination. One problem is that the final texture for different expressions may be averaged from a subset of photos that have different mean illuminations which results in an inconsistency in the overall color or illumination of the texture. This change in the color, however, is low-frequency and is mitigated in the multi-scale framework by preferring a uniform weight distribution in the lower frequency levels of the pyramid. We achieve this is by adding a uniform distribution term, which dominates the distribution in the coarser levels:
$$W^{i}_{jk}=\left(\exp\left(\frac{-\|F^{T}-F^{i}\|^{2}}{2\sigma^{2}}\right)+%
\tau l^{-\beta}\right)\cdot(L^{i}_{jk})^{\alpha}$$
(3)
where $l\in\{0,\ldots,p-1\}$ and $l=0$ represents the coarsest level of a pyramid with $p$ levels, and $\tau$ and $\beta$ are constants.
6 Experiments
In this section, we describe implementation details, runtime, and our results.
Implementation details
In Section 4, the 3D average models for both driver and puppet are reconstructed using [21] which outputs meshes as depth maps with a face size around 194 x 244 (width x height) pixels. To find correspondence between the driver and puppet for deformation transfer purpose, we project the 2D average of the puppet onto the rank-4 appearance subspace of the driver, then compute an optical flow using Ce Liu’s implementation based on Brox et al.[4] and Bruhn et al. [5] with parameters ($\alpha,$ ratio, minWidth, outer-,inner-,SOR-iterations) = $(0.02,0.85,20,4,1,40)$. The ROF denoising algorithm used for adjusting deformation magnitude has only two parameters: The weight constant for the TV regularization which is set to 1, and the Huber epsilon to 0.05. In Section 5, $\lambda$ in TPS warping objective is set to 10, and the dense warping step uses the same optical flow implementation but with $\alpha=0.3$. For Equation 3, $(\alpha,\beta,\tau)=(1,20,1)$, $\sigma$ is typically set to 10 but can vary around $6-12$ for different sizes and qualities of the photo collections (See 7).
Runtime We test our system on a single CPU core of a quad-core Intel [email protected]. Computing a deformed puppet mesh based on the driver sequence takes 0.2 second per frame with 0.13 second spent on denoising. Synthesizing a texture which includes TPS warping, dense warping, and multi-scale pyramid blending takes 0.34 second per frame on average.
Evaluating a puppetry system objectively are extremely hard, and there exists no accuracy metric or benchmark to evalute such system. Ground-truth shapes for evaluating deformation transfer across two people cannot be captured as this requires the puppet person whose shape will be captured, to perform exactly like a driver sequence, which is not possible unless the person is the driver themselves. However, such a setup of self puppetry to evalute the reconstructed geometry requires no deformation transfer and does not evaluate our system. Evaluating the synthesized textures is also qualitative in nature as the average texture we generate cannot be pixel-wise compared to the reference. We provide results and input references for qualitative comparisons and point out areas where further improvement can be done.
From Google Images, we gathered around 200 photos for celebrities and politicians in Figure 7. We generated output puppetry sequences of those people performing various facial expressions driven by YouTube Videos of Kevin Spacey and George W. Bush in the top row. These 3D models are generated by warping an average model of each person with 3D optical flow transfered from the driver (top). So, to render these texture-mapped models, we only synthesize textures in their neutral expressions for the average models but use the target expressions to calculate the blending weights. The identities of these puppets are well-preserved and remain recognizable even when driven by the same source, and the transformation provides plausible output for puppets with different genders, ethnicities, skin colors, or facial features. Facial details are enhanced and change dynamically according to the reference expressions, for example, in creases around the mouth in the last column. We strongly encourage the readers to watch our supplementary videos for these results.
In Figure 4, we show the capability to recreate consistent textures with similar expressions as reference photos in the photo collection. In other words, we are able to “regenerate” each photo in the entire collection so that they appear as if the person is performing different expressions within the same video or photograph captures. Note that each reference here is part of the photo collection used in the averaging process. Texture results for references outside the photo collection is in Figure 8. We compare our method with 3 baseline approaches: 1. A single static average is TPS warped to the reference. This approach produces textures that lack realistic changes such as wrinkles and creases, and shapes that only roughly match the reference (e.g. eyes in column (ii) second row which appear bigger than the reference) because the warping can only rely on sparse fiducial points. 2. A weighted average of the photo collection using identical weights as our method. With this approach, creases can be seen, but the overall texture colors appear inconsistent when there is a variation in the mean color of different high-weighted sets of photos. The overall textures look blurry as there is no alignment done for each photo, and the shapes (eyes in the third row) do not match the reference when the number of similar photos in the collection is small. 3. An improved weighted average with prewarping step which includes TPS and dense warping similar to our pipeline. The prewarping step improves the shapes and the sharpness of the faces, but the textures remain inconsistent. Our method in column (v) produces sharp, realistic, and consistent textures with expression-dependent details and is able to match references with strong illuminations or in black-and-white in Figure 4 (b). Since the references are part of the averaging process, some high-frequency details such as wrinkles are transfered to the output texture. However, the low-frequency details such as shading effects, soft shadow under the nose (in the last example, middle row), or highlights (in the second example, last row) are averaged out in the multi-scale blending and are not part of the final textures.
In Figure 8, we show self-puppetry results where we render output 3D models from [21] with our textures. Similarly to Figure 7, we only synthesize textures in neutral expressions for the average models with blending weights calculated based on the target expressions. The reference photos are excluded from the photo collection in the averaging process. Our textures remain consistent when the references have different lightings and look realistic from various angles. In the fourth reference in the last row, our textures have wrinkles but are less pronounced than the input reference, which is due partly to the fact that the number of photos with wrinkles in the collection is less than 5%.
7 Discussion
The quality of the synthesized textures highy depends on many aspects of the photo collection which include the number and resolutions of the photos, expression and light varations. Since the textures are synthesized based on the assumption that we can find photos with similar expressions, the results will degrade with smaller photo collection (less expression variation). In that situation, the method needs to take into account less-similar photos with a larger standard deviation in Equation 3 resulting in a less pronouced expression. If the standard deviation is kept small, high-frequency details can flicker when the rendered models from video input are played in sequence. Higher resolution photos directly contribute to a sharper average. Our method is less sensitive to having small light variations, in contrast to expression variations, because the shading differences are of low-frequency and can be shared across a wider range of photos in the coarser levels of pyramid.
When a photo collection contains in the order of thousands photos such as when we extract frames from all movies starring a particular actress, additional characteristics of photos can be used to fine-tune the similarity measure in the averaging process such as the directions of lights in the scene to enable a religthing capability or the age of the person (e.g. from a regressor) to synthesize textures at different ages. Only a small modification is needed to implement these changes in our framework. It is also useful to learn the association between the apperance of facial details and facial motions to help with unseen expressions that may share common facial details with already existing photos in the collection.
8 Conclusion
We presented the first system that allows reconstruction of a controllable 3D model of any person from a photo collection toward the goal of capturing persona. The reconstructed model has time-varying, expression-dependent textures and can be controlled by a video sequence of a different person. This capability opens up the ability to create puppets for any photo collection of a person, without requiring them to be scanned. Furthermore, we believe that the insights from this approach (i.e., using actor B’s shape and texture but A’s motion), will help drive future research in this area.
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1629–1637, 2011. |
Free homotopy algebras
Martin Markl
[email protected]
Mathematical Institute of the Academy
Žitná 25
115 67 Praha 1
The Czech Republic
Abstract
An explicit description of free strongly homotopy associative and
free strongly homotopy Lie algebras is given. A variant of the
Poincaré-Birkhoff-Witt theorem for the universal enveloping
A($m$)-algebra of a strongly homotopy Lie algebra is formulated.
keywords: Free algebra, A($m$)-algebra, L($m$)-algebra, PBW theorem
: 08B20
††thanks: This work was supported by the
grant GA ČR 201/96/0310 and MŠMT ME603.
Introduction
This note was originated many years ago as my reaction to questions of
several people how free strongly homotopy algebras can be described
and what can be said about the structure of the universal enveloping
A($m$)-algebra of an L($m$)-algebra constructed
in [3], and then circulated as a “personal
communication.” I must honestly admit that it contains no really deep
result and that everything I did was that I expanded definitions and
formulated a couple of statements with more or less obvious proofs.
A($m$)-algebras and their strict
homomorphisms [5, pages 147–148] form an equationally
given algebraic category ${\tt A}(m)$. It follows from general theory
that the forgetful functor to the category ${\tt gVect}$ of graded
vector spaces, $\Box:{\tt A}(m)\to{\tt gVect}$, has a left adjoint
${\raisebox{7.2pt}{\tiny$m$}\hskip-7.0pt{\mathfrak{A}}}:{\tt gVect}\to{\tt A}(m)$. Given a graded vector space $X\in{\tt gVect}$, the object ${\raisebox{7.2pt}{\tiny$m$}\hskip-7.0pt{\mathfrak{A}}}(X)\in{\tt A}(m)$ is the free
A($m$)-algebra on the graded vector space $X$. We will also, for
$n<m$, consider forgetful functors $\Box:{\tt A}(m)\to{\tt A}(n)$
and their left adjoints ${\raisebox{7.2pt}{\tiny$m\hskip-2.27622pt:\hskip-1.707165ptn$}\hskip-9.0pt{%
\mathfrak{A}}\hskip 2.0pt}:{\tt A}(n)\to{\tt A}(m)$; here
the case $n=1$ is particularly important, because ${\raisebox{7.2pt}{\tiny$m\hskip-2.27622pt:\hskip-1.707165pt1$}\hskip-9.0pt{%
\mathfrak{A}}\hskip 2.0pt}:{\tt dgVect}\to\mbox{{\tt A}($m$)}$ describes the free A($m$)-algebra
generated by a differential graded vector space.
We believe there is no need to emphasize the important rôle
of free objects in mathematics. Each A($m$)-algebra is a quotient of a
free one and free A($m$)-algebras were used in our definition of the
universal enveloping algebra of a strongly homotopy
Lie algebra [3, page 2154]. It
could be useful to have a concrete description of these free
algebras, as explicit as, for example,
the description of free associative algebras by tensor algebras.
This brief note gives such a description in terms of planar trees.
Replacing planar trees by
non-planar ones, one can equally easy represent also free
strongly homotopy Lie algebras.
Surprisingly, these free algebras are
simpler objects that their strict counterparts
and admit a nice linear basis
(Section 3), while
free (strict) Lie algebras are very complicated
objects
(see, for example, [9, Chapter IV]).
This might be explained by the fact that axioms of strongly
homotopy algebras are certain resolutions of
strict axioms, thus they are more amenable. They
manifest some properties of distributive
laws [6] – they are ‘directed,’ therefore one can
obtain unique representatives of elements of the corresponding
operads in terms of admissible trees, like
in Proposition 1.2, with obvious implications for the
structure of free algebras.
In the first section of this note we describe the operad ${\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-8.0pt\underline{\mathcal{A}}}$ for
A($m$)-algebras111underlining indicates that it is a
non-$\Sigma$ operad and then, in Section 2, free A($m$)-algebras. In
the third section we modify these constructions to free strongly
homotopy Lie algebras. In the last section we prove a
Poincaré-Birkhoff-Witt-type theorem for strongly homotopy Lie
algebras and formulate a problem related to the structure of the
category of modules. In the appendix we then recall the language of
trees used throughout the paper.
1 Operad for A($m$)-algebras
In this note we assume a certain familiarity with operads, a good
reference book for this subject is [8].
For simplicity we suppose the ground ring ${\bf k}$ to be a field of
characteristic zero, though the results of this and the following
section remain true over an arbitrary ring.
Let $m$ be a natural number or $\infty$. Recall
(see [10] for the original definition,
but we use the sign convention of [5])
that an A($m$)-algebra
is a graded ${\bf k}$-module $A$ together with a set $\{\mu_{k};1\leq k\leq m,\ k<\infty\}$
of degree $k-2$-linear maps, $\mu_{k}:\bigotimes^{k}A\to A$, such that
$$\sum_{\lambda=0}^{n-1}\sum_{k=1}^{n-\lambda}(-1)^{\omega}\cdot\mu_{n-k+1}(a_{1%
},\dots,a_{\lambda},\mu_{k}(a_{\lambda+1},\dots,a_{\lambda+k}),a_{\lambda+k+1}%
,\dots,a_{n})=0$$
for all homogeneous $a_{1},\ldots,a_{n}\in A$, $n\leq m$, where the sign
is given as
$$\omega:={k+\lambda+k\lambda+k(|a_{1}|+\dots+|a_{\lambda}|)}.$$
It easily follows from the above axiom that $\mu_{1}:A\to A$ is a
degree $-1$ differential. This operation will play a special rôle
and we denote it by $\partial:=\mu_{1}$. A($m$)-algebras are algebras
over a certain non-$\Sigma$ operad ${\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-8.0pt\underline{\mathcal{A}}}=\{%
{\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-8.0pt\underline{\mathcal{A}}}(n)%
\}_{n\geq 1}$ in the monoidal category of graded vector spaces. This operad
can be constructed as follows.
Let ${\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-7.2pt\underline{\mathcal{F}}}=\{%
{\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-7.2pt\underline{\mathcal{F}}}(n)%
\}_{n\geq 1}$ be the free graded non-$\Sigma$
operad $\underline{\Gamma}(\xi_{1},\xi_{2},\ldots,\xi_{m})$ generated by operations
$\xi_{i}$, $l\leq i\leq m$, where each $\xi_{i}$ has arity $i$ and degree
$i-2$. Alternatively, let ${\raisebox{7.2pt}{\tiny$m$}\hskip-5.8pt{\mbox{$\tt T$}}}(n)$ be the set of all connected
directed planar trees with $n$ input edges, all of whose vertices have
at most $m$ input edges; we admit also ‘unary’ vertices with only one
input edge.
Let ${\rm Vert}(T)$ denote the set of vertices of
a tree $T\in{\raisebox{7.2pt}{\tiny$m$}\hskip-5.8pt{\mbox{$\tt T$}}}(n)$. For $v\in{\rm Vert}(T)$, let ${\rm val}(v)$ be the number of
input edges of $v$. We assign to each $T\in{\raisebox{7.2pt}{\tiny$m$}\hskip-5.8pt{\mbox{$\tt T$}}}(n)$ a degree
$\deg(T)$ by the formula
$$\deg(T):=\sum_{v\in{\rm Vert}(T)}({\rm val}(v)-2).$$
Then ${\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-7.2pt\underline{\mathcal{F}}}(n)$ can be naturally identified with the ${\bf k}$-linear
span of the set ${\raisebox{7.2pt}{\tiny$m$}\hskip-5.8pt{\mbox{$\tt T$}}}(n)$,
$${\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-7.2pt\underline{\mathcal{F}}}(n)%
={\rm Span}_{{\bf k}}({\raisebox{7.2pt}{\tiny$m$}\hskip-5.8pt{\mbox{$\tt T$}}}%
(n)),$$
(1)
see [8, Section II.1.9]. The operadic
structure is, in this language, defined by grafting the corresponding
trees. This means that, for $T\in{\raisebox{7.2pt}{\tiny$m$}\hskip-5.8pt{\mbox{$\tt T$}}}(a)$ and $S\in{\raisebox{7.2pt}{\tiny$m$}\hskip-5.8pt{\mbox{$\tt T$}}}(b)$,
the operadic $i$-th circle
product [8, Definition II.1.16] $T\circ_{i}S$ is the tree obtained by grafting the tree $S$ at the $i$-th
input edge of $T$. Similarly, for $T\in{\raisebox{7.2pt}{\tiny$m$}\hskip-5.8pt{\mbox{$\tt T$}}}(l)$ and $S_{j}\in{\raisebox{7.2pt}{\tiny$m$}\hskip-5.8pt{\mbox{$\tt T$}}}(k_{j})$, $1\leq j\leq l$, the operadic
composition [8, Definition II.1.4]
$\gamma(T;S_{1},\ldots,S_{l})=T(S_{1},\ldots,S_{l})\in{\raisebox{7.2pt}{\tiny$m%
$}\hskip-5.8pt{\mbox{$\tt T$}}}(k_{1}+\cdots+k_{l})$ is the tree obtained by grafting $S_{j}$ at the
$j$-th input edge of $T$, for each $1\leq j\leq l$.
The operad ${\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-8.0pt\underline{\mathcal{A}}}$ is then defined as ${\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-7.2pt\underline{\mathcal{F}}}/{%
\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-6.0pt\underline{\mathcal{I}}}$, where ${\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-6.0pt\underline{\mathcal{I}}}$
is the operadic ideal generated by the elements
$$\Phi_{n}:=\sum_{\lambda=0}^{n-1}\sum_{k=1}^{n-\lambda}(-1)^{\lambda(k+1)+k}\xi%
_{n-k+1}\circ_{\lambda+1}\xi_{k}\in{\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}%
\hskip-7.2pt\underline{\mathcal{F}}}(n),\ n\leq m.$$
(2)
In the tree language of (1), the generators $\xi_{j}$
are represented by $j$-corollas (denoted by the same symbol)
$\xi_{j}\in{\raisebox{7.2pt}{\tiny$m$}\hskip-5.8pt{\mbox{$\tt T$}}}(j)$; recall that the $j$-corolla is the unique
tree having exactly one vertex. Relations (2) then represent
certain linear combinations of trees, indicated
in Figure 1.
Elements of ${\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-8.0pt\underline{\mathcal{A}}}(n)$ are equivalence classes of linear
combinations of trees from ${\raisebox{7.2pt}{\tiny$m$}\hskip-5.8pt{\mbox{$\tt T$}}}(n)$, modulo the relations of
Figure 1. Surprisingly, there exists, for any $n\geq 1$, a
very natural basis for the ${\bf k}$-vector space ${\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-8.0pt\underline{\mathcal{A}}}(n)$.
Definition 1.1.
We say that a tree $T\in{\raisebox{7.2pt}{\tiny$m$}\hskip-5.8pt{\mbox{$\tt T$}}}(n)$ is admissible if all its unary
vertices (i.e. vertices with just one input edge) are input vertices
of the tree $T$.
Let us denote by ${\hskip 8.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-14.0pt{{\rm Adm}}}(n)\subset{%
\raisebox{7.2pt}{\tiny$m$}\hskip-5.8pt{\mbox{$\tt T$}}}(n)$ the subset consisting of
admissible trees. Then the following proposition holds.
Proposition 1.2.
Admissible $n$-trees form, for $n\geq 1$ a linear basis of ${\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-8.0pt\underline{\mathcal{A}}}(n)$,
$${\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-8.0pt\underline{\mathcal{A}}}(n)%
\cong{\rm Span}_{{\bf k}}({\hskip 8.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-14.0pt%
{{\rm Adm}}}(n)).$$
The proposition follows from the following lemma.
Lemma 1.3.
There exists a unique map
$\Omega:{\raisebox{7.2pt}{\tiny$m$}\hskip-5.8pt{\mbox{$\tt T$}}}(n)\to{\rm Span%
}_{{\bf k}}({\hskip 8.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-14.0pt{{\rm Adm}}}(n))$
such that, for each $T\in{\raisebox{7.2pt}{\tiny$m$}\hskip-5.8pt{\mbox{$\tt T$}}}(n)$,
$$\Omega(T)\equiv T\mbox{ modulo the relations of Figure~{}\ref{picture}}.$$
Proof.
If $T$ is admissible, we put
$\Omega(T):=T$. If $T$ is not admissible, it contains a subtree of
the form $\xi_{1}\circ_{1}\xi_{j}$, with some $1\leq j\leq m$, or, pictorially,
$\bullet$$\bullet$$\cdots$$\xi_{1}$$\xi_{j}$
We may also assume that $j\geq 2$, otherwise $T\equiv 0$ modulo ${\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-6.0pt\underline{\mathcal{I}}}$.
Using the relations of Figure 1, we can replace the above
subtree
by a linear combination of trees of the form
$\cdots$$\cdots$$\cdots$$\bullet$$\bullet$$\xi_{s}$$\xi_{t}$
with $s\geq 2$. This enables us to move,
by local replacements, the vertex $\xi_{1}$
towards the inputs of the tree. Repeating this process sufficiently
many times, we get the requisite presentation $\Omega(T)$. It is
easy to see that this presentation is unique, compare also similar
arguments in the proof of [6, Theorem 2.3].
∎
Proposition 1.2 immediately implies that
$$\dim_{{\bf k}}({\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-8.0pt\underline{%
\mathcal{A}}}(n))=2^{n}\cdot b^{m}_{n},\ \mbox{ for }m\leq 2,\ n\leq 1.$$
(3)
where $b^{m}_{n}$ is the number of all planar rooted reduced (i.e. without
unary vertices) $n$-trees, all of whose vertices have arities $\leq m$.
Clearly, $b^{m}_{n}$ equals the number of cells of the $m$-skeleton of
Stasheff’s associahedron $K_{n}$, compare also Example 2.2. Another
kind of information about the size of the non-$\Sigma$-operad ${\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-8.0pt\underline{\mathcal{A}}}$
is given in:
Proposition 1.4.
Let us consider the generating function
$${\hskip 3.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-7.0pt{\varphi}}(t):=\sum_{n\geq 1%
}\dim_{{\bf k}}({\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-8.0pt\underline{%
\mathcal{A}}}(n))t^{n}.$$
Then ${\hskip 3.0pt\raisebox{7.2pt}{\tiny$1$}\hskip-7.0pt{\varphi}}(t)=2t$ while, for $m\geq 2$, $\varphi(t)={\hskip 3.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-7.0pt{\varphi}}(t)$ solves
the equation
$$\varphi(t)-2t=\varphi^{2}(t)(1+\varphi(t)+\cdots+\varphi^{m-2}(t)).$$
(4)
Proof.
Formula (4) is probably known, though we were not
able to find a reference in the literature. We give a proof that can
be easily modified to obtain the analogous
formula (10) for L($m$)-algebras.
Let $d_{n}^{m}:=\mbox{card}({\hskip 8.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-14.0pt{{%
\rm Adm}}}(n))$. By Proposition 1.2,
$d_{n}^{m}:=\dim_{{\bf k}}({\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-8.0pt%
\underline{\mathcal{A}}}(n))$. Let us denote, just in this proof,
by ${\tt T}_{n}$ the set of all planar reduced $n$-trees all of whose
vertices have at most $m$ input edges. Let $r_{n}:=\mbox{card}({\tt T}_{n})$. We claim that, for any $n\geq 2$,
$$r_{n}=\sum_{i_{1}+i_{2}=n}\hskip-4.0ptr_{i_{1}}r_{i_{2}}+\hskip-4.0pt\sum_{i_{%
1}+i_{2}+i_{3}=n}\hskip-6.0ptr_{i_{1}}r_{i_{2}}r_{i_{3}}+\cdots+\hskip-7.0pt%
\sum_{i_{1}+\cdots+i_{m}=n}\hskip-7.0ptr_{i_{1}}\cdots r_{i_{m}}$$
(5)
and
$$d_{n}^{m}=2^{n}r_{n}.$$
(6)
The first equation follows from the decomposition
$${\tt T}_{n}={\tt T}_{n,2}\sqcup{\tt T}_{n,3}\sqcup\cdots\sqcup{\tt T}_{n,m},$$
where ${\tt T}_{n,i}\subset{\tt T}_{n}$ is the subset of trees whose root
has $i$ input edges. The second equation follows from the
observation that each admissible tree $T\in{\hskip 8.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-14.0pt{{\rm Adm}}}(n)$ can be obtained
from a tree $S\in{\tt T}_{n}$ by grafting terminal unary trees at the
input legs of $S$; there are $2^{n}$ ways to do this. If we denote
$\psi(t):=\sum_{n\geq 1}r_{n}t^{n}$, then (5) implies
$$\psi(t)-t=\psi^{2}(t)(1+\cdots+\psi^{m-2}(t)),$$
while (6) implies that $\varphi(t)=\psi(2t)$. This finishes
the proof.
∎
Example 1.5.
For $m=\infty$, equation (4) can be rewritten as
$$2\varphi^{2}(t)-\varphi(t)(1+2t)+2t=0.$$
For $m=2$, (4)
gives $\varphi(t)-2t=\varphi^{2}(t)$,
therefore
$$\varphi(t)=\frac{1-\sqrt{(1-8t)}}{2}.$$
A little exercise in Taylor expansions results in
$$d^{2}_{n}=\frac{2^{2n-1}(2n-3)!!}{n!}\hskip 10.0pt\mbox{ for }n\geq 2,$$
where as usual
$$(2n-3)!!=1\cdot 3\cdot 5\cdots(2n-3).$$
2 Free A($m$)-algebras
The following theorem relies on the notation introduced in the previous
section. Recall namely that
${\hskip 8.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-14.0pt{{\rm Adm}}}(n)$ denotes the set of admissible $n$-trees in
the sense of Definition 1.1, ${\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-8.0pt\underline{\mathcal{A}}}(n)%
\cong{\rm Span}_{{\bf k}}({\hskip 8.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-14.0pt%
{{\rm Adm}}}(n))$ and
$\Omega$ is the map of Lemma 1.3.
Recall also that we denoted by $\xi_{j}$ the $j$-corolla.
If $T_{1},\ldots,T_{j}$ are trees, then
$\xi_{j}(T_{1},\ldots,T_{j})$ is the tree obtained by
grafting the tree
$T_{i}$ at the $i$-th input edge of the tree $\xi_{j}$, $1\leq i\leq j$.
Theorem 2.1.
Let $X$ be a graded vector space. Then the free A($m$)-algebra $\ {\raisebox{7.2pt}{\tiny$m$}\hskip-7.0pt{\mathfrak{A}}}(X)$
on $X$ can be described as ${\raisebox{7.2pt}{\tiny$m$}\hskip-7.0pt{\mathfrak{A}}}(X)=\bigoplus_{n\geq 1}{%
\raisebox{7.2pt}{\tiny$m$}\hskip-7.0pt{\mathfrak{A}}}{}^{n}(X)$, with
$${\raisebox{7.2pt}{\tiny$m$}\hskip-7.0pt{\mathfrak{A}}}{}^{n}(X):={\hskip 2.0pt%
\raisebox{7.2pt}{\tiny$m$}\hskip-8.0pt\underline{\mathcal{A}}}(n)\otimes X^{%
\otimes n}.$$
The structure operations $\mu_{1},\ldots,\mu_{m}$ are defined as follows. Let
$\vec{v}_{j}\in X^{\otimes k_{j}}$ and $T_{j}\in{\hskip 8.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-14.0pt{{\rm Adm}}}(k_{j})$, $1\leq j\leq k$. Then
$$\mu_{k}(T_{1}\otimes\vec{v}_{1},\ldots,T_{k}\otimes\vec{v}_{k}):=\Omega(\xi_{k%
}(T_{1},\ldots,T_{k}))\otimes(\vec{v}_{1}\otimes\cdots\otimes\vec{v}_{k}).$$
Proof.
It follows from general
theory [8, Section II.1.4] that the free
$\underline{\mathcal{P}}$-algebra $F_{\underline{\mathcal{P}}}(X)$
on a graded vector space $X$, where $\underline{\mathcal{P}}$ is a
non-$\Sigma$ operad, is given by $F_{\underline{\mathcal{P}}}(X)=\bigoplus_{n\geq 1}F_{\underline{\mathcal{P}}}^%
{n}(X)$, with
$F_{\underline{\mathcal{P}}}^{n}(X):=\underline{\mathcal{P}}(n)\otimes X^{%
\otimes n}$, with the algebra structure induced
from the operad structure of $\underline{\mathcal{P}}$. Our theorem
is then obtained by taking $\underline{\mathcal{P}}:={\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-8.0pt%
\underline{\mathcal{A}}}$ and
applying the results on the structure of this operad proved in the
previous section.
∎
Observe that the free algebra ${\raisebox{7.2pt}{\tiny$m$}\hskip-7.0pt{\mathfrak{A}}}(X)$ is bigraded. The internal
grading is the grading of the underlying vector space, while
the external grading is given by the decomposition
$${\raisebox{7.2pt}{\tiny$m$}\hskip-7.0pt{\mathfrak{A}}}(X)=\bigoplus_{n\geq 1}{%
\raisebox{7.2pt}{\tiny$m$}\hskip-7.0pt{\mathfrak{A}}}{}^{n}(X).$$
Because all main structures considered in this note (vector
spaces, A($m$)-algebras, L($m$)-algebras, etc.) have an implicit internal
grading, we will usually mean by saying that an object is graded the
presence of an external grading. We believe that the actual meaning
will always be clear from the context.
Example 2.2.
There exists a simple way to encode elements of
${\hskip 8.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-14.0pt{{\rm Adm}}}(n)$. Let ${{\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-8.0ptB}}(n)$ be the set of all meaningful bracketings
of strings $\epsilon_{1}\epsilon_{2}\cdots\epsilon_{n}$, where
$\epsilon_{i}\in\{\circ,\bullet\}$ and no pair of brackets
encompasses more than $m$ terms. A moment’s reflection shows that
${\hskip 8.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-14.0pt{{\rm Adm}}}(n)\cong{{%
\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-8.0ptB}}(n)$, $n\geq 1$. Thus
$$\displaystyle{ \raisebox{7.2pt}{\tiny$m$}\hskip-14.0pt{{\rm Adm}}}(1)$$
$$\displaystyle=$$
$$\displaystyle\{\circ,\bullet\},\ m\geq 1,$$
$$\displaystyle{ \raisebox{7.2pt}{\tiny$1$}\hskip-14.0pt{{\rm Adm}}}(2)$$
$$\displaystyle=$$
$$\displaystyle\emptyset,\ {\hskip 8.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-14.0pt{%
{\rm Adm}}}(2)=\{(\circ\circ),(\circ\bullet),(\bullet\circ),(\bullet\bullet)\}%
,\ m\geq 2,$$
$$\displaystyle{ \raisebox{7.2pt}{\tiny$1$}\hskip-14.0pt{{\rm Adm}}}(3)$$
$$\displaystyle=$$
$$\displaystyle\emptyset,$$
$$\displaystyle{ \raisebox{7.2pt}{\tiny$2$}\hskip-14.0pt{{\rm Adm}}}(3)$$
$$\displaystyle=$$
$$\displaystyle\{((\circ\circ)\circ),((\bullet\circ)\circ),((\bullet\bullet)%
\circ),\ldots,(\circ(\circ\circ)),(\bullet(\circ\circ)),(\bullet(\bullet\circ)%
),\ldots\},$$
$$\displaystyle{ \raisebox{7.2pt}{\tiny$m$}\hskip-14.0pt{{\rm Adm}}}(3)$$
$$\displaystyle=$$
$$\displaystyle {\hskip 8.0pt\raisebox{7.2pt}{\tiny$2$}\hskip-14.0pt{{\rm Adm}}}%
(3)\cup\{(\circ\circ\circ),(\bullet\circ\circ),(\bullet\bullet\circ),\ldots\},%
\ m\geq 3,\mbox{ etc.}$$
We conclude that ${\raisebox{7.2pt}{\tiny$m$}\hskip-7.0pt{\mathfrak{A}}}(X)$ is spanned by all meaningful bracketings of
strings of “variables” $x$ and $\partial x$, $\deg(\partial x)=\deg(x)-1$,
$x\in X$, with no pair of brackets encompassing more than $m$ terms.
Thus ${\hskip 2.0pt\raisebox{7.2pt}{\tiny$1$}\hskip-5.0pt{\mathfrak{A}}}(X)={\hskip 2%
.0pt\raisebox{7.2pt}{\tiny$1$}\hskip-5.0pt{\mathfrak{A}}}{}^{1}(X)$ is the free differential
complex generated by $X$, ${\hskip 2.0pt\raisebox{7.2pt}{\tiny$1$}\hskip-5.0pt{\mathfrak{A}}}(X)\cong X%
\oplus\partial X$. Similarly,
$$\displaystyle{\raisebox{7.2pt}{\tiny$m$}\hskip-7.0pt{\mathfrak{A}}}{}^{2}(X)$$
$$\displaystyle\cong$$
$$\displaystyle(X\otimes X)\oplus(\partial X\otimes X)\oplus(X\otimes\partial X)%
\oplus(\partial X\otimes\partial X),\ \mbox{for $m\geq 2$,}$$
$$\displaystyle{ \raisebox{7.2pt}{\tiny$2$}\hskip-5.0pt{\mathfrak{A}}}{}^{3}(X)$$
$$\displaystyle\cong$$
$$\displaystyle((X\otimes X)\otimes X)\oplus((\partial X\otimes X)\otimes X)%
\oplus((X\otimes\partial X)\otimes X)$$
$$\displaystyle \oplus\ ((X\otimes X)\otimes\partial X)\oplus((\partial X%
\otimes\partial X)\otimes X)\oplus\ ((\partial X\otimes X)\otimes\partial X)$$
$$\displaystyle \oplus\ ((X\otimes\partial X)\otimes\partial X)\oplus((%
\partial X\otimes\partial X)\otimes\partial X)\oplus(X\otimes(X\otimes X))$$
$$\displaystyle \oplus\ (\partial X\otimes(X\otimes X))\oplus(X\otimes(%
\partial X\otimes X))\oplus(X\otimes(X\otimes\partial X))$$
$$\displaystyle \oplus\ (\partial X\otimes(\partial X\otimes X))\oplus(%
\partial X\otimes(X\otimes\partial X))\oplus(X\otimes(\partial X\otimes%
\partial X))$$
$$\displaystyle \oplus\ (\partial X\otimes(\partial X\otimes\partial X)),$$
$$\displaystyle{{\raisebox{7.2pt}{\tiny$m$}\hskip-7.0pt{\mathfrak{A}}}}{}^{3}(X)$$
$$\displaystyle\cong$$
$$\displaystyle{ \raisebox{7.2pt}{\tiny$2$}\hskip-5.0pt{\mathfrak{A}}}{}^{3}(X)%
\oplus(X{\otimes}X{\otimes}X)\oplus(\partial X{\otimes}X{\otimes}X)\oplus(X{%
\otimes}\partial X{\otimes}X)\oplus(X{\otimes}X{\otimes}\partial X)$$
$$\displaystyle \oplus\ (\partial X{\otimes}\partial X{\otimes}X)\oplus(%
\partial X{\otimes}X{\otimes}\partial X)\oplus(X{\otimes}\partial X{\otimes}%
\partial X)$$
$$\displaystyle \oplus\ (\partial X{\otimes}\partial X{\otimes}\partial X),%
\ \mbox{for $m\geq 3$, etc.}$$
We see that
$$\dim(\hskip 2.0pt{\raisebox{7.2pt}{\tiny$m$}\hskip-7.0pt{\mathfrak{A}}}{}^{n}(%
X))=(2\dim(X))^{n}\cdot b^{m}_{n},\ n\geq 1,\ m\geq 2,$$
where $b^{m}_{n}$ is as in (3).
Unital A($m$)-algebras.
Recall [5, page 148] that an A($m$)-algebra
$A=(A,\{\mu_{n}\})$ is unital if there exists an element $1=1_{A}\in A$ such that $\mu_{2}(a,1_{A})=\mu_{2}(1_{A},a)=a$, for all $a\in A$, and that $\mu_{n}(a_{1},\dots,a_{i-1},1_{A},a_{i+1},\dots,a_{n})=0$, for all $n\not=2$ and $1\leq i\leq n$. The free unital A($m$)-algebra
can be constructed as ${\rm Span}_{{\bf k}}(1)\oplus{\raisebox{7.2pt}{\tiny$m$}\hskip-7.0pt{\mathfrak%
{A}}}(X)$, with the structure
operations defined in an obvious way.
Important modifications. One may as well consider, for any $n<m$, the forgetful functor $\Box:\mbox{{\tt A}($m$)}\to\mbox{{\tt A}($n$)}$.
This functor has a left adjoint ${\raisebox{7.2pt}{\tiny$m\hskip-2.27622pt:\hskip-1.707165ptn$}\hskip-9.0pt{%
\mathfrak{A}}\hskip 2.0pt}:\mbox{{\tt A}($n$)}\to\mbox{{\tt A}($m$)}$ which can be described as follows. Given an A($n$)-algebra
$(V,m_{1},m_{2},\ldots,m_{n})$, consider the free A($m$)-algebra ${\raisebox{7.2pt}{\tiny$m$}\hskip-7.0pt{\mathfrak{A}}}(V)$ on
the graded vector space $V$, and let $\mu_{1},\ldots,\mu_{m}$ be the
structure operations of ${\raisebox{7.2pt}{\tiny$m$}\hskip-7.0pt{\mathfrak{A}}}(V)$. Then
$${\raisebox{7.2pt}{\tiny$m\hskip-2.27622pt:\hskip-1.707165ptn$}\hskip-9.0pt{%
\mathfrak{A}}\hskip 2.0pt}(V,m_{1},m_{2},\ldots,m_{n})\cong{\raisebox{7.2pt}{%
\tiny$m$}\hskip-7.0pt{\mathfrak{A}}}(V)/(\mu_{i}=m_{i},\ i\leq n).$$
A very important special case is $n=1$ when we obtain a left adjoint
${\raisebox{7.2pt}{\tiny$m\hskip-2.27622pt:\hskip-1.707165pt1$}\hskip-9.0pt{%
\mathfrak{A}}\hskip 2.0pt}:\mbox{{\tt dgVect}}\to\mbox{{\tt A}($m$)}$ to the forgetful
functor $\Box:\mbox{{\tt A}($m$)}\to\mbox{{\tt dgVect}}$ to the category
of differential graded vector spaces. One can easily verify
that
$${\raisebox{7.2pt}{\tiny$m\hskip-2.27622pt:\hskip-1.707165pt1$}\hskip-9.0pt{%
\mathfrak{A}}\hskip 2.0pt}(V,\partial)\cong\bigoplus_{n\geq 1}({\hskip 2.0pt%
\raisebox{7.2pt}{\tiny$m$}\hskip-8.0pt\underline{\mathcal{A}}}(n),\partial)%
\otimes(V,\partial)^{\otimes n},$$
(7)
where $({\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-8.0pt\underline{\mathcal{A}}},\partial)$ is now the non-$\Sigma$ operad describing
A($m$)-algebras as algebras in the monoidal category of differential vector spaces. Let us
recall [7, page 1493] that ${\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-8.0pt\underline{\mathcal{A}}}=%
\underline{\Gamma}(\xi_{2},\ldots,\xi_{m})$ with the differential
$$d(\xi_{n}):=\sum(-1)^{(b+1)(i+1)+b}\cdot\xi_{a}\circ_{i}\xi_{b},$$
where the summation runs over all $a,b\geq 2$ with $a+b=n+1$ and $1\leq i\leq a$.
Formula (7) is in a perfect harmony
with [8, Definition II.1.24] describing
free operad algebras in a general monoidal category.
Example 2.3.
Clearly ${\raisebox{7.2pt}{\tiny$m\hskip-2.27622pt:\hskip-1.707165pt1$}\hskip-9.0pt{%
\mathfrak{A}}\hskip 2.0pt}{}^{1}(V,\partial)\cong(V,\partial)$, and,
for $m\geq 2$, ${\raisebox{7.2pt}{\tiny$m\hskip-2.27622pt:\hskip-1.707165pt1$}\hskip-9.0pt{%
\mathfrak{A}}\hskip 2.0pt}{}^{2}(V,\partial)\cong(V\otimes V)$ with the
induced differential. As graded vector spaces
$$\displaystyle{\raisebox{7.2pt}{\tiny$2\hskip-2.27622pt:\hskip-1.707165pt1$}%
\hskip-9.0pt{\mathfrak{A}}\hskip 2.0pt}{}^{3}(V,\partial)$$
$$\displaystyle\cong$$
$$\displaystyle((V\otimes V)\otimes V)\oplus(V\otimes(V\otimes V)),$$
$$\displaystyle{\raisebox{7.2pt}{\tiny$m\hskip-2.27622pt:\hskip-1.707165pt1$}%
\hskip-9.0pt{\mathfrak{A}}\hskip 2.0pt}{}^{3}(V,\partial)$$
$$\displaystyle\cong$$
$$\displaystyle{\raisebox{7.2pt}{\tiny$2\hskip-2.27622pt:\hskip-1.707165pt1$}%
\hskip-9.0pt{\mathfrak{A}}\hskip 2.0pt}{}^{3}(V)\oplus(V{\otimes}V{\otimes}V),%
\mbox{ for $m\geq 3$,}$$
$$\displaystyle{\raisebox{7.2pt}{\tiny$2\hskip-2.27622pt:\hskip-1.707165pt1$}%
\hskip-9.0pt{\mathfrak{A}}\hskip 2.0pt}{}^{4}(V,\partial)$$
$$\displaystyle\cong$$
$$\displaystyle(((V{\otimes}V){\otimes}V){\otimes}V)\oplus(V{\otimes}(V{\otimes}%
(V{\otimes}V)))\oplus((V{\otimes}V){\otimes}(V{\otimes}V))$$
$$\displaystyle \oplus\ ((V{\otimes}(V{\otimes}V)){\otimes}V)\oplus(V{%
\otimes}((V{\otimes}V){\otimes}V)),$$
$$\displaystyle{\raisebox{7.2pt}{\tiny$3\hskip-2.27622pt:\hskip-1.707165pt1$}%
\hskip-9.0pt{\mathfrak{A}}\hskip 2.0pt}{}^{4}(V,\partial)$$
$$\displaystyle\cong$$
$$\displaystyle{\raisebox{7.2pt}{\tiny$2\hskip-2.27622pt:\hskip-1.707165pt1$}%
\hskip-9.0pt{\mathfrak{A}}\hskip 2.0pt}{}^{4}(V,\partial)\oplus((V{\otimes}V{%
\otimes}V){\otimes}V)\oplus(V{\otimes}(V{\otimes}V{\otimes}V))\oplus((V{%
\otimes}V){\otimes}V{\otimes}V)$$
$$\displaystyle \oplus\ (V{\otimes}(V{\otimes}V){\otimes}V)\oplus(V{\otimes%
}V{\otimes}(V{\otimes}V)),$$
$$\displaystyle{\raisebox{7.2pt}{\tiny$m\hskip-2.27622pt:\hskip-1.707165pt1$}%
\hskip-9.0pt{\mathfrak{A}}\hskip 2.0pt}{}^{4}(V,\partial)$$
$$\displaystyle\cong$$
$$\displaystyle{\raisebox{7.2pt}{\tiny$3\hskip-2.27622pt:\hskip-1.707165pt1$}%
\hskip-9.0pt{\mathfrak{A}}\hskip 2.0pt}{}^{4}(V,\partial)\oplus(V{\otimes}V{%
\otimes}V{\otimes}V{\otimes}V)\mbox{ for $m\geq 4$, etc.}$$
The action of the differential in ${\raisebox{7.2pt}{\tiny$m\hskip-2.27622pt:\hskip-1.707165pt1$}\hskip-9.0pt{%
\mathfrak{A}}\hskip 2.0pt}(V,\partial)$ can be
easily read off from the axioms of A($m$)-algebras, for example
$$\displaystyle\partial(x{\otimes}y{\otimes}z)=((x{\otimes}y){\otimes}z)-(x{%
\otimes}(y{\otimes}z))\hskip 100.0pt$$
$$\displaystyle -(\partial x{\otimes}y{\otimes}z)-(%
-1)^{|a|}(x{\otimes}\partial y{\otimes}z)-(-1)^{|a|+|b|}(x{\otimes}y{\otimes}%
\partial z),$$
where $(x{\otimes}y{\otimes}z)\in{\raisebox{7.2pt}{\tiny$m\hskip-2.27622pt:\hskip-1.7%
07165pt1$}\hskip-9.0pt{\mathfrak{A}}\hskip 2.0pt}{}^{3}(V)$, $m\geq 3$. The above
formula illustrates a surprising fact that the differential in ${\raisebox{7.2pt}{\tiny$m\hskip-2.27622pt:\hskip-1.707165pt1$}\hskip-9.0pt{%
\mathfrak{A}}\hskip 2.0pt}(V,\partial)$ is nontrivial even if the differential $\partial$ on $V$ is
zero, because in this case still
$$\partial(x{\otimes}y{\otimes}z)=((x{\otimes}y){\otimes}z)-(x{\otimes}(y{%
\otimes}z))\ !$$
The dimension of the homogeneous components of ${\raisebox{7.2pt}{\tiny$m\hskip-2.27622pt:\hskip-1.707165pt1$}\hskip-9.0pt{%
\mathfrak{A}}\hskip 2.0pt}(V,\partial V)$ clearly equals
$$\dim{{\raisebox{7.2pt}{\tiny$m\hskip-2.27622pt:\hskip-1.707165pt1$}\hskip-9.0%
pt{\mathfrak{A}}\hskip 2.0pt}{}^{n}(V,\partial)}=(\dim(V))^{n}\cdot b^{m}_{n},%
\mbox{ for }n\geq 1,\ m\geq 2,$$
where the integers $b^{m}_{n}$ are as in (3).
3 Free strongly homotopy Lie algebras
In this section we indicate how to modify previous results to strongly
homotopy Lie algebras. Recall ([4]
or [3, Definition 2.1]) that an L($m$)-structure on a graded vector space $L$ is a system $\{l_{k};\ 1\leq k\leq m,\ k<\infty\}$ of degree $k-2$ linear maps $l_{k}:\otimes^{k}L\to L$ which are antisymmetric in the sense that
$$l_{k}(x_{\sigma(1)},\dots,x_{\sigma(k)})=\chi(\sigma)l_{k}(x_{1},\dots,x_{n})$$
(8)
for all $\sigma\in\Sigma_{n}$ and $x_{1},\dots,x_{n}\in L$.
Here $\chi(\sigma):={\rm sgn}(\sigma)\cdot\epsilon(\sigma)$, where
$\epsilon(\sigma)$ is the Koszul sign of the permutation $\sigma$,
see [3, page 2148] for details.
Moreover, the
following generalized form of the Jacobi identity is supposed to be
satisfied for any $n\leq m$:
$$\sum_{i+j=n+1}\sum_{\sigma}\chi(\sigma)(-1)^{i(j-1)}l_{j}(l_{i}(x_{\sigma(1)},%
\dots,x_{\sigma(i)}),x_{\sigma(i+1)},\dots,x_{\sigma(n)})=0,$$
(9)
where the summation is taken over all $(i,n-i)$-unshuffles with
$i\geq 1$.
The operad ${\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-7.0pt{\mathcal{L}}}=\{{\hskip 2.%
0pt\raisebox{7.2pt}{\tiny$m$}\hskip-7.0pt{\mathcal{L}}}(n)\}_{n\geq 1}$ describing L($m$)-algebras as algebras in the category of graded vector spaces can be
constructed as follows. Let ${\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-7.2pt{\mathcal{F}}}=\{{\hskip 2.%
0pt\raisebox{7.2pt}{\tiny$m$}\hskip-7.2pt{\mathcal{F}}}(n)\}_{n\geq 1}$ be the
free graded $\Sigma$-operad $\Gamma(\zeta_{1},\zeta_{2},\ldots,\zeta_{m})$
generated by the operations $\zeta_{i}$, $l\leq i\leq m$, where each
$\zeta_{i}$ has $i$ inputs, degree $i-2$ and spans the signum
representation of the symmetric group $\Sigma_{i}$. Using trees, this
operad can be described as follows. Let ${\raisebox{7.2pt}{\tiny$m$}\hskip-5.8pt{\mbox{$\tt T$}}}(n)$ be now the set of
all directed labeled (non-planar, abstract) trees with $n$-input
edges, all of whose vertices have at most $m$ input edges. We again
admit also ‘unary’ vertices with only one input edge and ‘labeled’
means that the input legs are labeled by natural numbers $1,\ldots,n$.
Then, as before,
$${\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-7.2pt{\mathcal{F}}}(n)\cong{\rm
Span%
}_{{\bf k}}({\raisebox{7.2pt}{\tiny$m$}\hskip-5.8pt{\mbox{$\tt T$}}}(n)).$$
We put ${\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-7.0pt{\mathcal{L}}}:={\hskip 2.0%
pt\raisebox{7.2pt}{\tiny$m$}\hskip-7.2pt{\mathcal{F}}}/{\hskip 2.0pt\raisebox{%
7.2pt}{\tiny$m$}\hskip-6.0pt{\mathcal{I}}}$, where the operadic ideal ${\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-6.0pt{\mathcal{I}}}$ is now
generated by relations whose pictorial presentation is indicated
in Figure 2.
By similar arguments as in the previous sections we easily infer that
there is a natural basis of the ${\bf k}$-vector space ${\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-7.0pt{\mathcal{L}}}(n)$, formed
by the set ${\hskip 8.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-14.0pt{{\rm Adm}}}(n)$ of admissible $n$-trees, whose definition is
formally the same as in Definition 1.1, only removing everywhere
the adjective “planar.” The size of the operad ${\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-7.0pt{\mathcal{L}}}$ is
described in the following proposition whose proof is similar to that
of Proposition 1.4.
Proposition 3.1.
The generating function
$\vartheta(t)={\buildrel\mbox{\scriptsize$m$}\over{\vartheta}}(t):=\sum_{n\geq 1%
}\frac{1}{n!}\dim_{{\bf k}}({\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-7.0%
pt{\mathcal{L}}}(n))t^{n}$
satisfies
the equation
$$\vartheta(t)-2t=\vartheta^{2}(t)\left(\frac{1}{2!}+\frac{1}{3!}\vartheta(t)+%
\cdots+\frac{1}{m!}\vartheta^{m-2}(t)\right).$$
(10)
Example 3.2.
For $m=\infty$, equation (10) becomes
$2\vartheta(t)-2t-1=\exp(\vartheta(t))$. For $m=2$,
equation (10) says that $\vartheta(t)=1-\sqrt{1-4t^{2}}$, therefore
$$\dim_{{\bf k}}({\buildrel\mbox{\scriptsize$2$}\over{\mathcal{L}}}(n))=2^{n}%
\cdot(2n-3)!!,\ n\geq 1.$$
As in Lemma 1.3, we have a map $\Omega:{\raisebox{7.2pt}{\tiny$m$}\hskip-5.8pt{\mbox{$\tt T$}}}(n)\to{\rm Span%
}_{{\bf k}}({\hskip 8.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-14.0pt{{\rm Adm}}}(n))$ with the property that $\Omega(T)\equiv T$ modulo
relations of Figure 2. Free L($m$)-algebras are then
described as follows.
Theorem 3.3.
Let $Y$ be a graded vector space. The free L($m$)-algebra ${\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-7.0pt{\mathfrak{L}}}(Y)$
generated by $Y$ can defined as ${\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-7.0pt{\mathfrak{L}}}(Y)=%
\bigoplus_{n\geq 1}{\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-7.0pt{%
\mathfrak{L}}}{}^{n}(Y)$, where
$${\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-7.0pt{\mathfrak{L}}}{}^{n}(Y):={%
\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-7.0pt{\mathcal{L}}}(n)\otimes_{%
\Sigma_{n}}Y^{\otimes n}.$$
The structure operations $l_{1},\ldots,l_{m}$ are given by the
following rule.
Let $\vec{v}_{j}\in Y^{\otimes k_{j}}$ and $T_{j}\in{\hskip 8.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-14.0pt{{\rm Adm}}}(k_{j})$, $1\leq j\leq k$. Then
$$l_{k}(T_{1}\otimes_{\Sigma_{k_{1}}}\vec{v}_{1},\ldots,T_{k}\otimes_{\Sigma_{k_%
{1}}}\vec{v}_{k}):=\Omega(\xi_{k}(T_{1},\ldots,T_{k}))\otimes_{\Sigma_{k_{1}+%
\cdots+k_{k}}}(\vec{v}_{1}\otimes\cdots\otimes\vec{v}_{k}).$$
Remark 3.4.
In exactly the same fashion as for A($m$)-algebras one may define
left adjoints ${\hskip 2.0pt\raisebox{7.2pt}{\scriptsize$m\hskip-2.27622pt:\hskip-1.707165ptn%
$}\hskip-9.0pt{\mathcal{L}}}:\mbox{{\tt L}($n$)}\to\mbox{{\tt L}($m$)}$ to
the forgetful functor $\Box:\mbox{{\tt L}($m$)}\to\mbox{{\tt L}($n$)}$, $n<m$, as
$${\hskip 2.0pt\raisebox{7.2pt}{\scriptsize$m\hskip-2.27622pt:\hskip-1.707165ptn%
$}\hskip-9.0pt{\mathcal{L}}}\ (L,\lambda_{1},\lambda_{2},\ldots,\lambda_{n})={%
\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-7.0pt{\mathfrak{L}}}(L)/(\lambda_%
{i}=l_{i},\ i\leq n),$$
where $l_{i}$ are the structure operations of ${\hskip 2.0pt\raisebox{7.2pt}{\tiny$m$}\hskip-7.0pt{\mathfrak{L}}}(L)$.
4 Reflections on the PBW theorem
As in the case of ordinary Lie algebras, there exists the
symmetrization functor from the category A($m$) of A($m$)-algebras to the
category L($m$) of L($m$)-algebras. More precisely, we proved
in [3, Theorem 3.1]:
Theorem 4.1.
Any A($m$)-structure $\{\mu_{n}:\bigotimes^{n}V\longrightarrow V\}$
on a differential graded vector space $V$ induces an L($m$)-structure
$\{l_{n}:\bigotimes^{n}V\longrightarrow V\}$ on the same
differential graded vector space, with
$$l_{n}(v_{1}\otimes\dots\otimes v_{n}):=\sum_{\sigma\in{\mathcal{S}}_{n}}\chi(%
\sigma)\mu_{n}(v_{\sigma(1)}\otimes\dots\otimes v_{\sigma(n)}),\ 2\leq n\leq m.$$
This correspondence defines a functor (the symmetrization)
$(-)_{L}:{\tt A}(m)\longrightarrow{\tt L}(m)$.
In [3, Theorem 3.3], we also proved that:
Theorem 4.2.
There exists a functor $\mathcal{U}_{m}:{\tt L}(m)\longrightarrow{\tt A}(m)$ that is left adjoint to the functor $(-)_{L}:{\tt A}(m)\longrightarrow{\tt L}(m)$.
The algebra $\mathcal{U}_{m}(L)$ of the previous theorem is called
the universal enveloping A($m$)-algebra
of the L($m$)-algebra $L$.
We gave, in [3, page 2154],
the following explicit construction of
${\mathcal{U}}_{m}(L)$.
Start with the free unital A($m$)-algebra ${\raisebox{7.2pt}{\tiny$m$}\hskip-7.0pt{\mathfrak{A}}}(L)$ generated by the
underlying vector space $L$ of the L($m$)-algebra $L=(L,\{l_{n}\})$ and let
$\{\mu_{n}\}$ be the A($m$)-structure maps of ${\raisebox{7.2pt}{\tiny$m$}\hskip-7.0pt{\mathfrak{A}}}(L)$. Let
$I$ denote the A($m$)-ideal in ${\raisebox{7.2pt}{\tiny$m$}\hskip-7.0pt{\mathfrak{A}}}(L)$ generated by the relations
$$\sum_{\sigma\in S_{n}}\chi(\sigma)\mu_{n}(\xi_{\sigma(1)},\dots,\xi_{\sigma(n)%
})=l_{n}(\xi_{1},\dots,\xi_{n}),\mbox{ for }\xi_{1},\dots,\xi_{n}\in L,\ n\leq
m.$$
Then we put $\mathcal{U}_{m}(L):={\raisebox{7.2pt}{\tiny$m$}\hskip-7.0pt{\mathfrak{A}}}(L)/I$. This universal
enveloping A($m$)-algebra is equipped with the canonical map (in fact, an
inclusion) $\iota:L\to\mathcal{U}_{m}(L)$.
Remark 4.3.
Hinich and Schechtman introduced
in [2], for any L($m$)-algebra
$L$ (they did it for $m=\infty$, but the general case is an
obvious modification), an associative algebra (which they
also called the universal enveloping algebra), having the property
that the category of modules (in a suitable sense, see for
example [3, Definition 5.1]) over an
L($m$)-algebra $L$ is equivalent to the category of left modules over
the universal enveloping associative algebra of $L$.
It is unclear whether there exists a functor ${\mathcal{G}}:\mbox{{\tt L}($m$)}\to\mbox{{\tt A}($m$)}$ such that the category of
modules over an L($m$)-algebra $L$ would be naturally equivalent to
the category of left modules, in the sense
of [5, page 157], over the A($m$)-algebra $\mathcal{G}(L)$. Our universal enveloping A($m$)-algebra functor whose
definition we recalled above probably does not have this property.
While it can be easily shown that each left module over the
A($m$)-algebra $\mathcal{U}_{m}(L)$ naturally defines a module over
$L$, we doubt that each $L$-module uniquely extends into a left
module over $\mathcal{U}_{m}(L)$. The classical universal enveloping
associative algebra of a strict Lie algebra is constructed as a
left adjoint to the symmetrization functor. By a miracle, this
functor also happens to describe the category of $L$-modules, but this
mystic phenomenon does not take place in the category of
L($m$)-algebras.
Moreover, there is no reason to believe that such a functor
$\mathcal{G}$ exits. The category of modules over a given
L($m$)-algebra $L$ is a well-behaved abelian category, therefore it
should be the category of modules over an associative
algebra, which turns out to be the one constructed by Hinich and
Schechtman. On the other hand, we do not see why this abelian
category should be at the same time the category of left modules
over some A($m$)-algebra. Let us close this remark with
Problem 4.4.
Let $L$ be an L($m$)-algebra. Does there exist an A($m$)-algebra $A$ with the property that the category of modules
over $L$ is equivalent to the category of left modules over $A$?
The simplest form of the P(oincaré)-B(irkhoff)-W(itt) theorem for an
ordinary Lie algebra $L=(L,[-,-])$ says that the associated graded
algebra $G^{*}(L)$ of the universal enveloping algebra ${\mathcal{U}}(L)$
of $L$ is isomorphic to the free commutative associative algebra
$S^{*}(L)$ (the polynomial ring) on the vector space $L$. More
precisely, recall that
$${\mathcal{U}}(L)=T(L)/(x{\otimes}y-y{\otimes}x=[x,y],\ x,y\in L),$$
where $T(L)$ is the free associative algebra (tensor algebra) on the
vector space $L$. The algebra ${\mathcal{U}}(L)$ is filtered, the ascending
filtration being given by ${\mathcal{U}}_{p}(L):=$ vector space generated by
linear combinations of elements which can be presented as a product of
$\ \leq p$ elements in the augmentation ideal of $T(L)$. Clearly
${\mathcal{U}}_{0}(L)={\bf k}$ and ${\mathcal{U}}_{1}(L)={\bf k}\oplus L$. If $G^{*}(L)$ denotes
the associated graded algebra, then one proves that the natural map
$L=G^{1}(L)\to G^{*}(L)$ is a monomorphism and $G^{*}(L)$ is generated by
$L$. It is immediate to see that $xy=yx$, for arbitrary $x,y\in L\subset G^{*}(L)$. We formulate the following trivial observation.
Observation 4.5.
Suppose that an associative algebra $A$ admits a set of mutually
commutative generators. Then $A$ is commutative.
From this observation we infer that $G^{*}(L)$ is commutative,
therefore there exists, by the universal property, a commutative
algebra map $S^{*}(L)\to G^{*}(L)$, induced by the inclusion $L\subset G^{*}(L)$. The PBW222not to be mistaken with
Praha-Berlin-Warszawa cycling race theorem says that this map is an
isomorphism of graded commutative associative algebras.
Let us try to guess which form the PBW theorem for the universal
enveloping A($m$)-algebra ${\mathcal{U}}_{m}(L)$ of an L($m$)-algebra $L$ may possibly
have. The classical PBW theorem compares the associated graded of
${\mathcal{U}}(L)$ to a commutative associative algebra. Commutative
associative algebras occur in the symbolic exact sequence (which may
be found in [1, page 228])
$$\mbox{Commutative associative algebras}\subset\mbox{Associative algebras}%
\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}\mbox{Lie algebras.}$$
Here the ‘projection’ $\pi$ is given by the symmetrization of the
associative product. Commutative associative
algebras are also the quadratic duals of Lie algebras
(and vice versa) [1, Theorem 2.1.11],
but this coincidence is misleading. Summing up,
the classical PBW theorem says that the associated graded $G^{*}(L)$
is the free algebra in the kernel of the symmetrization map $\pi$.
One would naturally expect something similar
also in the L($m$)-algebra case. The kernel
of the symmetrization map $(-)_{L}$ of
Theorem 4.1 clearly
consists of A($m$)-algebras
$A=(A,\{\mu_{n}\})$ such that
$$\sum_{\sigma\in S_{n}}\chi(\sigma)\mu_{n}(a_{\sigma(1)},\ldots,a_{\sigma(n)})=%
0,\ n\geq 2,\ a_{1},\dots,a_{n}\in A.$$
(11)
Therefore one tends to believe that A($m$)-algebras enjoying this form of
symmetry are analogs of the polynomial algebra $S^{*}(L)$ from the
classical PBW theorem. This is, however, not exactly so.
To see why, let $L=(L,\{l_{n}\})$ be an L($m$)-algebra and define
inductively an ascending filtration of the universal enveloping
A($m$)-algebra ${\mathcal{U}}_{m}(L)$ by ${\mathcal{U}}_{m,0}(L):={\bf k}$, ${\mathcal{U}}_{m,1}(L):={\mathcal{U}}_{m,0}\cup\iota(L)$ and ${\mathcal{U}}_{m,p}(L)$ being the vector space
generated by elements of the form $\mu_{k}(x_{1},\dots,x_{k})$, $k\geq 2$, $x_{i}\in{\mathcal{U}}_{m,p_{i}}(L)$ for $1\leq i\leq k$ and $\sum_{1}^{k}p_{i}\leq p$.
The associated graded A($m$)-algebra $G^{*}_{m}(L)$ is defined by
$$G^{*}_{m}(L)=\bigoplus_{q\geq 0}G^{q}_{m}(L),\ G^{q}_{m}(L):={\mathcal{U}}_{m,%
q}(L)/{\mathcal{U}}_{m,q-1}(L)$$
with the A($m$)-structure maps induced by that of ${\mathcal{U}}_{m}(L)$.
It is easily seen that $G^{*}_{m}(L)$
is generated by the image of the canonical
inclusion $\iota:L\hookrightarrow{\mathcal{U}}_{m,1}(L)\to G^{*}_{m}(L)$.
A moment’s reflection shows that the elements of $L\subset G_{m}^{*}(L)$
‘mutually commute’ in the sense that (11) is satisfied for
$a_{1},\ldots,a_{n}\in L$. Surprisingly,
this does not imply that (11)
is satisfied for all elements of $G^{*}_{m}(L)$ –
there is no analog of
Observation 4.5 for A($m$)-algebras!
One must instead consider A($m$)-algebras $S_{m}(V,\partial)$
defined as
$$S_{m}(V,\partial):={\raisebox{7.2pt}{\tiny$m\hskip-2.27622pt:\hskip-1.707165pt%
1$}\hskip-9.0pt{\mathfrak{A}}\hskip 2.0pt}(V,\partial)/{\mathfrak{I}}_{m},$$
where ${\mathfrak{I}}_{m}$ is the ideal generated by the relations
$$\sum_{\sigma\in S_{n}}\chi(\sigma)\mu_{n}(v_{\sigma(1)},\ldots,v_{\sigma(n)})=%
0\ \mbox{ for }\ 2\leq n\leq m\ \mbox{ and }\ v_{1},\dots,v_{n}\in V.$$
The algebra $S_{m}(V,\partial)$ is (externally) graded, with the grading induced from
the (external) grading of the free A($m$)-algebra ${\raisebox{7.2pt}{\tiny$m\hskip-2.27622pt:\hskip-1.707165pt1$}\hskip-9.0pt{%
\mathfrak{A}}\hskip 2.0pt}(V,\partial)$.
By a miracle, $S_{m}^{*}(V,\partial)$ has the
following very explicit description.
Choose a basis $\{f_{i}\}_{i\in I}$ of $V$ indexed by a linearly ordered
set $I$. Let ${\raisebox{7.2pt}{\tiny$m$}\hskip-5.8pt{\mbox{$\tt T$}}}{}^{n}(V)$ be the set of all planar rooted reduced
$n$-trees $S$ whose vertices have arity $\leq m$. We moreover
suppose that the input legs of $S$ are labeled by elements of the
basis $\{f_{i}\}_{i\in I}$ and that the labels satisfy the following
condition:
Let $v$ be an input vertex of $S$ and let $e_{1},\dots,e_{k}$ be the
input legs of $v$, written in the order induced by the imbedding of $S$
into the plane. Let $f_{i_{1}},\dots,f_{i_{k}}$ be the corresponding labels.
Then it is not true that ${i_{1}}\geq i_{2}\geq\cdots\geq i_{k}$.
Proposition 4.6.
For each $n\geq 1$, $S^{n}_{m}(V)\cong{\rm Span}_{{\bf k}}({\raisebox{7.2pt}{\tiny$m$}\hskip-5.8pt{%
\mbox{$\tt T$}}}{}^{n}(V))$.
Proof.
It follows from (7) that ${\raisebox{7.2pt}{\tiny$m\hskip-2.27622pt:\hskip-1.707165pt1$}\hskip-9.0pt{%
\mathfrak{A}}\hskip 2.0pt}(V,\partial)$ is spanned by the
set of planar directed reduced $n$-trees with all vertices of arity
$\leq m$ and the input legs labeled by the chosen basis
$\{f_{i}\}_{i\in I}$ of $V$. Let $v$ be an input vertex of $S$,
$e_{1},\ldots,e_{k}$ input legs of $v$ written in the order induced by
the plane and $f_{i_{1}},\dots,f_{i_{k}}$ their labels.
Suppose ${i_{1}}\geq i_{2}\geq\cdots\geq i_{k}$. If $i_{1}=i_{k}$, then
clearly $S\equiv 0$ modulo ${\mathfrak{I}}_{m}$. If $i_{1}\not=i_{k}$, we may replace $S$ modulo ${\mathfrak{I}}_{m}$ by a sum of trees
that differ from $S$ only by labels of $e_{1},\dots,e_{k}$ which are
changed to $i_{\sigma(1)},\dots,i_{\sigma(k)}$, for some $\sigma\in\Sigma_{k}$ such that it is not true that
${i_{\sigma(1)}}\geq i_{\sigma(2)}\geq\cdots\geq i_{\sigma(k)}$.
Repeating this process at any input vertex of $S$, we replace $S$
modulo ${\mathfrak{I}}_{m}$ by a sum of trees from ${\raisebox{7.2pt}{\tiny$m$}\hskip-5.8pt{\mbox{$\tt T$}}}{}^{n}(V)$.
This replacement is obviously unique.
∎
Let us formulate our version of the PBW theorem for the universal
enveloping A($m$)-algebra of an L($m$)-algebra.
Theorem 4.7.
The canonical map $\iota:L\hookrightarrow{\mathcal{U}}_{m,1}(L)\to G^{*}_{m}(L)$
induces an isomorphism
$$\rho:S^{*}_{m}(L)\cong G^{*}_{m}(L)$$
of (externally) graded A($m$)-algebras.
Proof.
The proof relies on the fact that $\{\rho(S);\ S\in{\raisebox{7.2pt}{\tiny$m$}\hskip-5.8pt{\mbox{$\tt T$}}}{}^{n}%
(V)\}$
forms a basis of $G^{n}_{m}(V)$, which easily follows from the above
analysis. We leave the details to the reader.
∎
Example 4.8.
Let us analyze the case $m=2$ when
L($m$)-algebras are just differential graded
anti-commutative non-associative algebras. For an L(2)-algebra $L=(L,[-,-],\partial)$, ${\raisebox{7.2pt}{\tiny$2\hskip-2.27622pt:\hskip-1.707165pt1$}\hskip-9.0pt{%
\mathfrak{A}}\hskip 2.0pt}(L)$ is the free
non-associative dg algebra generated by $(L,\partial)$, the universal enveloping
A(2)-algebra of $L$ is given by
$$\mathcal{U}_{2}(L)={\raisebox{7.2pt}{\tiny$2\hskip-2.27622pt:\hskip-1.707165pt%
1$}\hskip-9.0pt{\mathfrak{A}}\hskip 2.0pt}/(xy-(-1)^{|x||y|}yx=[x,y],\ x,y\in L),$$
and the associated graded $G^{*}_{2}(L)$ is the quotient of ${\raisebox{7.2pt}{\tiny$2\hskip-2.27622pt:\hskip-1.707165pt1$}\hskip-9.0pt{%
\mathfrak{A}}\hskip 2.0pt}(L)$
by the ideal generated by $xy=(-1)^{|x||y|}yx$, for $x,y\in L$.
Remark 4.9.
An ordinary strict Lie algebra $L=(L,[-,-])$ can be considered,
for any $m\geq 2$, as an L($m$)-algebra with $l_{2}=[-,-]$ and $l_{n}=0$ if $n\not=2$. Its universal enveloping A($m$)-algebra
${\mathcal{U}}_{m}(L)$ is a fully-fledged A($m$)-algebra, therefore an
object completely different from the ordinary universal enveloping
associative algebra ${\mathcal{U}}(L)$. Thus our $\mathcal{U}_{m}(L)$ cannot be
considered as a generalization of the classical universal
enveloping algebra functor.
Appendix: the tree language
The following definitions can be found for example
in [8, Section II.1.5]. By a tree we mean a connected graph $T$ without loops. Let ${\rm Vert}(T)$
denote the set of vertices of $T$. A valence of a vertex $v\in{\rm Vert}(T)$ is the number of edges adjacent to $v$. A leg of
$T$ is an edge adjacent to a vertex of valence one, other edges of $T$
are interior. We in fact discard vertices of valence one at
the endpoints of the legs, therefore the legs are “half-edges” having
only one vertex. This can be formalized by introducing (generalized)
graphs as certain sets with involutions,
see [8, Definition II.5.37].
By a rooted or directed tree we mean a tree with a
distinguished output leg called the root. The
remaining legs are called the input legs of the tree. A tree
with $n$ input legs is called an $n$-tree. A rooted tree is
automatically oriented, each edge pointing towards the root.
The edges pointing towards a given vertex $v$ are called the input edges of $v$, the number of these input edges is then the
arity of $v$. Vertices of arity one are called unary, vertices
of arity two binary, vertices of arity three ternary, etc. A tree is
reduced if it has no unary vertices. A vertex $v\in{\rm Vert}(T)$ is an input vertex of $T$ if all its input edges are
input legs of $T$. With the above conventions accepted, we must admit
also the unique reduced 1-tree which consist of one edge which is both
its root and the input leg, and no vertices (soul without body).
Each tree can be imbedded into the plane. By a planar tree we
mean a tree with a specified (isotopy class of) imbedding. Reading
off counterclockwise the input legs of a planar tree, starting from
the leftmost one, gives a linear order of input legs. In the same
manner, input edges of each vertex of a planar tree are linearly
oriented. Vice versa, specifying linear orders of input edges of each
vertex of a rooted tree $T$ uniquely determines an isotopy class of
imbedding of $T$ into the plane.
Example 4.10.
In Figure 3, $f$ is the root, $a$ the root vertex,
$l,m,n,o,p,q,r,s,t,u$ the input legs and $g,j,k,i$ the interior edges
of $T$. Vertices $b,c,d,v$ are the input vertices of $T$. The vertex $d$
is unary, $v$ binary and the remaining vertices are ternary. The tree
in Figure 3 is not reduced because it contains an unary
vertex.
References
[1]
V. Ginzburg and M.M. Kapranov.
Koszul duality for operads.
Duke Math. Journal, 76(1):203–272, 1994.
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V.A. Hinich and V.V. Schechtman.
Homotopy Lie algebras.
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T. Lada and M. Markl.
Strongly homotopy Lie algebras.
Communications in Algebra, 23(6):2147–2161, 1995.
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T. Lada and J.D. Stasheff.
Introduction to sh Lie algebras for physicists.
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M. Markl.
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Journ. Pure Appl. Algebra, 83:141–175, 1992.
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M. Markl.
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Annales de l’Institut Fourier, 46(4):307–323, 1996.
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M. Markl.
Models for operads.
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M. Markl, S. Shnider, and J. D. Stasheff.
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J.-P. Serre.
Lie Algebras and Lie Groups.
Benjamin, 1965.
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J.D. Stasheff.
Homotopy associativity of H-spaces I,II.
Trans. Amer. Math. Soc., 108:275–312, 1963. |
Coherent Multiple Scattering Effect in DIS
Xiaofeng Guo and Jun Li
Department of Physics and Astronomy, Iowa State University
Ames, Iowa 50011, U.S.A.
Abstract
We present a QCD calculation of the coherent multiple scattering effect
on single hadron production in semi-inclusive lepton-nucleus
deep inelastic scattering (DIS). We show that the destructive
interference of coherent multiple scattering leads to suppression
in hadron productions. The suppression is similar
to the expectations of energy loss calculations.
We compare our calculation with experimental data.
keywords:
Multiple scattering, energy loss, deep inelastic scattering
PACS: 12.38.Bx 13.85.Ni 24.85.+p 25.75.-q
††thanks: Supported in part by the National Science Fundation,
grant PHY-0340729.
1. Introduction The strong suppression of high transverse momentum hadron in
relativistic heavy ion collisons was considered to be
an evidence for the QCD quark-gluon plasma [1].
The suppression was believed to be the result of
medium induced
radiative energy loss of high energy partons [2].
However, recent data indicate that heavy quarks would have to lose
the same amount of energy as that of a light quark
if the radiative energy loss is the only source of the suppression
[3]. On the other hand, we expect heavy quarks to lose
much less energy than a light quark because of its mass [4].
In this talk, we present a new source of suppresion from
the destructive multiple scattering in a nuclear medium [5].
We calculated the single hadron production
in DIS in perturbative QCD, and
found that the coherent interference results into a suppression
in the hadron production rate.
2. Single hard scattering We consider the production of a single hadron of momentum $p_{h}$ in
lepton-nucleus deep inelastic scattering (DIS),
$e(k_{1})+A(p)\longrightarrow e(k_{2})+h(p_{h})+X$, with the target
atomic weight $A$ and average nucleon momentum $p=P_{A}/A$.
The semi-inclusive DIS cross section,
$$\frac{d{\sigma}_{eA\rightarrow ehX}}{dx_{B}dQ^{2}dz_{h}}=\frac{1}{8\pi}\,\frac%
{e^{4}}{x_{B}^{2}s^{2}Q^{2}}\,L^{\mu\nu}(k_{1},k_{2})\,\frac{dW^{A}_{\mu\nu}}{%
dz_{h}}\ ,$$
(1)
where $s=(p+k_{1})^{2}$ is the total invariant mass of the lepton-nucleon
system, and
$z_{h}\equiv{p\cdot p_{h}}/{p\cdot q}={2x_{B}p\cdot p_{h}}/{Q^{2}}$
is the fraction of the photon momentum carried by the observed hadron.
The Bjorken $x_{B}=Q^{2}/(2p\cdot q)$ with $Q^{2}=-q^{2}$.
The leptonic tensor $L^{\mu\nu}(k_{1},k_{2})={\rm Tr}(\gamma\cdot k_{1}\gamma^{\mu}\gamma\cdot k_{2%
}\gamma^{\nu})/2$. For single scattering
at the lowest order, the hadronic tensor can be factorized as
$$\frac{dW^{A}_{\mu\nu}}{dz_{h}}=\frac{1}{2}\,e^{T}_{\mu\nu}\sum_{q}\,e_{q}^{2}%
\,A\,\phi^{A}_{q}(x_{B},Q^{2})\,D_{q}(z_{h},Q^{2})\,,$$
(2)
where $e^{T}_{\mu\nu}=[p_{\mu}q_{\nu}+q_{\mu}p_{\nu}]/{p\cdot q}+p_{\mu}p_{\nu}({2x_{%
B}}/{p\cdot q})-g_{\mu\nu}$,
$\phi^{A}_{q}(x_{B},Q^{2})$ is nuclear quark distribution
per nucleon of flavor $q$, and $D_{q}(z_{h},Q^{2})$ is quark-to-hadron
fragmentation function.
The nuclear dependence of Eq. (2)
and Eq. (1) is
limited to the nuclear dependence of $\phi^{A}_{q}(x_{B},Q^{2})$.
3. Coherent multiple scattering Multiple scattering
introduces the medium size enhanced, or the $A^{1/3}$-type,
nuclear effects. We adopt the leading pole approaximation
to extract the leading contributions in powers of
$A^{1/3}$[6].
For double scattering at lowest order in $\alpha_{s}$,
we only need to consider two diagrams in Fig. 1.
When $x_{B}\geq x_{c}\sim 0.1$ [7],
we consider coherent multiple scattering effects
between the scattered quark and the partons of the nuclear target
at the same impact parameter in the Breit frame
and obtain [5]
$$\displaystyle\frac{dW_{\mu\nu}^{(1)}}{dz_{h}}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}e_{\mu\nu}^{T}\,\sum_{q}e_{q}^{2}\left[\frac{4\pi^{2}%
\alpha_{s}}{3}\right]\frac{z_{h}}{Q^{2}}\,T_{qg}^{A}(x_{B},Q^{2})\,\frac{dD(z_%
{h},Q^{2})}{dz_{h}}$$
(3)
$$\displaystyle\approx$$
$$\displaystyle\frac{1}{2}e_{\mu\nu}^{T}\,\sum_{q}e_{q}^{2}\left(A\phi^{A}_{q}(x%
_{B},Q^{2})\right)\left[\frac{z_{h}\xi^{2}(A^{1/3}-1)}{Q^{2}}\right]\frac{dD(z%
_{h},Q^{2})}{dz_{h}}$$
(4)
where $\sum_{q}$ runs over all quark and antiquark flavors.
The twist-4 quark-gluon correlation function, $T_{qg}^{A}(x_{B},Q^{2})$,
is defined in Ref. [6].
In deriving Eq. (4), we assumed that a large nucleus
is made of color neutral nucleons with a constant density.
The quantity $\xi^{2}$ defined in Ref. [7]
is proportional to a matrix element of a gluon density operator, and
represents a characteristic scale of quark interaction with
nuclear medium.
If $x_{B}<x_{c}$, the hadronic tensor in Eqs. (3) and
(4) has an additional term proportional to
$x_{B}\frac{d}{dx_{B}}\phi_{q}(x_{B},Q^{2})$ [5].
For $x_{B}\geq x_{c}$, we generalized our result in Eq. (4) to
$n$ scattering [5]
$$\displaystyle\frac{dW_{\mu\nu}^{(n)}}{dz_{h}}$$
$$\displaystyle\approx$$
$$\displaystyle\frac{1}{2}e_{\mu\nu}^{T}\,\sum_{q}e_{q}^{2}\left(A\phi^{A}_{q}(x%
_{B},Q^{2})\right)\frac{1}{n!}\left[\frac{z_{h}\xi^{2}(A^{1/3}-1)}{Q^{2}}%
\right]^{n}\frac{d^{n}D(z_{h},Q^{2})}{dz_{h}^{n}}\,.$$
(5)
We sum the $A^{1/3}$-type contributions in Eq. (5) to
$N$ scatterings, and take
$N\approx\infty$ because the effective value of $\xi^{2}$ is
relatively small. We obtain
$$\displaystyle\frac{dW^{A}_{\mu\nu}}{dz_{h}}$$
$$\displaystyle\approx$$
$$\displaystyle\frac{1}{2}e_{\mu\nu}^{T}\,\sum_{q}e_{q}^{2}\left(A\phi^{A}_{q}(x%
_{B},Q^{2})\right)\sum_{n=0}^{N}\frac{1}{n!}\left[\frac{z_{h}\xi^{2}(A^{1/3}-1%
)}{Q^{2}}\right]^{n}\frac{d^{n}D(z_{h},Q^{2})}{dz_{h}^{n}}$$
(6)
$$\displaystyle\approx$$
$$\displaystyle A\,\frac{1}{2}e_{\mu\nu}^{T}\,\sum_{q}e_{q}^{2}\,\phi^{A}_{q}(x_%
{B},Q^{2})\,D_{q}\left(z_{h}+\frac{z_{h}\xi^{2}(A^{1/3}-1)}{Q^{2}},Q^{2}\right%
)\,.$$
Eq. (6) is our main result.
The net effect of coherent multiple scattering of
a propagating quark in the medium,
is equivalent to a shift, $\Delta z$, in the variable $z_{h}$
of the quark fragmentation function $D_{q}(z_{h},Q^{2})$ with
$\Delta z=z_{h}\,\xi^{2}(A^{1/3}-1)/Q^{2},$
which leads to a suppression of the production rate
[5].
4. Comparison with HERMES data We compare our result with HERMES data [8] in Fig. 2.
$R_{M}$ is the ratio
of hadron production rate rate per DIS event for a nuclear target
$A$, $R^{A}$, to that of a deuterium target $D$, $R^{D}$ [5],
$$R_{M}\equiv\frac{R^{A}}{R^{D}}\approx\frac{\sum_{q}e_{q}^{2}\,\phi^{A}_{q}(x_{%
B},Q^{2})\,D_{q}(z_{h}+\Delta z,Q^{2})}{\sum_{q}e_{q}^{2}\,\phi^{D}_{q}(x_{B},%
Q^{2})\,D_{q}(z_{h},Q^{2})}\frac{\sum_{q}e_{q}^{2}\,\phi^{D}_{q}(x_{B},Q^{2})}%
{\sum_{q}e_{q}^{2}\,\phi^{A}_{q}(x_{B},Q^{2})}\ .$$
(7)
The dashed lines are from Eq. (7) and the dot-dashed lines
includes only the double scattering.
At large $z_{h}$, resummed theory curves are much
steeper than the data.
This is because the formation time for the hadron or
corresponding pre-hadron state is much shorter at large $z_{h}$,
and therefore, the number of scattering for the propagating parton
at large $z$, or the shift, should be reduced [5].
Since the hadron formation time is proportional to
$(1-z_{h})$ [9], we modify
$\Delta z$ by multiplying a factor $(1-z_{h})$ and
produce the solid lines in Fig. 2.
The suppression from the coherent multiple scattering, calculated here,
complements the suppression from medium-induced radiation
[10], which should give additional suppression and
bring down the curves in Fig. 2 [5].
5. Summary and outlook We showed that
the coherent multiple scattering of a propagating quark
in the medium, without the induced radiation,
can change the quark fragmentation or
hadron’s production rate. The net effect of
leading power contributions in medium length
is equivalent to a shift in the fragmentation
function’s $z_{h}\rightarrow z_{h}+\Delta z$, and
at the lowest order, $\Delta z$ is given by
an universal matrix element.
Our result is complementary to the
energy loss of induced radiation.
However, beyond the leading order, the separation
of the collisional energy loss and that of induced
radiation will depend on the factorization scheme and
is not unique, and needs further study.
Our approach can be generalized to hadron productions in $p+A$
and $A+A$ collisions.
References
[1]
S. S. Adler et al. [PHENIX Collaboration],
Phys. Rev. Lett. 91, 072301 (2003); and references therein.
[2]
M. Gyulassy and X. N. Wang,
Nucl. Phys. B 420, 583 (1994);
X. N. Wang, M. Gyulassy and M. Plumer,
Phys. Rev. D 51, 3436 (1995);
R. Baier, Y. L. Dokshitzer, A. H. Mueller, S. Peigne and D. Schiff,
Nucl. Phys. B 484, 265 (1997);
B. G. Zakharov,
JETP Lett. 63, 952 (1996); JETP Lett. 65, 615 (1997);
U. A. Wiedemann,
Nucl. Phys. B 588, 303 (2000)
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S. S. Adler et al. [PHENIX Collaboration],
Phys. Rev. Lett. 96, 032301 (2006);
J. Bielcik [STAR Collaboration],
Nucl. Phys. A 774, 697 (2006)
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Y. L. Dokshitzer and D. E. Kharzeev,
Phys. Lett. B 519, 199 (2001)
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X. F. Guo, and J. Li, in preparation.
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M. Luo, J.-W. Qiu and G. Sterman, Phys. Rev. D49, 4493 (1994).
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J. W. Qiu and I. Vitev,
Phys. Rev. Lett. 93, 262301 (2004).
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A. Airapetian et al. [HERMES Collaboration],
Eur. Phys. J. C 20, 479 (2001).
[9]
A. Accardi, in this proceedings.
[10]
X. F. Guo and X. N. Wang,
Phys. Rev. Lett. 85, 3591 (2000);
X. N. Wang and X. F. Guo,
Nucl. Phys. A 696, 788 (2001);
E. Wang and X. N. Wang,
Phys. Rev. Lett. 89, 162301 (2002). |
Theta Functions on Covers of Symplectic Groups
Solomon Friedberg
and
David Ginzburg
(corresponding author) [email protected]
Abstract.
We study the automorphic theta representation $\Theta_{2n}^{(r)}$ on the $r$-fold cover of the symplectic group $Sp_{2n}$. This representation is obtained from the residues of Eisenstein series on this group.
If $r$ is odd,
$n\leq r<2n$, then under a natural hypothesis on the theta representations, we show that
$\Theta_{2n}^{(r)}$ may be used to construct a generic representation
$\sigma_{2n-r+1}^{(2r)}$ on the $2r$-fold cover of $Sp_{2n-r+1}$. Moreover, when $r=n$ the
Whittaker functions of this representation attached to factorizable data
are factorizable, and the unramified local factors may be computed in terms of $n$-th order Gauss sums.
If $n=3$ we prove these results, which in that case pertain to the six-fold cover of $Sp_{4}$, unconditionally.
We expect that in fact the representation constructed here, $\sigma_{2n-r+1}^{(2r)}$, is precisely $\Theta_{2n-r+1}^{(2r)}$; that
is, we conjecture relations between theta representations on different covering groups.
Key words and phrases:Symplectic group, metaplectic cover, theta representation, descent integral, unipotent orbit, generic representation,
Whittaker function
2010 Mathematics Subject Classification: Primary 11F70; Secondary 11F27, 11F55
This work was supported by the BSF,
grant number 2012019, and by the NSF, grant number 1500977 (Friedberg).
1. Introduction
The classical metaplectic group is a double cover of a symplectic group, and may be defined over a local field or the ring of adeles of a number field. This group arises in the study of theta functions,
which may be constructed directly as sums over the global rational points of an isotropic subspace. This construction
of theta functions appears
to be special to the double cover case.
To generalize, let $r>1$ be any integer, and let
$F$ be a number field with a full set of $r$-th roots of unity. Then there is an $r$-fold cover of the symplectic
group $Sp_{2n}(\mathbb{A})$, where $\mathbb{A}$ is the adeles of $F$,
denoted $Sp_{2n}^{(r)}(\mathbb{A})$. When $r>2$ one may construct theta functions for such a group, but only indirectly – theta
functions are obtained
as the multi-residues of the minimal parabolic (Borel) Eisenstein series on $Sp_{2n}^{(r)}(\mathbb{A})$ at
the “right-most” (suitably normalized) pole. (This construction obtains when $r=2$ as well.)
As such, these theta functions are automorphic forms and give an
automorphic representation $\Theta_{2n}^{(r)}$, called the theta representation.
Since for a general cover the theta functions may only be defined via residues, it is considerably more challenging
to determine basic information such as their Whittaker or other Fourier
coefficients in this situation. Indeed, the Fourier coefficients of the theta function
have not been determined for covers of $SL_{2}$ of any degree higher than $4$. And when $n>1$ very little is known.
Moreover, except in the double cover case there are few examples of any automorphic functions on such a group, there
is no information about when their Whittaker coefficients might be factorizable (one expects rarely), and it is not clear
to what extent one might expect relations between theta functions or other automorphic
forms on different covers of different groups.
In a first example of such information, the authors [6] have established
relations between
the Whittaker coefficients of certain automorphic functions on different covering groups in two specific cases, related to the conjectures
of Patterson and Chinta-Friedberg-Hoffstein concerning the Fourier coefficients of theta functions on
covers of $GL_{2}$ . The key to doing so was to adapt
descent methods, originally used by Ginzburg-Rallis-Soudry [9] in the context of
algebraic groups or double covers, to higher degree covering groups. In particular, the treatment of
the adelic version of the conjecture
of Chinta-Friedberg-Hoffstein relied on the study of the theta function on the three-fold cover of $Sp_{4}$.
In this work we investigate theta functions and descent integrals on covers of symplectic groups of arbitrary rank.
As we shall explain, for $r$ odd a descent construction allows one to pass from
the theta representation $\Theta_{2n}^{(r)}$ to a representation $\sigma_{2n-r+1}^{(2r)}$ realized
inside the space of automorphic square-integrable functions on
$Sp^{(2r)}_{2n-r+1}(F)\backslash Sp^{(2r)}_{2n-r+1}(\mathbb{A})$. We expect that this representation is in fact the
theta representation $\Theta_{2n-r+1}^{(2r)}$ (Conjecture 2), and we show that $\sigma_{2n-r+1}^{(2r)}$ has non-zero
projection to this representation.
In Theorem 1 below we establish that if $r$ is odd,
$n\leq r<2n$, then the representation $\sigma_{2n-r+1}^{(2r)}$ is generic. Moreover, if $r=n$,
we show that the Whittaker coefficients of
$\sigma_{n+1}^{(2n)}$ arising from factorizable inputs are factorizable.
As a first new case, this is true of the Whittaker coefficients
of the descent representation on the six-fold cover of $Sp_{4}$.
Then in Theorem 2 we show that the unramified local
contributions to the Whittaker coefficients in the factorizable
case may be expressed as sums of Whittaker coefficients of the theta representation on the $n$-fold cover of $GL_{n}$.
These coefficients are $n$-th order Gauss sums by the work of Kazhdan and Patterson [12]. Thus we exhibit a new
class of automorphic forms on covering groups whose Whittaker coefficients are factorizable with algebraic factors at good places.
These results are proved on
two conditions. One concerns the specific unipotent orbit attached to $\Theta_{2n}^{(r)}$ (Conjecture 1).
This is an object of on-going study [4]; in the last Section we give a sketch of the proof of this Conjecture in the
first new case, $n=r=3$. A
second concerns the precise characters within a given unipotent orbit which support a nonzero Fourier coefficient
(Assumption 1). This is not needed in all cases, and in particular is not needed in the factorizable case
$\sigma_{n+1}^{(2n)}$.
Our results concerning Conjecture 1 are thus sufficient to establish unconditionally that
the Whittaker function of the descent to the six-fold cover of $Sp_{4}$ is Eulerian for factorizable data, and to compute its unramified
local factor in terms of cubic Gauss sums. We expect that this descent is in fact the theta function on the six-fold cover of $Sp_{4}$.
2. Definition and Properties of the Theta Representation
Let $n$ and $r$ be two natural numbers. Let $F$ be a number field containing a full set $\mu_{r}$ of $r$-th roots of unity,
and let $\mathbb{A}$ be the adeles of $F$. Let $Sp_{2n}$ denote the symplectic group consisting of the $2n\times 2n$ matrices leaving
invariant the standard symplectic form $<x,y>=\sum_{i=1}^{n}(x_{i}y_{2n-i+1}-x_{n+i}y_{n-i+1})$.
Denote by $Sp_{2n}^{(r)}(\mathbb{A})$ the metaplectic $r$-fold cover of the symplectic group $Sp_{2n}(\mathbb{A})$. This group
consists of pairs $(g,\zeta)$ with $g\in Sp_{2n}(\mathbb{A})$ and $\zeta\in\mu_{r}$; it
may be
obtained in the standard way from covers of the local groups $Sp_{2n}(F_{\nu})$ as $\nu$ ranges over the places of $F$,
identifying the copies of $\mu_{r}$. The multiplication in the local group is determined by a cocyle $\sigma$.
If $\sigma_{BLS}$ is the 2-cocycle of
Banks-Levy-Sepanski [1] for $SL_{2n}(F_{\nu})$ and $w$ is the permutation such that the conjugate of $Sp_{2n}$
by $w$ preserves the symplectic form $<x,y>^{\prime}=\sum_{i=1}^{n}x_{i}y_{n+i}-y_{i}x_{n+i}$, then we take
$\sigma(g_{1},g_{2})=\sigma_{BLS}(wg_{1}w^{-1},wg_{2}w^{-1})$.
We shall be concerned with the theta representation defined on the group $Sp_{2n}^{(r)}(\mathbb{A})$, denoted $\Theta_{2n}^{(r)}$. This representation is defined as
the space spanned by the residues of the Borel Eisenstein series on $Sp_{2n}^{(r)}({\mathbb{A}})$, similarly to the definition for the general linear
group in [12], p. 118. Our first task is to give a brief account of this construction.
Let $B_{2n}$ denote the standard Borel subgroup of $Sp_{2n}$, and let $T_{2n}$ denote the maximal torus of $Sp_{2n}$ which is a subgroup of $B_{2n}$. Working either locally at a completion of $F$ or globally over $\mathbb{A}$, if $H$ is any algebraic subgroup of $Sp_{2n}$, we let $H^{(r)}$ denote its inverse image in $Sp_{2n}^{(r)}$. We will call this inverse image parabolic or unipotent if $H$ has this property.
Let $Z(T_{2n}^{(r)})$ denote the center of $T_{2n}^{(r)}$.
Suppose first that $r$ is odd.
Given a character $\mu$ of $T_{2n}$, we may use it to define a genuine character,
again denoted $\mu$, of $Z(T_{2n}^{(r)})$.
(The notion of genuine depends on an embedding of the group of $r$-th roots of unity $\mu_{r}$ into $\mathbb{C}^{\times}$. We will fix this and omit it from the notation.)
Extending it trivially to any maximal abelian subgroup of $T_{2n}^{(r)}$ and then inducing up, we obtain a representation of $Sp_{2n}^{(r)}$ which we denote by $\text{Ind}_{B_{2n}^{(r)}}^{Sp_{2n}^{(r)}}\mu$. This representation is determined uniquely by the choice of $\mu$ and this procedure may be carried out both locally and globally. Here we consider
unnormalized induction, and shall include the modular character $\delta_{B_{2n}}^{1/2}$ when we require normalized induction.
Let $s_{i}$ be complex variables, and let $\mu$ be the character of $T_{2n}$ given by
$$\mu(\text{diag}(a_{1},\ldots,a_{n},a_{n}^{-1},\ldots,a_{1}^{-1}))=|a_{1}|^{s_{%
1}}\cdots|a_{n}|^{s_{n}}.$$
If this construction is carried out over a local field, then $|a|$ denotes the normalized local absolute value, while if it is
carried out over $\mathbb{A}$, then $|a|$ denotes the product of these over all places of $F$.
One may form the induced representation $\text{Ind}_{B_{2n}^{(r)}({\mathbb{A}})}^{Sp_{2n}^{(r)}({\mathbb{A}})}\mu\delta%
_{B_{2n}}^{1/2}$, and
for each vector in this space, one may construct the multi-variable Eisenstein series $E(h,s_{1},\ldots,s_{n})$ defined on the group $Sp_{2n}^{(r)}({\mathbb{A}})$. Computing the constant term as in [12], Prop. II.1.2, we deduce that the poles of the partial intertwining operator associated with the long Weyl element of $Sp_{2n}$ are determined by
(1)
$$\frac{\prod_{i<j}[\zeta_{S}(r(s_{i}-s_{j}))\zeta_{S}(r(s_{i}+s_{j}))]\prod_{i}%
\zeta_{S}(rs_{i})}{\prod_{i<j}[\zeta_{S}(r(s_{i}-s_{j})+1)\zeta_{S}(r(s_{i}+s_%
{j})+1)]\prod_{i}\zeta_{S}(rs_{i}+1)}.$$
Here $S$ is a finite set of all places including the archimedean places and all finite places $\mu$ such that $|r|_{\nu}\neq 1$
and $\zeta_{S}(s)$ is the partial global zeta function.
The expression (1) has a multi-residue at the point
$$s_{n}=\frac{1}{r},\qquad r(s_{i}-s_{i+1})=1.$$
From this we deduce that the Eisenstein series $E(h,s_{1},\ldots,s_{n})$ has a multi-residue at that point, and we denote the residue representation by $\Theta_{2n}^{(r)}$. If
$\mu_{0}$ denotes the character of $T_{2n}$ defined by
$$\mu_{0}(\text{diag}(a_{1},\ldots,a_{n},a_{n}^{-1},\ldots,a_{1}^{-1}))=|a_{1}|^%
{\frac{n}{r}}\cdots|a_{n-1}|^{\frac{2}{r}}|a_{n}|^{\frac{1}{r}},$$
then it follows that $\Theta_{2n}^{(r)}$ is a subquotient of the induced representation $\text{Ind}_{B_{2n}^{(r)}({\mathbb{A}})}^{Sp_{2n}^{(r)}({\mathbb{A}})}\mu\delta%
_{B_{2n}}^{1/2}$. From this we deduce that $\Theta_{2n}^{(r)}$ is also a subrepresentation of the induced representation
$\text{Ind}_{B_{2n}^{(r)}({\mathbb{A}})}^{Sp_{2n}^{(r)}({\mathbb{A}})}\chi_{Sp_%
{2n}^{(r)},\Theta}$ where
$$\chi_{Sp_{2n}^{(r)},\Theta}(\text{diag}(a_{1},\ldots,a_{n},a_{n}^{-1},\ldots,a%
_{1}^{-1}))=|a_{1}|^{\frac{n(r-1)}{r}}\cdots|a_{n-1}|^{\frac{2(r-1)}{r}}|a_{n}%
|^{\frac{r-1}{r}}.$$
We turn to even degree coverings. We shall only be concerned with the covers of degree twice an odd integer.
If $r$ is odd, the definition of the theta representation for $Sp_{2m}^{(2r)}$ is similar.
There is a small difference since the maximal parabolic subgroup of $Sp_{2m}$ whose Levi part is $GL_{m}$ splits under the double cover. Because of this, when computing the intertwining operator corresponding to the long Weyl element,
one finds that its poles are determined by
$$\frac{\prod_{i<j}[\zeta_{S}(r(s_{i}-s_{j}))\zeta_{S}(r(s_{i}+s_{j}))]\prod_{i}%
\zeta_{S}(2rs_{i})}{\prod_{i<j}[\zeta_{S}(r(s_{i}-s_{j})+1)\zeta_{S}(r(s_{i}+s%
_{j})+1)]\prod_{i}\zeta_{S}(2rs_{i}+1)}.$$
Accordingly we define $\Theta_{2m}^{(2r)}$ to be the multi-residue of the Eisenstein series at the point
$$s_{m}=\frac{1}{2r},\qquad r(s_{i}-s_{i+1})=1.$$
Then the representation $\Theta_{2m}^{(2r)}$ is a subquotient of the
induced representation $\text{Ind}_{B_{2m}^{(r)}({\mathbb{A}})}^{Sp_{2m}^{(2r)}({\mathbb{A}})}\mu_{e}%
\delta_{B_{2m}}^{1/2}$ where $\mu_{e}$ denotes the character of $T_{2m}$ defined by
$$\mu_{e}(\text{diag}(a_{1},\ldots,a_{m},a_{m}^{-1},\ldots,a_{1}^{-1}))=|a_{1}|^%
{\frac{2m-1}{2r}}\cdots|a_{m-1}|^{\frac{3}{2r}}|a_{m}|^{\frac{1}{2r}}.$$
It follows that $\Theta_{2m}^{(2r)}$ is also a subrepresentation of the induced representation $\text{Ind}_{B_{2m}^{(r)}({\mathbb{A}})}^{Sp_{2m}^{(2r)}({\mathbb{A}})}\chi_{Sp%
_{2m}^{(2r)},\Theta}$ where
$$\chi_{Sp_{2n}^{(2r)},\Theta}(\text{diag}(a_{1},\ldots,a_{m},a_{m}^{-1},\ldots,%
a_{1}^{-1}))=|a_{1}|^{\frac{2mr-(2m-1)}{2r}}\cdots|a_{m-2}|^{\frac{6r-5}{2r}}|%
a_{m-1}|^{\frac{4r-3}{2r}}|a_{m}|^{\frac{2r-1}{2r}}.$$
We now develop some properties of the theta representations that follow from induction in stages, or, equivalently,
by taking an $(n-1)$-fold residue of $E(h,s_{1},\dots,s_{n})$ to obtain a maximal parabolic Eisenstein series attached to
theta representations of lower rank groups.
Let $1\leq a\leq n$. Denote by $P_{2n,a}$ the maximal parabolic subgroup of $Sp_{2n}$ whose Levi part is $GL_{a}\times Sp_{2(n-a)}$, and let $L_{2n,a}$ denote its unipotent radical. We write $i$ for the inclusion of $GL_{a}$ in $P_{2n,a}$ in this paragraph
but we suppress $i$ afterwards.
From the block compatibility of the cocycle $\sigma_{BLS}$ (Banks-Levy-Sepanski [1]) and a short computation it follows
that if $g_{1},g_{2}\in GL_{a}$ over a local field then
$\sigma(i(g_{1}),i(g_{2}))=\sigma_{GL_{a}}(g_{1},g_{2})^{2}(\det g_{1},\det g_{%
2})^{-1}$, where $\sigma_{GL_{a}}$ is the metaplectic 2-cocycle
of [1] for $GL_{a}$.
Thus for both the $r$ and $2r$-fold covers, $i$ determines an $r$-fold cover of $GL_{a}$. We write this cover $GL_{a}^{(r)}$ (which case we are in will always be clear from context).
Suppose first that $r$ is odd.
Let $\Theta_{GL_{a}}^{(r)}$ denote the theta representation of the group $GL_{a}^{(r)}({\mathbb{A}})$, as constructed in [12]
(or the corresponding character if $a=1$).
Then it follows from induction in stages that
$\text{Ind}_{B_{2n}^{(r)}({\mathbb{A}})}^{Sp_{2n}^{(r)}({\mathbb{A}})}\mu_{0}%
\delta_{B_{2n}}^{1/2}$ is equal to $\text{Ind}_{P_{2n,a}^{(r)}({\mathbb{A}})}^{Sp_{2n}^{(r)}({\mathbb{A}})}(\Theta%
_{GL_{a}}^{(r)}\otimes\Theta_{2(n-a)}^{(r)})\delta_{P_{2n,a}}^{\frac{r+1}{2r}}$.
From this we deduce the following. Let $E_{a}(g,s)$ denote the Eisenstein series of $Sp_{2n}^{(r)}({\mathbb{A}})$ associated with the induced representation $\text{Ind}_{P_{2n,a}^{(r)}({\mathbb{A}})}^{Sp_{2n}^{(r)}({\mathbb{A}})}(\Theta%
_{GL_{a}}^{(r)}\otimes\Theta_{2(n-a)}^{(r)})\delta_{P_{2n,a}}^{s}$.
(For the construction of such Eisenstein series more generally, see Brubaker and Friedberg [2].)
Then the representation
$\Theta_{2n}^{(r)}$ is a residue of $E_{a}(g,s)$ at the point $s=(r+1)/2r$. Of course, this can also be verified directly by studying the corresponding intertwining operators.
In the case of a cover of degree $2r$, $r$ odd, the situation is roughly similar.
In this case, the representation $\text{Ind}_{B_{2m}^{(r)}({\mathbb{A}})}^{Sp_{2m}^{(2r)}({\mathbb{A}})}\mu_{e}%
\delta_{B}^{1/2}$ is equal to $\text{Ind}_{P_{2m,a}^{(r)}({\mathbb{A}})}^{Sp_{2m}^{(2r)}({\mathbb{A}})}(%
\Theta_{GL_{a}}^{(r)}\otimes\Theta_{2(m-a)}^{(2r)})\delta_{P_{2n,a}}^{\frac{(r%
+1)(2m-a)+r}{2r(2m-a+1)}}$. Hence if we let $E_{a}(g,s)$ denote the Eisenstein series associated with the induced representation $\text{Ind}_{P_{2m,a}^{(r)}({\mathbb{A}})}^{Sp_{2m}^{(2r)}({\mathbb{A}})}(%
\Theta_{GL_{a}}^{(r)}\otimes\Theta_{2(m-a)}^{(2r)})\delta_{P_{2n,a}}^{s}$, then we deduce that $\Theta_{2m}^{(2r)}$ is the residue of $E_{a}(g,s)$ at the point $s=\frac{(r+1)(2m-a)+r}{2r(2m-a+1)}$.
From these observations, we will deduce the following proposition. Here and below,
matrices are embedded in metaplectic groups by the trivial
section without additional notation, and we call metaplectic elements diagonal or unipotent when their projections to the
linear group are.
Proposition 1.
Suppose that $r$ is odd.
Let $\theta_{2n}^{(r)}$ be in the space of $\Theta_{2n}^{(r)}$. Then there exist functions $\theta_{GL_{a}}^{(r)}\in\Theta_{GL_{a}}^{(r)}$,
$\theta_{2(n-a)}^{(r)}\in\Theta_{2(n-a)}^{(r)}$
such that for all diagonal $g$ in $GL_{a}^{(r)}({\mathbb{A}})$ which lies in the center of the Levi part of the parabolic group $P_{2n,a}^{(r)}({\mathbb{A}})$ and for all unipotent $h\in Sp^{(r)}_{2(n-a)}({\mathbb{A}})$, $v\in GL_{a}^{(r)}({\mathbb{A}})$, one has
(2)
$$\int\limits_{L_{2n,a}(F)\backslash L_{2n,a}({\mathbb{A}})}\theta_{2n}^{(r)}(u(%
gv,h))du=\chi_{Sp_{2n}^{(r)},\Theta}(g)\,\theta_{GL_{a}}^{(r)}(v)\,\theta_{2(n%
-a)}^{(r)}(h).$$
A similar identity holds in the even-degree cover case.
The requirement that $h,v$ be unipotent could be dropped at the expense of a cocyle; being unipotent guarantees that it is $1$.
Proof.
Since $\Theta_{2n}^{(r)}$ is the residue of the Eisenstein series $E_{a}(\cdot,s)$, we first consider the constant term
(3)
$$\int\limits_{L_{2n,a}(F)\backslash L_{2n,a}({\mathbb{A}})}E_{a}(u(gv,h),s)\,du.$$
For $\text{Re}(s)$ large we can unfold the Eisenstein series and deduce that (3) is equal to
(4)
$$\sum_{w\in P_{2n,a}(F)\backslash Sp_{2n}(F)/P_{2n,a}(F)}\int\limits_{L_{2n,a}^%
{w}(F)\backslash L_{2n,a}({\mathbb{A}})}\sum_{\gamma\in(w^{-1}P_{2n,a}(F)w\cap
P%
_{2n,a}(F))\backslash P_{2n,a}(F)}f_{s}(w\gamma u(gv,h))\,du.$$
Here $L_{2n,a}^{w}=w^{-1}L_{2n,a}w\cap L_{2n,a}^{-}$ where $L_{2n,a}^{-}$ is the conjugate of $L_{2n,a}$ by the long Weyl element. Also, we have $f_{s}\in\text{Ind}_{P_{2n,a}^{(r)}({\mathbb{A}})}^{Sp_{2n}^{(r)}({\mathbb{A}})%
}(\Theta_{GL_{a}}^{(r)}\otimes\Theta_{2(n-a)}^{(r)})\delta_{P_{2n,a}}^{s}$.
Notice that all elements in $P_{2n,a}(F)\backslash Sp_{2n}(F)/P_{2n,a}(F)$ can be chosen to be Weyl elements.
Similarly to [13], Section 1.2, one can check that for all Weyl elements in $P_{2n,a}(F)\backslash Sp_{2n}(F)/P_{2n,a}(F)$
which are not the long Weyl element, the inner summation is a certain Eisenstein series or product
of such series. Moreover, one can also check that the Eisenstein series in
the corresponding summand in (4) is holomorphic at $s=(r+1)/2r$. Hence, taking the residue in (3) and (4) at $s=(r+1)/2r$, we are left only with the long
Weyl element, and for that element we obtain the identity
$$\int\limits_{L_{2n,a}(F)\backslash L_{2n,a}({\mathbb{A}})}\theta_{2n}^{(r)}(u(%
gv,h))du=\text{Res}_{s=(r+1)/2r}M_{w}f_{s}(gv,h)$$
where $M_{w}$ is the intertwining operator attached to $w$.
From this identity (2) follows.
∎
Similar statements for other groups are established in [3] Proposition 3.4
and [6] Proposition 1.
We end this section with a general conjecture about the unipotent orbit attached to the representation $\Theta_{2n}^{(r)}$,
and give a consequence of this conjecture.
Recall that if $\pi$ denotes an automorphic representation of a reductive group,
then the set ${\mathcal{O}}(\pi)$ was defined in [7]. It is the largest unipotent orbit that supports a nonzero coefficient
for this representation.
The extension of the definition of this set to the metaplectic groups is
clear.
In this paper we are interested in the set ${\mathcal{O}}(\Theta_{2n}^{(r)})$. The conjecture regarding this set is given as follows. Assume that $r<2n$. Write $2n=a(n,r)r+b(n,r)$ where $a(n,r)$ and $b(n,r)$ are both nonnegative numbers such that $0\leq b(n,r)\leq r-1$.
We recall that a partition is a symplectic partition if any odd number in the partition occurs with even multiplicity. As defined in [5], given a partition $\lambda$ of $2n$, we define the $Sp$ collapse of $\lambda$ to be the greatest symplectic partition which is smaller than $\lambda$. We have
Conjecture 1.
Let $r$ be an odd number.
If $r<2n$, then the set ${\mathcal{O}}(\Theta_{2n}^{(r)})$ consists of the partition which is the $Sp$ collapse of the partition $(r^{a(n,r)}b(n,r))$. If $r>2n$, then the representation $\Theta_{2n}^{(r)}$ is generic.
The conjecture is studied in [4]. We shall write Conjecture $1_{r,n}$ for this statement for a specific pair $(r,n)$.
We remark that an extension to even degree covers has not been formulated at this time.
To give a consequence of this conjecture, let $r$ be odd, let $a$ be an integer with $1\leq a\leq n-(r-1)/2$,
and let $b=n-(r-1)/2-a$.
Let $U_{2n,n-b}$ be the unipotent radical of the standard parabolic subgroup of $Sp_{2n}$ whose Levi part is $GL_{1}^{n-b}\times Sp_{2b}$. (In particular, $U_{2n,n}$ is the standard maximal unipotent subgroup of $Sp_{2n}$.)
Let $U_{2n,n-b,1}$ be the subgroup of $U_{2n,n-b}$ which consists of all matrices $u=(u_{i,j})\in U_{2n,n-b}$ such that $u_{n-b,i}=0$ for all $n-b+1\leq i\leq n$. Let $\underline{\alpha}=(\alpha_{1},\ldots,\alpha_{a-1})$ where for all $i$ we have $\alpha_{i}\in\{0,1\}$. Let $\psi$ be a nontrivial character of $F\backslash{\mathbb{A}}$,
which will be fixed throughout this paper.
Let
$\psi_{U_{2n,n-b,1},\underline{\alpha}}$ be the character of $U_{2n,n-b,1}$ given by
$$\psi_{U_{2n,n-b,1},\underline{\alpha}}(u)=\psi\left(\sum_{i=1}^{a-1}\alpha_{i}%
u_{i,i+1}+\sum_{j=a}^{n-b}u_{j,j+1}\right).$$
Then we have the following result, which will be used later.
Lemma 1.
Suppose $r$ is odd, $r<2n$, and Conjecture 1${}_{r,n-i_{0}}$ holds for all $0\leq i_{0}\leq a-1$. Then
the integral
(5)
$$\int\limits_{U_{2n,n-b,1}(F)\backslash U_{2n,n-b,1}({\mathbb{A}})}\theta_{2n}^%
{(r)}(ug)\,\psi_{U_{2n,n-b,1},\underline{\alpha}}(u)\,du$$
is zero for all choices of data, that is, for all $\theta_{2n}^{(r)}\in\Theta_{2n}^{(r)}$.
Proof.
Consider first the case where $\alpha_{i}=1$ for all $i$. Then the Fourier coefficient (5) is the Fourier coefficient which corresponds to the unipotent orbit $((2n-2b)1^{2b})$. From Conjecture 1${}_{r,n}$ we
have that ${\mathcal{O}}(\Theta_{2n}^{(r)})$ consists of the partition which is the $Sp$ collapse of the partition $(r^{a(n,r)}b(n,r))$. Hence $((2n-2b)1^{2b})$ is greater than or not related to ${\mathcal{O}}(\Theta_{2n}^{(r)})$. Indeed, this follows from the relation $2b=2n-2a-r+1$ which implies that $((2n-2b)1^{2b})=((r+2a-1)1^{2b})$. Thus, if $\alpha_{i}=1$ for all $i$, then the integral (5) is zero for all choices of data.
Next assume that at least one of the scalars $\alpha_{i}$ is zero. Let $i_{0}\leq a-1$ be the largest index such that $\alpha_{i_{0}}=0$. Then the integral (5) is equal to
(6)
$$\displaystyle\int\limits_{H(F)\backslash H({\mathbb{A}})}\int\limits_{L_{2n,i_%
{0}}(F)\backslash L_{2n,i_{0}}({\mathbb{A}})}\int\limits_{U_{2(n-i_{0}),n-i_{0%
}-b,1}(F)\backslash U_{2(n-i_{0}),n-i_{0}-b,1}({\mathbb{A}})}\\
\displaystyle\theta_{2n}^{(r)}(luhg)\,\psi_{U_{2(n-i_{0}),n-i_{0}-b,1}}(u)\,%
\psi_{H,\underline{\alpha}}(h)\,du\,dl\,dh.$$
Here $H$ is a certain unipotent subgroup of $GL_{i_{0}}$ which will not be important to us.
Notice that the integration over $L_{2n,i_{0}}(F)\backslash L_{2n,i_{0}}({\mathbb{A}})$ is the constant term along this unipotent group. Therefore, it follows from Proposition 1 above
that the integration along the quotient space $U_{2(n-i_{0}),n-i_{0}-b,1}(F)\backslash U_{2(n-i_{0}),n-i_{0}-b,1}({\mathbb{A}})$ is the coefficient of the representation $\Theta_{2(n-i_{0})}^{(r)}$ corresponding to the unipotent orbit $((2(n-i_{0}-b))1^{2b})$. Thus the vanishing of the integral (6) will follow if this unipotent orbit is greater than or not related to the
orbit ${\mathcal{O}}(\Theta_{2(n-i_{0})}^{(r)})$. From Conjecture 1${}_{r,n-i_{0}}$, this is the case if $2(n-i_{0}-b)>r$. Since this
inequality follows from the relations $2(n-a-b)=r-1$ and $a>i_{0}$, the result is proved.
∎
3. The Descent Construction
Recall that $U_{2m,k}$ denotes the unipotent radical of the standard parabolic subgroup of
$Sp_{2m}$ whose Levi part is $GL_{1}^{k}\times Sp_{2(m-k)}$. The quotient group $U_{2m,k-1}\backslash U_{2m,k}$ may be identified with the Heisenberg group in $2(m-k)+1$ variables, ${\mathcal{H}}_{2(m-k)+1}$. Indeed, there is a homomorphism $l:U_{2m,k}\mapsto{\mathcal{H}}_{2(m-k)+1}$ which is onto, and whose kernel is the group $U_{2m,k-1}$.
For the rest of this section we suppose that $r>1$ is odd. Set $r^{\prime}=(r-1)/2$.
For $\alpha\in F^{*}$ let $\psi_{U,\alpha}$ be the character
of $U_{r-1,r^{\prime}}(F)\backslash U_{r-1,r^{\prime}}({\mathbb{A}})$ given by
$$\psi_{U,\alpha}(u)=\psi(u_{1,2}+u_{2,3}+\cdots+u_{r^{\prime}-1,r^{\prime}}+%
\alpha u_{r^{\prime},r^{\prime}+1})\qquad u=(u_{i,j}).$$
From now on we shall suppose that Conjecture 1${}_{r,r^{\prime}}$ holds, that is, that
the representation $\Theta_{r-1}^{(r)}$ is generic. This is known if $r=3$ [12].
Thus there is an $\alpha\in F^{*}$ and a choice of data, i.e. an automorphic function $\theta_{r-1}^{(r)}$ in the space of $\Theta_{r-1}^{(r)}$, such that the integral
(7)
$$\int\limits_{U_{r-1,r^{\prime}}(F)\backslash U_{r-1,r^{\prime}}({\mathbb{A}})}%
\theta_{r-1}^{(r)}(ug)\,\psi_{U,\alpha}(u)\,du$$
is not zero.
Replacing $\psi$ by $\psi^{\alpha^{-1}}$ and changing the function, we may assume that $\alpha=1$. In other words,
there is a choice of data $\theta_{r-1}^{(r)}$ such that the integral
(8)
$$\int\limits_{U_{r-1,r^{\prime}}(F)\backslash U_{r-1,r^{\prime}}({\mathbb{A}})}%
\theta_{r-1}^{(r)}(ug)\,\psi_{U}(u)\,du$$
is not zero, where $\psi_{U}$ is defined as $\psi_{U,\alpha}$ with $\alpha=1$.
Remark: In fact we expect that for all $\alpha\in F^{*}$, the integral (7) will be nonzero for some choice of data.
Let $\Theta_{2m}^{\psi,\phi}$ denote the theta representation attached to the Weil representation and defined on the double cover of $Sp_{2m}({\mathbb{A}})$. Here $\phi$ is a Schwartz function on ${\mathbb{A}}^{m}$.
Since $r$ is odd, using the isomorphism
$\mu_{2r}\cong\mu_{2}\times\mu_{r}$ we may map $g\in Sp_{2n-r+1}^{(2r)}({\mathbb{A}})$ to its image in the double and $r$-fold covers;
we will not introduce separate notation for this.
To define the descent construction, consider the function of $g\in Sp_{2n-r+1}^{(2r)}({\mathbb{A}})$ given by
(9)
$$f(g)=\int\limits_{U_{2n,r^{\prime}}(F)\backslash U_{2n,r^{\prime}}({\mathbb{A}%
})}\theta_{2n-r+1}^{\psi,\phi}(l(u)g)\,\theta_{2n}^{(r)}(ug)\,\psi_{U_{2n,r^{%
\prime}}}(u)\,du.$$
Here $\theta_{2n-r+1}^{\psi,\phi}$ and $\theta_{2n}^{(r)}$ are vectors in the spaces of the representations
$\Theta_{2n-r+1}^{\psi,\phi}$ and $\Theta_{2n}^{(r)}$ (resp.), $\psi_{U_{2n,r^{\prime}}}$ is the character of $U_{2n,r^{\prime}}({\mathbb{A}})$ given by $\psi_{U_{2n,r^{\prime}}}(u_{i,j})=\psi(u_{1,2}+u_{2,3}+\cdots+u_{r^{\prime}-1,%
r^{\prime}})$, and $g\in Sp_{2n-r+1}$ is embedded in $Sp_{2n}$ as
$$g\mapsto\begin{pmatrix}I_{r^{\prime}}&&\\
&g&\\
&&I_{r^{\prime}}\end{pmatrix}.$$
(One could also consider descent integrals similar to (9) if $r$ is even but we shall not do so here.)
Then $f(g)$ is a genuine automorphic function defined on $Sp_{2n-r+1}^{(2r)}({\mathbb{A}})$.
Let $\sigma_{2n-r+1}^{(2r)}$ denote the representation of $Sp_{2n-r+1}^{(2r)}({\mathbb{A}})$ generated by all the functions $f(g)$.
4. Computation of the Constant Term of the Descent Integral
Let $V$ denote any standard unipotent subgroup of $Sp_{2n-r+1}$, and let $\psi_{V}$
be a character of $V(F)\backslash V({\mathbb{A}})$, possibly trivial.
In this paper we will compute integrals of the type
(10)
$$\int\limits_{V(F)\backslash V({\mathbb{A}})}f(vg)\,\psi_{V}(v)\,dv,$$
where $f(g)$ is given by (9).
Using (9), we arrive at an iterated integral. We first unfold the theta function $\theta_{2n-r+1}^{\psi,\phi}$. Collapsing summation and integration, and then using the formulas for the action of the Weil representation
$\omega_{\psi}$, the integral (10) is equal to
(11)
$$\int\limits_{{\mathbb{A}}^{n-r^{\prime}}}\int\limits_{V(F)\backslash V({%
\mathbb{A}})}\int\limits_{U_{2n,r^{\prime},1}(F)\backslash U_{2n,r^{\prime},1}%
({\mathbb{A}})}\omega_{\psi}(g)\phi(x)\,\theta_{2n}^{(r)}(uvj(x)g)\,\psi_{U_{2%
n,r^{\prime},1}}(u)\,\psi_{V}(v)\,du\,dv\,dx.$$
Here $x=(x_{1},\ldots,x_{n-r^{\prime}})\in\mathbb{A}^{n-r^{\prime}}$ is embedded in $Sp_{2n}$ via the map
$j(x)=I_{2n}+x_{1}e^{*}_{r^{\prime},r^{\prime}+1}+x_{2}e^{*}_{r^{\prime},r^{%
\prime}+2}+\cdots+x_{n-r^{\prime}}e^{*}_{r^{\prime},n}$,
where $e_{i,j}$ denotes the $(i,j)$-th elementary matrix and $e^{*}_{i,j}=e_{i,j}-e_{2n-j+1,2n-i}$.
Recall that the group $U_{2n,r^{\prime},1}$ is the subgroup of $U_{2n,r^{\prime}}$ consisting of all matrices $u=(u_{i,j})\in U_{2n,r^{\prime}}$ such that $u_{r^{\prime},k}=0$ for all $r^{\prime}+1\leq k\leq n$. The character $\psi_{U_{2n,r^{\prime},1}}(u)$ is defined as the product of $\psi_{U_{2n,r^{\prime}}}(u)$ and the character $\psi^{0}_{U_{2n,r^{\prime},1}}$ of $U_{2n,r^{\prime},1}$ given by $\psi^{0}_{U_{2n,r^{\prime},1}}(u)=\psi(u_{r^{\prime},2n-r^{\prime}+1})$.
At this point we consider the case when $V=L_{2n-r+1,a}$ where $1\leq a\leq n-r^{\prime}$ and $\psi_{V}$ is the trivial character. Thus,
$V$ is the unipotent radical of the standard maximal parabolic subgroup of $Sp_{2n-r+1}$ whose
Levi part is $GL_{a}\times Sp_{2n-2a-r+1}$. We recall that $2b=2n-2a-r+1$.
Let $w_{a}$ denote the Weyl group element of $Sp_{2n}$ defined by
$$w_{a}=\begin{pmatrix}&I_{a}&&&\\
I_{n-a-b}&&&&\\
&&I_{2b}&&\\
&&&&I_{n-a-b}\\
&&&I_{a}&\end{pmatrix}.$$
Since $\theta_{2n}^{(r)}$ is left-invariant under rational points, we may conjugate by $w_{a}$ from left to right.
Doing so, we see that the integral (11) is equal to
(12)
$$\displaystyle\int\limits_{{\mathbb{A}}^{n-r^{\prime}}}\int\limits_{Mat_{(n-a-b%
-1)\times a}(F)\backslash Mat_{(n-a-b-1)\times a}({\mathbb{A}})}\int\limits_{V%
_{0}(F)\backslash V_{0}({\mathbb{A}})}\int\limits_{U_{0}(F)\backslash U_{0}({%
\mathbb{A}})}\\
\displaystyle\omega_{\psi}(g)\phi(x)\,\theta_{2n}^{(r)}(u_{0}v_{0}k(y)w_{a}j(x%
)g)\,\psi_{V_{0}}(v_{0})\,du_{0}\,dv_{o}\,dy\,dx.$$
The notations are defined as follows. Recall that $L_{2n,a}$ denotes the unipotent radical of the standard maximal parabolic subgroup of $Sp_{2n}$ whose Levi part is $GL_{a}\times Sp_{2(n-a)}$. The group $U_{0}$ is defined to be the subgroup of $L_{2n,a}$ consisting of all matrices in $Sp_{2n}$ of the form
$$\begin{pmatrix}I_{a}&0&*&*&*\\
&I_{n-a-b}&0&0&*\\
&&I_{2b}&0&*\\
&&&I_{n-a-b}&0\\
&&&&I_{a}\end{pmatrix}.$$
The embedding of $Mat_{(n-a-b-1)\times a}$ into $Sp_{2n}$ is given by
$$k(y)=\begin{pmatrix}I_{a}&&&&\\
y&I_{n-a-b}&&&\\
&&I_{2b}&&\\
&&&I_{n-a-b}&\\
&&&y^{*}&I_{a}\end{pmatrix},$$
where $y^{*}$ is chosen to make the matrix symplectic. Here the embedding of $y$ is such that it consists of all matrices of size $(n-a-b)\times a$ such that its bottom row is zero. Finally $V_{0}$ is equal to the group $U_{2(n-a),n-a-b,1}$, defined similarly to the definition of $U_{2n,a,1}$ reviewed after (11) (in particular, the group $V_{0}$ is a subgroup of $Sp_{2(n-a)}$), and
the character $\psi_{V_{0}}:=\psi_{U_{2(n-a),n-a-b},1}$ as defined there. In coordinates,
$\psi_{V_{0}}(v)=\psi(v_{a+1,a+2}+v_{a+2,a+3}+\cdots+v_{n-b-1,n-b}+v_{n-b,n+b+1})$.
The next step is to carry out some root exchanges. The notion of root exchange was described in detail in the authors’
paper [6], Section 2.2; a more abstract formulation may be found in Ginzburg, Rallis and Soudry
[11], Section 7.1.
In our context, we perform the following root exchange. For $z\in Mat_{a\times(n-a-b)}$, let $m(z)$ in $Sp_{2n}$ be given by
$$m(z)=\begin{pmatrix}I_{a}&z&&&\\
&I_{n-a-b}&&&\\
&&I_{2b}&&\\
&&&I_{n-a-b}&z^{*}\\
&&&&I_{a}\end{pmatrix}.$$
We embed the group $Mat_{a\times(n-a-b-1)}(\mathbb{A})$ inside $Sp_{2n}(\mathbb{A})$ by considering all matrices $m(z)$ as above such that the first column of $z$ is zero. Expand the integral (12) along the group so obtained. Performing the root exchange with the group of all matrices $k(y)$ such that $y\in Mat_{(n-a-b-1)\times a}({\mathbb{A}})$, one sees that the integral (12) is equal to
(13)
$$\displaystyle\int\limits_{{\mathbb{A}}^{n-r^{\prime}}}\int\limits_{Mat_{(n-a-b%
-1)\times a}({\mathbb{A}})}\int\limits_{V_{0}(F)\backslash V_{0}({\mathbb{A}})%
}\int\limits_{L^{0}_{2n,a}(F)\backslash L^{0}_{2n,a}({\mathbb{A}})}\\
\displaystyle\omega_{\psi}(g)\phi(x)\,\theta_{2n}^{(r)}(uv_{0}k(y)w_{a}j(x)g)%
\,\psi_{V_{0}}(v_{0})\,du\,dv_{0}\,dy\,dx.$$
Here $L^{0}_{2n,a}$ is the subgroup of $L_{2n,a}$ which consists of all matrices $u=(u_{i,j})\in L_{2n,a}$ such that $u_{i,a+1}=0$ for all $1\leq i\leq a$.
Consider the quotient space of $L^{0}_{2n,a}\backslash L_{2n,a}$.
This quotient may be naturally identified with the column vectors of size $a$. This group embeds in $Sp_{2n}$ as the group of all matrices of the form $I_{2n}+z_{1}e^{*}_{1,a+1}+z_{2}e^{*}_{2,a+1}+\cdots+z_{a}e^{*}_{a,a+1}$. Expand the integral (13) along this group. The group $GL_{a}(F)$ acts on this expansion with two orbits.
The contribution of the nontrivial orbit to the integral (13) is the expression
(14)
$$\displaystyle\int\limits_{{\mathbb{A}}^{n-r^{\prime}}}\int\limits_{Mat_{(n-a-b%
-1)\times a}({\mathbb{A}})}\int\limits_{V_{0}(F)\backslash V_{0}({\mathbb{A}})%
}\int\limits_{L_{2n,a}(F)\backslash L_{2n,a}({\mathbb{A}})}\\
\displaystyle\omega_{\psi}(g)\phi(x)\,\theta_{2n}^{(r)}(uv_{0}k(y)w_{a}j(x)g)%
\,\psi_{V_{0}}(v_{0})\,\psi_{L_{2n,a}}(u)\,du\,dv_{0}\,dy\,dx,$$
where for $u=(u_{i,j})\in L_{2n,a}(F)\backslash L_{2n,a}({\mathbb{A}})$, $\psi_{L_{2n,a}}(u)=\psi(u_{a,a+1})$.
We claim that the integral (14) is zero for all choices of data.
To prove this, we carry out similar expansions repeatedly. To start, consider the quotient space $L_{2n,a-1}\backslash L_{2n,a}$. This space may be identified with the column vectors of size $a-1$, embedded in $Sp_{2n}$ as the group of all matrices of the form $I_{2n}+z_{1}e^{*}_{1,a}+z_{2}e^{*}_{2,a}+\cdots+z_{a-1}e^{*}_{a-1,a}$. Expand the integral (14) along this group. The group $GL_{a-1}(F)$ acts on this expansion with two orbits. If $a-1>1$ we continue in this way. Then the vanishing
of the integral (14) for all choices of data then follows from Lemma 1 above.
We conclude that the integral (10) with $V=U_{2n-r+1,a}$ and $\psi_{V}=1$ is equal to
(15)
$$\displaystyle\int\limits_{{\mathbb{A}}^{n-r^{\prime}}}\int\limits_{Mat_{(n-a-b%
-1)\times a}({\mathbb{A}})}\int\limits_{V_{0}(F)\backslash V_{0}({\mathbb{A}})%
}\int\limits_{L_{2n,a}(F)\backslash L_{2n,a}({\mathbb{A}})}\\
\displaystyle\omega_{\psi}(g)\phi(x)\,\theta_{2n}^{(r)}(uv_{0}k(y)w_{a}j(x)g)%
\,\psi_{V_{0}}(v_{0})\,du\,dv_{0}\,dy\,dx.$$
We shall prove momentarily that (15) is nonzero for some choice of data. Supposing this for the moment,
let us analyze this integral when $g=\text{diag}(tI_{a},I_{2n-r-2a+1},t^{-1}I_{a})$ is an element in the center of the Levi part of the maximal parabolic subgroup of $Sp_{2n-r+1}$ whose Levi part is $GL_{a}\times Sp_{2n-r-2a+1}$, embedded in the $r$-fold metaplectic group via the trivial section. Suppose also that $t$ is an $r$-th power. Inserting $g$ into (15), we
move it from right to left in the function $\theta_{2n}^{(r)}$. Moving it past $j(x)$ and making the corresponding variable change in $x$ gives a factor of $|t|^{-a}$,
as well as a factor of $\gamma(t^{a})|t|^{a/2}$ from the action of the Weil representation. Then moving it past $k(y)$ we obtain a factor of $|t|^{-a(n-a-b-1)}$ from the corresponding change of variables in $y$. Clearly, $g$ commutes with $V_{0}$, and finally, applying Proposition 1 above we also get a factor of
$$\delta_{P_{2n,a}}^{\frac{r-1}{2r}}(\text{diag}(tI_{a},I_{2(n-a)},t^{-1}I_{a}))%
=|t|^{a(r-1)(2n-a+1)/2r}$$
(here, recall that $P_{2n,a}$ is the maximal parabolic subgroup of $Sp_{2n}$ whose unipotent radical is $L_{2n,a}$).
We conclude that, for $g=\text{diag}(tI_{a},I_{2(n-a)},t^{-1}I_{a})$ as above and for $V=U_{2n-r+1,a}$, we have
$$\int\limits_{V(F)\backslash V({\mathbb{A}})}f(vg)\,dv=|t|^{\beta}\int\limits_{%
V(F)\backslash V({\mathbb{A}})}f(v)\,dv$$
where
$$\beta=\frac{a(r-1)(2n-a+1)}{2r}-\frac{a}{2}-a(n-a-b-1).$$
It follows from Section 2 that $\chi_{Sp_{2n-r+1}^{(2r)},\Theta}(g)=|t|^{\beta}$.
Finally, we prove that the integral (15) is not zero for some choice of data. This will
also imply that the representation $\sigma_{2n-r+1}^{(2r)}$ is not zero.
Suppose instead that (15) is zero for all choices of data. Arguing as in [6], Lemma 1, we may ignore the integration over the $x$ and $y$ variables, and deduce that the integral
$$\int\limits_{V_{0}(F)\backslash V_{0}({\mathbb{A}})}\int\limits_{L_{2n,a}(F)%
\backslash L_{2n,a}({\mathbb{A}})\,}\theta_{2n}^{(r)}(uv_{0})\,\psi_{V_{0}}(v_%
{0})\,du\,dv_{0}$$
is zero for all choices of data. From Proposition 1 we deduce that the integral
(16)
$$\int\limits_{V_{0}(F)\backslash V_{0}({\mathbb{A}})}\theta_{2(n-a)}^{(r)}(v_{0%
}h)\,\psi_{V_{0}}(v_{0})\,dv_{0}$$
is zero for all choice of data. Here $h\in Sp_{2(n-a)}^{(r)}({\mathbb{A}})$. From the definition of the group $V_{0}$ and the character $\psi_{V_{0}}$ it follows that the above Fourier coefficient corresponds to the unipotent orbit $((2(n-a-b))1^{2b})$. Since $2b=2n-2a-r+1$, this unipotent orbit is the same as $((r-1)1^{2n-2a-r+1})=((r-1)1^{2b})$.
At this point we would like to use instances of Conjecture 1, but we need to be careful. The Fourier coefficients attached to the orbit $((r-1)1^{2b})$ are given by the integrals
(17)
$$\int\limits_{V_{0}(F)\backslash V_{0}({\mathbb{A}})}\theta_{2b+r-1}^{(r)}(v_{0%
}h)\,\psi_{V_{0},\alpha}(v_{0})\,dv_{0}$$
with $\alpha\in F^{*}$ and $\psi_{V_{0},\alpha}$ defined on $V_{0}\subset Sp_{2(n-a)}$ by
$$\psi_{V_{0},\alpha}(v)=\psi(v_{1,2}+v_{2,3}+\cdots+v_{r^{\prime}-1,r^{\prime}}%
+\alpha v_{r^{\prime},2b+r^{\prime}+1}).$$
Conjecture 1 asserts that there is an $\alpha$ such that the integral (17) is not zero for some choice of data. However, we need to prove that the integral (16) is not zero for some choice of data, that is, the integral (17) with $\alpha=1$. This does not follow from Conjecture 1.
To prove that integral (16) is not zero for some choice of data, we will use the non-vanishing of the integral (8). Indeed, suppose instead that (16) is zero for all choices of data. Then the integral
(18)
$$\int\limits_{L_{2b,b}(F)\backslash L_{2b,b}({\mathbb{A}})}\int\limits_{V_{0}(F%
)\backslash V_{0}({\mathbb{A}})}\theta_{2b+r-1}^{(r)}(v_{0}l)\,\psi_{V_{0}}(v_%
{0})\,dv_{0}\,dl$$
is zero for all choices of data. We recall that $L_{2b,b}$ is the unipotent radical of
the maximal parabolic subgroup of $Sp_{2b}$ whose Levi part is $GL_{b}$. Arguing as in the computation of (11), we obtain that the vanishing of (18) implies the vanishing of the integral
(19)
$$\int\limits_{U_{r-1,r^{\prime}}(F)\backslash U_{r-1,r^{\prime}}({\mathbb{A}})}%
\int\limits_{L_{2b+r-1,b}(F)\backslash L_{2b+r-1,b}({\mathbb{A}})}\theta_{2b+r%
-1}^{(r)}(lv_{0})\,\psi_{V_{0}}(v_{0})\,dl\,dv_{0}.$$
But the vanishing of (19) for all choice of data implies that (8) is zero for all choice of data,
by Proposition 1. This is a contradiction.
Since the constant terms have been bounded and the descent is a compact integral of an automorphic form, the descent
and its derivatives have uniformly moderate growth. Then from truncation as in the Corollary in [14], I.2.12, together with
the above calculations, one
obtains the following result.
Proposition 2.
Let $r$ be odd, $r<2n$, and suppose that Conjecture 1${}_{r,i}$ holds for all $i$ with $r^{\prime}\leq i\leq n$.
Then the representation $\sigma_{2n-r+1}^{(2r)}$ is not zero and it is a sub-representation of $L^{2}(Sp_{2n-r+1}^{(2r)}(F)\backslash Sp_{2n-r+1}^{(2r)}({\mathbb{A}}))$. Moreover its projection to the residual spectrum is
nonzero.
One could also deduce this following [6], Propositions 4 and 5, by a sequence
of expansions along abelian unipotent radicals. We omit the details, as in fact
we expect that a stronger statement holds (compare [6], Conjecture 1).
Conjecture 2.
The representation $\sigma_{2n-r+1}^{(2r)}$ is equal to $\Theta_{2n-r+1}^{(2r)}$.
5. The Whittaker Coefficient of the Descent
In this section we study those cases in which the descent representation $\sigma_{2n-r+1}^{(2r)}$ is generic.
Let us explain how these cases are predicted by the
dimension equation of the second author [8]. It follows from this equation
together with (9) that the following equation should hold, where the dimension of a representation means
its Gelfand-Kirillov dimension:
(20)
$$\text{dim}\ \Theta_{2n-r+1}^{\psi,\phi}\ +\text{dim}\ \Theta_{2n}^{(r)}=\text{%
dim}\ U_{2n,r^{\prime}}+\text{dim}\ \sigma_{2n-r+1}^{(2r)}.$$
Since $\Theta_{2n-r+1}^{\psi,\phi}$ is the minimal representation of $Sp_{2n-r+1}^{(2)}({\mathbb{A}})$,
its dimension is $\frac{1}{2}(2n-r+1)$. An easy computation shows that $\text{dim}\ U_{2n,r^{\prime}}=\frac{1}{4}(r-1)(4n-r+1)$. If we want the representation $\sigma_{2n-r+1}^{(2r)}$ to be generic,
its dimension should equal $\frac{1}{4}(2n-r+1)^{2}$. Using these values, (20) gives
(21)
$$\text{dim}\ \Theta_{2n}^{(r)}=n^{2}-n+\frac{1}{2}(r-1).$$
It is not hard to check that for a representation $\Theta_{2n}^{(r)}$ which satisfies both
equation (21) and Conjecture 1${}_{r,n}$, we must have $n\leq r<2n$. If $n=2k$, then
for all $0\leq i\leq k-1$ the dimension of the unipotent orbit $((4k-2i-2)(2i+2))$ is equal to
$n^{2}-n+\frac{1}{2}(r-1)$ with $r=4k-2i-1$. If $n=2k+1$, then
for all $0\leq i\leq k-1$ the dimension of the unipotent orbit $((4k-2i)(2i+2))$ is equal to
$n^{2}-n+\frac{1}{2}(r-1)$ with $r=4k-2i+1$. In this case we also have the unipotent orbit
$((2k+1)^{2})$ whose dimension is $n^{2}-n+\frac{1}{2}(r-1)$ with $r=2k+1$. The above argument motivates the following
result.
Theorem 1.
Suppose that $r$ is odd, $n\leq r<2n$. Suppose that Conjecture 1${}_{r,i}$ holds for $r^{\prime}\leq i\leq n$. If
$r\neq n,n+1$, suppose that Assumption 1 below holds as well.
Then the representation $\sigma_{2n-r+1}^{(2r)}$ is generic. Moreover, in the special case where $r=n$, the Whittaker coefficient of
$\sigma_{n+1}^{(2n)}$ attached to factorizable data $\theta_{2n}^{(n)}\in\Theta_{2n}^{(n)}$ and $\phi$ is factorizable.
We shall see below that the hypotheses concerning Conjecture 1${}_{r,i}$ are satisfied when $r=n=3$.
Proof.
The proof is based on a computation of the integral (10)
with $V$ the maximal unipotent subgroup of $Sp_{2n-r+1}$ and $\psi_{V}$ the Whittaker character of $V$.
According to Conjecture 1${}_{r,n}$ the unipotent orbit attached to the
representation $\Theta_{2n}^{(r)}$ is computed as follows.
Let $r=2n-2l+1$ for convenience, and
write $2n=a(n,r)(2n-2l+1)+b(n,r)$. Then $a(n,r)=1$ and $b(n,2)=2l-1$. Thus, the unipotent orbit
attached to the representation $\Theta_{2n}^{(r)}$ is the $Sp$ collapse of
$((2n-2l+1)(2l-1))$. This is $((2n-2l)(2l))$ unless $n=2l-1$, in which case it is $((2l-1)^{2})$. This latter is the case $n=r$.
We consider first the case where the unipotent orbit attached to the representation $\Theta_{2n}^{(r)}$
is of the form $((2n-2l)(2l))$. (In the above notations we have $l=i+1$.)
We first describe the set of Fourier coefficients attached to this unipotent orbit.
Let $P^{\prime}_{l}$ denote the standard parabolic subgroup of $Sp_{2n}$ whose Levi part is $GL_{1}^{n-2l}\times GL_{2}^{l}$. We embed this group in $Sp_{2n}$ as the group of all matrices of the form $\text{diag}(a_{1},\ldots,a_{n-2l},g_{1},\ldots,g_{l},g_{l}^{*},\ldots,g_{1}^{*%
},a_{2n-l}^{-1},\ldots,a_{1}^{-1})$. Here $a_{i}\in GL_{1}$ and $g_{j}\in GL_{2}$ and the starred entries are determined by the requirement that the matrix be in $Sp_{2n}$. Let $U^{\prime}_{l}$ denote the
unipotent radical of $P^{\prime}_{l}$. Then the Fourier coefficients of an automorphic function $\varphi$ attached to the
unipotent orbit $((2n-2l)(2l))$ are given by
(22)
$$\int\limits_{U^{\prime}_{l}(F)\backslash U^{\prime}_{l}({\mathbb{A}})}\varphi(%
ug)\,\psi_{U^{\prime}_{l},a,b}(u)\,du$$
where $a,b\in F^{*}$ and the character $\psi_{U^{\prime}_{l},a,b}$ is defined by
(23)
$$\psi_{U^{\prime}_{l},a,b}(u)=\psi\left(\sum_{i=1}^{n-2l}u_{i,i+1}+\sum_{j=n-2l%
+1}^{n-2}u_{j,j+2}+au_{n-1,n+2}+bu_{n,n+1}\right).$$
Thus, to assert that the unipotent orbit $((2n-2l)(2l))$ is attached to a certain
representation, means first that there is a choice of $a$ and $b$ such that the Fourier
coefficient (22) is not zero for some choice of data, and second that for all orbits
which are greater than or not related to $((2n-2l)(2l))$ all corresponding
Fourier coefficients are zero. However, Conjecture 1${}_{r,n}$ does not specify for which values
of $a$ and $b$ the integral (22) is nonzero for some choice of data.
To prove the Theorem, we will need a compatibility assumption which is stated as follows.
Assumption 1.
Suppose that $r\neq n,n+1$. Then there is a choice of vector $\varphi$ in the space of $\Theta_{2n}^{(r)}$
such that the integral (22) with $a=1$ is not zero.
This assumption, which will not be needed when the orbit is of the form $(n^{2})$,
is used to guarantee the non-vanishing of the
Whittaker coefficient of the descent.
At this point we
begin the computation of the Whittaker coefficient when $n\neq r$; the case $n=r$ will be treated separately. We compute the integral (10) with $V=U_{2n-r+1,n-r^{\prime}}$, that is,
(24)
$$\int\limits_{U_{2n-r+1,n-r^{\prime}}(F)\backslash U_{2n-r+1,n-r^{\prime}}({%
\mathbb{A}})}f(ug)\,\psi_{U,b}(u)\,du.$$
As above $r^{\prime}=(r-1)/2$ and the character $\psi_{U,b}$, $b\in F^{*}$, is given by
$$\psi_{U,b}=\psi(u_{1,2}+u_{2,3}+\cdots+u_{n-r^{\prime}-1,n-r^{\prime}}+bu_{n-r%
^{\prime},n-r^{\prime}+1}).$$
Similarly to the analysis of (10) and (11), the integral (24)
is equal to
(25)
$$\displaystyle\int\limits_{{\mathbb{A}}^{n-r^{\prime}}}\int\limits_{U_{2n-r+1,n%
-r^{\prime}}(F)\backslash U_{2n-r+1,n-r^{\prime}}({\mathbb{A}})}\int\limits_{U%
_{2n,r^{\prime},1}(F)\backslash U_{2n,r^{\prime},1}({\mathbb{A}})}\\
\displaystyle\omega_{\psi}(g)\phi(x)\,\theta_{2n}^{(r)}(uvj(x)g)\,\psi_{U_{2n,%
r^{\prime},1}}(u)\,\psi_{U,b}(v)\,du\,dv\,dx.$$
Define the following Weyl element $w_{0}$ of $Sp_{2n}$. First, let $w$ denote the Weyl element of
$GL_{n}$ given by $w=\begin{pmatrix}I_{n-2l+1}&\\
&\nu\end{pmatrix}$ with $l=n-r^{\prime}$ and $\nu=(\nu_{i,j})\in GL_{2l-1}$ defined
as follows. Let $\nu_{i,j}=1$ for the pairs
$(i,j)=(1,l);(3,l+1);(5,l+2);\ldots;(2l-1,2l-1)$ and $(i,j)=(2,1);(4,2);(6,3)\ldots(2l-2,l-1)$ and $\nu_{i,j}=0$ otherwise.
Then let $w_{0}=\begin{pmatrix}w&\\
&w^{*}\end{pmatrix}$ where the star indicates that the matrix is in $Sp_{2n}$.
Since $\theta_{2n}^{(r)}$ is left-invariant under $w_{0}$, inserting $w_{0}$ into $\theta_{2n}^{(r)}$ and moving it rightward,
the integral (25) is equal to
(26)
$$\int\limits_{{\mathbb{A}}^{n-r^{\prime}}}\int\limits_{V_{1}(F)\backslash V_{1}%
({\mathbb{A}})}\int\limits_{U^{\prime}_{l,1}(F)\backslash U^{\prime}_{l,1}({%
\mathbb{A}})}\omega_{\psi}(g)\phi(x)\,\theta_{2n}^{(r)}(uvw_{0}j(x)g)\,\psi_{U%
^{\prime}_{l},1,b}(u)\,du\,dv\,dx.$$
Here $U^{\prime}_{l,1}$ is the subgroup consisting of $u=(u_{i,j})\in U^{\prime}_{l}$ such that
$u_{2m,2k+1}=0$ for all $1\leq m\leq l-1$ and $m\leq k\leq l-1$, and the character $\psi_{U^{\prime}_{l},1,b}$ is the
character $\psi_{U^{\prime}_{l},a,b}$ defined in (23) with $a=1$. Also, the group $V_{1}$ is the unipotent
subgroup of $Sp_{2n}$ consisting of all matrices of the form $I_{2n}+\sum_{i,j}r_{i,j}e_{i,j}^{*}$ where
the sum is over all pairs $(i,j)=(2k+1,2m)$ with $1\leq m\leq l-1$ and $m-1\leq k\leq l-2$.
Let $Z_{l}$ denote the unipotent subgroup of $U^{\prime}_{l}$ consisting of all matrices of the form
$I_{2n}+\sum_{i,j}r_{i,j}e_{i,j}^{*}$ where the sum is over all pairs $(i,j)=(2m,2k+1)$ with
$1\leq m\leq l-1$ and $m\leq k\leq l-1$.
Then $Z_{l}$ is a subgroup of $U^{\prime}_{l}$ which satisfies the condition
that $U^{\prime}_{l}=Z_{l}\cdot U^{\prime}_{l,1}$. We perform a root exchange in the integral (26).
Expanding the integral (26) along the quotient $Z_{l}(F)\backslash Z_{l}({\mathbb{A}})$
and exchanging roots with the group $V_{1}$, we deduce that this integral is equal to
(27)
$$\int\limits_{{\mathbb{A}}^{n-r^{\prime}}}\int\limits_{V_{1}({\mathbb{A}})}\int%
\limits_{U^{\prime}_{l}(F)\backslash U^{\prime}_{l}({\mathbb{A}})}\omega_{\psi%
}(g)\phi(x)\,\theta_{2n}^{(r)}(uvw_{0}j(x)g)\,\psi_{U^{\prime}_{l},1,b}(u)\,du%
\,dv\,dx.$$
Using [6], Lemma 1, we deduce that the integral (27) is not zero for some choice of data if and only if the integral (22) with $a=1$ is not zero for some choice of data.
When $r\neq n+1$, i.e. $n\neq 2l$, we use Assumption 1 to deduce the nonvanishing of integral (27), or equivalently,
the nonvanishing of a Whittaker coefficient of $\sigma_{2n-r+1}^{(r)}$.
When $r=n+1$, so $n=2l$, we may deduce the nonvanishing of the integral
(27) without requiring an analogue of Assumption 1.
Indeed, this situation is essentially the same as the cases studied in Ginzburg, Rallis and Soudry [9, 10].
In those works, the residue representation is defined on the linear group
$Sp_{2n}({\mathbb{A}})$ rather than a covering group. In our case, we are concerned with the $r=(n+1)$-fold cover, and the representation we use to define the descent is $\Theta_{2n}^{(n+1)}$. However, the crucial ingredient for the computations of the Whittaker coefficients is the unipotent orbit attached to the residue representations. In both cases it is the orbit $(n^{2})$. Since the computations for the proof of the Theorem involve only manipulations of unipotent groups, the cover does not enter.
Since the notations in those references are different, for the convenience of the reader we sketch the argument.
Starting with the integral (27), we choose $b=-1/4$. Let $\gamma=\begin{pmatrix}1&\\
-1&1\end{pmatrix}\begin{pmatrix}1&\tfrac{1}{2}\\
&1\end{pmatrix}$, and define
$\gamma_{0}=\text{diag}(\gamma,\ldots,\gamma,\gamma^{*},\ldots,\gamma^{*})\in Sp%
_{2n}(F)$. This is
analogous to the matrix $a$ defined in [9], (4.8).
Since $\theta_{2n}^{(n+1)}$ is invariant under $\gamma_{0}$, inserting it in the integral (27) and moving
it rightward we obtain
(28)
$$\int\limits_{{\mathbb{A}}^{n-r^{\prime}}}\int\limits_{V_{1}({\mathbb{A}})}\int%
\limits_{U^{\prime}_{l}(F)\backslash U^{\prime}_{l}({\mathbb{A}})}\omega_{\psi%
}(g)\phi(x)\,\theta_{2n}^{(n+1)}(u\gamma_{0}vw_{0}j(x)g)\,\psi_{U^{\prime}_{l}%
}(u)\,du\,dv\,dx$$
where $\psi_{U^{\prime}_{l}}$ is given by
$$\psi_{U^{\prime}_{l}}(u)=\psi(u_{1,3}+u_{2,4}+u_{3,5}+u_{4,6}+\cdots+u_{n-3,n-%
1}+u_{n-2,n}+u_{n-1,n+1}).$$
Define the following Weyl element $w_{0}^{\prime}$ of $Sp_{2n}$. If $w_{0}^{\prime}=(w^{\prime}_{0,i,j})$ then
$w^{\prime}_{0,i,2i-1}=1$ for all $1\leq i\leq n$. All other entries are $0,\pm 1$ and are determined uniquely such that $w_{0}^{\prime}\in Sp_{2n}$. This Weyl element was denoted by $\nu$ in [9], p. 881. Inserting $w_{0}^{\prime}$ in the integral (28), we obtain
(29)
$$\int\limits_{{\mathbb{A}}^{n-r^{\prime}}}\int\limits_{V_{1}({\mathbb{A}})}\int%
\limits_{Y(F)\backslash Y({\mathbb{A}})}\int\limits_{U_{2n,n,0}(F)\backslash U%
_{2n,n,0}({\mathbb{A}})}\omega_{\psi}(g)\phi(x)\,\theta_{2n}^{(n+1)}(uyw_{0}^{%
\prime}\gamma_{0}vw_{0}j(x)g)\,\psi_{U_{2n,n}}(u)\,du\,dy\,dv\,dx.$$
Here $Y$ is the unipotent group consisting of all matrices in $Sp_{2n}$ of the form $\begin{pmatrix}I_{n}&\\
y&I_{n}\end{pmatrix}$ such that $y_{i,j}=0$ if $i\geq j$. The group $U_{2n,n,0}$ is the
subgroup of $U_{2n,n}$ such that $u_{i,j}=0$ for all $1\leq i\leq n$ and $n+1\leq j\leq n+i$. Finally,
the character $\psi_{U_{2n,n}}$ is the character of $U_{2n,n}$ defined by $\psi_{U_{2n,n}}(u)=\psi(u_{1,2}+u_{2,3}+\cdots u_{n-1,n})$.
At this point we carry out a sequence of root exchanges. This process is described in detail in [9]
Section 5 so we will omit the details here. Carrying out this process, we deduce that the integral
(27) is equal to
(30)
$$\int\limits_{{\mathbb{A}}^{n-r^{\prime}}}\int\limits_{V_{1}({\mathbb{A}})}\int%
\limits_{Y({\mathbb{A}})}\int\limits_{U_{2n,n}(F)\backslash U_{2n,n}({\mathbb{%
A}})}\omega_{\psi}(g)\phi(x)\,\theta_{2n}^{(n+1)}(uyw_{0}^{\prime}\gamma_{0}vw%
_{0}j(x)g)\,\psi_{U_{2n,n}}(u)\,du\,dy\,dv\,dx.$$
Then once again [6], Lemma 1, implies that integral
(30) is not zero for some choice of data if and only if the inner integration along
the quotient $U_{2n,n}(F)\backslash U_{2n,n}({\mathbb{A}})$ is not zero for some choice of data. The
nonvanishing of this integral follows from Proposition 1
with $a=n$ and the genericity of the representation $\Theta_{GL_{n}}^{(n+1)}$ [12].
To complete the proof of the Theorem we need to consider the case when $n$ is odd and $r=n$. In
this case we have $l=(n+1)/2$. To use formulas from the previous case, we change the notation, and now
let $P^{\prime}_{l}$ be the standard parabolic subgroup of $Sp_{2n}$ whose Levi part is $GL_{1}\times GL_{2}^{{(n-1)}/{2}}$ and $U^{\prime}_{l}$ be its unipotent radical. (Above we would have written $P^{\prime}_{l-1}$, $U^{\prime}_{l-1}$.)
We again
start with the integral (24), define $w_{0}$ as above, and after conjugation we obtain the
integral (26), with the groups $U^{\prime}_{l,1}$ and $V_{1}$ changed as follows.
Now $U^{\prime}_{l,1}$ is the subgroup of $U^{\prime}_{l}$ consisting of $u=(u_{i,j})\in U^{\prime}_{l}$ such that $u_{2m-1,2k}=0$ for all $1\leq m\leq l-1$
and $m\leq k\leq l-1$, and $V_{1}$ is the unipotent subgroup of $Sp_{2n}$ consisting of all matrices of the form $I_{2n}+\sum_{i,j}r_{i,j}e_{i,j}^{*}$ where the sum ranges over all pairs $(i,j)=(2k,2m-1)$ where $1\leq m\leq l-1$ and $m-1\leq k\leq l-2$.
Let $Z_{l}$ now be the subgroup of $U^{\prime}_{l}$ consisting of all matrices of the form
$I_{2n}+\sum_{i,j}r_{i,j}e_{i,j}^{*}$ where the sum is over all pairs $(i,j)=(2m-1,2k)$ where
$1\leq m\leq l-1$ and $m\leq k\leq l-1$ together with the condition that $r_{1,2}=0$. Notice that
in this case it is not true that $U^{\prime}_{l}=Z_{l}\cdot U^{\prime}_{l,1}$. Performing root exchange similarly to our treatment of the
integral (26) we obtain
$$\int\limits_{{\mathbb{A}}^{n-r^{\prime}}}\int\limits_{V_{1}({\mathbb{A}})}\int%
\limits_{U^{\prime}_{l,0}(F)\backslash U^{\prime}_{l,0}({\mathbb{A}})}\omega_{%
\psi}(g)\phi(x)\,\theta_{2n}^{(n)}(uvw_{0}j(x)g)\,\psi_{U^{\prime}_{l},1,b}(u)%
\,du\,dv\,dx.$$
Here, $U^{\prime}_{l,0}$ is the subgroup of $U^{\prime}_{l}$ consisting of all matrices $u=(u_{i,j})\in U^{\prime}_{l}$
such that $u_{1,2}=0$, and the character $\psi_{U^{\prime}_{l},1,b}$ is given by
$$\psi_{U^{\prime}_{l},1,b}(u)=\psi\left(\sum_{i=1}^{n-2}u_{i,i+2}+u_{n-1,n+2}+%
bu_{n,n+1}\right).$$
Expand the above integral along the group $z(y)=I_{2n}+ye_{1,2}^{*}$. We obtain
$$\int\limits_{{\mathbb{A}}^{n-r^{\prime}}}\int\limits_{V_{1}({\mathbb{A}})}\sum%
_{\alpha\in F}\int\limits_{F\backslash{\mathbb{A}}}\int\limits_{U^{\prime}_{l,%
0}(F)\backslash U^{\prime}_{l,0}({\mathbb{A}})}\omega_{\psi}(g)\phi(x)\,\theta%
_{2n}^{(n)}(z(y)uvw_{0}j(x)g)\,\psi_{U^{\prime}_{l},1,b}(u)\,\psi(\alpha y)\,%
du\,dy\,dv\,dx.$$
Assume that $b=-1/4$. Then it is not hard to check that if $\alpha\neq-1/2$ then the two inner
integrations give a coefficient which is associated to the unipotent orbit $((n+1)(n-1))$.
Since $n$ is odd this is a symplectic partition. However, it follows from Conjecture 1${}_{n,n}$
that the unipotent orbit associated with $\Theta_{2n}^{(n)}$ is $(n^{2})$. Therefore, the only
nonzero contribution to the above integral is from $\alpha=1/2$. Let $\gamma=\begin{pmatrix}1&\\
1&1\end{pmatrix}\begin{pmatrix}1&-\tfrac{1}{2}\\
&1\end{pmatrix}$ and let
$\gamma_{0}=\text{diag}(\gamma,\ldots,\gamma,\gamma^{*},\ldots,\gamma^{*})\in Sp%
_{2n}(F)$. Using the invariance of
$\theta_{2n}^{(n)}$ by
this element and moving it rightward, we obtain the integral
(31)
$$\int\limits_{{\mathbb{A}}^{n-r^{\prime}}}\int\limits_{V_{1}({\mathbb{A}})}\int%
\limits_{U^{\prime}_{l}(F)\backslash U^{\prime}_{l}({\mathbb{A}})}\omega_{\psi%
}(g)\phi(x)\,\theta_{2n}^{(n)}(u\gamma vw_{0}j(x)g)\,\psi_{U^{\prime}_{l}}(u)%
\,du\,dv\,dx,$$
where $\psi_{U^{\prime}_{l}}$ is given by
$$\psi_{U^{\prime}_{l}}(u)=\psi(u_{1,2}+u_{2,4}+u_{3,5}+\cdots+u_{n-3,n-1}+u_{n-%
2,n}+u_{n-1,n+1}).$$
Next we introduce the Weyl element $w_{0}^{*}\in Sp_{2n}$ defined by
$$w_{0}^{*}=\begin{pmatrix}1&&\\
&w_{0}^{\prime}&\\
&&1\end{pmatrix}.$$
(Here $w_{0}^{\prime}$ was defined following (28).) Using invariance by $w_{0}^{*}$ and moving this element rightward,
(31)
is equal to
$$\int\limits_{{\mathbb{A}}^{n-r^{\prime}}}\int\limits_{V_{1}({\mathbb{A}})}\int%
\limits_{Y_{0}(F)\backslash Y_{0}({\mathbb{A}})}\int\limits_{U_{2n,n,0}(F)%
\backslash U_{2n,n,0}({\mathbb{A}})}\omega_{\psi}(g)\phi(x)\,\theta_{2n}^{(n)}%
(uy_{0}w_{0}^{*}\gamma_{0}vw_{0}j(x)g)\,\psi_{U_{2n,n}}(u)\,du\,dy_{0}\,dv\,dx.$$
Here $U_{2n,n,0}$ is now the subgroup of $U_{2n,n}$ of $u=(u_{i,j})\in U_{2n,n}$
such that $u_{i,j}=0$ for $2\leq i\leq n$ and $n+1\leq j\leq n+i-1$, and $Y_{0}$ is the group of all matrices of the form
$$y_{0}=\begin{pmatrix}1&&\\
&y^{\prime}&\\
&&1\end{pmatrix}$$
such that $y^{\prime}\in Y$ (where $Y$, as well as the character $\psi_{U_{2n,n}}$, was defined following (29)).
Carrying out root exchanges similarly to our treatment of (29), we obtain the integral
(32)
$$\int\limits_{{\mathbb{A}}^{n-r^{\prime}}}\int\limits_{V_{1}({\mathbb{A}})}\int%
\limits_{Y_{0}({\mathbb{A}})}\int\limits_{U_{2n,n}(F)\backslash U_{2n,n}({%
\mathbb{A}})}\omega_{\psi}(g)\phi(x)\,\theta_{2n}^{(n)}(uy_{0}w_{0}^{*}\gamma_%
{0}vw_{0}j(x)g)\,\psi_{U_{2n,n}}(u)\,du\,dy_{0}\,dv\,dx.$$
It is not hard to check that the integral (32) is not zero for some choice of data.
To complete the proof of the Theorem, observe that Proposition 1
allows us to express the integral of $\psi_{U_{2n,n}}$ times any right translate of $\theta_{2n}^{(n)}$ over
$U_{2n,n}(F)\backslash U_{2n,n}({\mathbb{A}})$ in terms of the Whittaker function of the representation $\Theta_{GL_{n}}^{(n)}$.
Since the Whittaker functional of this representation is unique up to scalars (see Kazhdan-Patterson [12], Theorem
II.2.1), we conclude that the integral (32) is
factorizable provided the data $\theta_{2n}^{(n)}\in\Theta_{2n}^{(n)}$ and $\phi$ are factorizable.
∎
6. Some Local Computations
We have just seen that for factorizable data the Whittaker coefficient of
$\sigma_{n+1}^{(2n)}$, $n$ odd, can be expressed as a product of local Whittaker functionals. We now analyze the corresponding
local integral in the unramified case. Consider a
place $\nu$ where all data is unramified. In this Section, $F$ denotes the corresponding local field, and in
all integrals we understand that we are taking the $F$-valued points of the algebraic groups that are shown.
We continue to use the convention that all matrices are embedded in the corresponding metaplectic groups via the trivial section, unless otherwise noted.
In this section, let $T_{2n}^{(n)}$ denote the inverse image in the local $n$-fold metaplectic group of the
standard maximal torus of $Sp_{2n}(F)$,
and let $Z(T_{2n}^{(n)})$ denote its center.
Let $p$ be a generator of the maximal ideal in the ring of integers ${\mathcal{O}}$ of $F$.
Since $n$ is odd, $-1$ is an $n$-th power
and hence $(p,p)=1$. This implies that $(p^{k_{1}},p^{k_{2}})=1$ for all integers $k_{1}$, $k_{2}$.
The subgroup of $T_{2n}^{(n)}$ generated by $Z(T_{2n}^{(n)})$ and by all matrices
$\text{diag}(p^{k_{1}},\ldots,p^{k_{n}},p^{-k_{n}},\ldots,p^{-k_{1}})$ is a maximal abelian subgroup of
$T_{2n}^{(n)}$. Denote this subgroup by $T_{2n,0}^{(n)}$. For a diagonal matrix
$t=\text{diag}(a,a^{*})\in Sp_{2n}(F)$ we shall denote $t_{0}=a\in GL_{n}(F)$. Also, recall that since we are in the unramified case the
maximal compact subgroup $Sp_{2n}({\mathcal{O}})$ of $Sp_{2n}(F)$ splits under the $n$-fold cover; fix an
injection $\iota$
of $Sp_{2n}({\mathcal{O}})$ into $Sp_{2n}^{(n)}(F)$.
Let $f_{W}^{(n)}$ be the function on $Sp_{2n}(F)$ defined as follows. First, it is right-invariant under
$\iota(Sp_{2n}({\mathcal{O}}))$.
Second, $f_{W}^{(n)}(ug)=\psi_{U_{2n,n}}(u)f_{W}^{(n)}(g)$ for all $u\in U_{2n,n}(F)$. (The character $\psi_{U_{2n,n}}$ was defined
before (30) above.) Finally, for all $(t,\zeta)\in T_{2n,0}^{(n)}$ ($t=\text{diag}(a,a^{*})\in Sp_{2n}(F)$, $\zeta\in\mu_{n})$ we have
$f_{W}^{(n)}((t,\zeta))=\zeta\,\delta_{Q}^{\frac{n-1}{2n}}(t)\,W_{GL_{n}}^{(n)}%
((t_{0},1))$. Here $W_{GL_{n}}^{(n)}$
is the normalized Whittaker function for the local theta representation of $GL_{n}^{(n)}$ that is unramified with respect to the
subgroup $GL_{n}(\mathcal{O})$ embedded in $GL_{n}^{(n)}(F)$ in a way that is compatible with $\iota$,
and $Q=P_{2n,n}$ is the maximal parabolic subgroup of $Sp_{2n}$ whose Levi part is $GL_{n}$.
(Recall from [12] that the space of
Whittaker
functionals for the theta function of $GL_{n}^{(n)}(F)$ is one dimensional.)
Consider the integral
(33)
$$W_{Sp_{n+1}}^{(2n)}(g)=\int\limits_{F^{n-r^{\prime}}}\int\limits_{V_{1}}\int%
\limits_{Y_{0}}\omega_{\psi}(g)\phi(x)\,f_{W}^{(n)}(y_{0}w_{0}^{*}\gamma_{0}vw%
_{0}j(x)g)\,dy_{0}\,dv\,dx,$$
where the function $\phi$ is the normalized unramified Schwartz function.
It may be desirable to compute $W_{Sp_{n+1}}^{(2n)}(g)$ for an arbitrary torus element; however,
a general formula would be quite complicated. We will concentrate on a special
case (though for $n=3$ it is general)
where the computations, though not trivial, are simpler. Let $n_{1},n_{2}$ be non-negative integers,
$a=p^{n_{1}}$ and $b=p^{n_{2}}$, and let
$$g=\text{diag}(ab,b,I_{n-3},b^{-1},a^{-1}b^{-1}).$$
Substituting this element in (33), changing variables
in $x$ and using the invariant properties of the functions involved, we obtain
$$\gamma(a)\,|ab^{2}|^{-1/2}\int\limits_{V_{1}}\int\limits_{Y_{0}}f_{W}^{(n)}(y_%
{0}w_{0}^{*}\gamma_{0}vw_{0}g)\,dy_{0}\,dv.$$
Here $\gamma(a)$ is the Weil factor and the factor $\gamma(a)\,|ab^{2}|^{-1/2}$ is obtained from the change
of variables in $x$ together with the properties of the Weil representation. We can ignore the
integration over $x$ since $\phi$ is the unramified Schwartz function.
The matrices $\gamma_{0}$ and $\gamma$ were defined before (5) above. We have
the factorization
$\gamma=\begin{pmatrix}1&1\\
&1\end{pmatrix}\begin{pmatrix}&-1\\
1&\end{pmatrix}\begin{pmatrix}1&\tfrac{1}{2}\\
&1\end{pmatrix},$ and this induces a factorization of $\gamma_{0}$:
$\gamma_{0}=\gamma_{0}^{\prime}w_{0}^{\prime}\gamma_{0}^{\prime\prime}$. Changing variables in $V_{1}$ we may ignore the matrix $\gamma_{0}^{\prime\prime}$.
Moving $\gamma_{0}^{\prime}$ to the left and using the left transformation of $f_{W}^{(n)}$ by matrices
in $U_{2n,n}$, we obtain the integral
$$\gamma(a)\,|ab^{2}|^{-1/2}\int\limits_{V_{1}}\int\limits_{Y_{0}}f_{W}^{(n)}(y_%
{0}w_{0}^{*}w_{0}^{\prime}vw_{0}g)\,\psi_{Y_{0}}(y_{0})\,dy_{0}\,dv.$$
Here we recall that $n$ is odd and that $Y_{0}$ consists of all matrices of
the form
$$y_{0}=\begin{pmatrix}1&&&\\
&I_{n-1}&&\\
&y&I_{n-1}&\\
&&&1\end{pmatrix}\quad\text{with~{}}y\in\text{Mat}_{(n-1)\times(n-1)},y_{i,j}=%
0~{}\text{for all~{}}i\geq j;$$
we have $\psi_{Y_{0}}(y_{0})=\psi(y_{\frac{n-1}{2},\frac{n+1}{2}})$.
Next we move the matrix $g$ to the left and the Weyl element $w_{0}^{*}w_{0}^{\prime}w_{0}$ to the right. Doing so, we
obtain the integral
(34)
$$\gamma(a)\,|ab^{2}|^{-\frac{n-2}{2}}\int\limits_{V_{2}}\int\limits_{Y_{0}}f_{W%
}^{(n)}(ty_{0}v_{2})\,\psi_{Y_{0}}(y_{0})\,dy_{0}\,dv_{2}$$
with $t=\text{diag}(ab,b,I_{2n-4},b^{-1},a^{-1}b^{-1})$. The group
$V_{2}$ consists of all matrices of the form $\left(\begin{smallmatrix}I_{n}&\\
v&I_{n}\end{smallmatrix}\right)$ where
$v=\sum_{i,j}r_{i,j}(e_{i,j}+e_{n-j+1,n-i+1})$ and the sum is over all pairs $(i,j)$ such that $1\leq i\leq\frac{n-3}{2}$ and $1\leq j\leq i+1$. The extra factor of
$|ab^{2}|^{-\frac{n-3}{2}}$ is obtained from the change of variables in $V_{1}$.
Integral (34) is the analogous to the integral that appears on the right hand side of
Ginzburg-Rallis-Soudry [10], p. 255, Eqn. (3.7). Indeed, the matrices $y_{0}$ and $v_{2}$ can be viewed
as matrices of $Sp_{2n-2}$ embedded in $Sp_{2n}$ in the obvious way. This comparison enables
us to proceed as in [10], pp. 256-260, and conclude that integral (34) is a sum
of two terms. First we have
$$\gamma(a)\,|ab^{2}|^{-\frac{n-2}{2}}f_{W}^{(n)}(t)=\gamma(a)|ab^{2}|^{\frac{2n%
-1}{2n}}W_{GL_{n}}^{(n)}(t_{0}).$$
This corresponds to the term which is denoted as Case 1 in [10], p. 260. The second term
we obtain is the one corresponding to Case 2 in that reference. It is equal to
(35)
$$\gamma(a)|ab^{2}|^{-\frac{n-2}{2}}\int\limits_{|h|>1}f_{W}^{(n)}(ty(h))\,\psi(%
h)\,|h|^{\frac{(n-1)(n-3)}{2}}\,dh,$$
with $y(h)=\text{diag}(I_{2},h^{-1}I_{n-2},hI_{n-2},I_{2})$. Write $h=p^{-m}\epsilon$ where $\epsilon$
is a unit. Then (35) is equal to
$$\gamma(a)\,|ab^{2}|^{-\frac{n-2}{2}}\sum_{m=1}^{\infty}q^{m\left(1+\frac{(n-1)%
(n-3)}{2}\right)}\int\limits_{|\epsilon|=1}f_{W}^{(n)}(ty(p^{-m}\epsilon))\,%
\psi(p^{-m}\epsilon)\,d\epsilon.$$
The factorization $y(p^{-m}\epsilon)=y(p^{-m})y(\epsilon)$ contributes, via the cocycle defining the metaplectic group, the factor
$(\epsilon,p)_{n}^{m(n-2)}$. Hence the above integral is equal to
(36)
$$\gamma(a)\,|ab^{2}|^{-\frac{n-2}{2}}\sum_{m=1}^{\infty}q^{m\left(1+\frac{(n-1)%
(n-3)}{2}\right)}f_{W}^{(n)}(ty(p^{-m}))\int\limits_{|\epsilon|=1}(\epsilon,p)%
_{n}^{m(n-2)}\,\psi(p^{-m}\epsilon)\,d\epsilon.$$
The inner integral is zero unless $m=1$, and when $m=1$ it is equal to $q^{-1/2}G_{n-2}^{(n)}(p)$
where the last term is a normalized $n$-th order Gauss sum, as in [6], Section 6. Substituting this into (36) we
obtain
$$\gamma(a)\,|ab^{2}|^{-\frac{n-2}{2}}q^{-\frac{(n-2)(2n-1)}{2n})}W_{GL_{n}}^{(n%
)}(t_{0}y(p)_{0}).$$
Thus we have established the following result, relating the Whittaker coefficients of the descent on a suitable
cover of the symplectic group to the Whittaker coefficients of the theta function on a cover of the general linear group.
Theorem 2.
Let $n\geq 3$ be an odd integer. Let $a=p^{n_{1}}$, $b=p^{n_{2}}$, with $n_{1},n_{2}\geq 0$. Let
$$t=\text{\rm diag}(ab,b,I_{n-3},b^{-1},a^{-1}b^{-1})\in Sp_{n+1}(F),$$
$$t_{0}=\text{\rm diag}(ab,b,I_{n-2})\in GL_{n}(F),\quad t_{1}=\text{\rm diag}(%
ab,b,pI_{n-2})\in GL_{n}(F).$$
Suppose that Conjecture 1${}_{n,i}$ holds for $(n-1)/2\leq i\leq n$. Then we have the identity
(37)
$$W_{Sp_{n+1}}^{(2n)}(t)=\gamma(a)\,|ab^{2}|^{\frac{2n-1}{2n}}\left(W_{GL_{n}}^{%
(n)}(t_{0})+q^{-\frac{(n-2)(2n-1)}{2n}}W_{GL_{n}}^{(n)}(t_{1})\right).$$
7. A Case of Conjecture 1
Suppose $n=3$. Then Conjecture 1${}_{3,i}$ is known for $i=1$ ([12], Corollary II.2.6) and $i=2$ ([6],
Proposition 3).
In this section we sketch the proof of Conjecture 1${}_{3,3}$.
Proposition 3.
Conjecture 1${}_{3,3}$ holds. That is, ${\mathcal{O}}(\Theta_{6}^{(3)})=(3^{2})$.
As a consequence, the hypotheses of Theorem 2 are satisfied for $n=3$, and we conclude unconditionally the
Whittaker function for the descent on the $6$-fold cover of $Sp_{4}$ is Eulerian, and
given at unramified places by formula (37).
Proof.
To prove the Proposition we need to prove that the representation $\Theta_{6}^{(3)}$ has no nonzero Fourier coefficients
corresponding to the unipotent orbits $(6)$ and $(42)$. Once we do so, an argument similar to the one given in
Ginzburg [7], Thm. 3.1, will imply that this representation also has no nonzero Fourier coefficients corresponding to the orbit $(41^{2})$. We also need to show that $\Theta_{6}^{(3)}$ has a nonzero Fourier coefficient
corresponding to the unipotent orbit $(3^{2})$. This last follows from Proposition 1. So the critical step is the
vanishing of Fourier coefficients attached to the two unipotent orbits $(6)$ and $(42)$.
For convenience, we drop the prior notation for unipotent subgroups and characters, and will introduce the ones needed anew.
To prove this claim for $\Theta_{6}^{(3)}$ we make use of the fact that this representation can be obtained as a residue of an Eisenstein series induced from a suitable theta function of a smaller rank group. This is explained in Section 2 above.
First we consider the unipotent orbit $(6)$.
Since $\Theta_{6}^{(3)}$ is a residue of $E_{\Theta_{4}}^{(3)}(g,s)$, in order to prove that
$\Theta_{6}^{(3)}$ is not generic it is enough to prove that $E_{\Theta_{4}}^{(3)}(g,s)$ is not generic. Moreover, let $U$ denote the standard maximal unipotent of $Sp_{6}$, and let $\psi_{U}$ denote a Whittaker character of $U$;
without loss we take $\psi_{U}(u)=\psi(u_{1,2}+u_{2,3}+au_{3,4})$ with
$a\in F^{*}$. Then a standard unfolding argument shows that the integral
$$\int\limits_{U(F)\backslash U({\mathbb{A}})}E_{\Theta_{4}}^{(3)}(ug,s)\,\psi_{%
U}(u)\,du$$
is not zero for some choice of data if and only if the representation
$\Theta_{4}^{(3)}$ is generic. However, it follows from Friedberg-Ginzburg
[6], Lemma 2, that this representation is not generic. Hence, for $\text{Re}(s)$ large, the above integral is zero for all choices of data, and by continuation it follows that $\Theta_{6}^{(3)}$ is not generic.
Next we consider the Fourier coefficients corresponding to the unipotent orbit $(42)$. Let $V\subseteq U$ denote the subgroup
$V=\{u=(u_{i,j})\in U\ :\ u_{2,3}=u_{4,5}=0\}$. This is the unipotent radical of the parabolic subgroup of $Sp_{6}$ whose Levi part is $L=GL_{1}\times GL_{2}$. The embedding of $L$ into $Sp_{6}$ is given by $L=\{\text{diag}(g_{1},g_{2},g_{2}^{*},g_{1}^{-1})\ :\ g_{1}\in GL_{1};\ g_{2}%
\in GL_{2}\}$. Let $\alpha_{1},\alpha_{2}\in F$ and
$\alpha_{3},\alpha_{4}\in F^{*}$. Consider the Fourier coefficient defined by
(38)
$$\int\limits_{V(F)\backslash V({\mathbb{A}})}\varphi(vg)\,\psi_{V,\alpha_{i}}(v%
)\,dv,$$
where $\psi_{V,\alpha_{i}}(v)=\psi(\alpha_{1}v_{1,2}+\alpha_{2}v_{1,3}+\alpha_{3}v_{3%
,4}+\alpha_{4}v_{2,5})$.
This Fourier coefficient is
associated with the unipotent orbit $(42)$ provided the connected component of the stabilizer of $\psi_{V,\alpha_{i}}$ inside
$L(F)$ is trivial (see for example Ginzburg [7]).
If $\alpha_{3}\alpha_{4}=-\epsilon^{2}$ for some $\epsilon\in F^{*}$, then one can find a suitable discrete element $\gamma_{L}\in L$
and $\alpha\in F^{*}$ such that the integral (38) is equal to
(39)
$$\int\limits_{V(F)\backslash V({\mathbb{A}})}\varphi(v\gamma_{L}g)\,\widetilde{%
\psi}_{V,\alpha}(v)\,dv,$$
where $\widetilde{\psi}_{V,\alpha}(v)=\psi(\alpha v_{1,2}+v_{1,3}+v_{2,4})$.
We will say that the Fourier coefficient (38) is of split type if $\alpha_{3}\alpha_{4}=-\epsilon^{2}$ for some $\epsilon\in F^{*}$,
and non-split otherwise.
Since at half of the places $\nu$ we have the condition $\alpha_{3}\alpha_{4}=-\epsilon^{2}$ for some $\epsilon\in F_{\nu}^{*}$,
to complete the proof it is enough to
prove that for every finite local unramified place, the Jacquet
module $J_{V,\widetilde{\psi}_{V,\alpha}}(\Theta_{6}^{(3)})$ is zero for all characters $\widetilde{\psi}_{V,\alpha}$
with $\alpha\in F^{*}$, where $\Theta_{6}^{(3)}$ now denotes the local theta representation at $\nu$.
To do so we consider the Jacquet module associated with the unipotent orbit $(3^{2})$.
Let $R$ denote the subgroup of $U$ defined by $R=\{u\in U\ :\ u_{1,2}=u_{3,4}=u_{5,6}=0\}$.
Then $R$ is the unipotent radical of the maximal parabolic subgroup
of $Sp_{6}$ whose Levi part is $GL_{2}\times SL_{2}$.
Let $\psi_{R}$ denote the character $\psi_{R}(r)=\psi(r_{1,3}+r_{2,4})$ of $R$ (that is, of $R(F_{\nu})$; from now on we take the $F_{\nu}$
points of the algebraic groups in question without changing the notation).
Next, we compute
$J_{RN,\psi_{R}\psi_{\alpha}}(\Theta_{6}^{(3)})$, where $N$ is the unipotent radical of $SL_{2}$ embedded in $Sp_{6}$ as the group of all matrices
$$n(x)=\begin{pmatrix}1&x&&&&\\
&1&&&&\\
&&1&x&&\\
&&&1&&\\
&&&&1&-x\\
&&&&&1\end{pmatrix},$$
and $\psi_{\alpha}$ is the
character of $N$ defined by $\psi_{\alpha}(n(x))=\psi(\alpha x)$, with
$\alpha\in F_{\nu}$.
Using the one-parameter subgroups $\{z(k)=I_{6}+ke_{3,4}\}$ and $\{y(m)=I_{6}+me_{2,3}-me_{4,5}\}$,
we perform root exchanges using the Lemma in Section 2 of [9].
This gives the isomorphism of Jacquet modules
(40)
$$J_{RN,\psi_{R}\psi_{\alpha}}(\Theta_{6}^{(3)})\cong J_{V,\widetilde{\psi}_{V,%
\alpha}}(\Theta_{6}^{(3)}).$$
Thus, to prove Prop. 3, it is enough to prove that for all
$\alpha\neq 0$, the Jacquet module $J_{RN,\psi_{R}\psi_{\alpha}}(\Theta_{6}^{(3)})$ is zero.
It is not hard to check that the stabilizer of $\psi_{R}$ inside $GL_{2}\times SL_{2}$ is $SL_{2}$ embedded diagonally. Thus,
$J_{R,\psi_{R}}(\Theta_{6}^{(3)})$, if not zero, defines an unramified
representation of $SL_{2}$.
Let $\sigma$ denote an irreducible unramified representation of
$SL_{2}$ such that the space
$$\text{Hom}_{SL_{2}}(\sigma,J_{R,\psi_{R}}(\Theta_{6}^{(3)}))$$
is not zero. Since $\sigma$ is unramified it is a sub-representation
of $\text{Ind}_{B}^{SL_{2}}\chi\delta_{B}^{1/2}$. Applying Frobenius reciprocity
we deduce that the space
$$\text{Hom}_{GL_{1}}(\chi,J_{RN,\psi_{R}}(\Theta_{6}^{(3)}))$$
is not zero. We shall show below that we must have $\chi=1$, which implies that $\sigma=1$.
Since the Jacquet module is always of finite length, we then conclude that all Jacquet modules
$J_{RN,\psi_{R}\psi_{\alpha}}(\Theta_{6}^{(3)})$ with $\alpha\neq 0$ are
zero. Thus once we have established that $\chi$ must be $1$ (and in particular,
that $J_{R,\psi_{R}}(\Theta_{6}^{(3)})$ is nonzero), the Proposition will be proved.
To do so, we use the isomorphism (40) above with $\alpha=0$.
If $\alpha=0$, then the Jacquet module $J_{RN,\psi_{R}\psi_{0}}(\Theta_{6}^{(3)})=J_{RN,\psi_{R}}(\Theta_{6}^{(3)})%
\cong J_{V,\psi_{V}}(\Theta_{6}^{(3)})$, if not zero, defines an unramified character of $GL_{1}$, which is embedded in $Sp_{6}$ as the
group of all matrices $\{t(a)\}$, with $t(a)={\text{diag}}(a,a^{-1},a,a^{-1},a,a^{-1})$.
Here $\psi_{V}=\widetilde{\psi}_{V,\alpha}$ with $\alpha=0$.
For $1\leq i\leq 3$, let $w_{i}$ denote the three simple reflections of the Weyl group of $Sp_{6}$. In matrices we have
$$w_{1}=\begin{pmatrix}&1&&&&\\
1&&&&&\\
&&1&&&\\
&&&1&&\\
&&&&&1\\
&&&&1&\end{pmatrix};\ \ w_{2}=\begin{pmatrix}1&&&&&\\
&&1&&&\\
&1&&&&\\
&&&&1&\\
&&&1&&\\
&&&&&1\end{pmatrix};\ \ w_{3}=\begin{pmatrix}1&&&&&\\
&1&&&&\\
&&&1&&\\
&&-1&&&\\
&&&&1&\\
&&&&&1\end{pmatrix}.$$
Let $Y=\{y_{1}(m_{1})=I_{6}+m_{1}e_{4,3}\}$, and let $U_{1}$ be the subgroup of the maximal unipotent subgroup $U$ of $Sp_{6}$ consisting
of all matrices $u=(u_{i,j})\in U$ such that $u_{2,4}=u_{3,5}=u_{3,4}=0$. Let $\psi_{U_{1}}(u_{1})=\psi(u_{1}(1,2)+u_{1}(2,3))$. Then
$U_{1}Y=(w_{3}w_{2})V(w_{3}w_{2})^{-1}$.
Hence, $J_{V,\psi_{V}}(\Theta_{6}^{(3)})$ is isomorphic to
$J_{U_{1},\psi_{U_{1}}}(\Theta_{6}^{(3)})$. Let $X=\{I_{6}+ke_{2,4}+ke_{3,5}\}$, and perform a root exchange of this unipotent subgroup with the
group $Y$. Then, since $\Theta_{6}^{(3)}$ is not generic,
we obtain that $J_{U_{1},\psi_{U_{1}}}(\Theta_{6}^{(3)})$ is isomorphic
to $J_{U,\psi_{U}}(\Theta_{6}^{(3)})$.
This Jacquet module is not zero. Indeed, using an argument similar to
[3] Theorem 2.3, the nonvanishing of this Jacquet module
follows from the genericity of the theta representation of $GL_{3}^{(3)}$.
From this we deduce that the Jacquet module
$J_{R,\psi_{R}}(\Theta_{6}^{(3)})$ is not zero.
Moreover, the Jacquet module $J_{U,\psi_{U}}(\Theta_{6}^{(3)})$ acts by the character $|a|^{4}$ under the torus
$\text{diag}(aI_{3},a^{-1}I_{3})\in Sp_{6}$. Taking into an account the
various root exchanges, we deduce that the group
$GL_{1}=\{t(a)\}$ acts trivially on the Jacquet module $J_{RN,\psi_{R}}(\Theta_{6}^{(3)})$.
Thus $\chi$ above must be $1$.
This completes the proof of the Proposition.
∎
References
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Block-compatible metaplectic cocycles.
J. Reine Angew. Math. 507 (1999), 131-163.
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Small representations for odd orthogonal groups. Int. Math. Res. Not. 2003,
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$GL_{2n}$ and $\widetilde{Sp}_{2n}$. J. Amer. Math. Soc. 12 (1999), no. 3, 849-907.
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Ginzburg, David; Rallis, Stephen; Soudry, David. Lifting cusp forms on $GL_{2n}$ to $\widetilde{Sp}_{2n}$: the unramified correspondence. Duke Math. J. 100 (1999), no. 2, 243-266.
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Ginzburg, David; Rallis, Stephen; Soudry, David. The Descent Map from Automorphic Representations of $GL(n)$ to Classical Groups. World Scientific Publishing Co. Pte. Ltd., Singapore, 2011.
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Kazhdan, David A.; Patterson, S. J. Metaplectic forms.
Publ. Math. Inst. Hautes Études Sci. 59 (1984), 35-142.
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S. S. Kudla; S. Rallis. Poles of Eisenstein series and $L$-functions. Isr. Math. Conf. Proc., Vol. 3, Weizmann Science Press of Isr., Jerusalem (1990), pp. 81-110.
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Une paraphrase de l’Écriture [A paraphrase of Scripture].
Cambridge Tracts in Mathematics, 113. Cambridge University Press, Cambridge, 1995. |
Value-Decomposition Networks For Cooperative Multi-Agent Learning
Peter Sunehag
DeepMind
[email protected] &Guy Lever
DeepMind
[email protected] &Audrunas Gruslys
DeepMind
[email protected] &Wojciech Marian Czarnecki
DeepMind
[email protected] &Vinicius Zambaldi
DeepMind
[email protected]
&Max Jaderberg
DeepMind
[email protected] &Marc Lanctot
DeepMind
[email protected] &Nicolas Sonnerat
DeepMind
[email protected] &Joel Z. Leibo
DeepMind
[email protected] &Karl Tuyls
DeepMind & University of Liverpool
[email protected] &Thore Graepel
DeepMind
[email protected]
Abstract
We study the problem of cooperative multi-agent reinforcement learning with a single joint reward signal. This class of learning problems is difficult because of the often large combined action and observation spaces. In the fully centralized and decentralized approaches, we find the problem of spurious rewards and a phenomenon we call the “lazy agent” problem, which arises due to partial observability. We address these problems by training individual agents with a novel value decomposition network architecture, which learns to decompose the team value function into agent-wise value functions. We perform an experimental evaluation across a range of partially-observable multi-agent domains and show that learning such value-decompositions leads to superior results, in particular when combined with weight sharing, role information and information channels.
Value-Decomposition Networks For Cooperative Multi-Agent Learning
Peter Sunehag
DeepMind
[email protected]
Guy Lever
DeepMind
[email protected]
Audrunas Gruslys
DeepMind
[email protected]
Wojciech Marian Czarnecki
DeepMind
[email protected]
Vinicius Zambaldi
DeepMind
[email protected]
Max Jaderberg
DeepMind
[email protected]
Marc Lanctot
DeepMind
[email protected]
Nicolas Sonnerat
DeepMind
[email protected]
Joel Z. Leibo
DeepMind
[email protected]
Karl Tuyls
DeepMind & University of Liverpool
[email protected]
Thore Graepel
DeepMind
[email protected]
1 Introduction
We consider the cooperative multi-agent reinforcement learning (MARL) problem (Panait and Luke, 2005, Busoniu et al., 2008, Tuyls and Weiss, 2012), in which a system of several learning agents must jointly optimize a single reward signal – the team reward – accumulated over time. Each agent has access to its own (“local”) observations and is responsible for choosing actions from its own action set. Coordinated MARL problems emerge in applications such as coordinating self-driving vehicles and/or traffic signals in a transportation system, or optimizing the productivity of a factory comprised of many interacting components. More generally, with AI agents becoming more pervasive, they will have to learn to coordinate to achieve common goals.
Although in practice some applications may require local autonomy, in principle the cooperative MARL problem could be treated using a centralized approach, reducing the problem to single-agent reinforcement learning (RL) over the concatenated observations and combinatorial action space. We show that the centralized approach consistently fails on relatively simple cooperative MARL problems in practice. We present a simple experiment in which the centralised approach fails by learning inefficient policies with only one agent active and the other being ‘‘lazy’’. This happens when one agent learns a useful policy, but a second agent is discouraged from learning because its exploration would hinder the first agent and lead to worse team reward.111For example, imagine training a 2-player soccer team using RL with the number of goals serving as the team reward signal. Suppose one player has become a better scorer than the other. When the worse player takes a shot the outcome is on average much worse, and the weaker player learns to avoid taking shots (Hausknecht, 2016).
An alternative approach is to train independent learners to optimize for the team reward. In general each agent is then faced with a non-stationary learning problem because the dynamics of its environment effectively changes as teammates change their behaviours through learning (Laurent et al., 2011). Furthermore, since from a single agent’s perspective the environment is only partially observed, agents may receive spurious reward signals that originate from their teammates’ (unobserved) behaviour. Because of this inability to explain its own observed rewards naive independent RL is often unsuccessful: for example Claus and Boutilier (1998) show that independent $Q$-learners cannot distinguish teammates’ exploration from stochasticity in the environment, and fail to solve even an apparently trivial, 2-agent, stateless, $3\times 3$-action problem and the general Dec-POMDP problem is known to be intractable (Bernstein et al., 2000, Oliehoek and Amato, 2016). Though we here focus on 2 player coordination, we note that the problems with individual learners and centralized approaches just gets worse with more agents since then, most rewards do not relate to the individual agent and the action space grows exponentially for the fully centralized approach.
One approach to improving the performance of independent learners is to design individual reward functions, more directly related to individual agent observations. However, even in the single-agent case, reward shaping is difficult and only a small class of shaped reward functions are guaranteed to preserve optimality w.r.t. the true objective (Ng et al., 1999, Devlin et al., 2014, Eck et al., 2016). In this paper we aim for more general autonomous solutions, in which the decomposition of the team value function is learned.
We introduce a novel learned additive value-decomposition approach over individual agents. Implicitly, the value decomposition network aims to learn an optimal linear value decomposition from the team reward signal, by back-propagating the total $Q$ gradient through deep neural networks representing the individual component value functions. This additive value decomposition is specifically motivated by avoiding the spurious reward signals that emerge in purely independent learners.The implicit value function learned by each agent depends only on local observations, and so is more easily learned. Our solution also ameliorates the coordination problem of independent learning highlighted in Claus and Boutilier (1998) because it effectively learns in a centralised fashion at training time, while agents can be deployed individually.
Further, in the context of the introduced agent, we evaluate weight sharing, role information and information channels as additional enhancements that have recently been reported to improve sample complexity and memory requirements (Hausknecht, 2016, Foerster et al., 2016, Sukhbaatar et al., 2016). However, our main comparison is between three kinds of architecture; Value-Decomposition across individual agents, Independent Learners and Centralized approaches. We investigate and benchmark combinations of these techniques applied to a range of new interesting two-player coordination domains. We find that Value-Decomposition is a much better performing approach than centralization or fully independent learners, and that when combined with the additional techniques, results in an agent that consistently outperforms centralized and independent learners by a big margin.
1.1 Other Related Work
Schneider et al. (1999) consider the optimization of the sum of individual reward functions, by optimizing local compositions of individual value functions learnt from them. Russell and Zimdars (2003) sums the $Q$-functions of independent learning agents with individual rewards, before making the global action selection greedily to optimize for total reward. Our approach works with only a team reward, and learns the value-decomposition autonomously from experience, and it similarly differs from the approach with coordination graphs (Guestrin et al., 2002) and the max-plus algorithm (Kuyer et al., 2008, van der Pol and Oliehoek, 2016).
Other work addressing team rewards in cooperative settings is based on difference rewards (Tumer and Wolpert, 2004), measuring the impact of an agent’s action on the full system reward. This reward has nice properties (e.g. high learnability), but can be impractical as it requires knowledge about the system state (Colby et al., 2016, Agogino and Tumer, 2008, Proper and Tumer, 2012). Other approaches can be found in Devlin et al. (2014), HolmesParker et al. (2016), Babes et al. (2008).
2 Background
2.1 Reinforcement Learning
We recall some key concepts of the RL setting (Sutton and Barto, 1998), an agent-environment framework (Russell and Norvig, 2010) in which an agent sequentially interacts with the environment over a sequence of timesteps, $t=1,2,3,\ldots$, by executing actions and receiving observations and rewards, and aims to maximize cumulative reward. This is typically modelled as a Markov decision process (MDP) (e.g. Puterman, 1994) defined by a tuple $\langle\mathcal{S,A},\mathcal{T}_{1},\mathcal{T},R\rangle$ comprising the state space $\mathcal{S}$, action space $\mathcal{A}$, a (possibly stochastic) reward function $R:\mathcal{S\times A\times S}\to\mathbb{R}$ start state distribution $\mathcal{T}_{1}\in\mathcal{P(S)}$ and transition function $\mathcal{T}:\mathcal{S\times A}\rightarrow\mathcal{P(S)}$, where $\mathcal{P(X)}$ denotes the set of probability distributions over the set $\mathcal{X}$. We use $\bar{R}$ to denote the expected value of $R$. The agent’s interactions give rise to a trajectory $(S_{1},A_{1},R_{1},S_{2},...)$ where $S_{1}\sim\mathcal{T}_{1}$, $S_{t+1}\sim\mathcal{T}(\cdot|S_{t},A_{t})$ and $R_{t}=R(S_{t},A_{t},S_{t+1})$, and we denote random variables in upper-case, and their realizations in lower-case. At time $t$ the agent observes $o_{t}\in\mathcal{O}$ which is typically some function of the state $s_{t}$, and when the state is not fully observed the system is called a partially observed Markov decision process (POMDP).
The agent’s goal is to maximize expected cumulative discounted reward with a discount factor $\gamma$, ${\cal R}_{t}:=\sum_{t=1}^{\infty}{\gamma^{t-1}R_{t}}$. The agent chooses actions according to a policy: a (stationary) policy is a function $\pi:\mathcal{S}\to\mathcal{P(A)}$ from states to probability distributions over $\mathcal{A}$. An optimal policy is one which maximizes expected cumulative reward. In fully observed environments, stationary optimal policies exist. In partially observed environments, the policy usually incorporates past agent observations from the history $h_{t}=a_{1}o_{1}r_{1},...,a_{t-1}o_{t-1}r_{t-1}$ (replacing $s_{t}$). A practical approach utilized here, is to parameterize policies using recurrent neural networks.
$V^{\pi}(s):=\mathbb{E}[\sum_{t=1}^{\infty}\gamma^{t-1}R(S_{t},A_{t},S_{t+1})|S%
_{1}=s;A_{t}\sim\pi(\cdot|S_{t})]$ is the value function and the action-value function is $Q^{\pi}(s,a):=\mathbb{E}_{S^{\prime}\sim\mathcal{T}(\cdot|s,a)}[R(S,a,S^{%
\prime})+\gamma V(S^{\prime})]$ (generally, we denote the successor state of $s$ by $s^{\prime}$). The optimal value function is defined by $V^{*}(s)=\sup_{\pi}V^{\pi}(s)$ and similarly $Q^{*}(s,a)=\sup_{\pi}Q^{\pi}(s,a)$. For a given action-value function $Q:\mathcal{S}\times\mathcal{A}\to\mathbb{R}$ we define the (deterministic) greedy policy w.r.t. $Q$ by $\pi(s):=\mathop{\arg\max}_{a\in\mathcal{A}}Q(s,a)$ (ties broken arbitrarily). The greedy policy w.r.t. $Q^{*}$ is optimal (e.g. Szepesvári, 2010).
2.2 Deep $Q$-Learning
One method for obtaining $Q^{*}$ is $Q$-learning which is based on the update $Q_{i+1}(s_{t},a_{t})=(1-\eta_{t})Q_{i}(s_{t},a_{t})+\eta_{t}(r_{t}+\gamma\max_%
{a}Q_{i}(s_{t+1},a))$, where $\eta_{t}\in(0,1)$ is the learning rate. We employ the $\varepsilon$-greedy approach to action selection based on a value function, which means that with $1-\varepsilon$ probability we pick $\mathop{\arg\max}_{a}Q_{i}(s,a)$ and with probability $\varepsilon$ a random action. Our study focuses on deep architectures for the value function similar to those used by Mnih et al. (2015), and our approach incorporates the key techniques of target networks and experience replay employed there, making the update into a stochastic gradient step. Since we consider partially observed environments our $Q$-functions are defined over agent observation histories, $Q(h_{t},a_{t})$, and we incorporate a recurrent network similarly to Hausknecht and Stone (2015). To speed up learning we add the dueling architecture of Wang et al. (2016) that represent $Q$ using a value and an advantage function, including multi-step updates with a forward view eligibility trace (e.g. Harb and Precup, 2016) over a certain number of steps. When training agents the recurrent network is updated with truncated back-propagation through time (BPTT) for this amount of steps. Although we concentrate on DQN-based agent architectures, our techniques are also applicable to policy gradient methods such as A3C (Mnih et al., 2016).
2.3 Multi-Agent Reinforcement Learning
We consider problems where observations and actions are distributed across $d$ agents, and are represented as $d$-dimensional tuples of primitive observations in $\mathcal{O}$ and actions in $\mathcal{A}$. As is standard in MARL, the underlying environment is modeled as a Markov game where actions are chosen and executed simultaneously, and new observations are perceived simultaneously as a result of a transition to a new state (Littman, 1994, 2001, Hu and Wellman, 2003, Busoniu et al., 2008).
Although agents have individual observations and are responsible for individual actions, each agent only receives the joint reward, and we seek to optimize ${\cal R}_{t}$ as defined above. This is consistent with the Dec-POMDP framework (Oliehoek et al., 2008, Oliehoek and Amato, 2016).
If we denote $\bar{h}:=(h^{1},h^{2},...,h^{d})$ a tuple of agent histories, a joint policy is in general a map $\pi:\mathcal{H}^{d}\to\mathcal{P}(\mathcal{A}^{d})$; we in particular consider policies where for any history $\bar{h}$, the distribution $\pi(\bar{h})$ has independent components in $\mathcal{P}(\mathcal{A})$. Hence, we write $\pi:\mathcal{H}^{d}\to\mathcal{P}(\mathcal{A})^{d}$. The exception is when we use the most naive centralized agent with a combinatorial action space, aka joint action learners.
3 A Deep-RL Architecture for Coop-MARL
Building on purely independent DQN-style agents (see Figure 2), we add enhancements to overcome the identified issues with the MARL problem. Our main contribution of value-decomposition is illustrated by the network in Figure 2.
The main assumption we make and exploit is that the joint action-value function for the system can be additively decomposed into value functions across agents,
$$Q((h^{1},h^{2},...,h^{d}),(a^{1},a^{2},...,a^{d}))\approx\sum_{i=1}^{d}\tilde{%
Q}_{i}(h^{i},a^{i})$$
where the $\tilde{Q}_{i}$ depends only on each agent’s local observations. We learn $\tilde{Q}_{i}$ by backpropagating gradients from the $Q$-learning rule using the joint reward through the summation, i.e. $\tilde{Q}_{i}$ is learned implicitly rather than from any reward specific to agent $i$, and we do not impose constraints that the $\tilde{Q}_{i}$ are action-value functions for any specific reward. The value decomposition layer can be seen in the top-layer of Figure 2. One property of this approach is that, although learning requires some centralization, the learned agents can be deployed independently, since each agent acting greedily with respect to its local value $\tilde{Q}_{i}$ is equivalent to a central arbiter choosing joint actions by maximizing the sum $\sum_{i=1}^{d}\tilde{Q}_{i}$.
For illustration of the idea consider the case with 2 agents (for simplicity of exposition) and where rewards decompose additively across agent observations222Or, more generally, across agent histories., $r({\mathbf{s}},{\mathbf{a}})=r_{1}(o^{1},a^{1})+r_{2}(o^{2},a^{2})$, where $(o^{1},a^{1})$ and $(o^{2},a^{2})$ are (observations, actions) of agents 1 and 2 respectively. This could be the case in team games for instance, when agents observe their own goals, but not necessarily those of teammates. In this case we have that
$$Q^{\pi}({\mathbf{s}},{\mathbf{a}})={\mathbb{E}}[\sum_{t=1}^{\infty}\gamma^{t-1%
}r({\mathbf{s}}_{t},{\mathbf{a}}_{t})|{\mathbf{s}}_{1}={\mathbf{s}},{\mathbf{a%
}}_{1}={\mathbf{a}};\pi]$$
$$={\mathbb{E}}[\sum_{t=1}^{\infty}\gamma^{t-1}r_{1}(o^{1}_{t},a^{1}_{t})|{%
\mathbf{s}}_{1}={\mathbf{s}},{\mathbf{a}}_{1}={\mathbf{a}};\pi]+{\mathbb{E}}[%
\sum_{t=1}^{\infty}\gamma^{t-1}r_{2}(o^{2}_{t},a^{2}_{t})|{\mathbf{s}}_{1}={%
\mathbf{s}},{\mathbf{a}}_{1}={\mathbf{a}};\pi]$$
$$=:\bar{Q}^{\pi}_{1}({\mathbf{s}},{\mathbf{a}})+\bar{Q}^{\pi}_{2}({\mathbf{s}},%
{\mathbf{a}})$$
where $\bar{Q}^{\pi}_{i}({\mathbf{s}},{\mathbf{a}}):={\mathbb{E}}[\sum_{t=1}^{\infty}%
\gamma^{t-1}r_{1}(o^{i}_{t},a^{i}_{t})|{\mathbf{s}}_{1}={\mathbf{s}},{\mathbf{%
a}}_{1}={\mathbf{a}};\pi],i=1,2$.
The action-value function $\bar{Q}^{\pi}_{1}({\mathbf{s}},{\mathbf{a}})$ – agent 1’s expected future return – could be expected to depend more strongly on observations and actions $(o^{1},a^{1})$ due to agent 1 than those due to agent 2. If $(o^{1},a^{1})$ is not sufficient to fully model $\bar{Q}^{\pi}_{1}({\mathbf{s}},{\mathbf{a}})$ then agent 1 may store additional information from historical observations in its LSTM, or receive information from agent 2 in a communication channel, in which case we could expect the following approximation to be valid
$$Q^{\pi}({\mathbf{s}},{\mathbf{a}})=:\bar{Q}^{\pi}_{1}({\mathbf{s}},{\mathbf{a}%
})+\bar{Q}^{\pi}_{2}({\mathbf{s}},{\mathbf{a}})\approx\tilde{Q}^{\pi}_{1}(h^{1%
},a^{1})+\tilde{Q}^{\pi}_{2}(h^{2},a^{2})$$
Our architecture therefore encourages this decomposition into simpler functions, if possible. We see that natural decompositions of this type arise in practice (see Section 4.4).
One approach to reducing the number of learnable parameters, is to share certain network weights between agents.
Weight sharing also gives rise to the concept of agent invariance, which is useful for avoiding the lazy agent problem.
Definition 1 (Agent Invariance).
If for any permutation (bijection) $p:\{1,...,d\}\to\{1,...,d\}$,
$$\pi(p(\bar{h}))=p(\pi(\bar{h}))$$
we say that $\pi$ is agent invariant.
It is not always desirable to have agent invariance, when for example specialized roles are required to optimize a particular system. In such cases we provide each agent with role information, or an identifier. The role information is provided to the agent as a 1-hot encoding of their identity concatenated with every observation at the first layer. When agents share all network weights they are then only conditionally agent invariant, i.e. have identical policies only when conditioned on the same role. We also consider information channels between agent networks, i.e. differentiable connections between agent network modules. These architectures, with shared weights, satisfy agent invariance.
4 Experiments
We introduce a range of two-player domains, and experimentally evaluate the introduced value-decomposition agents with different levels of enhancements, evaluating each addition in a logical sequence. We use two centralized agents as baselines, one of which is introduced here again relying on learned value-decomposition, as well as an individual agent learning directly from the joint reward signal. We perform this set of experiments on the same form of two dimensional maze environments used by Leibo et al. (2017), but with different tasks featuring more challenging coordination needs. Agents have a small $3\times 5\times 5$ observation window, the first dimension being an RGB channel, the second and third are the maze dimensions, and each agent sees a box 2 squares either side and 4 squares forwards, see Figures 2 and 2. The simple graphics of our domains helps with running speed while, especially due to their multi-agent nature and severe partial observability and aliasing (very small observation window combined with map symmetries), they still pose a serious challenge and is comparable to the state-of-the-art in multi-agent reinforcement learning (Leibo et al., 2017), which exceeds what is common in this area (Tuyls and Weiss, 2012).
4.1 Agents
Our agent’s learning algorithm is based on DQN (Mnih et al., 2015) and includes its signature techniques of experience replay and target networks, enhanced with an LSTM value-network as in Hausknecht and Stone (2015) (to alleviate severe partial observability), learning with truncated back-propagation through time, multi-step updates with forward view eligibility traces (Harb and Precup, 2016) (which helps propagating learning back through longer sequences) and the dueling architecture (Wang et al., 2016) (which speeds up learning by generalizing across the action space). Since observations are from a local perspective, we do not benefit from convolutional networks, but use a fully connected linear layer to process the observations.
Our network architectures first process the input using a fully connected linear layer with $32$ hidden units followed by a ReLU layer, and then an LSTM, with $32$ hidden units followed by a ReLU layer, and finally a linear dueling layer, with $32$ units. This produces a value function $V(s)$ and advantage function $A(s,a)$, which are combined to compute a $Q$-function $Q(s,a)=V(s)+A(s,a)$ as described in Wang et al. (2016). Layers of $32$ units are sufficiently expressive for these tasks with limited observation windows.
The architectures (see Appendix B for detailed diagrams) differ between approaches by what is input into each layer. For architectures without centralization or information channels, one observation of size $3\times 5\times 5$ is fed to the first linear layer of $32$ units, followed by the ReLU layer and the LSTM (see Figure 2). For the other (information channels and centralized) agents, $d$ such observations are fed separately to identical such linear layers and then concatenated into $64$ dimensional vectors before passing though ReLUs to an LSTM.
For architectures with information channels we concatenate the outputs of certain layers with those of other agents. To preserve agent invariance, the agent’s own previous output is always included first. For low-level communication, the signal’s concatenation is after the first fully connected layer, while for high-level communication the concatenation takes place on the output of the LSTM layer. Note, that this has the implication that what starts as one agent’s gradients are back-propagated through much of the other agents network, optimizing them to serve the purposes of all agents. Hence, representing in that sense, a higher degree of centralization than the lower-level sharing.
We have found a trajectory length of $8$, determining both the length of the forward view and the length of the back propagation through time is sufficient for these domains. We use an eligibility trace parameter $\lambda=0.9$. In particular, the individual agents learning directly from the joint reward without decomposition or information channels, has worse performance with lower $\lambda$. The Adam (Kingma and Ba, 2014) learning rate scheme initialized with $0.0001$ is uniformly used, and further fine-tuning this per agent (not domain) does not dramatically change the total performance. The agents that we evaluate are listed in the table above.
4.2 Environments
We use 2D grid worlds with the same basic functioning as Leibo et al. (2017), but with different tasks we call Switch, Fetch and Checkers. We have observations of byte values of size $3\times 5\times 5$ (RGB), which represent a window depending on the player’s position and orientation by extending $4$ squares ahead
and $2$ squares on each side. Hence, agents are very short-sighted. The actions are: step forward, step backward,
step left, step right, rotate left, rotate right, use beam
and stand still. The beam has no effect in our games, except for lighting up a row or column of squares straight ahead with yellow. Each player appears as a blue square in its own observation, and the other player, when in the observation window, is shown in red for Switch and Escape, and lighter blue for Fetch. We use three different maps shown in Figure 3 for both Fetch and Switch and a different one for Checkers, also shown in Figure 3 (bottom right). The tasks repeat as the agents succeed (either by full reset of the environment in Switch and Checkers or just by pickup being available again in Fetch), in training for 5,000 steps and 2,000 in testing.
Switch: The task tests if two agents can effectively coordinate their use of available routes on two maps with narrow corridors. The task is challenging because of strong observation aliasing. The two agents appear on different ends of a map, and must reach a goal at the other end. If agents collide in a corridor, then one agent needs to leave the corridor to allow the other to pass. When both players have reached their goal the environment is reset. A point is scored whenever a player reaches a goal.
Fetch: The task tests if two agents can synchronize their behaviour, when picking up objects and returning them to a drop point. In the Fetch task both players start on the same side of the map and have pickup points on the opposite side. A player scores $3$ points for the team for pick-up, and another $5$ points for dropping off the item at the drop point near the starting position. Then the pickup is available to either player again. It is optimal for the agents to cycle such that when one player reaches the pickup point the other returns to base, to be ready to pick up again.
Checkers:
The map contains apples and lemons. The first player is very sensitive and scores $10$ for the team for an apple (green square) and $-10$ for a lemon (orange square). The second, less sensitive player scores $1$ for the team for an apple and $-1$ for a lemon. There is a wall of lemons between the players and the apples. Apples and lemons disappear when collected, and the environment resets when all apples are eaten. It is important that the sensitive agent eats the apples while the less sensitive agent should leave them to its team mate but clear the way by eating obstructing lemons.
4.3 Results
We compare the eight approaches listed in Table 4.1, on the seven tasks. Each is run ten times, with different random seeds determining spawn points in the environment, as well as initializations of the neural networks.
We calculated curves of the average performance over 50,000 episodes (plots in Appendix A) for each approach on each task and we display the normalized area under the curve in Figure 4. Figure 6 displays the normalized final performance averaged over runs and the last 1,000 episodes. Average performance across tasks is also shown for both ways of evaluation.
The very clear conclusion is that architectures based on value-decomposition perform much better, with any combination of other techniques or none, than the centralized approach and individual learners. The centralized agent with value-decomposition is better than the combinatorially centralized as well as individual learners while worse than the more individual agents with value-decomposition.
We particularly see the benefit of shared weights on the hard task of Fetch with one corridor. Without sharing, the individual value-decomposition agent suffers from the lazy agent problem. The agent with weight sharing and role information also perfectly learns the one corridor Fetch task. It performs better than the agent just sharing weights on Switch, where coordination, in particular with one corridor, is easier with non-identical agents. Further, shared weights are problematic for the Checkers task because the magnitude of rewards (and hence the value function) from one agent is ten times higher than for the other agent.
Adding information channels does increase learning complexity because the input comes from more than one agent. However, the checkers task, designed for the purpose, shows that it can be very useful. Overall, the low-level channels where the agent’s LSTM processes the combined observations of both agents turned out to learn faster in our experiments than the more centralized high level communication (after the LSTM).
4.4 The Learned $Q$-Decomposition
Figure 6 shows the learned $Q$-decomposition for the value-decomposition network, using shared weights, in the game of Fetch. A video of the corresponding game can be seen at Video (2017). Spikes correspond to pick-up events (short spikes, 3 reward points), and return events (large spikes, 5 reward points). These are separated into events due to agent 1 (blue spikes) and agent 2 (red spikes). This disambiguation is for illustration purposes only: the environment gives a reward to the whole team for all of these events. The total $Q$-function is seen in yellow, clearly anticipating the team reward events, and dropping shortly afterwards. The component $Q$-functions $\tilde{Q}_{1}$ and $\tilde{Q}_{2}$ for agents 1 and 2 are shown in green and purple. These have generally disambiguated the $Q$-function into rewarding events separately attributable to either player. The system has learned to autonomously decompose the joint $Q$-function into sensible components which, when combined, result in an effective $Q$-function. This would be difficult for independent learners since many rewards would not be observed by both players, see e.g. the situation at 15-16 seconds in the corresponding video available at Video (2017).
5 Conclusions
We study cooperative multi-agent reinforcement learning where only a single joint reward is provided to the agents. We found that the two naive approaches, individual agents learning directly from team reward, and fully centralized agents, provide unsatisfactory solutions as previous literature has found in simpler environments, while our value-decomposition networks do not suffer from the same problems and shows much better performance across a range of more complex tasks. Further, the approach can be nicely combined with weight sharing and information channels, leading to agents that consistently optimally solve our new benchmark challenges.
Value-decomposition networks are a step towards automatically decomposing complex learning problems into local, more readily learnable sub-problems. In future work we will investigate the scaling of value-decomposition with growing team sizes, which make individual learners with team reward even more confused (they mostly see rewards from other agents actions), and centralized learners even more impractical. We will also investigate decompositions based on non-linear value aggregation.
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Appendix A: Plots
Appendix B: Diagrams |
Density-matrix renormalization group study of the linear conductance
in quantum wires coupled to interacting leads or phonons
Jan-Moritz Bischoff
[email protected]
Eric Jeckelmann
[email protected]
Leibniz Universität Hannover, Institut für Theoretische Physik, Appelstraße 2, D-30167
Hannover, Germany
(Draft of December 1, 2020)
Abstract
In a previous paper [J.-M. Bischoff and E. Jeckelmann, Phys. Rev. B 96, 195111 (2017)]
we introduced a density-matrix renormalization group method for calculating the linear conductance of
one-dimensional correlated quantum systems
and demonstrated it on homogeneous spinless fermion chains with impurities.
Here we present extensions of this method to inhomogeneous systems, models with phonons,
and the spin conductance of electronic models. The method is applied to a spinless fermion wire-lead model,
the homogeneous spinless Holstein model, and the Hubbard model. Its capabilities are demonstrated
by comparison with the predictions of Luttinger liquid theory combined with
Bethe Ansatz solutions and other numerical methods.
We find a complex behavior for quantum wires coupled to interacting leads
when the sign of the interaction (repulsive/attractive) differs in wire and leads.
The renormalization of the conductance given by the Luttinger parameter in purely fermionic systems
is shown to remain valid in the Luttinger liquid phase of the Holstein model with phononic degrees of freedom.
I Introduction
Confining electrons to one dimension results in various surprising properties because of the fermionic nature of the
particles [1; 2; 3].
In practice, one-dimensional electron systems can be realized in semiconductor wires [4],
atomic wires deposited on a substrate [5; 6], carbon nanotubes [7],
or atomic and molecular wire junctions [8; 9; 10; 11].
In contrast to the behavior of a three dimensional metal, which is well described by the Fermi liquid paradigm,
theory predicts that one-dimensional conductors should exhibit the behavior of a Luttinger liquid [2].
In particular, the transport properties have been studied and controversially discussed
for more than three decades [2; 3].
In a previous paper [12]
we introduced a density-matrix renormalization group method (DMRG) for calculating the linear conductance of
one-dimensional correlated quantum systems with short-range interactions at zero-temperature based on the Kubo formalism [13; 14].
By taking advantage of the area law for the entanglement entropy [15]
the DMRG method provides us with the most efficient method for calculating the properties of
these systems [16; 17; 18; 19].
Our method combines DMRG with a finite-size scaling of dynamical correlation functions to
compute the conductance in the thermodynamic limit.
The method was demonstrated on the homogeneous spinless fermion chains with impurities [12],
for which well-established results were available [2; 20; 21; 22].
In this paper we develop the method further to treat a variety of complexer models.
We present extensions to wire-lead systems with different interactions in the wire and leads,
to electron-phonon models, and to spinfull fermions (electrons).
These extensions are necessary steps toward future studies of more realistic models that
are able to describe quantitatively
the experimental realizations of one-dimensional electronic
conductors [4; 5; 6; 7; 8; 9; 10; 11].
Here, we apply our method to a spinless fermion wire-lead model, the homogeneous spinless Holstein model, and the Hubbard model
to demonstrate its current capabilities and limitations using the predictions of Luttinger liquid theory combined with
Bethe Ansatz solutions and numerical results found in the literature.
The method is summarized in the next section. The extension and results for inhomogeneous systems such as
the spinless fermion wire-lead system are presented in Sec. III.
The extension to systems with phonon degrees of freedom and the results for the spinless Holstein model
are shown in Sec. IV.
Section V describes the generalization to spinfull fermion (electron) model and to the calculation of the spin
conductance as well as our results for the spin conductance of the Hubbard model.
Finally, Sec. VI contains our conclusion and outlook.
II Method
In this section we summarize our DMRG method for calculating the conductance of one-dimensional correlated quantum systems.
Full details can be found in our previous paper [12].
We consider a one-dimensional lattice of $M$ sites. It consists of left and right segments (called leads)
with approximately $(M-M_{\text{W}})/2$ sites
each and a central segment (called wire) of $M_{\text{W}}$ sites. A current can be generated by applying a potential bias. We assume that the potential
remains constant in the leads and drop linearly in the wire, i.e. it has the shape
$$C(j)=\begin{cases}\hfill\hphantom{-}\frac{1}{2}\hfill&\text{ for $j\leq j_{1}$%
}\\
\hfill-\frac{j-j_{1}}{j_{2}-j_{1}}+\frac{1}{2}\hfill&\text{ for $j_{1}\leq j%
\leq j_{2}$}\\
\hfill-\frac{1}{2}\hfill&\text{ for $j\geq j_{2}$}\end{cases}$$
(1)
where $j_{1}$ and $j_{2}$ are the first and last site in the wire, respectively.
This is illustrated in Fig. 1.
The resulting linear conductance can be calculated directly from dynamical current-current correlation functions
in the thermodynamic limit using Kubo’s linear response theory [13].
The dynamical DMRG method [23; 24] can be used to compute the imaginary part of the frequency-resolved current-current correlation functions
for a finite lattice size $M$
$$G_{J,\eta}(\omega)=\left\langle 0\left|J\frac{\eta}{(E_{0}-H+\hbar\omega)^{2}+%
\eta^{2}}J\right|0\right\rangle$$
(2)
where $H$ is the Hamilton operator, $|0\rangle$ is its ground state, $E_{0}$ is the ground-state energy,
$J$ is the current operator in the wire, and $\eta>0$ is a small number.
The Hamilton operator and the current operator depend on the system studied and thus will be given explicitly
in the next sections where we present our results for specific models.
It is important to realize, that the wire segment (and consequently both lead segments) is defined by the potential profile
(1) and thus, in practice, by the lattice region, where the current operator $J$ acts.
It is possible but not necessary to include any information about the wire-lead layout in the Hamilton operator $H$,
such as different interactions or different degrees of freedom for wire and leads.
In fact, in our first work we tested
our method with homogeneous systems only, i.e. the degrees of freedom and interactions were identical for all sites,
but for the presence of up to two impurities (on-site potentials) in the middle of the wire.
From the DMRG data for (2) we can calculate the “finite-size conductance”
$$G(M)=\left.\frac{q^{2}}{\omega}\left[G_{J,\eta(M)}(\omega)-G_{J,\eta(M)}(-%
\omega)\right]\right|_{\omega=0}$$
(3)
for a given system size $M$ using the scaling $\eta(M)=C/M$, where
$q$ is the charge carried by each fermion.
The constant $C$ depends on the properties of the system investigated.
For all results presented here we used $C=48t$ or $96t$.
Extrapolating $G(M)$ to infinite system length $M$ yields our estimation of the conductance $G$ in the thermodynamic limit.
Here, we used chain lengths up to $M=2000$.
For a noninteracting tight-binding chain it can be shown analytically that $G(M)$ converges to the exact result for $M\rightarrow\infty$,
i.e. the quantum of conductance
$$G_{0}=n\frac{q^{2}}{h}$$
(4)
where $n=1$ for spinless fermions and $n=2$ for electrons.
For correlated systems we can only compute (2) and (3)
numerically.
In practice, there are two main practical issues. First, DMRG must be able to calculate
$G(M)$ with enough accuracy for large system sizes $M$. Second,
$G(M)$ should converge neatly to the actual value of the conductance for $M\gg 1$.
In our previous work we showed, that this method yields correct results for the renormalized
conductance of the one-component Luttinger liquid in the homogeneous spinless fermion chain
as well as for one and two impurities added to the chain.
In the next sections we will show that this approach can also be applied to
inhomogeneous systems, electronic Hamiltonians and models with phonons.
In all our numerical results and figures we use $q=1$, and $\hbar=1$.
This yields $G_{0}=\frac{n}{2\pi}$. Therefore, we show $2\pi G(M)$ in our figures.
III Inhomogeneous wire-lead system
As mentioned above our method was demonstrated in Ref. [12]
for homogeneous systems only.
In real systems, however, wire and leads are of different nature.
For instance, it is often assumed that metallic leads are Fermi liquids and thus they are modelled
by noninteracting tight-binding chains attached to the interacting
wire [25; 26; 14; 27; 28; 29; 30; 31; 32].
More generally, one may have to consider
different interactions in the leads and in the wire as well as additional degrees of freedom (such as phonons)
in the wires.
Therefore, we have generalized our method to this situation and tested it on a spinless fermion model with different nearest-neighbor interactions
in the wire ($V_{\text{W}}$) and in the leads ($V_{\text{L}}$). Explicitly, this corresponds
to the Hamiltonian
$$\displaystyle H=$$
$$\displaystyle-t\sum\limits_{j=2}^{M}\left(c^{\dagger}_{j}c_{j-1}+c^{\dagger}_{%
j-1}c_{j}\right)$$
(5)
$$\displaystyle+V_{\text{W}}\sum\limits_{j=j_{1}+1}^{j_{2}}\left(n_{j}-\frac{1}{%
2}\right)\left(n_{j-1}-\frac{1}{2}\right)$$
$$\displaystyle+V_{\text{L}}\sum\limits_{j=2}^{j_{1}}\left(n_{j}-\frac{1}{2}%
\right)\left(n_{j-1}-\frac{1}{2}\right)$$
$$\displaystyle+V_{\text{L}}\sum\limits_{j=j_{2+1}}^{M}\left(n_{j}-\frac{1}{2}%
\right)\left(n_{j-1}-\frac{1}{2}\right).$$
The operator $c^{\dagger}_{j}$ ($c_{j}$) creates (annihilates) a spinless fermion at position $j$ ($=1,\dots M$)
while $n_{j}=c^{\dagger}_{j}c_{j}$ counts the number of spinless fermions on this site.
The hopping amplitude is identical for wire and leads and sets the energy unit $t=1$.
We assume that the system has an even number of sites $M$ and is occupied by $M/2$ spinless fermions (half filling).
From the potential profile (1) follows that the current operator is given by
$$J=\frac{1}{M_{W}-1}\frac{it}{\hbar}\sum\limits_{j=j_{1}+1}^{j_{2}}\left(c^{%
\dagger}_{j}c_{j-1}-c^{\dagger}_{j-1}c_{j}\right).$$
(6)
In principle, the wire edge sites $j_{1}$ and $j_{2}$ in the definition of the Hamiltonian (5)
could be different from those used for the potential profile (and thus the current operator).
Here we assume that the screening of the potential difference between source and drain occurs
entirely (and linearly) in the wire and thus we use the same values for $j_{1}$ and $j_{2}$ in both the Hamilton operator (5)
and the current operator (6).
Field-theoretical approaches predict that
the conductance of a homogeneous Luttinger liquid is renormalized as
$$G=KG_{0}$$
(7)
where $K$ is the Luttinger parameter [20; 21; 22].
We could reproduce this result in our previous work for the homogeneous spinless fermion model,
i.e., for $V_{\text{W}}=V_{\text{L}}$ in (5).
On the contrary, field-theoretical approaches predict that
the conductance through a Luttinger liquid wire connected
to leads is not renormalized by interactions in the wire and is
entirely determined by the lead properties [33; 34; 35; 36; 37].
In particular, if the leads are Luttinger liquids their Luttinger parameter $K_{\text{L}}$
determines the conductance through (7) [34; 36].
However, field-theoretical approaches assume perfect contacts, i.e. without any single-particle
backscattering [36].
Numerical calculations for lattice models of quantum wires connected
to noninteracting leads show that the conductance deviates from the lead conductance $G_{0}$
when backscattering is taken into account.
In the lattice model (5)
single-particle backscattering is due to the sharp change of the on-site potential
at the boundaries between wire and leads
[$-V_{\text{W}}$ in the wire, $-V_{\text{L}}$ in the leads, and
$-(V_{\text{W}}+V_{\text{L}})/2$ on the boundary sites $j_{1}$ and $j_{2}$],
which is required to maintain a constant density in equilibrium.
We have found that the calculation of the frequency-resolved correlation
function (2) with dynamical DMRG is not more difficult for an inhomogeneous system
($V_{\text{W}}\neq V_{\text{L}}$) than for a homogeneous one ($V_{\text{W}}=V_{\text{L}}$).
To investigate the conductance of a Luttinger liquid between leads, however, we
would have to compute the conductance $G(M)$ in the limit of long wires $M_{\text{W}}\gg 1$
while maintaining much longer leads ($M\gg M_{\text{W}})$.
In our study of homogeneous systems [12] we found that the conductance scales
with $M_{\text{W}}/M$ and thus we were able to determine its value in the thermodynamic limit
using a fixed (and small) value of $M_{\text{W}}$. For the wire–lead system (5)
with $V_{\text{W}}\neq V_{\text{L}}$ considered here, we have found that the scaling
with $M$ and $M_{\text{W}}$ (up to $M_{\text{W}}=82$) is much more complicated. As a consequence, it is not possible
to compute the conductance of an infinitely long wire in most cases. Thus we discuss
here the results obtained for short wires and present only results for $M_{\text{W}}=10$.
In Fig. 2 we show the scaling of $G(M)$ with the system length $M$ for noninteracting
leads ($V_{\text{L}}=0$) and various interaction strengths $V_{\text{W}}$ in the wire.
The extrapolated values $G=\lim_{M\rightarrow\infty}G(M)$ are shown in Fig. 3.
For repulsive interactions ($V_{\text{W}}>0$) we observe a progressive decrease
of $G$ with increasing coupling $V_{\text{W}}$.
The conductance of the spinless fermion model connected to noninteracting
leads was investigated previously for repulsive interactions
using DMRG and the functional renormalization group
method [25; 26].
Our results for $G$ agree quantitatively with the (DMRG) results presented in Ref. [25]
for a slightly longer wire ($M_{\text{W}}=12$), as shown in Fig. 3.
In contrast, we observe in Fig. 2
that $G(M)$ tends to $G_{0}$ (i.e., the conductance of the noninteracting leads)
for attractive interactions $0>V_{\text{W}}>-2t$.
The non-monotonic behavior of $G(M)$ for $V_{\text{W}}=-1.9t$ in Fig. 2
and the resulting large deviation of the extrapolated value from $G_{0}$ in Fig. 3 illustrate
the problems that one encounters when analyzing finite-size effects in the wire-lead system.
For an interaction $V_{\text{W}}>2t$ the homogeneous half-filled spinless fermion model is in an insulating
charge-density-wave (CDW) phase and thus we expect that $G$ diminishes exponentially
with increasing wire length $M_{\text{W}}$.
In the Luttinger liquid phase with repulsive interactions ($2t\geq V_{\text{W}}>0$)
a power-law suppression of $G$ with increasing wire length was observed previously [26].
This deviation from the field-theoretical predictions mentioned above is due to the
single-particle backscattering caused by the sharp change of the interaction
at the wire boundaries.
The scaling of $G$ with the wire length in the attractive Luttinger liquid phase
($-2t<V_{\text{W}}<0$) is not known.
As explained above, we can not determine the conductance for infinitely long wire
($M_{\text{W}}\rightarrow\infty$) with the present method.
Nevertheless, our data suggest that for attractive wire interactions within the Luttinger liquid regime the noninteracting leads
determine the conductance of the system in agreement with field theory. This is in stark contrast to repulsive wire interactions,
for which even a small $V_{\text{W}}>0$ results in a reduction of $G$ from the noninteracting lead conductance $G_{0}$,
as already reported previously [36; 25; 26].
While there are numerous studies of the conductance of wires connected to noninteracting
leads [33; 34; 35; 36; 37; 25; 26; 14; 27; 28; 29; 30; 31; 32],
there seem to be only a few field-theoretical results for interacting leads [34; 36; 31].
Figure 4 shows the scaling of $G(M)$ with the system size $M$ for leads with a repulsive
interaction $V_{\text{L}}=2t$. From the Bethe Ansatz solution of the homogeneous spinless fermion model
we know that this corresponds to a Luttinger liquid parameter $K_{\text{L}}=1/2$ [38]
and thus to a conductance $G=G_{0}/2$ for perfect contacts according to field theory.
Clearly, all our results for $G(M)$ with $V_{\text{W}}>0$ converge toward this value and thus agree
with the field-theoretical predictions.
Surprisingly, this also holds for the wire with $V_{\text{W}}=3t$, which
would correspond to a CDW insulator in the thermodynamic limit $M_{\text{W}}\rightarrow\infty$.
Therefore, we expect that $G$ will decrease with increasing $M_{\text{W}}$
for $V_{\text{W}}=3t$.
For a noninteracting wire or an attractive interaction in the wire ($0\geq V_{\text{W}}>-2t$), however,
we observe in Fig. 4
a different, non-monotonic behavior of $G(M)$, which hinders the extrapolation of $G(M)$ to the limit $M\rightarrow\infty$.
Apparently, $G$ is lower than the lead conductance $G_{0}/2$ for $V_{\text{W}}=0$ and $-t$.
Figure 5 shows the scaling of $G(M)$ with system length $M$
for leads with an attractive interaction $V_{\text{L}}=-t$.
This corresponds to a Luttinger liquid parameter $K_{\text{L}}=3/2$ [38]
and thus to a conductance $G=(3/2)G_{0}$ for perfect contacts according to field theory.
We see in Fig. 5 that $G(M)$ seems to converge to different values for different
repulsive couplings in the wire ($V_{\text{W}}>0$).
Unfortunately, we cannot determine the limit $G=\lim_{M\rightarrow\infty}G(M)$
accurately even with system lengths up to $M=2000$.
The values of $G(M)$ for $M=2000$ are above the conductance of an homogeneous
Luttinger liquid with the corresponding interaction $V_{\text{W}}$. They seem to increase further with $M$
but to converge to lower values than the lead conductance $G=(3/2)G_{0}$.
Therefore, it appears that the conductance of the wire-lead system is lower than the lead conductance in
that case, similarly to what is found for noninteracting leads due to single-particle backscattering.
For $V_{\text{W}}=3t$ the conductance seems again to converge to a finite value. As for noninteracting leads
it should vanish in the limit $M_{\text{W}}\rightarrow\infty$ as this interaction strength leads to
a CDW insulating state in the wire.
For a noninteracting wire and for attractive interactions in the wire ($0\geq V_{\text{W}}\geq-2t$), however, we find
that $G\approx(3/2)G_{0}$ as predicted by field theory but it is difficult to extrapolate $G(M)$ reliably
close to the Luttinger phase boundary $V_{\text{W}}=-2t$.
In summary, our results agree with the known results for the conductance of the spinless
fermion wire-lead system and thus confirm the validity of our method.
Moreover, they also reveal complex qualitative differences in the attractive and repulsive
interaction regimes.
The conductance seems to be given by (7) with the lead Luttinger parameter
(as predicted by field theory) only when the interactions have the same sign in wire and leads.
The properties of short wires with attractive interactions connected to leads with repulsive
interactions (or vice versa) seem to be more complex than anticipated from the results reported so far
(using field theory) and should be investigated further.
IV Holstein chain
The interaction between charge carriers and phonons plays an important role for
transport properties [39; 40; 41], in particular
for atomic or molecular wire junctions between leads [8; 9; 10; 11].
Therefore,
we have to generalize our method to models that include phonon degrees of freedom coupled
to the charge carriers.
It is relatively straightforward to extend a DMRG program to models with phonons if one uses
a truncated eigenbasis of the boson number operator with at most $N_{\text{b}}$ states
to represent each phonon mode [42]. The only real challenge is the computational cost that
increases as $N^{3}_{\text{b}}$ and thus limit most applications to small $N_{\text{b}}$.
For Einstein phonons (i.e. dispersionless) and purely local couplings with the fermion degrees
of freedom
it is possible to speed up DMRG calculations using a pseudo-site representation for bosons described in [43].
Therefore, we have tested our method on the simplest model with these properties,
the spinless fermion Holstein model [44],
which is defined by the Hamiltonian
$$\displaystyle H=$$
$$\displaystyle-t\sum\limits_{j=2}^{M}\left(c^{\dagger}_{j}c_{j-1}+c^{\dagger}_{%
j-1}c_{j}\right)+\omega_{\text{b}}\sum\limits_{j=1}^{M}b^{\dagger}_{j}b_{j}$$
$$\displaystyle-g\,\omega_{\text{b}}\sum\limits_{j=1}^{M}\left(b^{\dagger}_{j}+b%
_{j}\right)n_{j},$$
(8)
where $\omega_{\text{b}}$ is the frequency of the Einstein phonons created (annihilated) by the boson operators $b^{\dagger}_{j}$ ($b_{j}$)
and $g$ is the dimensionless coupling between spinless fermions and phonons.
The spinless fermion operators $c^{\dagger}_{j}$, $c_{j}$, and $n_{j}$ have the same meaning as in the previous section
and the hopping term again sets the energy unit $t=1$, while we consider a system with an even number of sites $M$ filled
with $M/2$ spinless fermions.
This Hamiltonian is homogeneous and thus the wire and lead sections are defined by the potential profile (1) only.
The current operator is again given by (6).
We have found that calculating the conductance with our method
is more difficult for the Holstein model than for purely fermionic models such as (5).
This is due not only to the increase of the computational cost with the phonon cutoff $N_{\text{b}}$
but also to the fact that the dynamical DMRG algorithm [23; 24] often fails to converge in a reasonable time
for large $M$. Thus we can compute $G(M)$ systematically for smaller system sizes $M$ than in fermionic systems
and we show here results for $M$ up to $800$ sites only.
(One calculation for this system size requires about 600 CPU hours and 4GB of memory. Thus
one could certainly treat larger systems using supercomputer facilities.)
In contrast, the scaling of $G(M)$
with the system size is regular in the homogeneous Holstein wire as found previously for homogeneous
fermionic systems [12]. Thus we can estimate the conductance $G$ in the thermodynamic limit despite the short system lengths
as illustrated in Fig. 6 for the adiabatic regime ($\omega_{\text{b}}=0.1t$)
and in Fig. 7
for the intermediate regime ($\omega_{\text{b}}=t$).
The phase diagram of the half-filled spinless fermion Holstein model
was determined previously with DMRG [45] and quantum Monte Carlo computations [46].
It exhibits a Luttinger liquid phase at weak coupling or high phonon frequency.
The Luttinger liquid parameter $K$ was calculated from the long-wavelength limit
of the static structure factor [45; 46]. This corresponds to the
parameter $K$ determining the exponents in the power-law correlation functions
of the Luttinger liquid theory for purely fermionic systems [2].
As seen in Fig. 6,
we have found that the conductance corresponds to $K=G/G_{0}\approx 1$ in the adiabatic regime $\omega_{\text{b}}=0.1t$
for a coupling up to at least $g=1.5$ in agreement with the values determined from the
structure factor [45; 46].
In the intermediate regime $\omega_{\text{b}}=t$ shown in Fig. 7, the conductance $G(M)$
clearly decreases with increasing coupling $g$. The extrapolated values of $G$ for $1/M\rightarrow 0$
yield parameters $K=G/G_{0}$ that agree well with the values determined from
the structure factor in Ref. [45], as shown in Fig. 8.
In particular, we do not observe any sign of superconducting fluctuations ($K>1$ or $G>G_{0}$).
Unfortunately, we are not able to determine the conductance close
to the metal-insulator transition because this requires too much computational resources
(the phonon cutoff $N_{\text{b}}$ should be larger than the value $N_{\text{b}}=4$ used here).
In summary, our results for the conductance in the spinless fermion Holstein model
agree with the Luttinger parameter $K$ defined from the power-law decay of correlation functions
(or equivalently from the static structure factor).
This confirms the validity of our method for models with charge carriers
coupled to phonon degrees of freedom. It also shows that the relation (7)
for the conductance,
which was derived for purely fermionic systems using field theory [2; 20; 21; 22],
remains valid for the Luttinger liquid phase of models including phonons.
V Hubbard chain
A necessary but simple extension of our method is the generalization to electrons (i.e., fermions with spin $\frac{1}{2}$).
To illustrate this extension we consider the one-dimensional homogeneous Hubbard chain
with the Hamiltonian
$$H=-t\sum\limits_{j=2}^{M}\sum_{\sigma}\left(c^{\dagger}_{j,\sigma}c_{j-1,%
\sigma}+\text{h.c.}\right)+U\sum\limits_{j=1}^{M}n_{j,\uparrow}n_{j,\downarrow},$$
(9)
where $c^{\dagger}_{j,\sigma}$ ($c_{j,\sigma}$) creates (annihilates) an electron with spin $\sigma(=\uparrow,\downarrow)$
on site $j$ and $n_{j,\sigma}=c^{\dagger}_{j,\sigma}c_{j,\sigma}$ is the corresponding particle number operator.
The strength of the on-site coupling between electrons is given by the Hubbard parameter $U$
and the hopping term again sets the energy unit $t=1$.
We assume that the system length $M$ is even and that the system contains $M/2$ electrons of each spin (half filling).
The current operator for the electrons with spin $\sigma$ in the wire is
a simple generalization of (6)
$$J_{\sigma}=\frac{1}{M_{\text{W}}-1}\frac{it}{\hbar}\sum\limits_{j=j_{1}+1}^{j_%
{2}}\left(c^{\dagger}_{j,\sigma}c_{j-1,\sigma}-c^{\dagger}_{j-1,\sigma}c_{j,%
\sigma}\right).$$
(10)
The current operator for the charge transport is then given by
$$J_{\text{C}}=J_{\uparrow}+J_{\downarrow}.$$
(11)
Using this definition in Eqs. (2) and (3)
we obtain the (charge) conductance $G(M)$ of the electronic system.
The current operator for the spin transport is similarly given by
$$J_{\text{S}}=J_{\uparrow}-J_{\downarrow}.$$
(12)
Using this definition in Eqs. (2) and (3)
we obtain the spin conductance $G_{\text{S}}(M)$ of the electronic system.
This describes the linear response of the system to an external magnetic field with
the profile (1).
The one-dimensional Hubbard model is exactly solvable using the Bethe Ansatz method [47; 48].
Its low-energy properties are characterized by a separation of charge and spin excitations
and its gapless excitation modes are described by the Luttinger liquid theory.
At half filling there is an exact symmetry between the charge (spin) properties for $U$ and
the spin (charge) properties for a coupling $-U$. In particular, the ground state is a Mott insulator
with gapped charge excitations and gapless spin excitations for $U>0$, while
it is a Luther-Emery liquid [2] with gapless charge excitations and gapped spin excitations for $U<0$.
We have found that calculating the (charge or spin) conductance with our method is only slightly more difficult
for the homogeneous electronic model (9) with $U\neq 0$ than for the homogeneous spinless fermion model.
This is understandable because in both cases there is only one gapless excitation mode in the system.
As expected the charge conductance $G(M)$ for any $U$ is equal within the numerical errors
to the spin conductance $G_{\text{S}}(M)$ for an interaction $-U$. Thus we discuss only the latter case.
In Fig. 9 the scaling of $G_{\text{S}}(M)$ with $M$ is plotted for several values of the interaction $U$.
For noninteracting electrons $G_{\text{S}}(M)$ approaches $G_{0}$ as expected. [Note that the quantum of conductance (4)
for electrons ($n=2$) equals $2\pi G_{0}=2$ in our units].
The scattering of the $G_{\text{S}}(M)$ data and the resulting poor extrapolation are due to the lesser
accuracy of DMRG for a chain with two gapless excitation modes (i.e., for $U=0$) than for a single one (i.e. for $U\neq 0$).
For $U<0$, $G_{\text{S}}(M)$ vanishes in the thermodynamic limit as required for a phase with gapped spin excitations.
The qualitatively different behaviors for weak and strong attractive interactions is probably due to
the different correlation length, which is smaller than $M_{\text{W}}=10$ for $U=-8t$
but larger for $U=-2t$.
The Luttinger liquid parameter for the spinless spin excitations is $K_{\text{S}}=1$ in the Mott insulating phase at half
filling for any $U>0$.
Concordantly, we observe in Fig. 9 that $G_{\text{S}}(M)$ approaches $G_{0}$ for a weak repulsive interaction ($U=2t$).
For stronger interactions the convergence of $G_{\text{S}}(M)$ toward $G_{0}$ is less clear as shown by the results
for $U=8t$ in Fig. 9.
This is due to the small band width of the gapless spin excitations (of the order of $4t^{2}/U$ for large $U$)
which requires a smaller broadening $\eta\sim 4t^{2}/U$ and thus a larger $M$ to reach the same resolution
for the dynamical correlation functions (2).
VI Conclusion and Outlook
We have extended the DMRG method for the linear conductance of one-dimensional correlated lattice models to
more complex systems including interacting leads, coupling to phonons, and the electron spin properties.
The tests conducted in this work reveal
intriguing differences
when the sign of the interaction (attractive/repulsive) differs in wire and leads
and confirm that the renormalization of the conductance by the Luttinger parameter (7)
remains valid for the Luttinger liquid phase of homogeneous models with coupling to phonons.
Unfortunately, we have found that we cannot determine the conductance of a
Luttinger liquid wire between leads with a different interaction but can only investigate short
wires because of the complicated finite-size scaling.
Moreover, for models with coupling to phonon degrees of freedom the high computational cost of
dynamical DMRG calculations limits the system sizes that can be used.
Nevertheless,
the extensions presented in this work will certainly allows us to compute the conductance
of short electron-phonon-coupled wires connected to interacting leads (without phonons).
Therefore, we will be able to study more realistic models for realizations of quasi-one-dimensional
electronic conductors such as
atomic or molecular wire junctions [8; 9; 10; 11].
Moreover, we point out that our approach is not limited to the charge and spin conductance but could be extended to other
transport properties that are described by a local current operator, such as the energy current [49; 50; 51; 52].
Acknowledgements.
Jan Bischoff would like to thank the Lower Saxony PhD-Programme Contacts in Nanosystems for financial support.
We also acknowledge support from the DFG (Deutsche Forschungsgemeinschaft) through Grant No. JE 261/2-2 in the
Research Unit Advanced Computational Methods for Strongly Correlated Quantum Systems (FOR 1807).
The cluster system at the Leibniz Universität Hannover was used for the computations.
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\makeheadline
\summaryWe summarise our recent work on the faint galaxy contribution
to the cosmic X-ray background (XRB). At bright X-ray fluxes (in the
ROSAT pass band), broad line QSOs dominate the X-ray source
population, but at fainter fluxes there is evidence for a significant
contribution from emission-line galaxies. Here we present statistical
evidence that these galaxies can account for a large fraction
of the XRB. We also demonstrate that these galaxies have significantly
harder X-ray spectra than QSOs in the ROSAT band. Finally we present
preliminary findings from infra-red spectroscopy on the nature of this
X-ray emitting galaxy population. We conclude that a hybrid
explanation consisting of obscured/Type 2 AGN surrounded by starburst
activity can explain the properties of these galaxies and perhaps the
origin of the entire XRB. END
\keywActive galactic nuclei, X-ray background, deep surveys
END
\AAAclaEND
\kap
Introduction
Deep ROSAT surveys have revealed that broad-line QSOs account for at
least $30\%$ of the total $0.5-2$ keV XRB. Analyses of the number
count distribution and luminosity function of QSOs, however, suggests
that they are unlikely to form more than $\sim 50\%$ of the XRB at
these energies (Boyle et al 1994, Georgantopoulos et al 1996),
although we note that recent work by Schmidt et al (1997) and Hasinger
et al (1997) disputes this claim. Furthermore, it has been known for
many years that the X-ray spectra of QSOs are much steeper than the
spectrum of the XRB, leading to the so-called ‘spectral paradox’
(Gendreau et al 1995). A faint source population with a flatter X-ray
spectrum is required to account for the remainder of this background
radiation.
In recent years it has become clear that a population of X-ray
luminous narrow emission-line galaxies (NELGs) could provide a
possible explanation, perhaps dominating the X-ray source counts at
faint fluxes (Boyle et al 1995, McHardy et
al. 1997, Carballo et al 1995, Griffiths et al 1996). We refer
to these X-ray loud objects as ‘narrow-line X-ray galaxies’ (NLXGs)
to distinguish them from the field galaxy population.
The nature of the X-ray emitting mechanism in these galaxies will be
discussed further in Section 4, but we will will conclude
that many
of these objects contain low luminosity or obscured AGN (see
eg. Comastri et al 1995). In this sense the entire XRB could still be
due to ‘AGN’ rather than two distinct classes of X-ray source. The
most significant problem in assigning faint galaxies to X-ray sources
comes from the confusion associated with the relatively large X-ray
error box (see Hasinger et al 1997). To overcome this confusion, one
must adopt a statistical approach.
\kap
Probing the galaxy contribution by cross-correlation techniques
The clearest evidence that faint galaxies could be important
contributors to the XRB came from the study of Roche et al (1995), who
cross-correlated faint ($B<23$) galaxy catalogues with deep X-ray
observations, thus avoiding the problems associated with source
confusion. In Almaini et al (1997b) we performed an independent test
of the Roche et al results on new deep ($\sim$50ks) $\it ROSAT$
exposures and for the first time attempted to measure the evolution in
the X-ray emissivity of faint galaxies with redshift. The
cross-correlation of unidentified X-ray sources with faint galaxies
indicated that these could account for $20\pm 7\%$ of all X-ray
sources to a limiting flux of
$\sim 4\times 10^{-15}$erg$\,$s${}^{-1}$cm${}^{-2}$ in the $0.5-2.0$ keV
band. To probe deeper, cross-correlations were also performed with the
residual, unresolved XRB images. Significant signals were again
obtained on all 3 $\it ROSAT$ fields (see Figure 1), independently
confirming the results obtained by Roche et al (1995).
To constrain the evolution of X-ray emissivity with redshift, we used
the magnitudes of the thousands of catalogue galaxies as probes of
their redshift distribution, adapting the formalism of Treyer & Lahav
(1996). We found evidence for very strong evolution in the mean X-ray
luminosity of the form $\left\langle{L_{x}}\right\rangle\propto(1+z)^{3.22\pm 0.98}$. This represents the first evidence that the
X-ray emission from faint galaxies evolves as strongly as AGN. Similar
results were obtained by analysing small numbers of bright narrow
emission-line galaxies emerging from deep $\it ROSAT$ exposures
(Griffiths et al 1996, Boyle et al 1995). A simple extrapolation to
z=2 suggests that faint galaxies can account for $\sim 40\pm 10\%$
of the total XRB at 1keV.
\kap
The X-ray spectra of faint ROSAT sources
In Almaini et al (1996) we presented an analysis of the X-ray spectra
of all X-ray sources from our deep survey. The results confirmed the
trend found by Hasinger et al (1993) and Vikhlinin et al (1995) that
the average source spectra harden towards fainter fluxes. Hardness
ratios for the stacked spectra of all sources as a function of flux
are shown on Figure 2(a).
Using the optical identifications available for most of these sources,
it was found that ordinary, broad-line QSOs are not responsible for
this spectral hardening (Figure 2b). This implied that the change in
mean spectra was due to the emergence of another source
population. Our cross-correlation results suggest that $20\pm 7$% of
the remaining, unidentified sources are due to faint galaxies, but
confusion problems prevent us from determining exactly which X-ray
sources are responsible. We therefore selected a restricted sample of
the most likely galaxy candidates with brighter optical magnitudes
($B<21.5$) and lying within 20 arcsec of the X-ray source. The
hardness ratios for the 15 emission-line galaxies satisfying this
criteria are shown in Figure 2b. We expect approximately 3 of these
galaxies to be spurious identifications, but nevertheless it is clear
that their mean X-ray spectra are significantly harder than those of
QSOs (Almaini et al 1996, Romero-Colmenero et al 1996). Hence the
X-ray spectra provide further evidence that NLXGs may finally provide
the solution to the origin of the XRB. The key test is to examine
their mean spectral properties with a large sample at higher energies,
which will be possible with AXAF or XMM.
\kap
Understanding the nature of NLXGs
There is mounting evidence that a population of X-ray luminous
‘galaxies’ could finally provide a solution to the origin of the XRB.
The important question now is to understand the nature of this unusual
activity. Using standard optical emission line ratios, these objects
appear on the borderline between starburst and Seyfert 2
classification and in most cases the distinction is very ambiguous
(Boyle et al 1995, McHardy et al 1997). One possibility is that
these are a new type of evolved starburst galaxy in which the X-ray
emission comes from massive X-ray binaries (Griffiths and Padovani
1990). No local starburst galaxies have been found with such hard
X-ray spectra however. These galaxies are also significantly more
X-ray luminous than any known starburst galaxies, and their far
infra-red fluxes are generally too low for the X-ray emission to be
entirely due to starforming activity (Iwasawa et al 1997). The most
likely explanation is that the hard X-rays come from an obscured
active nucleus. Such models can readily reproduce the spectrum of the
X-ray background (Comastri et al 1995). Recent work by Hasinger et al
(1997; see also these proceedings) has suggested that many of
these galaxies would indeed be classified as AGN in higher signal to
noise optical spectra, albeit as low luminosity and/or type 2 AGN.
If these objects are obscured AGN, one might expect to detect broad
emission lines in the infra-red waveband (the dust extinction at $K$
is approximately 6 magnitudes lower than in the $V$ band). We have
recently obtained UK Infra-Red Telescope CGS4 spectroscopy for a small
number of NLXGs. Of the 5 objects studied so far, none show any
evidence for the expected broad Paschen emission lines. In most
cases, this implies an absorbing column of at least $5\times 10^{22}$
atom cm${}^{-2}$ (if an AGN is present). Such columns are inconsistent
with the large ROSAT luminosities below 1keV, unless there is some
contribution from an additional source of soft X-ray flux. In at
least one object, the detection of highly ionized Si[VI] clearly
indicates the presence of an AGN, despite the non-detection of broad
Paschen lines (Figure 3). As such, this is similar to the optical
narrow-line objects detected by Hasinger et al (1997). A plausible way
to account for these properties is to postulate a hybrid model,
consisting of an obscured AGN surrounded by starforming activity.
Such a model would certainly explain the ambiguous optical line ratios
seen in these objects. The hard ASCA NLXG detected by Iwasawa et al
(1997) shows clear evidence for this dual behaviour. As discussed in
Fabian et al (1997), obscuration by nuclear starburst activity may be
an inevitable consequence of triggering a low luminosity AGN, and
could perhaps account for the obscuration of the entire hard XRB.
\kap
Conclusions
There is now overwhelming evidence for the existence of a population
of X-ray luminous galaxies, which could explain the remainder of the
XRB. Our cross-correlation analysis suggests that these objects could
account for $40\pm 10\%$ of the XRB at 1keV. In addition, the
individually identified galaxies show
significantly harder X-ray spectra than QSOs, more consistent with
that of the XRB. Obscured and/or low luminosity AGN provide the most
natural explanation for this activity, although our infra-red
spectroscopy and the optical properties of these galaxies suggest
the presence of an additional soft X-ray component. A hybrid model
consisting of an obscured AGN surrounded by starburst activity
provides the most natural explanation for all observed properties.
\sect
References
\rf
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& Georgantopoulos I., 1996, MNRAS 282, 295
\rfAlmaini O. & Fabian A.C., 1997a, MNRAS 288, L19
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Georgantopoulos I., & Stewart G.C. 1997b, MNRAS 291, 372
\rfBoyle B.J., Griffiths R.E., Shanks T., Stewart G.C.,
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\rfBoyle B.J., McMahon R.G., Wilkes B.J., & Elvis M.,
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\rfCarballo R. et al 1995, MNRAS 277, 1312
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1995, A&A, 296, 1
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\rfGriffiths R.E. & Padovani P. 1990, ApJ 360, 483
\rfGriffiths R.E., Della Ceca R., Georgantopoulos I., Boyle B.J.,
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Trümper J., Zamorani G., 1993, A&A, 275, 1
\rfHasinger G., Burg R., Giacconi R., Schmidt M., Trumper J. & Zamorani G.,
1997, A&A in press
\rfIwasawa K., Fabian A.C., Brandt W.N., Crawford C.S.,
Almaini O., 1997, MNRAS, 291, L17
\rfMcHardy I. et al, 1997, MNRAS, In press
\rfRoche N., Shanks T., Georgantopoulos I., Stewart G.C., Boyle B.J., &
Griffiths R.E., 1995, MNRAS 273, L15
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& Griffiths R.E., 1991 Nat 353, 315
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1995 ApJ 451, 564
Omar Almaini, Institute of Astronomy, Madingley Road, Cambridge, CB3 OHA, e-mail: [email protected]
END |
Soft Open Charm Production in Heavy-Ion Collisions.
V. Topor Pop
McGill, University, Montreal, Canada, H3A 2T8
J. Barrette
McGill, University, Montreal, Canada, H3A 2T8
M. Gyulassy
Physics Department,
Columbia University, New York, N.Y. 10027
(February 23, 2009)
Abstract
Effects of strong longitudinal color
electric fields (SCF) on the open charm production in nucleus-nucleus
(A + A) collisions at 200A GeV are investigated
within the framework of
the HIJING/BB̄ v2.0 model.
A three fold increase of the effective string tension
due to in medium effects in A + A collisions,
results in a sizeable ($\approx$ 60-70 %) enhancement
of the total charm production cross sections
($\sigma^{\rm NN}_{{\it c}\,\bar{{\it c}}}$).
The nuclear modification factors show a
suppression at moderate transverse momentum (${\rm p}_{\rm T}$)
consistent with RHIC data.
At Large Hadron Collider energies
the model predicts an increase of $\sigma^{\rm NN}_{{\it c}\,\bar{{\it c}}}$
by approximately an order of magnitude.
pacs: 25.75.Dw, -25.75.Cj, 25.75.+r, 24.85.+p
The phase transition from hadronic degrees of freedom to
partonic degrees of freedom in ultra-relativistic nuclear collisions
is a central focus of experiments at the Relativistic
Heavy Ion Collider (RHIC).
Heavy-flavor quarks are an ideal probe to study early dynamics
in these nuclear collisions.
Several theoretical studies predict
wang_prl92 ; shuryak_prl92 ; geiger_prc93
a substantial enhancement of open
charm production associated to plasma formation of the deconfined
parton matter relative to the case of a purely hadronic scenario
without plasma formation.
A recent analysis shows that the dynamics of heavy-quarks at
RHIC are dominated by partonic or “pre-hadronic” interactions
in the strongly coupled plasma (sQGP) stage and can neither
be modeled by “hadronic interactions” nor described appropriately
by color screening alone cassing_08 .
Therefore, these quarks are key observables in the study
of thermalization of the initially created hot nuclear matter
nuxu_07 .
A review of heavy-flavor production in heavy-ion collisions
has been recently published vogt08_1 .
Direct reconstructed $D^{0}$ mesons
via hadronic channel ($D^{0}\,\rightarrow K\pi$)
in d + Au star_05 , Cu + Cu star_08_1 and
Au + Au star_08_2 collisions have been measured.
Due to the difficulty to reconstruct D-mesons hadronic decay vertex,
both STAR and PHENIX have studied open charm indirectly
via semileptonic decay to non-photonic electrons (NPE) or muons
star_05 ; star_08_2 ; star_07 ; yzhang_08 ; phenix_05 ; phenix_06_1 ; phenix_06_2 ; phenix_07 . Theory predict that charm quarks are produced by initial
gluon fusion Lin95 and
their production rates are expected to be well described
by perturbative Quantum Cromodynamics (pQCD)
at Fixed Order plus Next to-leading Logarithms (FONLL) cacciari_05 .
Total charm cross sections reported by both experiments
are however only compatible with the upper limit
of the FONLL predictions.
In addition, the data indicate a suppression
as large as that of light quarks star_07 ,phenix_07
while, due to their large mass and to the dead cone effect
charm quarks are predicted to loose
less energy than light quarks by gluon radiation in the medium
kharzeev_01 .
Recent model calculations based on in-medium charm
resonances/diffusion or collisional dissociation
Rapp_06 ; Vitev_07 ; Moore_05 ,
radiative energy loss via few hard scatterings Magdalena_06
or radiative energy loss via multiple soft collisions
Armesto_06 , have been applied to describe the non-photonic
electrons (NPE) spectra. They all predict less suppression than
that observed in experiments. On the other hand,
a good description of the nuclear modification factor
(NMF), ${\rm R}_{\rm AA}^{\rm npe}({\rm p}_{\rm T})$, was obtained in
non-perturbative time-dependent heavy-quark diffusion in the
Quark-Gluon Plasma (QGP) rapp_prl08 .
In previous papers prc72_top05 ; prc75_top07 we have shown
that the dynamics of strangeness production
deviates considerably from calculations based on Schwinger-like
estimates for homogeneous and constant color fields schwinger
and point to the
contribution of fluctuations of transient strong
color fields (SCF).
These fields are similar with those which could appear in a
“glasma” mclerran_08 at initial stage of the collisions.
In a scenario with QGP phase
transitions the typical field strength of SCF at RHIC energies
was predicted to be about 5-12 GeV/fm csernai_npa02 .
Recently Schwinger mechanism has been revisited cohen_jul08 and
pair production in time-dependent
electric fields has been studied gies_jul08 .
It is concluded that particles with large momentum are likely to have
been created earlier than particle with small momentum and
for very short temporal widths
($\Delta\tau\approx 10{\rm t}_{\rm c}$, where the Compton time
${\rm t}_{\rm c}=1/{\rm m}_{\rm c}$) the Schwinger formula strongly
underestimates the reachable particle number density.
In this paper we extend our study in the framework of
HIJING/BB̄ v2.0 model prc75_top07
to open charm productions.
We explore dynamical effects associated with
long range coherent fields (i.e strong color fields, SCF),
including baryon junctions and loops prc72_top05 , with emphasis
on the novel open charm observables measured at RHIC
in p+p and heavy-ion collisions.
Using this model we analyze the enhancement of total charm production
at 200A GeV energy.
For a uniform chromoelectric flux tube with field (E)
the pair production rate Biro:1984cf ; Gyulassy:1986jq ; cohen_jul08
per unit volume for a heavy quark is given by:
$$\Gamma=\frac{\kappa^{2}}{4\pi^{3}}{\text{exp}}\left(-\frac{\pi\,m_{Q}^{2}}{%
\kappa}\right)$$
(1)
where for $Q=c$ or $b$ , $m_{\rm Q}=1.27,\;{\rm or}\;4.16$
GeV (with $\pm 1\%$
uncertainty Steinhauser:2008pm )
Note that $\kappa=|eE|_{eff}=\sqrt{C_{2}(A)/C_{2}(F)}\,\kappa_{0}$
is the effective string tension
in terms of the vacuum string tension $\kappa_{0}\approx 1$ GeV/fm
and $C_{2}(A)$, $C_{2}(F)$ are the second order Casimir operators
(see Ref. Gyulassy:1986jq ).
In a nuclear collisions, the local longitudinal
field strength increases with the square root of the
number of color exchanges proportional to
the number of binary collisions per unit area
($\kappa(x_{\perp},b)\propto\sqrt{T_{A}A(x_{\perp},b)}$),
where for a given impact parameter
$b$ and transverse coordinate $x_{\perp}$, $T_{A}A\propto A^{2/3}$
is the Glauber A+A
binary collision distribution. Therefore, the effective string tension
$\kappa\propto A^{1/3}$.
A measurable rate for spontaneous pair production requires
“strong chromo electric fields”, such
that $\kappa/m_{\rm Q}^{2}\,\,>$ 1 at least some of the time.
On the average, longitudinal electric field “string” models
predict for heavier flavor a very suppressed production rate per unit
volume $\gamma_{Q}$ via the well known Schwinger formula schwinger ,
since
$$\gamma_{Q\bar{Q}}=\frac{\Gamma_{Q\bar{Q}}}{\Gamma_{q\bar{q}}}={\text{exp}}%
\left(-\frac{\pi(m_{Q}^{2}-m_{q}^{2})}{\kappa_{0}}\right)\ll 1$$
(2)
for $Q=c$ and $q=u,d$.
For a color rope on the other hand,
if the average string tension value ($<\kappa>$)
increases from 1.0 GeV/fm to 3.0 GeV/fm,
the rate $\Gamma$ for charm pairs to tunnel through the longitudinal field
increases from $\approx 1.4\,\cdot 10^{-12}$ to
$\approx 3.5\,\cdot 10^{-4}$ fm${}^{-4}$,
and this can lead to a net “soft” tunneling production
comparable to the initial “hard” FONLL pQCD production.
The conventional hard pQCD mechanism, mainly
gluon fusion wang_prl92 , is calculated via the PYTHIA
subroutines in HIJING/BB̄ v2.0.
The advantage of HIJING over PYTHIA
is the ability to include novel SCF color rope effects
that arise from longitudinal fields amplified by the random walk
in color space of the high x valence partons in
A+A collisions.
This random walk could induce
a very broad fluctuation spectrum of the effective string tension.
Thus, if the average or mean $<\kappa>=n\,\kappa_{0}$, then the typical
fluctuation is of order $1/\sqrt{n}$ which is large because
$n\approx 6$ for Au nuclei.
A Poisson fluctuation of effective $\kappa$ about the mean, gives a
strong bias toward less probable but larger
than the average value $<\kappa>$. This is amplified for heavy
quarks.
Here we do not investigate in details such fluctuations, but we will
estimate the effects of a larger effective value $\kappa>$ 3 GeV/fm
on the enhancement of $\sigma^{\rm NN}_{{\it c}\,\bar{{\it c}}}$.
Both STAR and PHENIX experiments have measured charm production cross
sections in several collision systems.
Figure 1 shows the measured total charm production cross
sections at mid-rapidity,
$d\,\sigma^{\rm NN}_{{\it c}\,\bar{{\it c}}}/dy$ (left
panel) and in all phase space,
$\sigma^{\rm NN}_{{\it c}\,\bar{{\it c}}}$ (right panel).
The data from both experiments seems to indicate a scaling
with number of binary collisions (${\rm N}_{\rm bin}$),
as expected because of the high mass of charm pairs produced in initial
nucleon-nucleon collisions Lin95 .
However, there is still an unresolved discrepancy of the order of
a factor of two between STAR and PHENIX data.
The predictions of HIJING/BB̄ v2.0 model without SCF (open crosses)
and including SCF effects (filled squares) are shown in the figure.
For completeness the results with SCF but no gluon
shadowing effects (open triangles) are also included.
However, in this scenario multiplicities at
mid-rapidity are strongly overestimated topor_prc03 .
The main parameters used in the calculations are given in
Table II of reference prc75_top07 ,
and corresponds to
strengths of strong color (electric) field dependent on collision system
($\kappa$ = 1.5; 2.0; 3.0 GeV/fm for p + p, d + Au, and
A + A collisions respectively).
In our calculations we estimate the total open charm
production (c + $\bar{\it c}$) cross section considering
the 12 lightest D-mesons (${\rm D}^{0}$, ${\bar{\rm D}}^{0}$,
${\rm D}^{0*}$, ${\bar{\rm D}}^{0*}$,
${\rm D}^{+}$, ${\bar{\rm D}}^{+}$, ${\rm D}^{+*}$, ${\bar{\rm D}}^{+*}$,
${\rm D}_{\rm s}$, ${\bar{\rm D}}_{\rm s}$,
${\rm D}^{*}_{\rm s}$, ${\bar{\rm D}}^{*}_{\rm s}$), and the hyperons
$\Lambda_{\rm c}$ and $\bar{\Lambda_{\rm c}}$.
The contribution of higher mass charm hyperons is negligible.
For calculations which take into consideration SCF effects (filled
squares) we obtain an increase of $60-70\,\%$ in comparison
with a scenario without SCF effects (open crosses).
These results describe well the PHENIX data within statistical
and systematical errors and are close to the upper limit of
uncertainty band of the pQCD FNOLL predictions cacciari_05 .
Our calculations also show that the scaling with ${\rm N}_{\rm bin}$
is only approximately satisfied, the reason being
an interplay between the mass dependent SCF and
shadowing effects, which act in opposite directions.
In fact, we calculate that only $60\,\%$ of total open charm production
(c + $\bar{\it c}$) comes from partons embedded within the target
and projectile.
The study of open charm production in d + Au collisions
allow to separate “cold nuclear matter” effects.
The initial production
of c $\bar{\it c}$ pairs by gluon fusion might be suppressed
due to gluon shadowing.
We recall that shadowing is a depletion of the low-momentum parton
distribution in a nucleon embedded in a nucleus compared to a free
nucleon; this leads to a lowering in the (scaled)
c + $\bar{\it c}$ production relative to p + p
collisions.
The shadowing in the regular HIJING parameterization hij_94
implemented also in our model seems to be too strong wang_pl527_02 .
There is a considerable uncertainty
(up to a factor of 3) in the amount of shadowing
predicted at RHIC energies by the different models with HIJING
predicting the strongest effect cms_enteria_08 .
This could explain
why the results for scaled cross sections in d + Au collisions
are smaller than those obtained for p + p
collisions (see Fig. 1 left panel).
We study if we can find scenarios that would give larger enhancement
of total cross sections for open charm production, than
those reported in Fig. 1 (filled squares),
and that would be consistent with the STAR data.
The random walk in color space of heavy quark could induce a broad spectrum
of the effective string tension.
Therefore, we study the effects of
a further increase of mean value of the string tension from
3.0 GeV/fm to 5.0 GeV/fm on ${\it c}\,\bar{{\it c}}$ pair production.
This results in only a modest increase of scaled
cross sections, $\sigma^{\rm sNN}_{{\it c}\,\bar{{\it c}}}=\sigma^{\rm AA}_{{\it c}\,\bar{{\it c%
}}}/{\rm N}_{\rm bin}$,
by approximately $20\,\%$ for central collisions.
For values between 5 - 10 GeV/fm a saturation
sets in, as an effect of energy and momentum conservation constraints.
In our model the multiplicative factor that accounts for
next to leading order corrections eskola_03 in calculations of
hard or semihard parton
scattering processes via pQCD is set to K = 2.
Increasing this factor to K = 3.5, as suggested
in reference eskola_03 results in an increase of
$\sigma^{\rm sNN}_{{\it c}\,\bar{{\it c}}}$ by approximately $70\%$
in central Au + Au collisions, but also overpredicts by $40\,\%$
the total charged particle production at
mid-rapidity (N${}_{\rm ch}$ (y=0)).
Therefore, we conclude that the large charm cross sections obtained by
the STAR collaboration cannot be explained within our phenomenology.
The ${\it D}^{0}$-mesons spectra are sensitive to
the dynamics of produced charm particles.
In Fig. 2 we present the calculated
$D^{0}$-mesons spectra for systems where data are available star_05 ,star_08_1 ,star_08_2 .
In all cases, the calculated yield is much smaller than
the STAR data, consistent with the results shown in Fig. 1.
The calculated spectra show little shoulder at low ${\rm p}_{\rm T}$
indicating small radial flow of ${\it D}^{0}$-mesons consistent with
the results of STAR yzhang_08 .
Figure 3 shows our predictions for
the Nuclear Modification factor (NMF),
${\rm R}_{\rm AA}({\rm p}_{\rm T})$ for $D^{0}$ and $\pi^{0}$ mesons.
Data (filled symbols) are NMF for non photonic
electrons, ${\rm R}_{\rm AA}^{\rm npe}({\rm p}_{\rm T})$
star_07 ,phenix_07 .
The data for $\pi^{0}$ meson (open symbols) are from reference
phenix_03 .
Note, that non photonic electrons include also electrons
from bottom (b) production (B $\rightarrow$ lX)
and the yields of $D^{0}$ mesons could
be affected by the decay (B $\rightarrow$ D).
For central (0-10 %) Au + Au
collisions we calculate a scaled total cross section for
bottom production with (without) SCF of
$\sigma^{\rm sNN}_{{\it b}\,\bar{{\it b}}}$ = 17.8 $\mu$b
($\sigma^{\rm sNN}_{{\it b}\,\bar{{\it b}}}$ = 0.86 $\mu$b).
These values are
few orders of magnitude lower than $\sigma^{\rm sNN}_{{\it c}\,\bar{{\it c}}}$
and this contribution is estimated to be
negligible for ${\rm p}_{\rm T}\,<6.0$ GeV/c.
In our calculations for low
${\rm p}_{\rm T}$ ($0\,<\,{\rm p}_{\rm T}\,<2.5$ GeV/c),
non-perturbative production mechanism via SCF
results in a split between $D^{0}$ and $\pi^{0}$ mesons.
The charged and $\pi^{0}$ mesons are suppressed due to
conservation of energy prc72_top05 .
The yields of the ${\it D}^{0}$-mesons are enhanced due to an increase
of $c\,\bar{c}$ pair production rate (see Eq. 1).
In central ($0-10\,\%$) Au + Au collisions a suppression
at moderate $p_{\rm T}$ ($4\,<\,{\rm p}_{\rm T}\,<\,6$ GeV/c)
as large as that of light quarks is observed
in contrast to previous theoretical studies kharzeev_01 ,
Magdalena_06 ; Rapp_06 ; Armesto_06 ; wicks_npa07 .
Our model predicts a suppression consistent with the data.
We can interpret this results as experimental evidence for
“in-medium mass modification” of charm quark, due to
possible induced chiral symmetry restoration kharzeev_05 .
An in-medium mass modification has also been predicted
near the phase boundary (i.e. at lower energy) in tolos_06 .
In contrast statistical hadronization model andronic_08
predicts no medium effects at top RHIC energy.
We performed calculations at the much higher
Large Hadrons Collider (LHC) energy
using parameters from reference top_jpg08 ,
i.e $\kappa$ = 2.0; 5.0 GeV/fm for p + p
and central (0-10 %) Pb + Pb collisions respectively.
The predicted charm production cross section
is approximately an order of magnitude larger than at RHIC energy.
We obtain $\sigma^{\rm NN}_{{\it c}\,\bar{{\it c}}}$ = 6.4 mb
in p+p collisions and a (scaled) cross section
$\sigma^{\rm sNN}_{{\it c}\,\bar{{\it c}}}$ = 2.8 mb
for central Pb + Pb collisions (${\rm N}_{\rm bin}$ = 960
and N${}_{\rm ch}$ (y=0) = 2500).
This indicates a clear violation of scaling with ${\rm N}_{\rm bin}$
at the LHC.
These values increase by a factor of 2 to 3 if the effects
of shadowing are not included
( N${}_{\rm ch}$ (y=0) $\approx$ 5000 and
$\sigma^{\rm sNN}_{{\it c}\,\bar{{\it c}}}$ $\approx$ 8.4 mb).
In summary, we studied the influence of possible strong homogenous
constant color fields in open charm production
in heavy-ion collisions by varying the effective string tension that
control QQ̄ pair creation rates. This is equivalent with
assuming an in medium mass modification of charm quark.
We show that this approach is an important
dynamical mechanism that can explain the observed
D-mesons enhancement production observed by the PHENIX experiments.
Our model is based on the time-independent color field while in
reality the production of QQ̄ pairs is a far-from-equilibrium,
time-dependent phenomenon. Thus to achieve more quantitative
conclusions, such mechanisms gies_jul08
should be considered in future generation of Monte Carlo codes.
The large cross sections reported by the STAR collaboration
remain unexplained within our study.
Solving the discrepancy between the measurements is important, since
confirmation of the STAR results
may indicate the importance of other dynamical mechanisms such
as pre-equilibrium production
from secondary parton cascades wang_prl92 ,
or hot-glue scenario shuryak_prl92 .
Acknowledgments: We thank C. Gale and S. Jeon for useful discussions.
This work was supported by the Natural Sciences and Engineering
Research Council of Canada.
This work was supported also by the Division of Nuclear Science,
of the U. S. Department of Energy under Contract No. DE-AC03-76SF00098 and
DE-FG02-93ER-40764.
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On the Geometry of Spaces of Oriented Geodesics
Dmitri V. Alekseevsky$\,$,
School of Mathematics and Maxwell Insitute for
Mathematical Sciences, The Kings Buildings, JCMB, University of
Edinburgh,
Mayfield Road, Edinburgh, EH9 3JZ,UK.
[email protected]
Brendan Guilfoyle$\,$,
Department of Computing and Mathematics,
IT Tralee, Clash, Tralee,
County Kerry, Ireland.
[email protected]
Wilhelm Klingenberg$\,$.
School of Mathematical Sciences, University of Durham, Durham DH1
3LE, UK.
[email protected]
Abstract:
Let $M$ be either a simply connected pseudo-Riemannian space of constant curvature or a rank one Riemannian symmetric space other than $\mathbb{O}H^{2}$, and consider the space
$L(M)$ of oriented geodesics of $M$. The space $L(M)$ is a smooth
homogeneous manifold and in this paper we describe all invariant
symplectic structures, (para)complex structures, pseudo-Riemannian metrics and
(para)Kähler structure on $L(M)$.
Keywords: space of geodesics, rank one symmetric spaces, homogeneous Riemannian manifolds, pseudo-Riemannian metrics, symplectic structures, complex structures, Kähler structures, flag manifolds,
Cayley projective plane.
MSC 2000: Primary 53A25, Secondary 53B35.
This
work was supported by Leverhulme Trust, EM/9/2005/0069.
Contents
1 Introduction
1.1 Background
1.2 Main results
1.3 Outline of paper
2 The Space of Oriented Geodesics on a Riemannian Manifold
2.1 Tangent and unit sphere bundle
2.2 Smooth structure in the space of geodesics $L(M)$
2.3 Canonical symplectic structure on $L(M)$
3 Pseudo-Riemannian Spaces of Constant Curvature
3.1 Spaces of zero curvature
3.2 Spaces of constant non-zero curvature
4 Rank One Symmetric Spaces of Non-Constant Curvature
4.1 Complex projective and hyperbolic spaces
4.2 Quaternionic projective and hyperbolic spaces
4.3 Cayley projective plane
1 Introduction
1.1 Background
The geometry of the set of straight lines of projective space $P^{3}$ and Euclidean space $E^{3}$ is a classical
subject of investigations of such 19th century geometers as Grassmann, Plücker, F. Klein and Study.
They studied natural correspondences between submanifolds (i.e. points, curves and surfaces) in $E^{3}$ and submanifolds of $L(E^{3})$, the
space of oriented lines of $E^{3}$.
For example, a point $p\in E^{3}$ defines a surface in $L(E^{3})$ which consists of the oriented lines through $p$, a curve $C\subset E^{3}$
defines three curves in $L(E^{3})$ associated with the Frenet frame
of $C$, and a surface $S\subset E^{3}$ defines a surface in $L(E^{3})$ by its oriented normal lines. Conversely, distinguished (from the point of view of
intrinsic geometry of $L(E^{3})$) submanifolds determine special families of oriented lines in $E^{3}$.
In [14], Study identified the space $L(E^{3})$ of oriented lines in $E^{3}$ with the 2-sphere (the Study sphere) over the dual numbers
and defined and studied a notion of distance between oriented lines.
For a nice modern exposition and generalization of these results with applications to computational geometry, computer graphics
and visualization, see [9].
A natural complex structure in the space $L(E^{3})$ of oriented lines has been considered by Hitchin, who used it for a description of monopoles [7].
In [5], two of the authors defined a neutral Kähler structure in the space $L(E^{3})$ and gave its geometric description.
Recently, it has been used in the solution of a long-standing conjecture of Carathéodory [6].
In general, however, the space $L(M)$ of oriented geodesics of a (complete) Riemannian manifold is not a smooth manifold and has very bad topology.
But $L(M)$ is a smooth manifold if $(M,g)$ is either an Hadamard manifold (i.e. complete simply connected Riemannian manifold of non-positive curvature)
or a manifold with closed geodesics of the same length (aufwiederseen manifolds).
Symplectic and Riemannian structures in the space of geodesics $L(M)$ of aufwiederseen manifolds are discussed by Besse
[1], see also Reznikov [10] [11].
The symplectic form on the space of geodesics of a Hadamard manifold is described in [2].
Geometric structures in the space of oriented geodesics of hyperbolic 3-space are studied in [3], while Salvai has addressed the existence
and uniqueness of pseudo-Riemannian metrics in the spaces $L(E^{n})$ and $L(H^{n})$ [12] [13]. Note that both spaces are homogeneous manifolds
of the corresponding isometry group. Salvai proved that $L(E^{n})$ admits a pseudo-Riemannian metric invariant under
a transitive subgroup $G$ of the isometry group $I(E^{n})=E(n)=SO(n)\cdot E^{n}$ only for $n=3,7$ and gave an explicit description of the
corresponding metric.
1.2 Main results
The aim of the paper is to describe the natural geometric structures
in the space $L(M)$ of oriented geodesics of $M$, where $M$ is either a simply connected pseudo-Riemannian
space of constant curvature, or a rank one Riemannian symmetric space other than $\mathbb{O}H^{2}$. In these cases the space of geodesics $L(M)$ is a smooth
homogeneous manifold and we use Lie groups and Lie algebras to describe all invariant
symplectic structures, (para)complex structures, pseudo-Riemannian metrics and
(para)Kähler structure on $L(M)$.
More specifically, let $S^{p,q}=\{x\in E^{p+1,q},x^{2}=1\}$, where $E^{p+1,q}$ is ${\mathbb{R}}^{p+q+1}$ endowed with the flat metric of signature
$(p+1,q)$. Here, for $p=0$ we
assume in addition that $x^{0}>0$ so that $S^{p,q}$ is connected. The induced metric on $S^{p,q}$
has signature $(p,q)$ and is of constant curvature $1$.
Let $L^{+}(S^{p,q})$ (respectively $L^{-}(S^{p,q})$)
be the set of spacelike (respectively, timelike) geodesics in $S^{p,q}$ and similarly for $L^{\pm}(E^{p+1,q})$. We prove:
Main Theorem 1
For the flat pseudo-Euclidean spaces $E^{p+1,q}$
i)
The space $L^{-}(E^{p+1,q})=E(p+1,q)/SO(p,q)\cdot{\mathbb{R}^{+}}$ is a symplectic symmetric space with an invariant
Grassmann structure defined by a decomposition $\mbox{\goth m}=W\otimes{\mathbb{R}}^{2}$. Moreover, if $n=p+1+q=3$,
it has an invariant Kähler structure $(g,J)$ of neutral signature $(2,2)$. In addition, $L^{+}(E^{p+1,q})=L^{-}(E^{q,p+1})$.
while for the non-flat constant curvature manifolds $S^{p,q}$
ii)
Suppose that $p+q>3$. Then there exists a unique (up to scaling)
invariant symplectic structure $\omega$ and a unique (up to a sign)
invariant complex structure $I^{+}=J$ on $L^{+}(S^{p,q})$
and a unique (up to sign) invariant para-complex structure $K=I^{-}$ on $L^{-}(S^{p,q})$.
There exists unique (up to scaling) invariant
pseudo-Riemannian metric $g^{\varepsilon}=\omega\circ I^{\varepsilon}$ on $L^{\varepsilon}(S^{p,q})$ which is Kähler of signature
$(2(p-1),2q)$ for
$\varepsilon=+$ and para-Kähler (of neutral signature) for $\varepsilon=-$.
iii)
Suppose that $p+q=3$. Then there are 2-linearly independent invariant (parallel and closed)
2-forms $\omega,\omega^{\prime}$ on $L^{\varepsilon}(S^{p,q})$ with values $\omega_{\mbox{\goth m}}=\omega_{H}\otimes g_{V},\,\omega^{\prime}_{\mbox{\goth
m%
}}=g_{H}\otimes\omega_{V}$.
Any invariant metric has the form $h=\lambda g+\mu g^{\prime}$ where $g^{\prime}$ is a neutral metric with
value $g^{\prime}_{\mbox{\goth m}}=\omega_{H}\otimes\omega_{V}$. Any metric $h$ is Kähler for $\varepsilon=1$ (respectively,
para-Kähler for $\varepsilon=-$) with respect to the complex (respectively, para-complex) structure
$I^{\varepsilon}=I^{\varepsilon}_{H}\otimes 1$ with the Kähler form $h\circ I^{\pm}=\lambda\omega+\mu\omega^{\prime}$.
Moreover, the endomorphism $I^{\prime}=1\otimes I_{V}$ of m defines an invariant parallel
$h$-skew-symmetric complex structure of $L^{\varepsilon}(S^{p,q})$ if $\varepsilon=1,(p-1,q)=(2,0)$ or $(0,2)$ or
$\varepsilon=-$ and $(p,q-1)=(2,0)$ or $(0,2)$ and skew-symmetric parallel para-complex structure
if $\varepsilon=+,(p-1,q)=(1,1)$ or $\varepsilon=-,(p,q-1)=(1,1)$. The Kähler or para-Kähler structure
$(h,I^{\prime})$ has the Kähler form $h\circ I^{\prime}=\lambda\omega+\mu\omega^{\prime}$.
Consider now the rank one Riemannian symmetric spaces of non-constant
curvature. That is, $M=G/K$ is one of the projective spaces
$${\mathbb{C}}P^{n}=SU_{n+1}/U_{n},\qquad{\mathbb{H}}P^{n}=Sp_{n+1}/Sp_{1}\cdot
Sp%
_{n},\qquad{\mathbb{O}}P^{2}=F_{4}/Spin_{9}$$
or one of the dual hyperbolic spaces
$${\mathbb{C}}H^{n}=SU_{1,n}/U_{n},\qquad{\mathbb{H}}H^{n}=Sp_{1,n}/Sp_{1}\cdot
Sp%
_{n},\qquad{\mathbb{O}}H^{2}=F^{non-comp}_{4}/Spin_{9}.$$
For the projective spaces we prove:
Main Theorem 2
i)
The space $L({\mathbb{C}}P^{n})=SU_{n+1}/T^{2}\cdot SU_{n-1}$ has a one–parameter
family $\omega^{t}=\omega_{1}+t\omega_{0}$ of invariant symplectic forms (up to scaling) and four invariant
almost complex structures, up to sign, two of them being integrable.
All (almost) complex structures $J$ are compatible with $\omega^{t}$ i.e. they define an (almost)
Kähler or pseudo-Kähler metric $g=\omega^{t}\circ J$.
ii)
The spaces $L({\mathbb{H}}P^{n})=Sp_{n+1}/T^{1}\cdot Sp_{1}\cdot Sp_{n-1}$ and $L({\mathbb{O}}P^{2})=F_{4}/T^{1}\cdot Spin_{7}$
has a unique (up to scaling) invariant symplectic forms $\omega$ and
unique (up to sign) invariant complex structure $J$.
The pair $(\omega,J)$ defines a unique (up to scaling) invariant Kähler metric
$g=\omega\circ J$.
iii)
None of the above spaces have any invariant almost para-complex structures.
iv)
The canonical
symplectic structure on $L({\mathbb{C}}P^{n})$ is identified with $\omega_{1}$.
while for the hyperbolic spaces (other than $\mathbb{O}H^{2}$) we show
Main Theorem 3
i)
The space $L({\mathbb{C}}H^{n})=SU_{1,n}/T^{2}\cdot SU_{n-1}$ has a one–parameter family of
invariant symplectic structures $\omega^{t}=\omega_{1}+t\omega_{0}$,
and two (up to sign) invariant almost para-complex structures $K^{\pm}$, one of them
being integrable; and both are consistent
with $\omega^{t}$
i.e. $(K^{\pm},\omega^{t})$ defines a para-Kähler metric $g=\omega\circ K^{\pm}$.
ii)
The space $L({\mathbb{H}}H^{n})=Sp_{1,n}/T^{1}\cdot Sp_{1}\cdot Sp_{n-1}$ admits a unique (up to scaling)
invariant symplectic form $\omega$ and two (up to sign) invariant almost para-complex
structures, one of which is integrable. Both are compatible with $\omega$.
iii)
None of the above spaces have any invariant almost complex structures.
iv)
The canonical
symplectic structure on the space geodesics $L({\mathbb{C}}H^{n})$ is $\omega_{1}$.
In Table 1 we summarize the results of Main Theorems 1, 2 and 3.
1.3 Outline of paper
This paper is organised as follows. In the following section we consider the set of oriented geodesics $L=L(M)$ of a general complete
Riemannian manifold $M$.
Though $L$ may not be a manifold, we can still define
differential geometric objects on $L$ in terms of $G_{1}$-invariant objects on $SM$, where $SM$ is the unit sphere bundle and $G_{1}=\{expt\Gamma\}$ is the
geodesic flow. For example, the algebra of smooth functions ${\mathcal{F}}(L)$ on $L$ is defined as the algebra
of $G_{1}$-invariant functions on $SM$ (that is first integrals of the geodesic flow).
We sketch this approach in section 2 and define the canonical symplectic form on $L$. In the case when
$L$ is a manifold this coincides with
the standard symplectic form defined in [1].
Section 3 considers the oriented geodesics of pseudo-Riemannian manifolds of constant curvature, taking the flat and non-flat cases separately.
Main Theorem 1 follows from Theorems 1 and 2 of sections 3.1 and 3.2, respectively.
In section 4 we turn to rank one Riemannian symmetric spaces with non-constant curvature, dealing separately with the complex projective
and hyperbolic spaces,
the quaternionic projective and hyperbolic spaces and the Cayley plane. In particular, Main Theorem 2
follows from Theorem 3 of section 4.1, Theorem 5 of section 4.2 and Theorem 7 of section 4.3, while
Main Theorem 3
follows from Theorem 4 of section 4.1 and Theorem 6 of section 4.2.
2 The Space of Oriented Geodesics on a Riemannian Manifold
2.1 Tangent and unit sphere bundle
Let $M$ be an $n$-dimensional manifold and $\pi:TM\to M$ be the tangent bundle of $M$. Local coordinates $(x^{i})$ on $M$ give rise
to local coordinates
$(x^{i},y^{i})$ on $TM$ via $v_{x}=y^{i}\partial/\partial x^{i}$. Then $T_{(x,y)}TM\ni(x,y,\dot{x},\dot{y})$.
The vertical subspace $V_{(x,y)}TM$ is given by $(x,y,0,\dot{y})$ and we denote by $N$
the canonical endomorphism
$$N:T_{(x,y)}TM\to V_{(x,y)}TM\to 0:\;\;(x,y,\dot{x},\dot{y})\to(x,y,0,\dot{x})%
\to 0,$$
with $N^{2}=0$. Note that
$$\mathrm{Ker}\;N=\mathrm{Im}\;N=T^{v}(TM)=\{(x,y,0,\dot{y})\},$$
is the vertical subbundle of $T(TM)$. The restriction of the projection $\pi_{*}:T(TM)\to TM$
onto the vertical subspace is an isomorphism
$$\pi_{*}:T^{v}_{(x,y)}(TM)\to T_{x}M,\,\,(x,y,\dot{x},\dot{y})\mapsto(x,\dot{x}).$$
Assume now that $(M,g)$ is
Riemannian and $SM$ is the unit sphere bundle:
$$SM=\{(x,y)\in TM,\,g_{ij}(x)y^{i}y^{j}=1\}.$$
The metric $g$ induces an isomorphism of the tangent bundle
onto cotangent bundle $T^{*}M=\{\alpha=(x,p),\,\alpha=p_{i}dx^{i}\}$ given by
$$(x^{i},y^{i})\to(x^{i},p_{i}=g_{ij}y^{j}).$$
The pull-back of the canonical
one-form $\alpha=p_{i}dx^{i}$ and symplectic form $\omega=d\alpha$ of $T^{*}M$ are given by
$$\alpha^{g}=g^{*}\alpha=g_{ij}y^{i}dx^{j},\,\,\omega^{g}=g\circ\omega=g\circ d%
\alpha=d(g_{ij}y^{i})\wedge dx^{j}.$$
The tangent bundle of the unit sphere bundle has the canonical decomposition
$T_{(x,y)}SM=V_{(x,y)}SM+H^{g}_{(x,y)}SM$
into vertical space and horizontal subspaces.
Lemma 1
The sphere bundle of the Riemannian manifold $(M,g)$
admits the canonical contact structure
$$\theta=\alpha^{g}|SM,\;\;\;\alpha^{g}=g^{*}\alpha=g_{ij}y^{i}dx^{j},$$
where the associated Reeb vector field is the geodesic vector field given by
$$\Gamma=y^{i}\partial/\partial x^{i}-\Gamma^{i}_{jk}y^{j}y^{k}\partial/\partial
y%
^{i}.$$
The horizontal lift of $\partial/\partial x^{i}=\partial_{i}$ into $SM$ is given by
$$\nabla_{i}=\partial_{i}-\Gamma^{k}_{ij}(x)y^{j}\partial/\partial y^{k},(x,y)%
\in SM.$$
In particular, $\Gamma$ preserves $\theta$ and $d\theta$.
Proof. The last claims are verified as follows. A vector field along $x(t)$ given by $X(t)=y^{i}(x(t))\partial_{i}$ is parallel if
$$\dot{X}^{i}+\Gamma^{i}_{jk}(x(t))\dot{x}^{i}X^{j}=0.$$
Then $(x(t),X(t))$ is the horizontal lift of the curve $x(t)$. The horizontal space is spanned by
the tangent vectors of such lifts, namely
$$H_{(x,y)}={\rm{span}}\{(\dot{x}^{i},-\Gamma^{i}_{jk}\dot{x}^{j}y^{k})\}.$$
The vector field $\Gamma=y^{i}\partial/\partial x^{i}-\Gamma^{i}_{jk}y^{j}y^{k}\partial/\partial
y%
^{i}$ on $TM$ is tangent to $SM$ since
$$\nabla_{k}(g_{ij}y^{i}y^{j})=g_{ij,k}-g_{im}\Gamma^{m}_{kl}-g_{jm}\Gamma^{m}_{%
ki}=g_{ij,k}-\Gamma_{i,kj}-\Gamma_{j,ki}=0.$$
Moreover, $\Gamma$ is a geodesic vector field since its integral curves satisfy the geodesic equation. Therefore we have
the decomposition into vertical and horizontal parts
$$T_{(x,y)}SM=V_{(x,y)}SM+H_{(x,y)}SM=\{v^{i}\partial/\partial y^{i}:g_{ij}y^{i}%
v^{j}=0\}+\{u^{k}\nabla_{k}\}.$$
We compute
$$d\theta=g_{ij,k}y^{j}dx^{i}\wedge dx^{k}+g_{ij}dy^{j}\wedge dx^{i}.$$
One easily checks that $\theta(\Gamma)=1$ and $d\theta(\Gamma)=0$.
Clearly the form $d(g^{*}\alpha)=g^{*}d\alpha=g^{*}\omega$ is non-degenerate on $TM$. Therefore its restriction
to $SM$ has one-dimensional kernel spanned by $\Gamma$ and so $\Gamma$ is the Reeb vector field of the
contact form $\theta$ and it preserves $\theta$ and $d\theta$.
$\square$
We associate with the vector field $X=X^{i}\partial_{i}$ the function $f_{X}$ on $SM$ given by
$f_{X}(x,y)=g_{ij}X^{i}y^{j}$.
Lemma 2
The covariant derivative $\nabla_{i}X$ corresponds to the Lie derivative of $f_{X}$ in direction of the vector field
$\nabla_{i}$ :
$$f_{\nabla_{i}X}=\nabla_{i}f_{X}.$$
Proof. Applying the vector field $\nabla_{i}$ to the function $f_{X}$ on $SM$ we get
$$\begin{array}[]{rl}\nabla_{i}f_{X}&=(g_{kl}X^{k})_{,i}y^{l}-g_{kl}X^{k}\Gamma^%
{l}_{ij}y^{j},\\
&=g_{kl}X^{k}_{,i}y^{l}+(g_{kj,i}-\Gamma_{k,ij})X^{k}y^{j},\\
&=g_{kl}X^{k}_{,i}y^{l}-\frac{1}{2}(g_{kj,i}+g_{ki,j}-g_{ij,k})X^{k}y^{j},\\
&=g_{ka}(X^{a}_{,i}+\Gamma^{a}_{,li}X^{l})y^{k},\\
&=f_{\nabla_{i}X}.\end{array}$$
$\square$
Denote the tangent bundle without the zero section by $T^{\prime}M$. Then $T^{\prime}M=SM\times{\mathbb{R}}^{+}$ with coordinate
$r$ on the second factor. We denote $E=r\partial/r=y^{i}\partial/\partial y^{i}$ the Euler vector field.
Note that the symplectic form $\omega^{g}$ is homogeneous of degree one: $E\cdot\omega^{g}=\omega^{g}$ and $E\cdot(\omega_{g})^{-1}=-\omega_{g}$.
Denote the homogeneous functions of degree $k$ by ${\cal{F}}_{k}(T^{\prime}M)$ and the extension of
the function $f\in{\cal F}(SM)$ to ${\cal F}_{k}$ by $f_{(k)}$. Furthermore, denote the Poisson
structure by $\{f,g\}=\omega_{g}^{-1}(df,dg)$. Since $\omega_{g}^{-1}$ has degree $-1$, we have
$$\{{\cal F}_{k},{\cal F}_{l}\}\subset{\cal F}_{k+l-1}.$$
We identify ${\cal F}(SM)$ with ${\cal{F}}_{1}(T^{\prime}M)$, $f\mapsto\tilde{f}=f_{1}=f\otimes{\mathbb{R}}$
and define the Legendrian bracket in the space ${\cal F}(SM)$ by
$$\{f,h\}:=\{\tilde{f},\tilde{h}\}_{SM}.$$
$\square$
2.2 Smooth structure in the space of geodesics $L(M)$
We first consider the topology of $L(M)$.
Let $(M,g)$ be a smooth complete Riemannian $n$-dimensional
manifold. By geodesic we mean an oriented maximally extended
geodesic on $M$ and such a geodesic $\gamma$ has a natural
parameterization by arc-length $\gamma=\gamma(s)$ defined up to a
shift $s\rightarrow s+C$. We denote by $\Gamma$ the canonical geodesic vector
field on the unit sphere bundle $SM$.
The maximal integral curves of $\Gamma$ through $(x,v)\in SM$ have the form
$(\gamma(s),\gamma^{\prime}(s))=(\exp_{x}(sv),d/ds\;\exp_{x}(sv))$, where $\gamma(s)=\exp_{x}(sv)$ is the maximal geodesic defined by $(x,v)\in SM$.
The set $L(M)$ is identified with the set of orbits of the flow
generated by $\Gamma$, i.e. the maximal integral curves of $\Gamma$. Denote by $\pi:SM\rightarrow L(M)=SM/\Gamma$ the natural projection,
equip $L(M)$ with the weakest topology such that $\pi$ is
continuous, and call the resulting topological space $L(M)$ the
space of geodesics of M. In general, $L(M)$ is not Hausdorff, but if $M$ is complete and compact, then
$L(M)$ is compact.
Lemma 3
The projection $\pi:SM\rightarrow L(M)$ is an open map if $(M,g)$ is
a complete Riemannian manifold.
Proof. We have to prove that if $U\subset SM$ is an open set then $\pi(U)$ is open, i.e.
$V:=\pi^{-1}(\pi(U))\subset SM$ is open.
It is clear since $V=\cup_{t\in{\mathbb{R}}}\varphi_{t}U$ where $\varphi_{t}=\exp(t\Gamma)$ is
the 1-parameter group of transformations which
is generated by $\Gamma$.
$\square$
To define a smooth structure in $L(M)$ we consider:
Definition 1
i) A function $f$ on an open subset $U\subset L(M)$ is called smooth if its pull back
$\pi^{*}f$ is a smooth function on $\pi^{-1}U$. We identify the algebra
${\cal F}(U)$ of smooth functions on $U$ with the algebra
${\cal F}(U)^{\Gamma}$ of $\Gamma$-invariant functions on $\pi^{-1}U$. In particular,
${\cal F}(L(M))={\cal F}(SM)^{\Gamma}$. We denote by ${\cal F}_{\gamma}(L(M))$
the germ of smooth functions at $\gamma$.
ii) A tangent vector $v$ of $L(M)$ at $\gamma$ is a derivation $v:{\cal F}_{\gamma}(L(M))\rightarrow{\mathbb{R}}$,
i.e. $v:f\mapsto v\cdot f$ such that $v\cdot(fh)=h(\gamma)v\cdot f+f(\gamma)v\cdot(h).$
They give rise to a vector space denoted by $T_{\gamma}L(M).$
iii) A vector field on $U\subset L$ is a derivation of the algebra ${\cal F}(U)$.
We identify the Lie algebra $\mbox{\mcal X}(U)$ of vector fields with
the Lie algebra $\mbox{\mcal X}(\pi^{-1}U)^{\Gamma}$ of $\Gamma$-invariant vector fields on $\partial^{-1}U\subset SM$.
iv) A k-form $\omega\in\Omega^{k}(U)$ is a ${\cal F}(U)$- polylinear skew-symmetric map
$$\omega:\mbox{\mcal X}(U)\times.....\times\mbox{\mcal X}(U)\rightarrow\mbox{%
\mcal X}(U).$$
The standard definition of the exterior differential holds on $\Omega^{k}.$
Note in particular that if the manifold $M$ has a
dense geodesic, then $C^{\infty}(L)={\mathbb{R}}$ and there are no
non-trivial vectors and vector fields on $L$. But if we restrict $M$ to a sufficiently small neighborhoods
$M^{\prime}$ of a point, the algebra $\mathcal{F}(M^{\prime})$ will be non-trivial and we get a non-trivial Lie algebra of vector fields.
2.3 Canonical symplectic structure on $L(M)$
We now give another, more general, definition of tangent vectors in $T_{\gamma}L(M)$ in terms of Jacobi fields along
$\gamma$.
Definition 2
A Jacobi tangent vector $v\in T_{\gamma}L(M)$ at a point $\gamma\in L(M)$
is a Jacobi vector field $Y$ along $\gamma$ which is normal to $\gamma$. The space $T^{J}_{\gamma}(L(M)):=Jac_{\gamma}^{\perp}$ of such
vector fields is called the Jacobi tangent vector space of $L(M)$ at $\gamma$.
Note that $\dim T^{J}_{\gamma}(L(M))=2n-1$. It is useful to give a relation between the tangent vector space $T_{\gamma}L(M)$
and the Jacobi tangent vector space.
For a tangent vector $Y\in T_{m}M$ we denote by $Y_{x,y}^{v}\in V_{x,y}TM$
the vertical lift and by $Y_{x,y}^{h}\in H_{x,y}TM$ the horizontal lift. We denote by
$\gamma^{S}=(\gamma,\dot{\gamma})$ the natural lift of a geodesic $\gamma$ to $SM$.
Lemma 4
The horizontal lift $Y^{h}$ of a Jacobi field $Y\in Jac^{\perp}_{\gamma}=T^{J}_{\gamma}L(M)$ is a horizontal $\Gamma$-invariant vector field along
$\gamma^{S}\subset SM$. It defines a tangent vector
$$\hat{Y}:\mathcal{F}(L(M))\to\mathbb{R},\,\,f\mapsto Y^{h}(f).$$
The map $Y\mapsto\hat{Y}$ is a homomorphism of $T^{J}_{\gamma}(L(M))$ into $T_{\gamma}(L(M))$.
Note that Proposition 1.90 in [1] is incorrect: not every Jacobi field is the transverse field
of a geodesic variation, a counterexample being the vertical constant vector field along the
minimal geodesic of $x_{1}+x_{2}^{2}-x_{3}^{2}=1$.
We now define the canonical symplectic 2-form $\omega$ on $L(M)$.
Lemma 5
The 2-form $\omega=d\theta$ on $SM$ is $\Gamma$-horizontal $(\iota_{\Gamma}\omega=0)$ and $\Gamma$-invariant.
Proof. We have $\theta(\Gamma)=1$ and $\iota_{\Gamma}\omega=0$, therefore
$\Gamma\cdot\omega=(d\iota_{\Gamma}+\iota_{\Gamma}d)\omega=0$ and $\Gamma\in ker\;\omega$.
Therefore $d\theta$ pushes down to a closed 2-form $\omega=\omega_{L}$
on $L(M)=SM/\Gamma.$
$\square$
Now we describe a Poisson structure on $L(M)$, namely
Lemma 6
${\cal F}(L(M))={\cal F}(SM)^{\Gamma}$ is a subalgebra of the Lie algebra $({\mathcal{F}}(SM),\{,\}).$
Proof. It is known that the Hamiltonian field preserves the symplectic form. If $\Gamma\cdot f=0$, then $\Gamma\cdot\tilde{f}=0$
since $\Gamma\cdot r=0$. We then have $\Gamma\cdot\{f,h\}=\Gamma\cdot\{\tilde{f},\tilde{h}\}=\Gamma\cdot\omega_{g}^{-%
1}(d\tilde{f},d\tilde{h})|SM=0.$
We conclude that $L(M)$ has a canonical Poisson structure.
$\square$
One can check that if the form $\omega_{L}$ is non-degenerate, then the Poisson structure on $L(M)$ is associated with the
symplectic structure $\omega_{L}$, i.e.
$\{f,h\}=\omega_{L}^{-1}(df,dh),f,h\in{\cal F}(L)={\mathcal{F}}_{1}(SM)^{\Gamma}$ if $\Gamma\cdot f=\Gamma\cdot h=0$,
where
$\omega_{L}^{-1}(df,dh):=\omega^{-1}(d\tilde{f},d\tilde{h})$.
3 Pseudo-Riemannian Spaces of Constant Curvature
3.1 Spaces of zero curvature
Let $E=E^{p+1,q}$ be a pseudo-Euclidean vector space of signature
$(p+1,q)$ with basis $(e_{0}^{+},...,e_{p}^{+},e_{1}^{-},....,e_{q}^{-})$. The
scalar product is given by
$$g(X,Y)=<x,y>=\sum_{0}^{p}x_{+}^{i}y_{+}^{i}-\sum_{1}^{q}x_{-}^{j}y_{-}j.$$
Definition 3
A vector $v\in E^{p+1,q}$ is said to be timelike (respectively spacelike, null) if
its norm is negative (positive, vanishes). A straight line is said to be timelike
(respectively spacelike, null) if its tangent vector has that type.
The space of oriented timelike (respectively spacelike) geodesics of $E^{p+1,q}$ is denoted by $L^{-}(E^{p+1,q})$ (respectively $L^{+}(E^{p+1,q})$).
Note that by changing the sign of the metric $L^{+}(E^{p+1,q})=L^{-}(E^{q,p+1})$.
We denote the unit pseudosphere by
$S^{p,q}=\{x\in E^{p+1,q},\,\,x^{2}=1\}$. Here, for $p=0$ we
assume in addition that $x^{0}>0$ so that $S^{p,q}$ is connected. The induced metric
has signature $(p,q)$ and constant curvature $1$.
We denote by
$SO(E)=SO^{0}(p+1,q)$ the connected pseudo-orthogonal group which
preserves the scalar product and by $SO(E)_{e}=SO^{0}(p,q)$ the connected
subgroup which preserves the vector $e=e_{0}^{+}$. The group $SO^{0}(p,q)$
acts transitively on $S^{p,q}$ and we can identify $S^{p,q}$ with the quotient
$SO^{0}(p+1,q)/SO^{0}(p,q)$. Note that the
tangent space $T_{e}S^{p,q}$ has the orthonormal basis $(e_{1}^{+},...,e_{p}^{+},e_{1}^{-},....,e_{q}^{-})$
Any non-null oriented straight line in $E$ can be canonically written
in the form
$$\ell_{e,v}(t)=\{v+te\},$$
where $e$ is the unit tangent vector (s.t. $e^{2}=\pm 1$) and $v$ is a vector orthogonal to $e$.
So we can identify the space $L^{-}(E)$ of timelike lines in $E$ with the tangent bundle
$TS^{p,q}$. The group $E(p+1,q)=SO^{0}(p+1,q)\cdot E^{p+1,q}$ of pseudo-Euclidean motions acts in the space
$L^{-}(E)=TS^{p,q}$ of timelike lines $(e,v):=l_{e,v}$ by
$$T_{a}(e,v)=(e,v+a_{e^{\perp}}),\,\,A(e,v)=(Ae,Av),\,\,a\in E,\,\,A\in SO^{0}(p%
+1,q),$$
(3.1)
where $a_{e^{\perp}}=a-<a,e>e$.
The group $E(p+1,q)$ also naturally acts on $S^{p+1,q}$ with the kernel of effectivity $E^{p+1,q}$.
Proposition 1
The isometry group $E(p+1,q)$ acts transitively on the space $L^{-}(E)$ of straight lines with stabilizer
$SO(n-1)\cdot R^{+}$ and this action commutes with the projection $\pi:L^{-}(E)\to S^{p.q},(e,v)\to e.$
The proof follows from equation (3.1).
Proposition 2
A necessary condition that a subgroup $G\subset E(p+1,q)$ acts transitively on the space
$L^{-}(E^{p+1,q})$ of timelike geodesics is that its linear part $LG=G/{G\cap E}$ acts transitively on $S^{p,q}$.
If the group $G$ contains the group $T_{E}$ of parallel translations this condition is also sufficient.
Proof. The first claim follows from the previous Proposition. Assume now that $G$ contains $E$ and $LG$ acts transitively on $S^{p,q}$. Let $\ell=\ell_{e,v},\,\ell^{\prime}=\ell_{e^{\prime},v^{\prime}}$ be two lines. Using a transformation from $G$ we transform
$\ell_{e^{\prime},v^{\prime}}$ into a line $\ell^{\prime\prime}=\ell_{e,v^{\prime\prime}}$ with the tangent vector $e$ and then using parallel translation we
transforms $\ell^{\prime\prime}$ into $\ell$.
Corollary 1
Let $E=E^{n}$ be the Euclidean vector space. Then any connected subgroup $G$ of the group $E(n)=SO(E)\cdot E$ of Euclidean motions has the form $G=LG\cdot E$ where $LG\subset SO(E)$ is a connected orthogonal group which acts transitively on $S^{n-1}$, that is $LG$ is one of the groups
$$SO(n),\,U(n/2),\,SU(n/2),\,Sp(1)\cdot Sp(n/4),\,Sp(n/4),$$
$$G_{2},(n=7),\,Spin(7),\,(n=8),\,Spin(9),\,(n=16).$$
We identify the space $L^{-}(E^{p+1,q})$ with the homogeneous space
$$G/H=E(p+1,q)/SO(p,q)\cdot R^{*}$$
where $H=SO(p,q)\cdot R^{+}$ is the stabilizer of the line $\ell_{0}=\ell_{e_{0},0}$. Let $E=\mathbb{R}e_{0}+W$ be the orthogonal decomposition. We write
the corresponding reductive decomposition of the homogeneous space $G/H$ as
$$\mbox{\goth g}=\mathfrak{e}(p+1,q)={\mbox{\goth h}}+\mbox{\goth m}=(\mbox{%
\goth so}(W))+\mathbb{R}e_{0})+(U+W),$$
where $\mbox{\goth so}(E)=\mbox{\goth so}(W)+U$ is the reductive decomposition of $\mbox{\goth so}(E)$. In matrix notation,
elements from the algebra of Euclidean isometries $\mathfrak{e}(p+1,q)\subset\mbox{\goth gl}(n+1)$ can be written as
$$(A,\lambda e_{0},u,w)=\left(\begin{array}[]{ccc}A&u&w\\
-u^{t}&0&\lambda\\
0&0&0\end{array}\right),\,\lambda\in\mathbb{R},\,u,w\in\mathbb{R}^{p+q}.$$
The adjoint action of $(A,\lambda e_{0})$ on $\mbox{\goth m}=U\oplus W=\{(u,w)\}$ is given by
$$\mathrm{ad\;}_{(A,\lambda)}(u,w)=(Au,Aw+\lambda u),$$
and the bracket of two elements of $\rm=U\oplus W$ is given by
$$[(u,w),(u^{\prime},w^{\prime})]=-(u\wedge u^{\prime},(u\cdot w^{\prime}-u^{%
\prime}\cdot w)e_{0})\in{\mbox{\goth h}},$$
where $u\wedge u^{\prime}$ is an element from $\mbox{\goth so}(W)$ and dot means the standard scalar product of vectors from $\mathbb{R}^{n-1}=U=W$.
Note that $L^{-}(E^{p+1,q})=G/H$ is a symmetric manifold since $\mbox{\goth g}={\mbox{\goth h}}+\mbox{\goth m}$ is a symmetric decomposition.
The isomorphism $\mathrm{ad\;}_{e_{0}}:U\to W$ allows one to identify $U$ with $W$ and the tangent space $\mbox{\goth m}=U+W$ with a tensor product
$\mbox{\goth m}=W\otimes{\mathbb{R}}^{2}$, where $U=W\otimes f_{1}$, $W=W\otimes f_{2}$ and $f_{1},f_{2}$ is the standard basis of $\mathbb{R}^{2}$.
The isotropy representation $\mathrm{ad\;}_{{\mbox{\goth h}}}$ preserves the Grassmann structures $\mbox{\goth m}=W\otimes\mathbb{R}^{2}$
and $\mbox{\goth so}(W)$ act on the first factor $W$ and $\lambda e_{0}$ acts on the second factor $\mathbb{R}^{2}$ by the matrix
$$\lambda\;\mathrm{ad\;}_{e_{0}}=\left(\begin{array}[]{cc}0&0\\
-\lambda&0\end{array}\right).$$
Moreover, the isotropy action $\mathrm{ad\;}_{{\mbox{\goth h}}}|_{\mbox{\goth m}}$ preserves the metric $g^{W}=g|_{W}$
of signature $(p,q)$ in $W$ and the symplectic structure $\omega_{0}=f_{1}\wedge f_{2}$ in $\mathbb{R}^{2}$.
The tensor product $\omega^{\mbox{\goth m}}=g^{W}\otimes\omega_{0}$
defines a non-degenerate $\mathrm{ad\;}_{{\mbox{\goth h}}}$-invariant 2-form in
m which is extended to an invariant symplectic form $\omega$ in $L^{-}(E^{p+1,q})$. The form $\omega$ is closed since it is invariant and
the manifold $L^{-}(E^{p+1,q})$ is a symmetric space.
In the case of dimension $n=3$ the action $\mathrm{ad\;}_{\mbox{\goth so}(W)}$ on $W$ also preserves a 2-form $\omega^{W}$ (which is the volume form of $W$).
Hence we get an invariant metric $g^{\mbox{\goth m}}=\omega^{W}\otimes\omega_{0}$ on m which extends to an invariant pseudo-Riemannian metric
$g$ of signature $(2,2)$ on $L^{-}(E^{p+1,q})$. The quotient $J=g^{-1}\circ\omega$ is an invariant (hence, integrable)
complex structure and the pair $(g,J)$ is an invariant Kähler structure.
Summarizing, we get (cf. [12]):
Theorem 1
The space $L^{-}(E^{p+1,q})=E(p+1,q)/SO(p,q)\cdot{\mathbb{R}^{+}}$ is a symplectic symmetric space with an invariant Grassmann structure defined by a decomposition $\mbox{\goth m}=W\otimes{\mathbb{R}}^{2}$. Moreover, if $n=p+1+q=3$,
it has an invariant Kähler structure $(g,J)$ of neutral signature $(2,2)$.
3.2 Spaces of constant non-zero curvature
We now describe the space of oriented timelike and spacelike geodesics of the pseudo-Riemannian space $S^{p,q}$
of constant curvature 1. Any such geodesic
through $e\in S^{p,q}$ in direction of a unit vector $e_{1}^{\pm}$ with $(e_{1}^{\pm})^{2}=\pm 1$ is given by
$$\gamma^{+}=\gamma_{e_{1}^{+}}^{+}=cos(s)e+sin(s)e_{1}^{+}\;,\;\;\;\gamma^{-}=%
\gamma_{e_{1}^{-}}^{-}=ch(s)e+sh(s)e_{1}^{-}.$$
The subgroup of the stability group $SO^{0}(p,q)$
preserving the spacelike geodesic $\gamma^{+}$ is $SO(p-1,q)$ and the timelike
geodesic $\gamma^{-}$ is $SO(p,q-1)$. The one-parameter subgroup $SO(2)$ generated by the
element $e\wedge e_{1}^{+}$ preserves $\gamma^{+}$ and the one-parameter subgroup $SO(1,1)$ generated by $e\wedge e_{1}^{-}$ preserves $\gamma^{-}$. Since the group $SO^{0}(p+1,q)$ acts
transitively on the space $L^{+}(S^{p,q})$ of spacelike geodesics
and on the space $L^{-}(S^{p,q})$ of timelike geodesics, we can
represent these spaces as
$$L^{+}(S^{p,q})=SO^{0}(p+1,q)/SO(2)\cdot SO(p-1,q),$$
$$L^{-}(S^{p,q})=SO^{0}(p+1,q)/SO^{0}(1,1)\cdot SO(p,q-1).$$
To get the reductive decomposition associated with
these spaces, fix the orthogonal decomposition
$$E^{p+1,q}={\mathbb{R}}e\oplus{\mathbb{R}}e^{\pm}_{1}\oplus V^{\pm},$$
where $V^{+}$ is the vector space of signature $(p-1,q)$ with basis
$(e_{2}^{+},...,e_{p}^{+},e_{1}^{-},...,e_{q}^{-})$ and $V^{-}$ is the vector space
of signature $(p,q-1)$ with basis
$(e_{1}^{+},...,e_{p}^{+},e_{2}^{-},...,e_{q}^{-})$.
Using the metric, we identify the Lie
algebra $\mbox{\goth so}(p+1,q)$ of $SO^{0}(p+1,q)$ with the space of
bivectors $\Lambda^{2}(E^{p+1,q})$. Then the reductive decomposition
associated with the unit sphere bundles
$$S^{\pm}(S^{p,q})=\{e_{1}^{\pm}\in T_{e}S^{\partial,q},\,<e_{1}^{\pm},e_{1}^{%
\pm}>=\pm 1\}$$
is given by
$$\mbox{\goth so}(p+1,q)=\Lambda^{2}(V^{\pm})\oplus(e\wedge V^{\pm}\oplus e_{1}^%
{\pm}\wedge V^{\pm})\oplus{\mathbb{R}}(e\wedge e_{1}^{\pm}).$$
The bivector $e\wedge e_{1}^{+}$ (resp.,$e\wedge e_{1}^{+}$ ) is invariant under the stability subgroup $SO(p-1,q)$
(resp., $SO(p,q-1)$ )and defines
an invariant vector field $\Gamma$ on $S^{\pm}(S^{p,q})$, which is the geodesic field. It
is the velocity field of the right action of the subgroup $SO^{\pm}(2)=SO(2),\,SO(1,1)$ of
$SO(p+1,q)$. The space of geodesics is the quotient
$$L^{+}(S^{p,q})=SO(p+1,q)/SO(p-1,q)\cdot SO^{\pm}(2),\,\,L^{-}(S^{p,q})=SO(p+1,%
q)/SO(p,q-1)\cdot SO(1,1).$$
The corresponding reductive decomposition may be
written as
$$\mbox{\goth so}(p+1,q)={\mbox{\goth h}}^{\pm}\oplus\mbox{\goth m}^{\pm}={%
\mathbb{R}}(e\wedge e_{1}^{\pm})\oplus\Lambda^{2}(V^{\pm})\oplus(e\wedge V^{%
\pm}\oplus e_{1}^{\pm}\wedge V^{\pm}).$$
We identify $\mbox{\goth m}^{\pm}$ with the tangent space
$T_{\gamma^{\pm}}(L^{\pm}S^{p,q})$. There is also a natural identification with the tensor product
$\mbox{\goth m}^{\pm}=H\otimes V^{\pm}$, where $H={\rm span}(e,e_{1}^{\pm})\simeq{\mathbb{R}}^{2}$
is the 2-dimensional oriented pseudo-Euclidean vector space.
Then the action of the isotropy subalgebra ${\mbox{\goth h}}^{\pm}$ takes the form:
$$ad_{e\wedge e_{1}^{\pm}}:e\otimes x\mapsto(e\wedge e_{1}^{\pm})e\otimes x=-e_{%
1}^{\pm}\otimes x,$$
$$ad_{e\wedge e_{1}^{\pm}}:e_{1}^{\pm}\otimes x\mapsto(e\wedge e_{1}^{\pm})e_{1}%
^{\pm})\otimes x=\pm e\otimes x,$$
$$ad_{a\wedge b}:e^{\prime}\otimes x\mapsto(e^{\prime}\otimes(a\wedge b)x=e^{%
\prime}\otimes<b,x>a-<a,x>b,$$
for all $a,b,x\in V^{\pm}$ and $e^{\prime}\in{\mathbb{R}}^{2}$.
Note that $L^{\pm}(S^{p,q})$ is identified with the Grassmanian
$Gr_{2}^{\pm}({\mathbb{R}}^{p+1,q})$ of two-planes of signature $(2,0)$ or $(1,1)$ and the decomposition
$$T_{\gamma^{\pm}}(L^{\pm}S^{p,q})=H\otimes V^{\pm},$$
defines an invariant Grassmann
structure in $L^{\pm}(S^{p,q})$.
Denote by $H^{\varepsilon}$ two-dimensional vector space with a scalar product $g_{H}=g_{H}^{\varepsilon}$ of signature
$(2,0)$ for $\varepsilon=+$ and $(1,1)$ for $\varepsilon=-$, and by $I_{H}=I_{H}^{\varepsilon}$ the $SO(H^{\varepsilon})$-invariant
endomorphism
of $H^{\varepsilon}$ with $I_{H}^{2}=-\varepsilon 1$ and by $\omega_{H}=\omega_{H^{\varepsilon}}=g_{\varepsilon}\circ I_{H}$ the invariant volume form.
Let $(V,g_{V})$ be a pseudo-Euclidean vector space of dimension $m$. If $m=2$, we denote by $I_{V}$ the
$SO(V)$-invariant endomorphism with $I_{V}^{2}=-1$ for signature $(2,0)$ or $(0,2)$ and with $I_{V}^{2}=1$
for the signature $(1,1)$. Denote also by $\omega_{V}=g_{V}\circ I_{V}$ the volume form of $V$.
Lemma 7
i)
Any $SO(H^{\varepsilon})\times SO(V)$-invariant endomorphism of the space $W^{\varepsilon}=H^{e}\otimes V$ has the form $A=1\otimes A+I_{H}^{\varepsilon}\otimes B$ where $A,B\in\mbox{\goth gl}(V)^{SO(V)}$ are invariant endomorphisms of $V$.
ii)
Any invariant endomorphism $I$ of $W^{\varepsilon}$ different from $1$ with $I^{2}=\pm 1$ is given
(up to a sigh) by
$I_{\varepsilon}=I_{H}\otimes 1$ if $m>2$ and by
$I_{\varepsilon},I^{\prime}_{\varepsilon}:=1\otimes I_{V},I^{\prime\prime}_{%
\varepsilon}=I_{H}\otimes I_{V}$ if $m=2.$
iii)
Any invariant metric on $W^{\varepsilon}$ is proportional to $g:=g_{H}\otimes g_{V}$ if $m>2$ and is a
linear combination of the metric $g$ and the neutral metric $g^{\prime}:=\omega_{H}\otimes\omega_{V}$ otherwise.
iv)
The space of invariant 2-forms has the basis $\omega=\omega_{H}\otimes g_{V}$ if $m>2$ and
$\omega,\omega^{\prime}:=g_{H}\otimes\omega_{V}$ if $m=2$.
v)
The endomorphisms $I,I^{\prime}$ are skew-symmetric with respect to any invariant metric $h$ on $W^{\varepsilon}$,
hence define a Hermitian or para-Hermitian structure, and the endomorphism $I^{\prime\prime}$ is symmetric
with respect to any invariant metric $h$.
Note that the tensor product of two complex or two para-complex structures is a para-complex structure
and the tensor product of a complex and a para-complex structures is a complex structure.
Proof. To prove part i), it is sufficient to write the endomorphism $A$ in block matrix form with respect to
the decomposition $W=h_{1}\otimes V+h_{2}\otimes V$, where $h_{1},h_{2}$ is an orthonormal basis of
$H^{+}$ or isotropic basis of $H^{-}$ and write the conditions that it is
$SO(H)\times SO(V)$-invariant. Since the only invariant endomorphism of $V$ is a scalar if $m>2$
and is a linear combination of $1,I_{V}$ if $m=2$, part ii) follows from part i). Parts iii) and iv) follow
from the fact that the space of symmetric bilinear forms
$$S^{2}(H\otimes V)=S^{2}(H)\otimes S^{2}(V)+\Lambda^{2}H\otimes\Lambda^{2}(V),$$
and that the space of 2-forms
$\Lambda^{2}(H\otimes V)=\Lambda^{2}(H)\otimes S^{2}(V)+S^{2}(H)\otimes\Lambda^%
{2}V$.
Now part v) follows from parts iii) and iv).
$\square$
Since the spaces
$$L^{+}(S^{p,q})=SO^{0}(p+1,q)/SO(2)\cdot SO^{0}(p-1,q)$$
$$L^{-}(S^{p,q})=SO^{0}(p+1,q)/SO^{0}(1,1)\cdot SO^{0}(p,q-1)$$
of spacelike and timelike geodesics
are symmetric spaces, any Hermitian pair $(h,I)$ which consists of invariant pseudo-Euclidean
metric on $\mbox{\goth m}^{\varepsilon}$ and skew-symmetric invariant complex or para-complex structure $I$ (such that
$I^{2}=-1$ or $I^{2}=1$) defines an invariant Kähler or para-Kähler structure on $L^{\pm}S^{p,q}$.
We get the following theorem.
Theorem 2
i)
Let $L^{+}(S^{p,q})$, (respectively $L^{-}(S^{p,q})$)
be the space of spacelike (respectively timelike) geodesics in $S^{p,q}$ and
$p+q>3$. Then there exists a unique (up to scaling)
invariant symplectic structure $\omega$ and a unique (up to sign)
invariant complex structure $I^{+}=J$ on $L^{+}(S^{p,q})$
(respectively para-complex structure $K=I^{-}$ on $L^{-}(S^{p,q})$).
There exists unique (up to a scaling ) invariant
pseudo-Riemannian metric $g^{\varepsilon}=\omega\circ I^{\varepsilon}$ on $L^{\varepsilon}(S^{p,q})$ which is Kähler of signature
$(2(p-1),2q)$ for
$\varepsilon=+$ and para-Kähler (of neutral signature) for $\varepsilon=-$.
ii)
If $p+q=3$, then there are two linearly independent invariant (parallel and closed)
2-forms $\omega,\omega^{\prime}$ on $L^{\varepsilon}(S^{p,q})$ with values $\omega_{\mbox{\goth m}}=\omega_{H}\otimes g_{V},\,\omega^{\prime}_{\mbox{\goth
m%
}}=g_{H}\otimes\omega_{V}$.
Any invariant metric has the form $h=\lambda g+\mu g^{\prime}$ where $g^{\prime}$ is a neutral metric with
value $g^{\prime}_{\mbox{\goth m}}=\omega_{H}\otimes\omega_{V}$. Any metric $h$ is Kähler for $\varepsilon=1$ (respectively,
para-Kähler for $\varepsilon=-$) with respect to the complex (respectively, para-complex) structure
$I^{\varepsilon}=I^{\varepsilon}_{H}\otimes 1$ with the Kähler form $h\circ I^{\pm}=\lambda\omega+\mu\omega^{\prime}$.
Moreover, the endomorphism $I^{\prime}=1\otimes I_{V}$ of m defines an invariant parallel
$h$-skew-symmetric complex structure of $L^{\varepsilon}(S^{p,q})$ if $\varepsilon=1,(p-1,q)=(2,0)$ or $(0,2)$ or
$\varepsilon=-$ and $(p,q-1)=(2,0)$ or $(0,2)$ and skew-symmetric parallel para-complex structure
if $\varepsilon=+,(p-1,q)=(1,1)$ or $\varepsilon=-,(p,q-1)=(1,1)$. The Kähler or para-Kähler structure
$(h,I^{\prime})$ has the Kähler form $h\circ I^{\prime}=\lambda\omega+\mu\omega^{\prime}$.
One can easily check that the form $\omega$ is the canonical symplectic form of the space of
geodesics $L^{\varepsilon}(S^{p,q})$.
4 Rank One Symmetric Spaces of Non-Constant Curvature
In this section we discuss the invariant geometric structures on
the space $L(M)$ of oriented geodesics of a rank one Riemannian symmetric space of non-constant
curvature $M=G/K$, that is for the projective spaces
$${\mathbb{C}}P^{n}=SU_{n+1}/U_{n},\qquad{\mathbb{H}}P^{n}=Sp_{n+1}/Sp_{1}\cdot
Sp%
_{n},\qquad{\mathbb{O}}P^{2}=F_{4}/Spin_{9},$$
and the dual hyperbolic spaces
$${\mathbb{C}}H^{n}=SU_{1,n}/U_{n},\qquad{\mathbb{H}}H^{n}=Sp_{1,n}/Sp_{1}\cdot
Sp%
_{n},\qquad{\mathbb{O}}H^{2}=F^{non-comp}_{4}/Spin_{9}.$$
In all of these cases, the space of geodesics is a homogeneous manifold
$L(M)=G/H$ where the stability subgroup $H$ is the same for the compact and dual non-compact
case, and is given by
$$H=T^{2}\cdot SU_{n-1},\,\,T^{1}\cdot Sp_{1}\cdot Sp_{n-1},T^{1}\cdot Spin_{7}.$$
Moreover, in the case of a classical Lie group $G$, the space $L(M)$ is the adjoint orbit
$L(M)=\mathrm{Ad\;}_{G}I^{\varepsilon}$ of the element
$I^{\varepsilon}=h_{1}^{\varepsilon}={\rm{diag}}(I^{\varepsilon}_{2},0,0,...,0),$ where
$$I_{2}^{\varepsilon}=\left(\begin{array}[]{cc}0&-\varepsilon\\
1&0\end{array}\right),$$
and $\varepsilon=1$ in the compact case and $\varepsilon=-1$ otherwise.
Main Theorems 2 and 3 describe all invariant structures
(symplectic structures, complex and para-complex, Kähler and para-Kähler
structures) on the space of geodesics $L(M)$.
We prove these in three stages: first for the complex and
quaternionic projective spaces, then for their hyperbolic
counterparts and finally for the Cayley projective
plane.
4.1 Complex projective and hyperbolic spaces
We now describe the space of real geodesics in complex projective
space
$$M^{1}={\mathbb{C}}P^{n}=SU_{n+1}/U_{n}$$
and in complex hyperbolic space
$$M^{-1}={\mathbb{C}}H^{n}=SU_{1,n}/U_{n}.$$
We set $\mathfrak{g}^{1}=\mathfrak{su}_{n+1}$ and
$\mathfrak{g}^{-1}=\mathfrak{su}_{1,n}$. We then choose the
associated reductive decompositions $\mathfrak{g}^{\epsilon}=j(\mathfrak{u}_{n})+\mathfrak{p}^{\epsilon}$, $\epsilon=\pm 1$, where
$j(\mbox{\goth u}_{n})=\{\left(\begin{array}[]{cc}-trA&0\\
0&A\end{array}\right)\left|\quad A\in\mbox{\goth u}_{n}\right.\}$, $\mbox{\goth p}_{\epsilon}=\{\left(\begin{array}[]{cc}0&-\epsilon X^{*}\\
X&0\end{array}\right)\left|\quad\quad X\in\mathbb{C}^{n}\right.\}$,
and where $X$ is a column vector and $X^{*}$ denotes the
Hermitian conjugate. We identify $\mathfrak{p}^{\epsilon}$ with the
tangent space $T_{o}M^{\epsilon},o=eU_{n}$.
We next describe the stability subalgebra ${\mbox{\goth h}}^{\epsilon}$ of the
geodesic $\gamma=\rm{exp}(th_{1}^{\epsilon})(o)$, where the
element $h_{1}^{\varepsilon}$ is represented by the matrix
$$h_{1}^{\varepsilon}=I^{\epsilon}=\left(\begin{array}[]{cc}I^{\epsilon}_{2}&0\\
0&0\end{array}\right),\qquad\qquad\rm{and}\qquad\qquad I^{\epsilon}_{2}=\left(%
\begin{array}[]{cc}0&-\epsilon\\
1&0\end{array}\right).$$
We have that
${\mbox{\goth h}}^{\varepsilon}=Z_{\mbox{\goth g}}(h^{\epsilon}_{1})={\mathbb{R%
}}h^{\epsilon}_{1}+Z_{\mbox{\goth u}_{n}}(h^{\epsilon}_{1})$,
where $Z_{\mbox{\goth u}_{n}}(h^{\epsilon}_{1})$ is the centraliser of $h^{\epsilon}_{1}$
in $\mbox{\goth u}_{n}$. We now describe the reductive decomposition
$\mbox{\goth g}^{\epsilon}={\mbox{\goth h}}^{\epsilon}+\mbox{\goth l}^{\epsilon}$.
We have
$${\mbox{\goth h}}^{\epsilon}=\{A=\left(\begin{array}[]{ccc}i\alpha&-\epsilon%
\beta&0\\
\beta&i\alpha&0\\
0&0&A_{n-1}\end{array}\right)|\quad\alpha,\beta\in{\mathbb{R}},\quad A_{n-1}%
\in\mbox{\goth u}_{n-1},trA_{n-1}+2i\alpha=0\},$$
and the complimentary subspace is
$$\mbox{\goth l}^{\varepsilon}=\{X=(x_{1},x_{2},X_{1},X_{2})=\left(\begin{array}%
[]{ccc}ix_{1}&\varepsilon ix_{2}&-\varepsilon X_{1}^{*}\\
ix_{2}&-ix_{1}&-X_{2}^{*}\\
X_{1}&X_{2}&0\end{array}\right)|\quad x_{1},x_{2}\in{\mathbb{R}},\quad X_{1},X%
_{2}\in{\mathbb{C}}^{n-1}\}.$$
We may write
$${\mbox{\goth h}}^{\varepsilon}={\mathbb{R}}h_{0}+{\mathbb{R}}h^{\varepsilon}_{%
1}+\mathfrak{su}_{n-1},$$
where
$$h_{0}=\left(\begin{array}[]{ccc}i&0&0\\
0&i&0\\
0&0&\frac{-2i}{n-1}{\mathrm{I}d}_{n-1}\end{array}\right),\quad h^{\varepsilon}%
_{1}=I^{\varepsilon}=\left(\begin{array}[]{ccc}0&-\varepsilon&0\\
1&0&0\\
0&0&0\end{array}\right).$$
Similarly,
$$\mbox{\goth l}^{\varepsilon}=V_{0}^{\varepsilon}+V_{+}^{\varepsilon}+V_{-}^{%
\varepsilon},$$
where
$$V_{0}^{\varepsilon}=\{(x_{1},x_{2},0,0)\}={\mathbb{R}}E_{1}+{\mathbb{R}}E_{2}^%
{\varepsilon},\;\;V_{\pm}^{\varepsilon}=\{X_{\pm}=(0,0,X,\pm X),X\in{\mathbb{C%
}}^{n-1}\},$$
$$E_{1}=(1,0,0,0)=\left(\begin{array}[]{ccc}i&0&0\\
0&-i&0\\
0&0&0\end{array}\right),\quad E_{2}^{\varepsilon}=(0,1,0,0)=\left(\begin{array%
}[]{ccc}0&\varepsilon i&0\\
i&0&0\\
0&0&0\end{array}\right).$$
We denote the canonical Hermitian form in the
space of vector columns ${\mathbb{C}}^{n-1}$ by $\eta(X,Y)=X^{*}Y$ . Then
$$g(X,Y)=Re\;\eta(X,Y)=\frac{1}{2}(X^{*}Y+Y^{*}X),$$
$$\rho(X,Y)=Im\;\eta(X,Y)=\frac{1}{2i}(X^{*}Y-Y^{*}X).$$
For any $X\in{\mathbb{C}}^{n-1}$ we set $X_{\pm}=(0,0,X,\pm X).$
Lemma 8
We have the following commutator relations:
$$[E_{1},E_{2}^{\varepsilon}]=2h_{1}^{\varepsilon},$$
(4.1)
$$[E_{1},(0,0,X_{1},X_{2})]=(0,0,-iX_{1},iX_{2}),$$
(4.2)
$$[E_{2}^{\varepsilon},(0,0,X_{1},X_{2})]=(0,0,-iX_{2},-\varepsilon iX_{1}).$$
(4.3)
The isotropy action of ${\mbox{\goth h}}^{\varepsilon}$ on $\mbox{\goth l}^{\varepsilon}$ is given by
$${\rm ad}_{h_{0}}(x_{1},x_{2},X_{1},X_{2})=(0,0,-iX_{1},-iX_{2}),$$
$${\rm ad}_{h_{1}^{\varepsilon}}(x_{1},x_{2},X_{1},X_{2})=(-2\varepsilon x_{2},2%
x_{1},-X_{2},\varepsilon X_{1}),$$
$${\rm ad}_{A_{n-1}}(x_{1},x_{2},X_{1},X_{2})=(0,0,A_{n-1}X_{1},A_{n-1}X_{2}).$$
Moreover
$$[X_{\pm}^{1},Y_{\pm}^{1}]=2\rho(X,Y)(-h_{0}\mp E_{2}^{1})\quad\rm{mod}\quad%
\mbox{\goth su}_{n-1},$$
(4.4)
$$[X_{\pm}^{-1},Y_{\pm}^{-1}]=2\rho(X,Y)(E_{1}\pm E_{2}^{-1})\quad\rm{mod}\quad%
\mbox{\goth su}_{n-1},$$
(4.5)
$$[X_{+}^{1},Y_{-}^{-1}]=-2\rho(X,Y)E_{1}+2g(X,Y)h_{1}\quad\rm{mod}\quad\mbox{%
\goth su}_{n-1},$$
(4.6)
$$[X_{+}^{-1},Y_{-}^{-1}]=2\rho(X,Y)h_{0}+2g(X,Y)h_{-1}\quad\rm{mod}\quad\mbox{%
\goth su}_{n-1},$$
(4.7)
for all
$X_{\pm}^{\varepsilon},Y_{\pm}^{\varepsilon}\in V_{\pm}^{\varepsilon}.$
Proposition 3
The $\mathfrak{h}$-module $\mathfrak{l^{\varepsilon}}$ has the
following decomposition into irreducible components.
a) For $\varepsilon=1$
$$\begin{array}[]{cccc}\mathfrak{l}_{+}^{1}=&\quad V_{0}^{1}&\quad+\quad V_{+}^{%
1}&\quad+\quad V_{-}^{1}\\
\mbox{ad}_{h_{0}}:&0&-i\mathrm{Id\;}&\quad-i\mathrm{Id\;}\\
\mbox{ad}_{h_{1}^{\varepsilon}}:&2J_{0}&i\mathrm{Id\;}&\quad-i\mathrm{Id\;}\\
A_{n-1}:&0&A_{n-1}&\quad A_{n-1}.\end{array}$$
b) For $\varepsilon=-1$
$$\begin{array}[]{ccccc}\mathfrak{l}_{-}^{-1}=&\quad\mathbb{R}E_{+}&\quad+\quad%
\mathbb{R}E_{-}&\quad+\quad V_{+}^{-1}\quad+&\quad V_{-}^{-1}\\
\mbox{ad}_{h_{0}}:&0&0&-i\mathrm{Id\;}&-i\mathrm{Id\;}\\
\mbox{ad}_{h_{1}^{\varepsilon}}:&2&-2&\mathrm{Id\;}&-\mathrm{Id\;}\\
A_{n-1}:&0&0&A_{n-1}&A_{n-1},\end{array}$$
where
$$E_{\pm}=(1,\pm 1,0,0)=E_{1}\pm E_{2}^{-1}.$$
With this notation in case b), the commutation relations read as follows :
$$[E_{\pm},V_{\pm}]=0,\;[E_{+},E_{-}]=-4h_{1}^{-1},$$
(4.8)
$$[E_{+},X_{-}]=-2iX_{+},\;[E_{-},X_{+}]=-2iX_{-}\;$$
(4.9)
$$[X_{\pm},Y_{\pm}]=2\rho(X,Y)E_{\pm},\;\;\;\;[X_{+},Y_{-}]=2\rho(X,Y)h_{0}+2g(X%
,Y)h_{1}\quad\rm{mod}\;\mathfrak{su}_{n-1},$$
(4.10)
$\mathrm{for}\;\;X_{\pm}\in V_{\pm}^{-1}$.
Recall that any invariant 2-form on $L(M)$ is generated by an
$\mathrm{ad\;}_{\mbox{\goth h}}$-invariant two form $\omega$ on the tangent space
$\mbox{\goth l}=T_{0}L(M)$. Any such form may be represented as $\omega=d(B\circ h)$,
where $h\in Z({\mbox{\goth h}})$ is a central element and $B$ is the Killing form.
Theorem 3
i)
The only invariant almost complex structures on
$L({\mathbb{C}}P^{n})$ are defined by
$$J_{\varepsilon_{0}\varepsilon_{1}\varepsilon_{2}}=\varepsilon_{0}\frac{1}{2}{%
\rm ad}_{h_{1}^{1}}|_{V^{0}}\oplus\varepsilon_{1}{\rm ad}_{h_{1}^{1}}|_{V^{+}}%
\oplus\varepsilon_{2}{\rm ad}_{h_{1}^{1}}|_{V^{-}}=\varepsilon_{0}J^{V_{0}}%
\oplus\varepsilon_{1}J^{V_{+}}\oplus\varepsilon_{2}J^{V_{-}},$$
where $\varepsilon_{k}=\pm 1$. The integrable ones among these are (up to a sign) $J_{\pm}=(\pm J^{V_{0}})\oplus J^{V_{-}}\oplus J^{V_{+}}.$
ii)
Any $SU(n+1)$-invariant closed 2-form on
$L({\mathbb{C}}P^{n})$ is a linear combination $\omega=\lambda_{0}\omega_{0}+\lambda_{1}\omega_{1}$ of the invariant differential forms
defined by
$\omega_{0}=d(B\circ h_{0})|\mbox{\goth l}^{1}$ and $\omega_{1}=d(B\circ h_{1}^{1})|\mbox{\goth l}^{1}$ where $B$ is the Killing form.
Moreover
$$ker\;\omega_{0}=V_{0},\quad\omega_{0}(X_{\pm},Y_{\pm})=2\rho(X,Y),\quad\omega_%
{0}(V_{+}^{1},V_{-}^{1})=0,$$
$$\omega_{1}(E_{1},E_{2})=-2,\quad\omega_{1}(V_{\pm},V_{\pm})=0,\quad\omega_{1}(%
X_{+},Y_{-})=-\omega_{1}(Y_{-},X_{+})=2g(X,Y).$$
Here $X_{\pm}=(0,0,X,\pm X)$, $Y_{\pm}=(0,0,Y,\pm Y)\in\mathfrak{l^{1}}$
and
$$g(X,Y)=\mathrm{Re}\;\eta(X,Y),\omega(X,Y)=\mathrm{Im}\;\eta(X,Y)$$
are the real and imaginary parts of the standard Hermitian form
$\eta(X,Y)=X^{*}Y$ on $\mathbb{C}^{n-1}.$
iii)
The canonical
symplectic structure on the space oriented geodesics is $\omega_{1}$.
Corollary 2
Up to scaling, any invariant symplectic form on
$L({\mathbb{C}}P^{n})$ may written as
$$\omega^{t}=\omega_{1}+t\omega_{0},\;\;t\in\mathbb{R}$$
It is compatible with any invariant complex structure $J_{\pm}$.
That is, the pair $(\omega^{t},\pm J_{\pm})$ is a Kähler structure on
$L$.
Proof of the Theorem: The description in part i) follows directly from the previous
Proposition. For integrability, we calculate the Niejenhuis bracket
$$N_{J}(X,Y)=[JX,JY]-J[X,JY]-J[JX,Y]-[X,Y],$$
for
$X,Y\in\mbox{\goth l}=\mathbb{R}E_{1}+\mathbb{R}E_{2}^{\varepsilon}+V_{+}+V_{-}$. For
example, for $X_{+}\in V_{+},Y_{-}\in V_{-}$, $J=J_{\varepsilon_{0}1\epsilon}$ we
calculate :
$$\begin{array}[]{cc}N_{J}(X_{+},Y_{-})&=[JX_{+},JY_{-}]-J[X_{+},JY_{-}]-J[JX_{+%
},Y_{-}]-[X_{+},Y_{-}]\\
&=[iX_{+},\varepsilon iY_{-}]-J[X_{+},\varepsilon iY_{-}]-J[iX_{+},Y_{-}]-[X_{%
+},Y_{-}]\\
&=(-\varepsilon-1)[X_{+},Y_{-}]+(-\varepsilon-1)iJ[X_{+},Y_{-}].\\
\end{array}$$
Hence $N_{J}(X_{+},Y_{-})=0$ iff $J=J_{\varepsilon_{0}11}$ i.e. $\varepsilon=1$.
Similarly we calculate $N_{J}(X_{+},E_{1})$ :
$$\begin{array}[]{cc}N_{J}(X_{+},E_{1})&=[JX_{+},JE_{1}]-J[X_{+},JE_{1}]-J[JX_{+%
},E_{1}]-[X_{+},E_{1}]\\
&=[iX_{+},E_{2}]-J[X_{+},E_{2}]-J[-iX_{+},E_{1}]-[X_{+},E_{1}]\\
&=(i-J)[X_{+},E_{2}]+(iJ-1)[X_{+},E_{1}]\\
&=(-i+J)iX_{+}(iJ-1)X_{+}\end{array}$$
This always vanishes. The proof of part ii) follows from the fact that the center
$Z({\mbox{\goth h}}^{1})$ has basis $h_{0},h_{1}^{1}.$ To verify part iii) we consider
$h_{1}^{1}\in\mbox{\goth p}^{1}=T_{0}{\mathbb{C}}P^{n}.$ The stability subalgebra of this element is
${\mbox{\goth h}}_{h_{1}^{1}}={\mathbb{R}}h_{0}+\mathfrak{su}_{n-1}\subset{%
\mbox{\goth h}}^{1}$.
The sphere bundle $S{\mathbb{C}}P^{n}$ is identified with $SU_{n+1}/T^{1}\cdot SU_{n}$ with the
reductive decomposition $\mathfrak{su}_{n+1}=({\mathbb{R}}h_{0}+\mathfrak{su}_{n-1})+({\mathbb{R}}h_{1}%
^{1}+\mbox{\goth l}^{1}).$
The geodesic vector field $\Gamma$ on $S{\mathbb{C}}P^{n}$ is the invariant vector field
generated by the element $h_{1}^{1}$ and the contact form $\theta$ is the invariant form
associated the one-form $\theta_{0}=cB\circ h_{1}^{1}$. This shows that the canonical
form coincides with $\omega^{1}$ (up to scaling).
$\square$
Theorem 4
i)
There is no invariant almost
complex structure on the space $L(\mathbb{C}H^{n})$. There exist two
(up to sign) almost para-complex structures $K^{\pm}$ with
$(\pm 1)$-eigenspace decompositions given by
$$K^{+}:\mbox{\goth l}^{-1}=\mbox{\goth l}_{+}+\mbox{\goth l}_{-}=(\mathbb{R}E_{%
+}+V_{+})+(\mathbb{R}E_{-}+V_{-}),$$
$$K^{-}:\mbox{\goth l}^{-1}=\mbox{\goth l}_{+}+\mbox{\goth l}_{-}=(\mathbb{R}E_{%
+}+V_{-})+(\mathbb{R}E_{-}+V_{+}).$$
Only $K^{+}$ is integrable.
ii)
Any closed invariant two-form is a linear combination
of the form defined by
$$\omega_{0}=d(B\circ h_{0})|\mbox{\goth l}^{-1},\;\;\omega_{1}=d(B\circ h_{1}^{%
-1})|\mbox{\goth l}^{-1},$$
where $B$ is the Killing form.
Moreover, we have
$$\rm{ker}\,\omega_{0}=V_{0},\,\,\omega_{0}(V_{\pm},V_{\pm})=0,\;\;\omega_{0}(X_%
{+},Y_{-})=2\rho(X,Y),$$
$$\omega_{1}(E_{+},E_{-})=4,\;\omega_{1}(X_{+},Y_{-})=-\omega_{1}(Y_{-},X_{+})=2%
g(X,Y),$$
where $X_{+}=(0,0,X,X),\;Y_{-}=(0,0,Y,-Y).$
Proof: An invariant almost complex structure on $L(\mathbb{C}H^{n})$
preserves the one-dimensional $ad_{h_{1}^{-1}}$ - eigenspaces
$\mathbb{R}E_{\pm}$, which is impossible. Since $\mbox{\goth l}_{\pm}=\mathbb{R}E_{\pm}+V_{\pm}$
are subalgebras, the endomorphisms $K$ of $\mbox{\goth l}^{-1}$ with $K|\mbox{\goth l}_{\pm}=\pm\mathrm{Id}$ define an invariant para-complex structure. It is unique up
to sign since $[V_{\pm},V_{\pm}]=\mathbb{R}E_{\pm}$. This proves i).
The first claim of ii) follows from the remark that
$h_{0},h_{1}^{-1}$ form a basis of the center $Z({\mbox{\goth h}}^{-1})$. The explicit
formulas for $\omega_{0},\omega_{1}$ follow from equations (4.8), (4.9) and (4.10).
$\square$
Corollary 3
Any invariant symplectic form on $L\mathbb{C}H^{n}$ may
be written as
$$\omega^{t}=\omega_{1}+t\omega_{0}.$$
They are compatible with the para-complex structures $K^{\pm}$, i.e.
$(\omega^{t},K^{\pm})$ is a para-Kähler structure and, in particular, $g^{t}=\omega^{t}\circ K^{\pm}$ is a para-Kähler metric.
4.2 Quaternionic projective and hyperbolic spaces
Consider now the spaces $M^{+}=\mathbb{H}P^{n}=Sp_{n+1}/Sp_{1}\cdot Sp_{n}$ and $M^{-}=\mathbb{H}H^{n}=Sp_{1,n}/Sp_{1}\cdot Sp_{n}$. The
reductive decomposition $\mbox{\goth g}^{\varepsilon}=(sp_{1}+sp_{n})+\mbox{\goth p}^{\varepsilon}$ associated
to the homogeneous space $M^{\varepsilon}$ may be written as
$$sp_{1}+sp_{n}=\{\left(\begin{array}[]{cc}a&0\\
0&A_{n}\end{array}\right)\left|\;a\in{\rm{Im}}{\mathbb{H}}=\mbox{\goth sp}_{1}%
,\;\;A_{n}\in\mbox{\goth sp}_{n}\right.\},\;\mbox{\goth p}^{\varepsilon}=\{%
\left(\begin{array}[]{cc}0&-\varepsilon X^{*}\\
X&0\end{array}\right)\left|\quad X\in{\mathbb{H}}^{n}\right\}.$$
We next describe the stability subalgebra ${\mbox{\goth h}}^{\epsilon}$ of the
geodesic $\gamma=\rm{exp}(th_{1}^{\epsilon})(o)$, $o=e(Sp_{1}\cdot Sp_{n})\in M^{\varepsilon}$, which is the orbit of the one-parameter group
$\rm{exp}(th_{1}^{\varepsilon})$, where $h_{1}^{\varepsilon}=\mathrm{diag}(I_{2}^{\varepsilon},0)$.
We have ${\mbox{\goth h}}=Z_{\mbox{\goth g}}(h_{1}^{\epsilon})={\mathbb{R}}h_{1}^{%
\epsilon}+Z_{(sp_{1}+sp_{n})}(h_{1}^{\epsilon})$,
where $Z_{(sp_{1}+sp_{n})}(h_{1}^{\epsilon})$ is the
centraliser of $h_{1}^{\epsilon}$.
The reductive decomposition associated to $L(M)^{\varepsilon}$ may be written
as
$${\mbox{\goth h}}^{\epsilon}=\{A=\left(\begin{array}[]{ccc}a&-\epsilon\alpha&0%
\\
\alpha&a&0\\
0&0&A_{n-1}\end{array}\right)=ah_{0}+\alpha h_{1}^{\varepsilon}+A_{n-1},a\in sp%
_{1},\;\alpha\in{\mathbb{R}},A_{n-1}\in sp_{n-1}\},$$
where
$h_{0}=\rm{diag}(1,1,0),h_{1}^{\varepsilon}=\rm{diag}(I^{\varepsilon}_{2},0)$,
and the complimentary subspace is
$$\mbox{\goth l}^{\varepsilon}=\{X=(x_{1},x_{2},X_{1},X_{2})=\left(\begin{array}%
[]{ccc}x_{1}&\varepsilon x_{2}&-\varepsilon X_{1}^{*}\\
x_{2}&-x_{1}&-X_{2}^{*}\\
X_{1}&X_{2}&0\end{array}\right)|\quad x_{1},x_{2}\in sp_{1},\quad X_{1},X_{2}%
\in{\mathbb{H}}^{n-1}\}.$$
We set
$$E_{1}=\rm{diag}(1,-1,0)\;\;,\;\;E_{2}^{\varepsilon}=h_{1}^{\varepsilon}.$$
Then
$$\mbox{\goth l}^{\varepsilon}=\mathfrak{sp}_{1}E_{1}+\mathfrak{sp}_{1}E_{2}^{%
\varepsilon}+\{(0,0,X_{1},X_{2}),X_{i}\in{\mathbb{H}}^{n-1}\}.$$
The isotropy action of ${\mbox{\goth h}}^{\varepsilon}$ on $\mbox{\goth l}^{\varepsilon}$ is given by
$${\rm ad}_{ah_{0}}(x_{1},x_{2},X_{1},X_{2})=([a,x_{1}],[a,x_{2}],-X_{1}a,-X_{2}%
a),$$
$${\rm ad}_{h_{1}^{\varepsilon}}(x_{1},x_{2},X_{1},X_{2})=(-2\varepsilon x_{2},2%
x_{1},-X_{2},\varepsilon X_{1}),$$
$${\rm ad}_{A_{n-1}}(x_{1},x_{2},X_{1},X_{2})=(0,0,A_{n-1}X_{1},A_{n-1}X_{2}).$$
As in the complex case, we introduce the canonical Hermitian form $\eta(X,Y)=X^{*}Y$ on
${\mathbb{H}}^{n-1}$. We have the following commutator relations:
$$[x_{1}E_{1},x_{2}E_{2}^{\varepsilon}]=-(x_{1}x_{2}+x_{2}x_{1})h_{1}^{%
\varepsilon}=-2Re(x_{1}x_{2})h_{1}^{\varepsilon},$$
$$[x_{1}E_{1},y_{1}E_{1}]=[x_{1},y_{1}]E_{1},\;[x_{2}E_{2}^{\varepsilon},y_{2}E_%
{2}^{\varepsilon}]=\varepsilon[x_{2},y_{2}]h_{0},$$
$\begin{array}[]{cc}&[(0,0,X_{1},X_{2}),(0,0,Y_{1},Y_{2})]=-(g(X_{2},Y_{1})+g(Y%
_{2},X_{1}))h_{1}^{\varepsilon}\\
&-\big{(}\varepsilon\rho(X_{1},Y_{1})+\rho(X_{2},Y_{2})\big{)}h_{0}+\big{(}-%
\varepsilon\rho(X_{1},Y_{1})+\rho(X_{2},Y_{2})\big{)}E_{1}\;\rm{(mod}\;\;sp_{n%
-1}),\end{array}$
$$[x_{1}E_{1},(0,0,X_{1},X_{2})]=(0,0,-X_{1}x_{1},X_{2}x_{1}),$$
$$[x_{2}E_{2},(0,0,X_{1},X_{2})]=(0,0,-X_{2}x_{2},-\varepsilon X_{1}x_{2}).$$
For $\varepsilon=1$ we have the following decomposition of the ${\mbox{\goth h}}^{1}$-module
$\mbox{\goth l}^{1}$ into two irreducible submodules: $\mbox{\goth l}^{1}=V_{0}+V_{1},V_{0}=Im\mathbb{H}+Im\mathbb{H}=\mathbb{R}^{6},%
V_{1}=\mathbb{H}^{n-1}+\mathbb{H}^{n-1}$. Define an ${\mbox{\goth h}}^{1}$-invariant complex structure
$J^{V_{0}},J^{V_{1}}$ by
$$J^{V_{0}}=1/2ad_{h_{1}}|V_{0}:(x_{1},x_{2})\to(-x_{2},x_{1})\quad J^{V_{1}}=ad%
_{h_{1}}|V_{1}:(X_{1},X_{2})\to(-X_{2},X_{1}).$$
The commutator relations imply the following Theorem in a similar
way as in the complex case.
Theorem 5
i)
There exist (modulo sign) two invariant almost complex structures on
$L(\mathbb{H}P^{n})$ defined by $J_{\pm}=J^{V_{0}}\oplus\pm J^{V_{1}}$.
Only one of them, namely $J_{+}$, is integrable.
ii)
Up to scaling, there exists a unique invariant symplectic form $\omega$
on $L(\mathbb{H}P^{n})$ defined by $\omega=d(h_{1}^{\varepsilon})^{*}$. More precisely
we have
$$\omega((x_{1},x_{2},X_{1},X_{2}),(y_{1},y_{2},Y_{1},Y_{2}))=$$
$$+2Re(y_{1}x_{2})+2Re(y_{1}x_{2})+Reh(X_{1},Y_{2})-Reh(Y_{1},X_{2}).$$
iii)
The symplectic structure $\omega$ consistent with the complex structure
$J_{+}$ and gives rise to a Kähler structure.
For $\varepsilon=-1$ we have the decomposition into irreducible $(\mathrm{ad\;}_{{\mbox{\goth h}}^{-1}})$ -submodules
$$\mbox{\goth l}^{-1}=V_{2}+V_{-2}+V_{1}+V_{-1}$$
such that $\mathrm{ad\;}_{{\mbox{\goth h}}^{-1}}|V_{k}=k\mathrm{Id}$,
$V_{2}\simeq V_{-2}\simeq\mathrm{Im}{\mathbb{H}}\simeq\mathfrak{sp}_{1}$ and
$V_{1}\simeq V_{-1}\simeq\mathrm{Im}{\mathbb{H}}^{n-1}$ with the standard action of
${\mbox{\goth h}}^{-1}=\mathfrak{sp}_{1}+\mathfrak{sp}_{n-1}$.
We finally define two $\mathrm{ad}_{{\mbox{\goth h}}}$- invariant para-complex structures
$K_{\pm}$ on ${\mbox{\goth h}}_{-1}$ by
$$K_{+}|_{V_{1}+V_{2}}=1,\;\;K_{+}|_{V_{-1}+V_{-2}}=-1,\;\;K_{-}|_{V_{1}}=1,\;$$
$$K_{-}|_{V_{2}}=-1,\;\;K_{-}|_{V_{2}}=-1,\;\;K_{-}|_{V_{-1}}=-1,\;\;K_{-}|_{V_{%
-2}}=1.$$
We then have the following
Theorem 6
On $L(\mathbb{H}H^{n})$ there exist no invariant almost complex
structures and two (up to sign) unique invariant almost para-complex
structures $K_{\pm}$, with $K_{+}$ being integrable and $K_{-}$ being non-integrable.
4.3 Cayley projective plane
Let $M=\mathbb{O}P^{2}=F_{4}/Spin_{9}$ be the octonian projective
plane and $\mbox{\goth g}=\mbox{\goth f}_{4}=spin_{9}+\mbox{\goth p}$ the associated reductive
decomposition.
The isotropy group $Spin_{9}$ acts on the $16$-dimensional tangent space
$\mbox{\goth p}=T_{o}M$ by the spinor
representation with $15$-dimensional spheres as orbits. Let $\gamma=\rm{exp(t\,h_{1}))o},h_{1}\in\mbox{\goth p}$ be the geodesic through the point
$o=e(Spin_{16})\in M$.
The stability subgroup of $\gamma$ is $H=SO_{2}\cdot Spin_{7}$ and the stability subalgebra
${\mbox{\goth h}}=\mathbb{R}h_{1}+Z_{\mathfrak{spin_{9}}}(h_{1})=\mathbb{R}h_{1%
}+{\mathfrak{spin}}_{7}$.
We
identify the space of
geodesics in $M=\mathbb{O}P^{2}$ with $L(M)=F_{4}/SO_{2}\cdot Spin_{7}$.
Following [4] we choose the root system
$R=\{\pm\varepsilon_{i},\pm\varepsilon_{i}\pm\varepsilon_{j},1/2(\pm\varepsilon%
_{1}\pm\varepsilon_{2}\pm\varepsilon_{3}\pm\varepsilon_{4})\}$
of the complex Lie algebra $\mbox{\goth f}_{4}$ with
respect to a Cartan subalgebra $\mathfrak{a}$ and a system of
simple roots as follows
$$\alpha_{1}=1/2(\varepsilon_{1}-\varepsilon_{2}-\varepsilon_{3}-\varepsilon_{4}%
),\,\alpha_{2}=\varepsilon_{4},\,\alpha_{3}=\varepsilon_{3}-\varepsilon_{4},\,%
\alpha_{4}=\varepsilon_{4}-\varepsilon_{3}.$$
Here, $\varepsilon_{i},\;\;i=1,...,4$ is an orthonormal basis of the real space
$\mbox{\goth a}_{{\mathbb{R}}}=B^{-1}\circ\mathrm{span}_{\mathbb{R}}R$.
We may assume that $\frac{1}{2}d=-ih_{1}\in\mathfrak{a}^{{\mathbb{R}}}$ is the vector dual to the fundamental weight $\pi_{1}=\varepsilon_{1}$. Then the adjoint operator $ad_{d}$ defines a gradation
$$\mathfrak{f}_{4}=\mathfrak{g}_{-2}+\mathfrak{g}_{-1}+\mathfrak{g}_{0}+%
\mathfrak{g}_{1}+\mathfrak{g}_{2},$$
where $\mathfrak{g}_{0}=Z_{\mathfrak{f}_{4}}(h_{1})=\mathbb{C}h_{1}+\mathfrak{spin}_{%
7}^{\mathbb{C}},$ and
$\mathfrak{spin}_{7}^{\mathbb{C}}$ has the root system given by
$$\{\pm\varepsilon_{i},\pm\varepsilon_{i}\pm\varepsilon_{j},i,j=2,3,4\}.$$
The space $\mathfrak{g}_{\pm 1}$
is spanned by the root vectors with roots $1/2(\varepsilon_{1}\pm\varepsilon_{2}\pm\varepsilon_{3}\pm\varepsilon_{4})$ and
$\mathfrak{g}_{\pm 2}$ is spanned by the root vectors
with roots $\pm\varepsilon_{1},\pm(\varepsilon_{1}\pm\varepsilon_{i}),\;i=2,3,4.$
Let $\tau$ be the standard
compact involution of $\mathfrak{f}_{4}$ such that
$\mathfrak{f}_{4}^{\tau}$ is the compact real form of $\mathfrak{f}_{4}$.
Then the reductive decomposition associated with the space of
geodesics can be
written as
$\mathfrak{f}_{4}^{\tau}={\mbox{\goth h}}+\mbox{\goth l}=\mathfrak{g}_{0}^{\tau%
}+\big{(}(\mathfrak{g}_{-1}+\mathfrak{g}_{1})^{\tau}+(\mathfrak{g}_{-2}+%
\mathfrak{g}_{2})^{\tau}$ ).
The decomposition
$$\mbox{\goth l}^{\mathbb{C}}=\mbox{\goth l}^{10}+\mbox{\goth l}^{01}=(\mathfrak%
{g}_{1}+\mathfrak{g}_{2})+(\mathfrak{g}_{-1}+\mathfrak{g}_{-2})$$
defines a unique (up to sign) invariant complex structure $J$ on the space
of geodesics defined by $J|_{\mbox{\goth l}^{10}}=i\mathrm{Id\;},\,\,J|_{\mbox{\goth l}^{01}}=-i\mathrm%
{Id\;}$.
The 2-form $\omega=d(B\circ h_{1})$ associated with the central
element $h_{1}\in Z({\mbox{\goth h}})=\mathbb{R}h$ defines a unique
(up to scaling) symplectic form compatible with $J$, where $B$ is the Killing form. We get
Theorem 7
The space $L(\mathbb{O}P^{2})=F_{4}/SO_{2}\cdot Spin_{7}$ admits a
unique (up to a sign) invariant complex structure $J$, unique (up to a
scaling) invariant symplectic structure $\omega=dB\circ h_{1},h_{1}=\in Z({\mbox{\goth h}})$ and a unique
invariant Kähler structure $(\omega,J)$.
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Abstract
We study the local geometry of testing a mean vector within a
high-dimensional ellipse against a compound alternative. Given samples
of a Gaussian random vector, the goal is to distinguish whether the
mean is equal to a known vector within an ellipse, or equal to some
other unknown vector in the ellipse. Such ellipse testing problems
lie at the heart of several applications, including non-parametric
goodness-of-fit testing, signal detection in cognitive radio, and
regression function testing in reproducing kernel Hilbert spaces.
While past work on such problems has focused on the
difficulty in a global sense, we study difficulty in a way that is
localized to each vector within the ellipse. Our main result is to
give sharp upper and lower bounds on the localized minimax testing
radius in terms of an explicit formula involving the Kolmogorov width
of the ellipse intersected with a Euclidean ball. When applied to
particular examples, our general theorems yield interesting rates that
were not known before: as a particular case, for testing in Sobolev
ellipses of smoothness $\alpha$, we demonstrate rates that vary from
$(\sigma^{2})^{\frac{4\alpha}{4\alpha+1}}$, corresponding to the
classical global rate, to the faster rate $(\sigma^{2})^{\frac{8\alpha}{8\alpha+1}}$, achievable for vectors at favorable
locations within the ellipse. We also show that the optimal test for
this problem is achieved by a linear projection test that is based on
an explicit lower-dimensional projection of the observation vector.
The local geometry of testing in ellipses:
Tight control
via localized Kolmogorov widths
Yuting Wei$${}^{\dagger}$$
Martin J. Wainwright$${}^{\dagger,\star}$$
January 13, 2021
Department of Statistics$${}^{\dagger}$$, and
Department of Electrical
Engineering and Computer Sciences$${}^{\star}$$
UC Berkeley, Berkeley,
CA 94720
1 Introduction
Testing and estimation are two fundamental classes of problems in
statistics [24, 25]. In a classical
decision-theoretic framework, different methods are compared via their
risk functions, and under the minimax formalism, methods are compared
by their worst-case behavior over the entire parameter space. Such
global minimax risk calculation are reflective of typical behavior
when the risk function is close to constant over the entire parameter
space. On the other hand, when the risk function varies
substantially, then it is possible that a global minimax calculation
will be unduly conservative. In such settings, one is motivated to
study the notion of an adaptive or localized minimax risk (e.g., see
references [10, 4] for early work
in this vein).
Recent years have witnessed a rapidly evolving line of work on
studying notions of local minimax risk for both estimation and
testing. In the context of shape-constrained regression, Meyer and
Woodruffe [28] introduced a notion of degrees of
freedom that adapts to the geometry of the function being estimated.
Focusing on the problem of isotonic regression, Chatterjee et
al. [7] proved a range of convergence rates,
depending on the structure of the true regression function. For the
task of estimating a convex function at a point, Cai and
Low [6] proposed a local minimax criterion to
evaluate the performance of different estimators. This criterion was
adapted to establish a form of instance-dependent optimality for
convex set estimation by Cai et al. [5]. In the
context of hypothesis testing, Valiant and
Valiant [32] studied a class of compound
testing problems for discrete distributions, and characterized the
local minimax testing radius in the TV norm. Balakrishnan and
Wasserman [2] studied an analogous
problem for testing density functions, and also characterized the
local testing radius in TV norm.
In this paper, we study the local geometry of testing a mean vector
inside an ellipse against a compound alternative. More precisely,
consider an ellipse of the form $\mathcal{E}=\{\theta\in{}^{d}\mid\sum_{j=1}^{d}\theta_{j}^{2}/\mu_{j}\leq 1\}$,
where $\{\mu_{j}\}_{j=1}^{d}$ is a non-negative sequence that
defines the aspect ratios of the ellipse. Although we focus on
finite-dimensional ellipses, all of our theory is non-asymptotic and
explicit, so that we can obtain results for infinite-dimensional
ellipses by taking suitable limits. Given an observation of a
Gaussian random vector $y\in{}^{d}$, our goal is to test
whether its mean is equal to some known vector $\theta^{*}\in\mathcal{E}$, or
equal to some unknown vector $\theta\in\mathcal{E}$ that is suitably
separated from $\theta^{*}$ in Euclidean norm. As we discuss in more detail
below, such ellipse testing problems lie at the heart of a number of
applications, including non-parametric goodness-of-fit testing, signal
detection in cognitive radio, and testing regression functions in
kernel spaces.
Study of the ellipse testing problems date back to the seminal work of
Ingster [19, 20, 21] and
Ermakov [12], who focused their attention on the
special case when the null is zero ($\theta^{*}=0$), and the ellipse
is induced by a Sobolev function class. For this particular class of
testing problems, they provided a sharp characterization of the
minimax testing radius. A more general question is whether it is
possible to provide a geometric characterization of the minimax
testing radius, both as a function of the ellipse parameters
$\{\mu_{j}\}_{j=1}^{d}$ as well as of the location of null vector
$\theta^{*}$ within the ellipse. The main contribution of this paper
is to answer this question in the affirmative. In particular, we show
how for any ellipse—including the Sobolev ellipses as special
cases—the localized minimax testing radius is characterized by a
formula that involves the Kolmogorov width of a particular set. The
Kolmogorov width is a classical notion from approximation
theory [29], which measures the “size” of a set in
terms of how well it can be approximated by a lower-dimensional linear
subspace of
fixed dimension. Our formula involves the Kolmogorov width of the
ellipse intersected with an Euclidean ball around the null vector
$\theta^{*}$, which leads to an equation involving both $\mathcal{E}$ and
$\theta^{*}$ that can be solved to determine the localized minimax
risk. We show that the zero case ($\theta^{*}=0$) is the most
difficult, in that the localized minimax radius is largest at
this instance. Conversely, we exhibit other more favorable locations
within ellipses for which the local minimax testing radius can be
substantially smaller.
1.1 Some motivating examples
Before proceeding, let us consider some concrete examples so as to
motivate our study.
Example 1 (Non-parametric goodness-of-fit testing).
Consider a set of samples $\{x_{i}\}_{i=1}^{n}$ drawn from an
unknown distribution supported on a compact set $\mathcal{X}$. Assuming
that the unknown distribution has a density, the goodness-of-fit
problem is to test whether the samples have been drawn from some fixed
density $p^{*}$, in which case the null hypothesis holds, or according
to some other density $p\neq p^{*}$, referred to as the alternative
hypothesis. In the non-parametric version of this problem, the
alternative density $p$ is allowed to vary over a broad function
class; one example might be the class of all twice continuously
differentiable densities with second derivative $p^{\prime\prime}$ belonging to the
unit ball in $L^{2}(\mathcal{X})$. There is a very broad literature on this
topic, with the book of Ingster and
Suslina [22] covering many different
classes of alternative densities.
One way to approach non-parametric goodness-of-fit is via orthogonal
series expansions. In particular, if we let $\{\phi_{j}\}_{j=1}^{\infty}$
be an orthonormal basis for $L^{2}(\mathcal{X})$, then the null density
$p^{*}$ and alternative $p$ can be described in terms of their basis
expansion coefficients
$$\displaystyle\theta^{*}_{j}=\int_{\mathcal{X}}\phi_{j}(x)p^{*}(x)dx\;=\;%
\mathbb{E}_{p^{*}}[\phi_{j}(X)],\quad\mbox{and}\quad\theta_{j}=\int_{\mathcal{%
X}}\phi_{j}(x)p(x)dx\;=\;\mathbb{E}_{p}[\phi_{j}(X)],$$
respectively, for $j=1,2,\ldots$. Note that the sample averages
$y_{j}:\,=\frac{1}{n}\sum_{i=1}^{n}\phi_{j}(x_{i})$ are
unbiased estimates of coefficients of the true underlying density, so
the testing problem can be written as
$$\displaystyle\mathcal{H}_{0}:\quad y=\theta^{*}+w\quad\mbox{versus}\quad%
\mathcal{H}_{1}:\quad y=\theta+w,\quad\mbox{for some $\theta\neq\theta^{*}$}$$
where $w=\{w_{j}\}_{j=1}^{\infty}$ is a sequence of noise variables. A
typical smoothness constraint on the alternative—for instance, the
condition $\int_{\mathcal{X}}(p^{\prime\prime}(x))^{2}dx\leq 1$—amounts to requiring
that, in a suitably chosen sinusoidal basis, the vector $\theta$
belongs an ellipse with parameters of the form $\mu_{j}=cj^{-4}$.
Example 2 (Detection of unknown signals in noise).
In cognitive radio and other wireless applications
(e.g., [1, 8, 30]),
it is frequently of interest to test for the presence of unknown
signals that might potentially interfere with transmission. More
concretely, given an observation period $[0,T]$, one observes a
continuous-time waveform $\{y(t)\mid t\in[0,T]\}$, and the goal is to
determine whether it was generated by pure noise, or a by combination
of some band-limited signal $\theta(t)$ plus noise. See
Figure 1 for an illustration.
After applying the Fourier transform, the waveform and signal can be
represented by sequences $y\in\ell^{2}(\mathbb{N})$ and $\theta\in\ell^{2}(\mathbb{N})$, respectively. Assuming that the Fourier frequencies
are ordered from smallest to largest, a hard band-width constraint on
the signal corresponds to the restriction that $\theta_{j}=0$ for all
$j>B$, where the integer $B$ corresponds to the bandlimit cut-off.
Note that this bandwidth constraint defines a (degenerate) ellipse in
$\ell^{2}(\mathbb{N})$. More generally, a soft bandwidth constraint can be
imposed by requiring that the weighted sum $\sum_{j=1}^{\infty}\frac{\theta_{j}^{2}}{\mu_{j}}$ for some summable sequence of positive
coefficients $\{\mu_{j}\}_{j=1}^{\infty}$. As before, this constraint
means that the unknown signal vector $\theta$ must belong to an
ellipse $\mathcal{E}$.
Given this set-up, the problem of detecting the presence of an unknown
signal versus noise corresponds to testing the null that $y\sim N(0,\sigma^{2}I_{d})$ versus the alternative $y\sim N(\theta,\sigma^{2}I_{d})$ for some $\theta\in\mathcal{E}$. More generally, one
might be interested in testing between the null hypothesis $y\sim N(\theta^{*},\sigma^{2}I_{d})$, for some fixed signal vector
$\theta^{*}\in\mathcal{E}$, versus the same alternative. We refer the
reader to the
papers [1, 8, 30]
for further details on this application.
Example 3 (Regression function testing in kernel spaces).
In the problem of fixed design regression, we observe pairs $(x_{i},z_{i})\in\mathcal{X}\times\real$ of the form
$$\displaystyle z_{i}$$
$$\displaystyle=f(x_{i})+\xi_{i}\quad\mbox{for $i=1,\ldots,n$}.$$
(1)
Here $f:\mathcal{X}\rightarrow\real$ is an unknown function, and
$\{\xi_{i}\}_{i=1}^{n}$ are a sequence of zero-mean noise variables
with variance $\sigma^{2}$. Based on the observation vector $z\in{}^{n}$, we might be interested in testing various hypotheses
related to the unknown function $f$. For instance, is $f$ a non-zero
function? More generally, does $f$ deviate significantly from
some fixed class of parametric functions?
These questions are meaningful only if we impose some structures on the
underlying function space, and given the
observations (1) based on pointwise evaluation, one
of the most natural is based on reproducing kernel Hilbert spaces
(e.g., [34, 14]). These function spaces are defined by a
symmetric positive semidefinite kernel function $\mathcal{K}:\mathcal{X}\times\mathcal{X}\rightarrow\real$; as some simple examples, when
$\mathcal{X}=[0,1]$, the kernel function $\mathcal{K}(x,x^{\prime})=1+\min\{x,x^{\prime}\}$ defines a class of Lipschitz functions, whereas the Gaussian
kernel $\mathcal{K}(x,x^{\prime})=\exp(-\frac{1}{2t}(x-x^{\prime})^{2})$ with bandwidth
$t>0$ defines a class of infinitely differentiable functions.
For any choice of kernel, the pointwise observation
model (1) can be transformed to a sequence model
form. In particular, define the kernel matrix $K\in{}^{n\times n}$ with entries $K_{ij}=\mathcal{K}(x_{i},x_{j})/n$, and
let $\mu_{1}\geq\mu_{2}\geq\cdots\geq\mu_{n}\geq 0$ represent
its ordered eigenvalues. In terms of this notation, the pointwise
model with any function $f$ in the unit norm Hilbert ball can be
transformed to the model $y=\theta+w$, where the vector $\theta\in{}^{n}$ satisfies the ellipse constraint
$\sum_{j=1}^{n}\frac{\theta_{j}^{2}}{\mu_{j}}\leq 1$, and each entry
of the noise vector is zero-mean with variance $\frac{\sigma^{2}}{n}$.
We have thus arrived at another instance of our general ellipse
problem.
1.2 Problem formulation
Having understood the range of motivations for our problem,
let us set up the problem more precisely. Given a sequence of
positive scalars $\mu_{1}\geq\ldots\geq\mu_{d}>0$, consider the
associated ellipse
$$\displaystyle\mathcal{E}:\,=\Big{\{}\theta\in{}^{d}\mid\sum_{i=1}^{d}\frac{%
\theta_{i}^{2}}{\mu_{i}}\leq 1\Big{\}}.$$
(2)
Suppose that we make observations of the form $y=\theta+\sigma g$.
Here $\theta\in\mathcal{E}$ is an unknown vector, whereas $g\sim N(0,I_{d})$ is a noise vector, and the variance level $\sigma^{2}$ is known. For
a given vector $\theta^{*}\in\mathcal{E}$, our goal is to test the null
hypothesis $\theta=\theta^{*}$ versus the alternative $\theta\in\mathcal{E}\backslash\{\theta^{*}\}$.
Under this formulation, it is not possible to make any non-trivial
assertion about the power of any test, since the alternative allows
for vectors $\theta$ that are arbitrarily close to $\theta^{*}$. In
order to make quantitative statements, we need to exclude a certain
$\epsilon$-ball around $\theta^{*}$ from the alternative. Doing so
leads to the notion of the minimax testing radius associated
this composite decision problem. This minimax formulation was
introduced in the seminal work of Ingster and
co-authors [18, 22]; since
then, it has been studied by many other researchers
(e.g., [12, 31, 26, 27, 3]).
More precisely, for a given radius $\epsilon>0$, we consider the
compound testing problem specified by the observation model $y\sim N(\theta,\sigma^{2}I_{d})$, and the null and alternative
$$\displaystyle\mathcal{H}_{0}:\theta=\theta^{*}\in\mathcal{E}~{}~{}~{}\text{%
versus}~{}~{}~{}\mathcal{H}_{1}:\|\theta-\theta^{*}\|_{2}\geq\epsilon,\theta%
\in\mathcal{E}.$$
(3)
We thus have a sequence of testing problems indexed by $\epsilon>0$,
and our goal is to calibrate their difficulty in terms of the noise
level $\sigma^{2}$, the null vector $\theta^{*}$, and the local geometry
of the ellipse $\mathcal{E}$ around $\theta^{*}$. To be clear, all the
tests that we analyze in this paper are not given knowledge of
$\epsilon$; rather, it is a unknown quantity that is used to titrate
the difficulty of the problem. See Figure 2 for an
illustration of the testing problem (3).
Letting $\psi:{}^{d}\rightarrow\{0,1\}$ be any measurable
test function, we characterize its performance on the testing
problem (3) in terms of the sum of its (type I)
error under the null, and its (type II) error uniformly over the
alternative. In particular, we define the uniform error
$$\displaystyle\operatorname{Err}(\psi;\theta^{*},\epsilon):\,=\mathbb{E}_{%
\theta^{*}}[\psi(y)]+\sup_{\theta\in\mathcal{E},\|\theta-\theta^{*}\|_{2}\geq%
\epsilon}\mathbb{E}_{\theta}[1-\psi(y)],$$
(4)
where $\mathbb{E}_{\theta}$ denotes expectation over $y$ under the $N(\theta,\sigma^{2}I_{d})$ distribution.
For a given error level $\rho\in(0,1)$, we are interested in the
smallest setting of $\epsilon$ for which there exists some test with
uniform error at most $\rho$. More precisely, we define
$$\displaystyle\epsilon_{\text{{\tiny OPT}}}(\theta^{*};\rho,\mathcal{E}):\,=%
\inf\Big{\{}\epsilon\,\mid\,\inf_{\psi}\;\operatorname{Err}(\psi;\theta^{*},%
\epsilon)\leq\rho\Big{\}},$$
(5)
a quantity which we call the $\theta^{*}$-local minimax testing
radius. By definition, the local minimax testing radius
$\epsilon_{\text{{\tiny OPT}}}(\theta^{*};\rho,\mathcal{E})$ corresponds to the smallest separation
$\epsilon$ at which there exists some test that distinguishes
between the hypotheses $\mathcal{H}_{0}$ and $\mathcal{H}_{1}$ in
equation (3) with uniform error at most $\rho$.
Thus, it provides a fundamental characterization of the statistical
difficulty of the hypothesis testing as a function of $\theta^{*}.$
Our results also have consequences for the more classical global
minimax testing radius, given by
$$\displaystyle\epsilon_{\text{{\tiny OPT}}}(\rho):\,=\sup_{\theta^{*}\in%
\mathcal{E}}\epsilon_{\text{{\tiny OPT}}}(\theta^{*};\rho,\mathcal{E}).$$
As we discuss in the sequel, this global quantity has been
characterized for various Sobolev ellipses in past
work [22].
1.3 Overview of our results
Having set up the problem, let us now provide a high-level overview of
the main results of this paper. We will show that the testing radius is
fully described by a purely geometric quantity, namely the
Kolmogorov $k$-width of a set that localized at
$\theta^{*}$. Given a set $A$, the Kolmogorov width characterizes how
well the set is approximated by a $k$-dimensional linear subspace. It
is known that the Kolmogorov width provides risk upper bounds for the
truncated series estimator
(e.g. [23, 11]) and it also turns
out to be a key quantity in problems such as density estimation, as
well as compressed sensing (see the
papers [15, 9]). We discuss
the definition of the Kolmogorov width and its property at length in
Section 2.2.
We will also show that the optimal test, as will be discussed in sequel, is given by a linear projection test that is based on
the projection of observation vector to a particular lower dimensional
subspace. More discussions on its formulation and background can be
found in our Section 2.1. We state our main results and
some of their important consequences in Section 3.
Section 4 is devoted to the proofs of our main
theorems and corollaries. We defer the auxiliary lemmas for our main
proofs to the appendices.
2 Background
Before proceeding to the statements of our main results, we introduce some
background on linear projection tests, as well as the notion of Kolmogorov
width.
2.1 Linear projection tests
In this paper, we prove some upper bounds on $\epsilon_{\text{{\tiny OPT}}}(\theta^{*};\rho,\mathcal{E})$ via a concretely defined class of tests, in particular
those based on linear projections. Given an integer $k\in[d]:\,=\{1,\ldots,d\}$, let $\Pi_{k}$ denote an orthogonal
projection operator, mapping any vector $v\in{}^{d}$ to a
particular $k$-dimensional subspace. Any projection $\Pi_{k}$
defines a family of linear projection tests, indexed by the threshold
$\beta\geq 0$, of the following form
$$\displaystyle\psi_{\beta,\Pi_{k}}(y):\,=\mathbb{I}\big{[}\|\Pi_{k}(y-\theta^{*%
})\|_{2}\geq\beta\big{]}\;=\;\begin{cases}1&\mbox{if
$\|\Pi_{k}(y-\theta^{*})\|_{2}\geq\beta$}\\
0&\mbox{otherwise.}\end{cases}$$
(6)
We use $\Psi_{\text{{\tiny LPT}}}$ to denote the collection of all linear projection
tests
$$\displaystyle\Psi_{\text{{\tiny LPT}}}$$
$$\displaystyle:\,=\Big{\{}\psi_{\beta,\Pi_{k}}\;\mid k\in[d],\;\Pi_{k}\in%
\mathcal{P}_{k},\;\beta\in{}_{+}\Big{\}},$$
obtained by varying the dimension $k\in[d]$, the projection
$\Pi_{k}$ over the space $\mathcal{P}_{k}$ of all $k$-dimensional
orthogonal projections, and the threshold $\beta\geq 0$.
Given the family $\Psi_{\text{{\tiny LPT}}}$, we then define
$$\displaystyle\epsilon_{\text{{\tiny LPT}}}(\theta^{*};\rho,\mathcal{E}):\,=%
\inf\Big{\{}\epsilon\,\mid\,\inf_{\psi\in\Psi_{\text{{\tiny LPT}}}}\;%
\operatorname{Err}(\psi;\theta^{*},\epsilon)\leq\rho\Big{\}}.$$
(7)
a quantity which we call
$\theta^{*}$-local LPT testing radius. When the error level
$\rho$ is clear from the context, we adopt the shorthand notation
$\epsilon_{\text{{\tiny LPT}}}(\theta^{*};\mathcal{E})$ and $\epsilon_{\text{{\tiny OPT}}}(\theta^{*};\mathcal{E})$, with the implicit
understanding that all of our analysis depends on the choice of
$\rho$.
2.2 Kolmogorov width
Our characterization of testing difficulty involves a classical
geometric notion known as the Kolmogorov width, which we introduce
here; we refer the reader to the book by Pinkus [29] for
more details.
For a given compact set $C\subset{}^{d}$ and integer $k\in[d]$, the Kolmogorov $k$-width is a measure of how well the set
can be approximated by a $k$-dimensional subspace. More precisely,
recalling that $\mathcal{P}_{k}$ denotes the set of all $k$-dimensional
orthogonal linear projections, the Kolmogorov $k$-width,
denoted by $\omega_{k}$, is given by
$$\displaystyle\omega_{k}(C)=\min_{\Pi_{k}\in\mathcal{P}_{k}}\max_{\theta\in C}%
\|\theta-\Pi_{k}\theta\|_{2}.$$
(8)
Any $\Pi_{k}$ achieving the minimum in the saddle point
problem (8) is said to be an optimal
projection for $\omega_{k}(C)$. By definition, the Kolmogorov
widths are non-increasing as a function of the integer $k$—in
particular, we have
$$\displaystyle r(C)=\omega_{0}(C)\geq\omega_{1}(C)\geq\ldots\geq\omega_{d}(C)=0,$$
where $r(C):\,=\max_{\theta\in C}\|\theta\|_{2}$ is the diameter
of the compact set $C$.
3 Main results and their consequences
We now turn to the statement of our main results, along with a
discussion of some of their consequences.
3.1 Upper bound on the local minimax testing radius
In this section, we establish an upper bound on the minimax testing
radius for problem (3) localized to a given vector
$\theta^{*}$. In order to do so, we study the behavior of the linear
projection tests previously described in Section 2.1, It
turns out that LPT testing radius $\epsilon_{\text{{\tiny LPT}}}(\theta^{*};\mathcal{E})$ can be
characterized by the Kolmogorov $k$-width—as defined in
expression (8)—of a particular set.
Before stating the main result, let us first introduce some notation.
For each $\epsilon>0$, we define the upper critical dimension
$$\displaystyle k_{u}(\epsilon,\theta^{*},\mathcal{E}):\,=\arg\min_{1\leq k\leq d%
}\Big{\{}\omega_{k}(\mathcal{E}_{\theta^{*}}\cap\mathbb{B}(\epsilon))\leq\frac%
{\epsilon}{\sqrt{2}}\Big{\}}.$$
(9)
Here set $\mathcal{E}_{\theta^{*}}:\,=\{\theta-\theta^{*}\mid\theta\in\mathcal{E}\}$
for the ellipse defined in expression (2) and
$\mathbb{B}(\epsilon):\,=\{v\in{}^{d}\mid\|v\|_{2}\leq\epsilon\}.$
For any fixed $\epsilon>0$, the Kolmogorov width
$\omega_{k}(\mathcal{E}_{\theta^{*}}\cap\mathbb{B}(\epsilon))$ is non-increasing
with index $k$ and it goes to zero as $k$ approaches $d$ (where
$d$ can grow as well, then we have an infinite dimensional
ellipse) and $\omega_{d}(\mathcal{E}_{\theta^{*}}\cap\mathbb{B}(\epsilon))=0$.
Therefore, the critical dimension is always well defined.
This dimension is used to define the optimal linear projection test at
$\theta^{*}$. We also use it to define for each error level $\rho\in(0,1/2]$, the upper critical radius
$$\displaystyle\epsilon_{u}(\theta^{*};\rho,\mathcal{E})$$
$$\displaystyle:\,=\inf\{\epsilon\mid\epsilon\geq\frac{8}{\sqrt{\rho}}\sigma^{2}%
\frac{\sqrt{k_{u}(\epsilon,\theta^{*},\mathcal{E})}}{\epsilon}\Big{\}}.$$
(10)
For each choice of error level $\rho$, this radius defines the
separation between null and alternative that can be distinguished with
error controlled at level $\rho$. From the fact that the function
$\epsilon\mapsto k_{u}(\epsilon,\theta^{*},\mathcal{E})$ is
non-increasing, it follows that $\epsilon_{u}(\theta^{*};\rho,\mathcal{E})$
always exists, and is guaranteed to be unique and strictly positive.
More details can be found in Section D.1.
With these two ingredients, we are ready to state our upper
bound for the LPT testing radius localized to $\theta^{*}$:
Theorem 1.
For any vector $\theta^{*}\in\mathcal{E}$ and any error level $\rho\in(0,1/2]$, we have
$$\displaystyle\inf_{\psi\in\Psi_{\text{{\tiny LPT}}}}\;\operatorname{Err}(\psi;%
\theta^{*},\delta)\leq\rho\qquad\text{ for all }\delta\geq\epsilon_{u}(\theta^%
{*};\rho,\mathcal{E}).$$
(11)
We prove this theorem in Section 4.1.
Recall the definition of the $\theta^{*}$-localized minimax radius
$\epsilon_{\text{{\tiny OPT}}}$ and $\theta^{*}$-localized LPT radius $\epsilon_{\text{{\tiny LPT}}}$ from
equations (5)
and (7), respectively. In terms of this
notation, Theorem 1 guarantees that
$$\displaystyle\epsilon_{\text{{\tiny OPT}}}(\theta^{*};\rho,\mathcal{E})\;%
\stackrel{{\scriptstyle(i)}}{{\leq}}\;\epsilon_{\text{{\tiny LPT}}}(\theta^{*}%
;\rho,\mathcal{E})\;\stackrel{{\scriptstyle(ii)}}{{\leq}}\;\epsilon_{u}(\theta%
^{*};\rho,\mathcal{E}).$$
To be clear, the first inequality (i) is a trivial consequence of the
definitions, whereas inequality (ii) is the substance of
Theorem 1.
Structure of optimal linear test:
Although we have stated
the theorem in an existential way, our analysis also gives an explicit
form of the test that achieves the error
guarantee (11). The construction of this test
consists of three steps:
•
For a given error level $\rho$, first compute the critical
radius $\epsilon_{u}(\theta^{*};\rho,\mathcal{E})$ using
expression (10).
•
Second, using equation (9), compute the induced
critical upper dimension
$$\displaystyle k^{*}_{u}:\,=k_{u}(\epsilon_{u}(\theta^{*};\rho,\mathcal{E}),%
\theta^{*},\mathcal{E}),$$
as well as the associated projection matrix $\Pi_{k^{*}_{u}}$ that
defines the Kolmogorov width indexed by $k^{*}_{u}$.
•
Third, apply the linear projection test
$$\displaystyle y\mapsto\mathbb{I}\Big{[}\|\Pi_{k^{*}_{u}}(y-\theta^{*})\|_{2}^{%
2}\geq\sigma^{2}(k^{*}_{u}+\sqrt{4k^{*}_{u}/\rho})\Big{]}.$$
(12)
The proof of the theorem is based on analyzing this linear
test (12) directly and explicitly controlling its type I
and type II error.
In order to illustrate the use of Theorem 1,
let us consider a very simple example.
Example 4 (Circular constraints).
Suppose that the ellipse $\mathcal{E}$ is simply a circle in
${}^{d}$, say with $\mu_{j}=d$ for all $j=1,\ldots,d$. Let us consider the case of testing at zero (so
that $\theta^{*}=0$) with a noise level $\sigma\in(0,1)$. For
any $\epsilon\in(0,\sqrt{d})$, we claim that the upper
critical dimension (9) is equal to $d$.
Indeed, for any projection matrix of dimension $k<d$, there
is at least one dimension that is missing, and this dimension
contains a vector of Euclidean length $\sqrt{d}$ inside the
sphere. Consequently, solving for the upper critical radius from
equation (10) yields that $\epsilon_{u}^{2}(0;\mathcal{E})=\frac{8}{\sqrt{\rho}}\sigma^{2}\sqrt{d}$.
Note that this rate matches the minimax testing radius for a subspace
of dimension $d$. This calculation makes sense, because for
this particular example—that is, testing at the zero vector for a
circle with the chosen scalings of $\mu_{j}$ and $\sigma$—the geometry
and boundary of the circle actually plays no role.
In Section 3.3, we consider some more substantive
applications of Theorem 1, and in particular,
demonstrate that the local minimax testing radius varies substantially
as a function of the location of $\theta^{*}$ within the ellipse.
3.2 Lower bound on the local minimax testing radius
Thus far, we have derived an upper bound for the localized testing
radius for a particular procedure—namely, the LPT. Of course, it is
of interest to understand when the LPT is actually an optimal test,
meaning that there is no other test that can discriminate between the
null and alternative for smaller separations. In this section, we use
information-theoretic methods to prove a lower bound on the localized
minimax testing radius $\epsilon_{\text{{\tiny OPT}}}(\theta^{*};\mathcal{E})$. Whenever the
radius $\epsilon_{\text{{\tiny LPT}}}(\theta^{*};\mathcal{E})$ achievable by a linear projection
test matches the lower bound for every $\theta^{*}$, we can claim the
linear projection tests are locally minimax optimal.
For the testing problem (3), it turns out that the
local minimax testing radius $\epsilon_{\text{{\tiny OPT}}}(\theta^{*};\mathcal{E})$ can also be
lower bounded via the Kolmogorov $k$-width, as previously defined in
equation (8), of a particular set. In order to
state this lower bound, we fix a triplet of positive constants $a,b,c$
linked by the relations
$$\displaystyle a>3b,\mbox{ and }c=c(a,b):\,=\frac{b}{8\sqrt{2}}-\frac{\sqrt{a^{%
2}-9b^{2}}}{12\sqrt{2}}>0.$$
(13)
Recall the recentered ellipse $\mathcal{E}_{\theta^{*}}:\,=\{\theta-\theta^{*}\mid\theta\in\mathcal{E}\}$, and Euclidean ball of radius $\epsilon$ centered at zero $\mathbb{B}(\epsilon):\,=\{v\in{}^{d}\mid\|v\|_{2}\leq\epsilon\}$.
For our upper bound, the upper critical dimension $k_{u}$ from
equation (9) played a central role. As the lower
analogue to this quantity, let us define the lower critical
dimension
$$\displaystyle k_{\ell}(\epsilon,\theta^{*},\mathcal{E}):\,=\arg\min_{1\leq k%
\leq d}\left\{\omega_{k}(\mathcal{E}_{\theta^{*}}\cap\mathbb{B}(a\epsilon))%
\leq 3b\epsilon\right\},$$
(14)
where the fixed choice of constants $(a,b)$ should be understood
implicitly. We then define the lower critical radius
$$\displaystyle\epsilon_{\ell}(\theta^{*};\mathcal{E})$$
$$\displaystyle:\,=\sup\Big{\{}\epsilon\mid\epsilon\leq\frac{1}{4}\sigma^{2}%
\frac{\sqrt{k_{\ell}(\epsilon,\theta^{*},\mathcal{E})}}{\epsilon}\Big{\}},$$
(15)
From the fact that the function $\epsilon\mapsto k_{\ell}(\epsilon,\theta^{*},\mathcal{E})$ is non-increasing, it follows that $\epsilon_{\ell}$ is
well-defined and unique. More details can be found in
Appendix D.1.
We prove a lower bound as a function of the quantity
$\epsilon_{\ell}(\theta^{*};\mathcal{E})$ and also a second quantity, one which
measures its proximity to the boundary of the ellipse. More
precisely, for a given vector $\theta^{*}$ and constant $a>0$, we
define the mapping $\Phi:{}_{+}\rightarrow{}_{+}$ via
$$\displaystyle\Phi(\delta)$$
$$\displaystyle=\begin{cases}1&\mbox{if $\delta>\|\theta^{*}\|_{2}/a$}\\
1\wedge\min\Big{\{}r\geq 0\,\mid a^{2}\delta^{2}\leq\sum_{i=1}^{d}\frac{r^{2}}%
{(r+\mu_{i})^{2}}(\theta^{*}_{i})^{2}\Big{\}}&\mbox{otherwise.}\end{cases}$$
(16)
As shown in Appendix D.2, this mapping is well-defined, and
has the limiting behavior $\Phi(\delta)\rightarrow 0$ as $\delta\rightarrow 0^{+}$. Let us denote $\Phi^{-1}(x)$ as the largest
positive number of $\delta$ such that $\Phi(\delta)\leq x$. Note
that by this definition, we have $\Phi^{-1}(1)=\infty$.
We are now ready to state our lower bound for the optimal
testing radius localized to $\theta^{*}$:
Theorem 2.
Consider constants $a,b,c\in(0,1)$ satisfying the
conditions (13). Then for any $\theta^{*}\in\mathcal{E}$,
we have
$$\displaystyle\inf_{\psi}\;\operatorname{Err}(\psi;\theta^{*},\delta)\geq\frac{%
1}{2}\qquad\text{ whenever }\quad\delta\,\leq\,c\min\Big{\{}\epsilon_{\ell}(%
\theta^{*};\mathcal{E}),~{}~{}\Phi^{-1}\big{(}(\|\theta^{*}\|_{\mathcal{E}}^{-%
1}-1)^{2}\big{)}\Big{\}}.$$
(17)
See Section 4.2 for the proof of this theorem.
Remarks:
Regarding concrete choices of the constants $(a,b,c)$
that underlie this theorem, if we set $a=\frac{\sqrt{43}}{12}$ and
$b=\frac{1}{4}$, then $c=\frac{1}{288\sqrt{2}}$. Since our main interest
is to understand the scaling of the testing radius with respect to
$\sigma$ and the geometric parameters of the problem, we have made no
efforts to obtain the sharpest constants in the theorem statement.
Second, to clarify the role of the term $\Phi^{-1}\big{(}(\|\theta^{*}\|_{\mathcal{E}}^{-1}-1)^{2}\big{)}$, it is not of primary interest
and serves to measure whether or not $\theta^{*}$ is sufficiently far
inside the ellipse. Concretely, if we assume that $\|\theta^{*}\|_{\mathcal{E}}\leq 1/2$, then $(\|\theta^{*}\|_{\mathcal{E}}^{-1}-1)^{2}\geq 1$ therefore
$\Phi^{-1}\big{(}(\|\theta^{*}\|_{\mathcal{E}}^{-1}-1)^{2}\big{)}=\infty$ which
means that
$$\displaystyle\min\Big{\{}\epsilon_{\ell}(\theta^{*};\mathcal{E}),~{}~{}\Phi^{-%
1}\big{(}(\|\theta^{*}\|_{\mathcal{E}}^{-1}-1)^{2}\big{)}\Big{\}}=\epsilon_{%
\ell}(\theta^{*};\mathcal{E}).$$
A final remark on the proof of Theorem 2: it is
based on constructing a prior distribution on the alternative $\mathcal{H}_{1}$
that is independent of observation $y$. This distribution is
supported on precisely those points in $\mathcal{H}_{1}$ that are relatively
hard to distinguish from $\mathcal{H}_{0}$. Then the testing error can be
lower bounded through controlling the total variation (TV) distance between
two marginal likelihood functions.
3.3 Some consequences of our results
One useful consequence of Theorems 1
and 2 is in providing a sufficient condition for the
optimality of the LPT. In particular, suppose that the upper and
lower dimensions defined in equations (9)
and (14) differ only by constant pre-factors, which we
write as
$$\displaystyle k_{u}(\epsilon,\theta^{*},\mathcal{E})\asymp k_{\ell}(\epsilon;%
\theta^{*},\mathcal{E}).$$
(18)
Under this condition, it follows that the critical upper and lower
radii match—namely, that $\epsilon_{u}(\theta^{*};\mathcal{E})\asymp\epsilon_{\ell}(\theta^{*};\mathcal{%
E})$. Theorem 1 then
guarantees that the linear projection test at $\theta^{*}$ is locally
minimax optimal.
As previously discussed, the local optimality studied here provides a
finer-grained distinction than the usual notion of global minimax
optimality. All of the rates depend on the vector $\theta^{*}$ via
the set $\mathcal{E}_{\theta}^{*}\cap\mathbb{B}(\epsilon)$, whose shape changes as
$\theta^{*}$ moves across the ellipse. The resulting changes in the local
minimax radii can be quite substantial, as we now illustrate with some
concrete examples.
3.3.1 Testing at zero
We begin our exploration by considering the testing
problem (3) with $\theta^{*}=0$. In order to
characterize the $\theta^{*}$-localized minimax testing
radius (5), we define the functions
$$\displaystyle m_{u}(\delta;\mathcal{E}):\,=\arg\max_{1\leq k\leq d}\{\mu_{k}%
\geq\frac{1}{2}\delta^{2}\},~{}\text{ and }~{}m_{\ell}(\delta;\mathcal{E}):\,=%
\arg\max_{1\leq k\leq d}\{\mu_{k+1}\geq\frac{9}{16}\delta^{2}\}.$$
(19)
In order to ensure that both functions are well-defined, we restrict
our attention to $\delta$ in the interval $\big{(}0,\min\{\sqrt{2\mu_{1}},\frac{4}{3}\sqrt{\mu_{2}}\}\big{)}$. We now state
a consequence of Theorems 1 and 2 in
terms of these quantities:
Corollary 1.
Given an error level $\rho\in(0,1)$, the local minimax testing
radius $\epsilon_{\text{{\tiny OPT}}}(\theta^{*};\mathcal{E})$ at $\theta^{*}=0$ is bounded as
$$\displaystyle\sup\Big{\{}\delta\,\mid\,\delta\leq\frac{1}{4}\sigma^{2}\frac{%
\sqrt{m_{\ell}(\delta;\mathcal{E})}}{\delta}\Big{\}}\;\leq\;\epsilon_{\text{{%
\tiny OPT}}}(0;\mathcal{E})\;\leq\;\inf\Big{\{}\delta\,\mid\,\delta\geq\frac{8%
}{\sqrt{\rho}}\sigma^{2}\frac{\sqrt{m_{u}(\delta;\mathcal{E})}}{\delta}\Big{\}},$$
(20)
where we only consider $\delta\leq\min\{\sqrt{2\mu_{1}},\frac{4}{3}\sqrt{\mu_{2}}\}$.
We prove Corollary 1 in
Section 4.3.
The quantities defined on the left and right hand sides of
expression (20) exist due to the fact that
$m_{u}(\delta;\mathcal{E})$ and $m_{\ell}(\delta;\mathcal{E})$ are
non-increasing functions of $\delta.$ Putting inequalities in
(20) together, whenever $m_{u}$ and $m_{\ell}$ differ
only by constant factors, we have a tight characterization of the
testing radius $\epsilon_{\text{{\tiny OPT}}}(0;\mathcal{E})$ localized to $\theta^{*}=0$.
In order to illustrate, we now apply Corollary 1 to some
special cases of the ellipse $\mathcal{E}$, and see how the minimax testing
radius localized to zero varies as a function of the ellipse
parameters. In particular, let us consider two broad classes of decay
rates for the ellipse parameters $\{\mu_{j}\}_{j}$.
$\gamma$-exponential decay:
For some $\gamma>0$, suppose the scalars $\{\mu_{j}\}_{j}$ satisfies a decay
condition of the form $\mu_{j}=c_{1}\exp(-c_{2}j^{\gamma})$, where
$c_{1},c_{2}$ are universal constants. Such exponential decay rates arise
the problem of non-parametric regression testing
(Example 3), in particular for reproducing kernel
Hilbert spaces based on Gaussian kernels. From
expression (19), we know $k_{u}=C\log^{1/\gamma}(1/\epsilon)$ and $k_{B}=c\log^{1/\gamma}(1/\epsilon)$
and solving the two inequalities from equation (20)
yields
$$\displaystyle\epsilon_{\text{{\tiny OPT}}}^{2}(0)\asymp\sigma^{2}(\log(\frac{1%
}{\sigma^{2}}))^{1/2\gamma}.$$
(21)
Here $\asymp$ denotes equality up to a factor involving constants and
terms of the order $\log\log(1/\sigma^{2})$.
$\alpha$-polynomial decay:
Now suppose that the ellipse parameters $\{\mu_{j}\}_{j}$ satisfy a
decay condition of the form $\mu_{j}=c_{1}j^{-2\alpha}$ for some
$\alpha>1/2$. As previously discussed, polynomial decay of this
type arises in ellipses that are induced by Sobolev function spaces of
smoothness $\alpha$. From expression (19), we know
$k_{u}=C\epsilon^{-1/\alpha}$ and $k_{B}=c\epsilon^{-1/\alpha}$ and
solving the two inequalities in equation (20) yields
$$\displaystyle\epsilon_{\text{{\tiny OPT}}}^{2}(0)\asymp(\sigma^{2})^{\frac{4%
\alpha}{1+4\alpha}}.$$
(22)
Here our notation $\asymp$ denotes equality up to constants
independent of $(\sigma,d)$, so that
inequality (22) provides a characterization of the
testing radius localized at zero that is tight up to constant factors.
It is worth noticing that that rate (22) matches the
(non-localized) globally minimax optimal testing radius established by
Ingster [19, 20, 21]. Thus, testing at
zero is as hard as the hardest testing problem in the ellipse.
Intuitively, this correspondence makes sense because the ellipse
contains a relatively large volume around zero, meaning that the
compound alternative contains a larger number of possible vectors that
need to be distinguished from the null.
Ingster [19] proves the lower bound by the direct
analysis of a $\chi^{2}$-type test, which is then shown to be optimal.
For the estimation analogue of this problem, Ibragimov and
Khasminskii [16, 17] showed that
the global minimax rate for estimation scales as
$\epsilon_{\text{est}}\asymp(\sigma^{2})^{2\alpha/(1+2\alpha)}$. This slower rate confirms
the intuition that testing is an easier problem than estimation:
we can detect a vector of length much smaller than the accuracy
at which we can estimate.
Remark:
As a final comment,
Corollary 1 can be generalized to the $\ell_{p}$-ellipse
with $p\geq 2$, given by $\mathcal{E}_{p}:\,=\{\theta\in{}^{d}\mid\sum_{i=1}^{d}\frac{\theta_{i}^{p}}{%
\mu_{i}}\leq 1\}$. Let us again consider testing at $\theta^{*}=0.$ Using the
same technique to prove Corollary 1, it is easy to check
that inequalities in equation (20) still hold with the
definitions of $m_{u}$ and $m_{\ell}$ in
expression (19) replaced by
$$\displaystyle m_{u}(\delta;\mathcal{E}):\,=\arg\max_{1\leq k\leq d}\big{\{}\mu%
_{k+1}^{2/p}\geq\frac{1}{2}\delta^{2}\big{\}},\quad\mbox{and}\quad m_{\ell}(%
\delta;\mathcal{E}):\,=\arg\max_{1\leq k\leq d}\big{\{}\mu_{k}^{2/p}\geq\delta%
^{2}\big{\}}.$$
3.3.2 Testing at extremal vectors
In the previous section, we studied the behavior of the minimax
testing radius for $\theta^{*}=0$, and recovered some known results from
the classical literature. In this section, we study some non-zero
cases of the vector $\theta^{*}$. For concreteness, we restrict our
attention to vectors that are non-zero some coordinate $s\in[d]=\{1,\ldots,d\}$, and zero in all other
coordinates. Even for such simple vectors, our analysis reveals some
interesting scaling of the minimax testing radius that is novel.
Given an integer $s\in[d]$ and a positive scalar $\delta$, we
define
$$\displaystyle m_{u}(\delta;s)$$
$$\displaystyle:\,=\arg\max_{1\leq k\leq d}\Big{\{}\mu_{k}^{2}\geq\frac{1}{64}%
\delta^{2}\mu_{s}\Big{\}},~{}\text{ and }~{}m_{\ell}(\delta;s):\,=\arg\max_{1%
\leq k\leq d}\Big{\{}\mu^{2}_{k}\geq\delta^{2}\mu_{s}\Big{\}}.$$
(23)
So as to ensure that these functions are well-defined, we restrict our
attention to $\delta$ in the interval $\big{(}0,\mu_{1}/\sqrt{\mu_{s}}\big{)}$. By definition, we have $m_{u}(\delta;s)\geq m_{\ell}(\delta;s)$.
Before stating our corollary, let us define the upper and lower
critical radii as follows
$$\displaystyle t^{*}_{u}(s,\mathcal{E})$$
$$\displaystyle=\inf\Big{\{}\delta>0\mid\delta\geq\frac{8}{\sqrt{\rho}}\sigma^{2%
}\frac{\sqrt{m_{u}(\delta;s)}}{\delta}\Big{\}}$$
(24a)
$$\displaystyle\text{ and
}~{}~{}t^{*}_{\ell}(s,\mathcal{E})$$
$$\displaystyle=\sup\Big{\{}\delta>0\mid\delta\leq\frac{1}{4}\sigma^{2}\frac{%
\sqrt{m_{\ell}(\delta;s)}}{\delta}\Big{\}}.$$
(24b)
Using the fact that $\rho\in(0,1/2]$ and non-increasing functions
satisfy $m_{u}(\delta;s)\geq m_{\ell}(\delta;s)$, we have $t^{*}_{\ell}(s,\mathcal{E})\leq t^{*}_{u}(s,\mathcal{E})$. Again these two quantities do not depend
on the $\theta^{*}$ that we are testing at. We then define the set
$$\displaystyle C(s;\mathcal{E})$$
$$\displaystyle:\,=\Big{\{}\theta\in{}^{d}\mid\theta_{j}=0\quad\mbox{for all $j%
\neq s$, and}\quad\theta_{s}\in\big{[}\sqrt{\mu_{s}}-t^{*}_{u}(s,\mathcal{E}),%
\sqrt{\mu_{s}}-t^{*}_{\ell}(s,\mathcal{E})\big{]}\Big{\}}.$$
We are ready to state our corollary:
Corollary 2.
Consider any positive integer $s$ such that $t^{*}_{u}(s,\mathcal{E})\leq\sqrt{\mu_{s}}$. Then for any testing error level $\rho>0$, we
have
$$\displaystyle t^{*}_{\ell}(s,\mathcal{E})\,\leq\,\epsilon_{\text{{\tiny OPT}}}%
(\theta^{*};\mathcal{E})\,\leq\,t^{*}_{u}(s,\mathcal{E})\qquad\mbox{for all $%
\theta^{*}\in C(s;\mathcal{E})$.}$$
(25)
See Section 4.4 for the proof of
Corollary 2.
Corollary 2 characterizes the optimal testing radius for
vectors that are sufficiently close to the boundary of the ellipse, In
order to compare with the results of Section 3.3.1 for
testing at zero, let us apply Corollary 2 to some
special examples of the ellipse $\mathcal{E}$, and see how the testing
radius varies as we vary the index $s\in[d]$.
For these comparisons, we restrict our attention to ellipses
for which the parameters $\{\mu_{j}\}$ decay at a polynomial rate.
$\alpha$-Polynomial decay:
Suppose that the scalars $\{\mu_{j}\}_{j}$ satisfy a decay condition of the form
$\mu_{j}=c_{1}j^{-2\alpha}$ for some $\alpha>1/2$. Recalling
the definition (23), some calculations yield
$$\displaystyle\frac{\mu_{m_{u}}^{2}}{\mu_{s}}$$
$$\displaystyle=\frac{m_{u}^{-4\alpha}}{s^{-2\alpha}}~{}\geq~{}\frac{\delta^{2}}%
{64}~{}\Longrightarrow~{}m_{u}=\lfloor(64s^{2\alpha})^{\frac{1}{4\alpha}}%
\delta^{-\frac{1}{2\alpha}}\rfloor,\qquad\mbox{and}$$
$$\displaystyle\frac{\mu_{m_{\ell}}^{2}}{\mu_{s}}$$
$$\displaystyle=\frac{m_{\ell}^{-4\alpha}}{s^{-2\alpha}}~{}\geq~{}\delta^{2}~{}%
\Longrightarrow~{}m_{\ell}=\lfloor s^{\frac{1}{2}}\delta^{-\frac{1}{2\alpha}}\rfloor.$$
Solving the fixed point equations yields
$$\displaystyle t^{*}_{u}(s,\mathcal{E})=\Big{(}8^{\frac{4\alpha+1}{4\alpha}}%
\rho^{-\frac{1}{2}}s^{\frac{1}{4}}\sigma^{2}\Big{)}^{\frac{4\alpha}{8\alpha+1}%
}\qquad\text{and }~{}~{}t^{*}_{\ell}(s,\mathcal{E})=\Big{(}\frac{1}{4}s^{\frac%
{1}{4}}\sigma^{2}\Big{)}^{\frac{4\alpha}{8\alpha+1}}.$$
If we omit the dependence on the level $\rho$ and constants, the
optimal testing radius is given by
$$\displaystyle\epsilon_{\text{{\tiny OPT}}}^{2}(\theta^{*};\mathcal{E})\asymp(%
\sigma^{2}s^{1/4})^{\frac{8\alpha}{1+8\alpha}}\qquad\text{and }~{}~{}m_{u},m_{%
\ell}\asymp(\sigma^{-2}s^{2\alpha})^{\frac{2}{8\alpha+1}}$$
(26)
whenever111The constraint on $s$ arises from the fact that
$t^{*}_{u}(s,\mathcal{E})\leq\sqrt{\mu_{s}}$. $s\lesssim(\sigma^{2})^{-2/(4\alpha+1)}$. As the choice of $s$ is varied,
inequality (26) actually provides us with a range
of different results.
(a)
First considering $s=1$, suppose that $\theta^{*}$ is zero at
every coordinate except the first: we then have
$$\displaystyle\epsilon_{\text{{\tiny OPT}}}^{2}(\theta^{*};\mathcal{E})\asymp(%
\sigma^{2})^{\frac{8\alpha}{1+8\alpha}},$$
(27)
where $\asymp$ means equal up to a constant that is independent of
problem parameters $d$ and $\sigma$. To the best of our
knowledge, this is a novel result on ellipse testing. It shows that
that the local minimax rate at this extremal vector is very different
from the corresponding rate at zero, which as we have established in
our Section 3.3.1 is equal $(\sigma^{2})^{\frac{4\alpha}{1+4\alpha}}$.
(b)
On the other hand, let us consider a sequence (indexed by
$\sigma$) of the form $s=(\sigma^{2})^{-\beta}$ for some $\beta\in\Big{(}0,\frac{2}{(4\alpha+1)}\Big{)}$. Over such a sequence,
inequality (26) shows that the minimax
testing radius scales as
$$\displaystyle\epsilon_{\text{{\tiny OPT}}}^{2}(\theta^{*};\mathcal{E})\asymp(%
\sigma^{2})^{\frac{2\alpha(4-\beta)}{1+8\alpha}}.$$
(28)
As $\beta$ ranges from $0$ to $\frac{2}{(4\alpha+1)}$, the testing
radius $\epsilon_{\text{{\tiny OPT}}}^{2}(\theta^{*};\mathcal{E})$ ranges from
$(\sigma^{2})^{\frac{4\alpha}{1+4\alpha}}$ to
$(\sigma^{2})^{\frac{8\alpha}{1+8\alpha}}$.
4 Proofs of main results
We now turn to the proofs of our main results, with the
proof of Theorems 1 and 2
given in Sections 4.1 and 4.2
respectively, followed by the proofs of
Corollary 1 and 2 in
Sections 4.3 and 4.4, respectively.
In all cases, we defer the proofs of certain more technical lemmas to the appendices.
4.1 Proof of Theorem 1
From our discussion following the statement of
Theorem 1, recall that the optimal linear
test (12) is based on the statistic
$$\displaystyle\psi(y):\,=\mathbb{I}\big{[}\|\Pi_{k^{*}_{u}}(y-\theta^{*})\|_{2}%
^{2}\geq\sigma^{2}(k^{*}_{u}+\sqrt{4k^{*}_{u}/\rho})\big{]},$$
(29)
where the critical upper dimension $k^{*}_{u}:\,=k_{u}(\epsilon_{u}(\theta^{*};\rho,\mathcal{E}),\theta^{*},%
\mathcal{E})$ is
computed from equation (9).
From here onwards, we adopt $\epsilon$ and $k$ as convenient
shorthands for the quantities $\epsilon_{u}$ and $k^{*}_{u}$ respectively.
We let $\Pi_{k}$ denote the projection matrix that defines the
Kolmogorov $k$-width of $\mathcal{E}_{\theta}^{*}\cap\mathbb{B}(\epsilon)$—that is
$$\displaystyle\Pi_{k}:\,=\arg\min_{\Pi_{k}\in\mathcal{P}_{k}}\max_{\Delta\in%
\mathcal{E}_{\theta^{*}}\cap\mathbb{B}(\epsilon)}\|\Delta-\Pi_{k}\Delta\|_{2}^%
{2}.$$
(30)
Controlling the type I error:
We first show that the
test statistic (29) has type I error at most
$\rho/2$. Under the null hypothesis $\mathcal{H}_{0}$, we have $y\sim N(\theta^{*},\sigma^{2}I_{d})$. Since $\Pi_{k}$ is an orthogonal
projection matrix, the rotation invariance of the Gaussian
distribution guarantees that
$$\displaystyle T_{k}$$
$$\displaystyle:\,=\|\Pi_{k}(y-\theta^{*})\|_{2}^{2}\stackrel{{\scriptstyle d}}{%
{=}}\|\tilde{w}\|_{2}^{2},\qquad\mbox{where $\tilde{w}\sim N(0,\sigma^{2}%
\mathbb{I}_{k})$.}$$
Consequently, the statistic $T_{k}/\sigma^{2}$ follows a
$\chi_{k}^{2}$-distribution, and hence
$$\displaystyle\mathbb{P}_{0}\big{[}\psi(y)=1\big{]}$$
$$\displaystyle=\mathbb{P}_{\theta^{*}}\big{[}T_{k}\geq\sigma^{2}(k+\sqrt{4k/%
\rho})\big{]}\leq\frac{\rho}{2},$$
(31)
where the last step follows from Chebyshev’s inequality.
Controlling the type II error:
Under the alternative
hypothesis $\mathcal{H}_{1}$, we have $y\sim N(\theta,\sigma^{2}I_{d})$
for some vector $\theta\in\mathcal{E}$ with $\|\theta-\theta^{*}\|_{2}\geq\epsilon$. Since $\Pi_{k}$ is a linear operator, we have
$$\displaystyle T_{k}=\|\Pi_{k}(y-\theta^{*})\|_{2}^{2}=\|\Pi_{k}w+\Pi_{k}(%
\theta-\theta^{*})\|_{2}^{2},\qquad\mbox{where $w\sim N(0,\sigma^{2}I_{d})$.}$$
Using the fact that Gaussian distribution is rotation invariant, the
statistic $T_{k}/\sigma^{2}$ follows a non-central $\chi^{2}_{k}(c_{0})$
distribution, defined by the recentering $c_{0}=\|\Pi_{k}(\theta-\theta^{*})\|_{2}^{2}$.
For each vector $\theta\in\mathcal{E}$ such that $\|\theta-\theta^{*}\|_{2}\geq\epsilon$, we claim that
$$\displaystyle c_{0}=\|\Pi_{k}(\theta-\theta^{*})\|^{2}_{2}\geq\min_{u\in%
\mathcal{E},\|u-\theta^{*}\|_{2}=\epsilon}\{\|\Pi_{k}(u-\theta^{*})\|^{2}_{2}\}.$$
(32)
We take inequality (32) as given for the moment,
returning to prove it at the end of this section. Since $\Pi_{k}$
is an orthogonal projection onto a subspace, the Pythagorean theorem
guarantees that
$$\displaystyle\|\Pi_{k}(u-\theta^{*})\|^{2}_{2}=\|(u-\theta^{*})\|_{2}^{2}-\|(u%
-\theta^{*})-\Pi_{k}(u-\theta^{*})\|^{2}_{2}.$$
Consequently, we have
$$\displaystyle c_{0}=\|\Pi_{k}(\theta-\theta^{*})\|^{2}_{2}$$
$$\displaystyle\stackrel{{\scriptstyle(i)}}{{\geq}}\min_{u\in\mathcal{E},\|u-%
\theta^{*}\|_{2}=\epsilon}\,\left\{\|(u-\theta^{*})\|_{2}^{2}-\|(u-\theta^{*})%
-\Pi_{k}(u-\theta^{*})\|^{2}_{2}\right\}$$
$$\displaystyle=\epsilon^{2}-\max_{u\in\mathcal{E},\|u-\theta^{*}\|_{2}=\epsilon%
}\|(u-\theta^{*})-\Pi_{k}(u-\theta^{*})\|_{2}^{2}$$
$$\displaystyle\geq\epsilon^{2}-\max_{u\in\mathcal{E},\|u-\theta^{*}\|_{2}\leq%
\epsilon}\|(u-\theta^{*})-\Pi_{k}(u-\theta^{*})\|_{2}^{2},$$
where inequality (i) makes use of inequality (32).
Recalling the definition of $\Pi_{k}$ from
equation (30), we find that
$$\displaystyle c_{0}$$
$$\displaystyle~{}\geq\epsilon^{2}-\min_{\Pi_{k}\in\mathcal{P}_{k}}\max_{\theta%
\in\mathcal{E}_{\theta^{*}}\cap\mathbb{B}(\epsilon)}\|\Delta-\Pi_{k}\Delta\|_{%
2}^{2}\;=\;\epsilon^{2}-\omega_{k}^{2}(\mathcal{E}_{\theta^{*}}\cap\mathbb{B}(%
\epsilon))\stackrel{{\scriptstyle(ii)}}{{\geq}}\frac{\epsilon^{2}}{2},$$
where inequality (ii) follows since the critical dimension $k$ was
chosen so that $\omega_{k}(\mathcal{E}_{\theta^{*}}\cap\mathbb{B}(\epsilon))\leq\frac{\epsilon%
}{\sqrt{2}}$.
Now we are ready to bound the type II error. For any vector $\theta$
in the alternative, we have
$$\displaystyle\mathbb{P}_{\theta}[\psi(y)=0]$$
$$\displaystyle=\mathbb{P}_{\theta}\big{[}T_{k}\leq\sigma^{2}(k+\sqrt{4k/\rho})%
\big{]}=\mathbb{P}_{\theta}\Biggr{[}\frac{T_{k}-\sigma^{2}k-c_{0}}{\sigma^{2}%
\sqrt{k}}\leq\frac{2}{\sqrt{\rho}}-\frac{c_{0}}{\sigma^{2}\sqrt{k}}\Biggr{]}.$$
Together, the two inequalities $c_{0}\geq\epsilon^{2}/2$ and $\epsilon^{2}\geq 8\rho^{-1/2}\sigma^{2}\sqrt{k}$ imply that $c_{0}/\sigma^{2}\sqrt{k}\geq 4\rho^{-1/2}$, whence
$$\displaystyle\mathbb{P}_{\theta}[\psi(y)=0]$$
$$\displaystyle\leq\mathbb{P}_{\theta}\left[\frac{T_{k}-\sigma^{2}k-c_{0}}{%
\sigma^{2}\sqrt{k}}\leq\frac{2}{\sqrt{\rho}}-\frac{4}{\sqrt{\rho}}\right]\leq%
\frac{\rho}{2},$$
where the last inequality is again due to Chebyshev’s inequality.
Therefore whenever $\epsilon^{2}\geq 8\rho^{-1/2}\sigma^{2}\sqrt{k}$,
then type II error is upper bounded by
$$\displaystyle\sup_{\theta\in\mathcal{H}_{1}}\mathbb{P}_{\theta}\big{[}\psi(y)=%
0\big{]}\leq\frac{\rho}{2}.$$
(33)
Combining inequalities (31)
and (33) yields the claim.
Proof of inequality (32):
The only remaining detail in the proof of Theorem 1 is
to establish inequality (32). It suffices to show that
the minimum
$$\displaystyle\min_{u\in\mathcal{E},\;\|u-\theta^{*}\|_{2}\geq\epsilon}\|\Pi_{k%
}(\theta-\theta^{*})\|^{2}_{2}$$
(34)
is always achieved by a vector $u^{*}\in\mathcal{E}$ with $\|u^{*}-\theta^{*}\|_{2}=\epsilon$.
Noting the continuity of the objective and the compactness of the
constraint set, Weierstrass’ theorem ensures that the minimum is
always achieved. Leting $\widehat{u}$ be any vector that achieves the
minimum, we define the new vector
$$\displaystyle u^{*}$$
$$\displaystyle:\,=(1-\alpha)\theta^{*}+\alpha\widehat{u}\qquad\mbox{where $%
\alpha:\,=\frac{\epsilon}{\|\widehat{u}-\theta^{*}\|_{2}}\in(0,1]$.}$$
The convexity of $\mathcal{E}$ ensures that $u^{*}\in\mathcal{E}$, and
moreover, by construction, we have $\|u^{*}-\theta^{*}\|_{2}=\epsilon$. Thus, the vector $u^{*}$ is feasible for the original
problem (34), and by linearity of projection, we have
$$\displaystyle\|\Pi_{k}(u^{*}-\theta^{*}\|_{2}$$
$$\displaystyle\leq(1-\alpha)\|\Pi_{k}(0)\|_{2}+\alpha\|\Pi_{k}(\widehat{u}-%
\theta^{*})\|_{2}\;\leq\;\|\Pi_{k}(\widehat{u}-\theta^{*})\|_{2},$$
showing that $u^{*}$ is also a minimizer, as claimed.
4.2 Proof of Theorem 2
We now turn to the proof of the lower bound stated in
Theorem 2. This proof, as well as subsequent ones,
makes use of the Bernstein width as a lower bound on the Kolmogorov
width. Accordingly, we begin by introducing some background on it, as
well stating an auxiliary lemma that shows how to obtain testing error
lower bounds via Bernstein width.
4.2.1 Lower bounds via the Bernstein width
The Bernstein $k$-width of a compact set $C$ is the radius of the
largest $k+1$-dimensional ball that can be inscribed into $C$.
Typically, we use the Euclidean ($\ell_{2}$) norm to define the ball,
but more generally, we can define widths respect to the $\ell_{p}$-norms
with $p\in[1,\infty]$. More precisely, for a given integer $k\geq 1$, let $\mathcal{S}_{k+1}$ denote the set of all $(k+1)$-dimensional
subspaces. With this notation, the Bernstein $k$-width in
$\ell_{p}$-norm is given by
$$\displaystyle b_{k,p}(C)$$
$$\displaystyle:\,=\arg\max_{r\geq 0}\Big{\{}\mathbb{B}^{k+1}_{p}(r)\subseteq C%
\cap S\quad\mbox{for some subspace $S\in\mathcal{S}_{k+1}$}\Big{\}},$$
(35)
where $\mathbb{B}^{k+1}_{p}(r):\,=\{u\in{}^{k+1}\mid\|u\|_{p}\leq r\}$ denotes a $(k+1)$-dimensional $\ell_{p}$-ball of radius $r$
centered at zero.
An important fact is that the Kolmogorov $k$-width from
equation (8) can always be lower bounded by the
Bernstein $k$-width in $\ell_{2}$—that is,
$$\displaystyle\omega_{k}(C)$$
$$\displaystyle\geq b_{k,2}(C)\quad\mbox{for any $k=1,2,\ldots$.}$$
(36)
See Pinkus [29] for more details of this property.
Moreover, comparing the Bernstein widths in $\ell_{2}$ and $\ell_{\infty}$
norm, we have the sandwich relation
$$\displaystyle b_{k,\infty}(C)\;\leq\;b_{k,2}(C)\;\leq\;\sqrt{k+1}\;b_{k,\infty%
}(C),$$
(37)
which is an elementary consequence of the fact that $1\leq\frac{\|u\|_{2}\;\,}{\|u\|_{\infty}}\leq\sqrt{k+1}$ for any non-zero vector
$u\in{}^{k+1}$.
We now use these definitions to define a lower bound on the testing
error in terms of Bernstein width in $\ell_{\infty}$. Define the
$\ell_{\infty}$-Bernstein lower critical dimension as
$$\displaystyle k_{B}(\epsilon,\theta^{*},\mathcal{E}):\,=\arg\max_{1\leq k\leq d%
}\Big{\{}kb^{2}_{k-1,\infty}(\mathcal{E}_{\theta^{*}})\geq\epsilon^{2}\Big{\}},$$
(38a)
and the corresponding lower critical radius
$$\displaystyle\epsilon_{B}(\theta^{*};\mathcal{E})$$
$$\displaystyle:\,=\sup\Big{\{}\epsilon\mid\epsilon\leq\frac{1}{4}\sigma^{2}%
\frac{\sqrt{k_{B}(\epsilon,\theta^{*},\mathcal{E})}}{\epsilon}\Big{\}}.$$
(38b)
In terms of these quantities, we have the following auxiliary result:
Lemma 1.
For any vector $\theta^{*}\in\mathcal{E}$, we have
$$\displaystyle\inf_{\psi}\operatorname{Err}(\psi;\theta^{*},\delta)\geq\frac{1}%
{2}\qquad\text{for all }\delta\leq\epsilon_{B}(\theta^{*};\mathcal{E}).$$
(39)
See Appendix A for the proof of this lemma.
Lemma 1 is useful because evaluating the Bernstein
width is often easier than evaluating the Kolmogorov width. Note that
we also have a (possibly weaker) lower bound in term of
$\ell_{2}$-Bernstein widths. In particular, suppose that we define the
$\ell_{2}$-Bernstein critical dimension
$$\displaystyle\widetilde{k}_{B}(\epsilon,\theta^{*},\mathcal{E}):\,=\arg\max_{1%
\leq k\leq d}\Big{\{}b^{2}_{k-1,2}(\mathcal{E}_{\theta^{*}})\geq\epsilon^{2}%
\Big{\}}$$
(40)
and then define the critical radius $\epsilon_{B}$ using $\widetilde{k}_{B}$. By
the sandwich relation (37), the lower
bound (39) still holds.
4.2.2 Main portion of proof
Let $D:\,=\max_{\theta\in\mathcal{E}}\|\theta-\theta^{*}\|_{2}$
denote the diameter of the ellipse with respect to $\theta^{*}$. We
need only consider testing radii $\epsilon\leq D$, since otherwise,
the alternative is empty. Moreover, we need only consider noise
standard deviations $\sigma\leq D$, since otherwise even the simple
(non-compound) testing problem is hard.
Next, we claim that $\epsilon_{\text{{\tiny OPT}}}^{2}(\theta^{*};\mathcal{E})\geq\sigma^{2}$.
In order to establish this claim, we consider a simple hypothesis test
between a pair $(\theta^{*},\theta)$ both in the ellipse, and such
that $\|\theta^{*}-\theta\|_{2}=\delta$ for some $\delta\in[\sigma,D]$. This single-to-single test requires separation
$\delta^{2}\geq\sigma^{2}$, a classical fact that can be proved by
various arguments, for example, by analyzing the likelihood ratio and
using Neyman-Pearson lemma. Since our composite testing
problem (3) is at least as hard, it follows that
$\epsilon_{\text{{\tiny OPT}}}^{2}(\theta^{*};\mathcal{E})\geq\sigma^{2}$.
For the remainder of our analysis, we assume that $\sqrt{k_{\ell}}\geq 18$. (Otherwise, the squared critical radius $\epsilon_{u}$ from in
equation (15)) is upper bounded by $9\sigma^{2}/2$, so
that the claim of Theorem 2 follows by adjusting the
constant $c$, combined with the lower bound $\epsilon_{\text{{\tiny OPT}}}^{2}(\theta^{*};\mathcal{E})\geq\sigma^{2}$.) We now divide our analysis into two cases,
depending on whether or not $\|\theta^{*}\|_{\mathcal{E}}\leq 1/2$.
Case 1:
First, suppose that $\|\theta^{*}\|_{\mathcal{E}}\leq 1/2$, which
implies that $\Phi(\delta)\leq(\|\theta^{*}\|_{\mathcal{E}}-1)^{2}\leq 1$. In this
case, we apply the $\ell_{2}$-Bernstein-width based lower bound from
Lemma 1 to establish our theorem. Define the
quantities $\widetilde{k}_{B}(\epsilon,\theta^{*},\mathcal{E}):\,=\arg\max\limits_{1\leq k%
\leq d}\Big{\{}b_{k-1,2}(\mathcal{E}_{\theta^{*}}\cap\mathbb{B}(a\epsilon))%
\geq b\epsilon\Big{\}}$, and
$$\displaystyle\epsilon_{B}(\theta^{*};\mathcal{E})$$
$$\displaystyle:\,=\sup\Big{\{}\epsilon\mid b^{2}\epsilon\leq\frac{1}{4}\sigma^{%
2}\frac{\sqrt{\widetilde{k}_{B}(\epsilon,\theta^{*},\mathcal{E})}}{\epsilon}%
\Big{\}}.$$
With this notation, an equivalent statement of 1
is that $\inf_{\psi}\operatorname{Err}(\psi;\theta^{*},\delta)\geq\frac{1}{2}$ for all $\delta\leq b\epsilon_{B}(\theta^{*};\mathcal{E})$.
(In this statement, we have replaced $b_{k-1,2}(\mathcal{E}_{\theta^{*}})$ by $b_{k-1,2}(\mathcal{E}_{\theta^{*}}\cap\mathbb{B}(a\epsilon))$ for later theoretical convenience; doing so does
not change the definition as $a>b$.)
In order to prove Theorem 2, we need to show that
$c\epsilon_{\ell}(\theta^{*};\mathcal{E})\leq b\epsilon_{B}(\theta^{*};\mathcal%
{E})$, and we claim that it suffices to show that $k_{\ell}(\epsilon,\theta^{*},\mathcal{E})\leq\widetilde{k}_{B}(\epsilon,\theta%
^{*},\mathcal{E})$. Indeed, supposing that the latter inequality holds, the
definition of the lower critical radius in
equation (15) implies that
$$\displaystyle\epsilon_{\ell}(\theta^{*};\mathcal{E})$$
$$\displaystyle:\,=\sup\Big{\{}\epsilon\mid\epsilon\leq\frac{1}{4}\sigma^{2}%
\frac{\sqrt{k_{\ell}(\epsilon,\theta^{*},\mathcal{E})}}{\epsilon}\Big{\}}.$$
Comparing with the above definition of $\epsilon_{B}(\theta^{*};\mathcal{E})$ and using222 It should also be noted that the right
hand sides of both fixed point equations are non-increasing function
of $\epsilon$ while the left hand side is non-decreasing. the fact
that $b\in(0,1)$, we see that $\epsilon_{\ell}(\theta^{*};\mathcal{E})\leq\epsilon_{B}(\theta^{*};\mathcal{E})$. Finally, since $c<b$, it follows
that $c\epsilon_{\ell}(\theta^{*};\mathcal{E})\leq b\epsilon_{B}(\theta^{*};\mathcal%
{E})$.
Now it is only left for us to show that $k_{\ell}(\epsilon,\theta^{*},\mathcal{E})\leq\widetilde{k}_{B}(\epsilon,\theta%
^{*},\mathcal{E})$. By definition of
the critical lower dimension $k_{\ell}(\epsilon,\theta^{*},\mathcal{E})$ in
equation (14), we have
$\omega_{k_{\ell}-1}(\mathcal{E}_{\theta^{*}}\cap\mathbb{B}(a\epsilon))>3b\epsilon$. So our proof for Case 1 will be complete once we establish
that $b_{k_{\ell},2}(\mathcal{E}_{\theta^{*}}\cap\mathbb{B}(a\epsilon))\geq b\epsilon$. In order to do so, we use the following lemma.
Lemma 2.
The following chain of inequalities hold
$$\displaystyle\min\{a\epsilon,\frac{1}{2}\sqrt{\mu_{k+1}}\}\,\stackrel{{%
\scriptstyle(i)}}{{\leq}}\,b_{k,2}(\mathcal{E}_{\theta^{*}}\cap\mathbb{B}(a%
\epsilon))\leq\omega_{k}(\mathcal{E}_{\theta^{*}}\cap\mathbb{B}(a\epsilon))\,%
\stackrel{{\scriptstyle(ii)}}{{\leq}}\,\frac{3}{2}\omega_{k}(\mathcal{E}\cap%
\mathbb{B}(a\epsilon))\,\stackrel{{\scriptstyle(iii)}}{{=}}\,\frac{3}{2}\min\{%
a\epsilon,\sqrt{\mu_{k+1}}\}.$$
(41)
See Appendix B.1 for the proof of this claim.
From Lemma 2, we have the string of inequalities
$$\displaystyle 3b\epsilon<\omega_{k_{\ell}-1}(\mathcal{E}_{\theta^{*}}\cap%
\mathbb{B}(a\epsilon))\leq\frac{3}{2}\min\{a\epsilon,\sqrt{\mu_{k_{\ell}}}\},$$
and hence $b_{k_{\ell},2}(\mathcal{E}_{\theta^{*}}\cap\mathbb{B}(a\epsilon))\geq\min\{a%
\epsilon,\frac{1}{2}\sqrt{\mu_{k_{\ell}}}\}\geq b\epsilon$,
which completes the proof in Case 1.
Case 2:
Otherwise, we may assume that $\|\theta^{*}\|_{\mathcal{E}}>1/2$,
in which case $\Phi(\delta/c)\leq(\|\theta^{*}\|_{\mathcal{E}}-1)^{2}<1$, and
hence by definition of the function $\Phi$, we have $\delta<c\|\theta^{*}\|_{2}/a$.
Our analysis in this case makes use of the following auxiliary result
(stated as Lemma 6 in Wei et al. [35]). Suppose
that we observe a Gaussian random vector $y\sim N(\Delta,\sigma^{2}I_{d})$. For a set $C$, we use $\operatorname{Err}(\psi;\{0\},C,\delta)$ to denote the uniform testing error defined by the compound
testing problem (in the Gaussian sequence model) of distinguishing the
null $\Delta=0$ from the compound alternative $\Delta\in C\cap\mathbb{B}^{c}(\delta)$, where $\mathbb{B}^{c}(\delta):\,=\{v\mid\|v\|_{2}\geq\delta\}$.
Lemma 3.
For every set $C$ and probability measure $\mathbb{Q}$ supported on $C\cap\mathbb{B}^{c}(\delta)$, we have
$$\displaystyle\inf_{\psi}\operatorname{Err}(\psi;\{0\},C,\delta)\geq 1-\frac{1}%
{2}\sqrt{\mathbb{E}_{\eta,\eta^{\prime}}\,\exp\Big{(}\frac{\langle\eta,\,\eta^%
{\prime}\rangle}{\sigma^{2}}\Big{)}-1},$$
(42)
where $\mathbb{E}_{\eta,\eta^{\prime}}$ denotes expectation with respect to an
i.i.d pair $\eta,\eta^{\prime}\sim\mathbb{Q}$.
In order to apply Lemma 3, we need to specify a
suitable set $C$ and probability distribution $\mathbb{Q}$. For our
application, the relevant set is $C=\mathcal{E}_{\theta}^{*}$. In
addition, we need to construct an appropriate distribution $\mathbb{Q}$
with which to apply Lemma 3. To this end, we
introduce two more auxiliary results. In the following statement, we
fix a constant $\epsilon>0$, as well as pair $a,b\in(0,1)$ such
that $a>3b$, and recall that $k_{\ell}(\epsilon)$ denotes the lower
critical dimension from equation (14). Also recall that
the ellipse norm $\|\theta\|_{\mathcal{E}}^{2}=\theta^{T}M\theta$ is
defined in terms of the diagonal matrix $M$ with diagonal
entries $\{\frac{1}{\mu_{i}}\}_{i=1}^{d}$.
Lemma 4.
For any vector $\theta^{*}\in\mathcal{E}$ such that
$\|\theta^{*}\|_{2}>a\epsilon$, there exists a vector $\theta^{\dagger}\in\mathcal{E}$, a collection of $d$-dimensional orthonormal
vectors $\{u_{i}\}_{i=1}^{k_{\ell}}$ and an upper triangular matrix of the
form
$$\displaystyle H:\,=\begin{bmatrix}1&h_{3,2}&h_{4,2}&\cdots&h_{k_{\ell},2}\\
&1&h_{4,3}&\cdots&h_{k_{\ell},3}\\
&&1&\cdots&h_{k_{\ell},4}\\
&&&\ddots&\vdots\\
&&&&1\end{bmatrix}\in{}^{(k_{\ell}-1)\times(k_{\ell}-1)},$$
(43)
with ordered singular values $\nu_{1}(H)\geq\nu_{2}(H)\geq\cdots\geq\nu_{k_{\ell}-1}(H)\geq 0$ such that:
(a)
The vectors $u_{1}$, $M\theta^{\dagger}$, and $\theta^{\dagger}-\theta^{*}$ are all scalar multiples of one another.
(b)
We have $\|\theta^{\dagger}-\theta^{*}\|_{2}=a\epsilon$.
(c)
For each $i\in[k_{\ell}-1]$, the $i^{th}$ column $H_{i,\cdot}$ of the matrix $H$ has its squared Euclidean norm upper
bounded as $\|H_{\cdot,i}\|_{2}^{2}\leq a^{2}/9b^{2}$.
(d)
For each $i\in[k_{\ell}-1]$, each of the vectors $\theta^{\dagger}\pm b\epsilon\underbrace{\begin{bmatrix}u_{2}&\cdots u_{k_{%
\ell}}\end{bmatrix}}_{:\,=U}H_{\cdot,i}$ belongs to the ellipse $\mathcal{E}$.
(e)
For each pair of integers $(s,t)\in[k_{\ell}-1]\times[k_{\ell}-2]$, we have
$$\displaystyle\nu_{s}\stackrel{{\scriptstyle(i)}}{{\leq}}\frac{a}{3b}\sqrt{%
\frac{k_{\ell}-1}{s}},~{}~{}\text{ and }~{}~{}\nu_{t+1}\stackrel{{\scriptstyle%
(ii)}}{{\geq}}1-\frac{t}{k_{\ell}-1}-\sqrt{\frac{a^{2}-9b^{2}}{9b^{2}}}.$$
(44)
See Appendix B.2 for the proof of this lemma.
Let us write the SVD of the matrix $H$ from
equation (43) in the form $Q\Sigma V^{T}$. Since $H$
is square and full rank, the matrices $Q$ and $V$ are square and
orthogonal, and $\Sigma$ is a square matrix of singular values. By
construction, we have $H^{T}H=V\Sigma^{2}V^{T}$.
Define matrix $A:\,=V^{T}(UH)^{T}M(UH)V$ and write its diagonal
as $(A_{1},\ldots,A_{k_{\ell}-1}).$
We introduce index set $\mathcal{M}(r):\,=\{i\mid A_{i}\leq A_{(r)}\}$
which contains those elements that lie below the $r$-th quantile of $A$’s diagonals elements.
We assume that $(k_{\ell}-1)/8$ is an integer; otherwise, the same argument can be applied by taking its floor function.
Let $m_{1}:\,=(k_{\ell}-1)/4$ and $m_{2}:\,=(k_{\ell}-1)/2$.
Fixing a sparsity level $s\leq m_{2}-m_{1}+1$, let $\mathcal{S}$ be the set
of all subsets of
set $F:\,=\{m_{1},m_{1}+1,\ldots,m_{2}\}\cap\mathcal{M}(7(k_{\ell}-1)/8)$ of size $s$.
Note that from definitions, the cardinality of $F$ is lower bounded by $(k_{\ell}-1)/8.$
Given a subset $S\in\mathcal{S}$
and a sign vector $z^{S}=(z^{S}_{1},\ldots,z^{S}_{k_{\ell}-1})\in\{-1,0,1\}^{k_{\ell}-1}$, define the vector
$$\displaystyle\theta^{S}:\,=\theta^{*}+b\epsilon\frac{1}{4\sqrt{2s}}XVz^{S},%
\qquad\Delta^{S}:\,=\theta^{S}-\theta^{*},$$
(45)
where the matrix $X=UH$ was previously defined in
4. We then make the following claim:
Lemma 5.
Under the assumption of Lemma 4 and $\Phi(\epsilon)\leq(\|\theta^{*}\|_{\mathcal{E}}-1)^{2}$, for any subset $S\in\mathcal{S}$, there is
a sign vector $z^{S}$ supported on $S$ such that:
$$\displaystyle\theta^{S}(z^{S})$$
$$\displaystyle\;\stackrel{{\scriptstyle(i)}}{{\in}}\;\mathcal{E},\qquad\text{%
and }~{}~{}\|\Delta^{S}\|_{2}\;\stackrel{{\scriptstyle(ii)}}{{\geq}}\;c\epsilon.$$
(46)
See Appendix B.3 for the proof of this lemma.
We now apply Lemma 3 with $\mathbb{Q}$ as the
uniform distribution on those perturbation vectors $\Delta^{S}=\theta^{S}-\theta^{*}$ that are defined via
expression (45). Note that the sign vector $z^{S}$ is
taken to be the one whose existence is guaranteed by
Lemma 5. For any $S\in\mathcal{S}$, due to
inequality (46), we have $\|\Delta^{S}\|_{2}\geq\epsilon$,
and thus, with the choice $\delta:\,=c\epsilon$, the
distribution $\mathbb{Q}$ is supported on set $\mathcal{E}_{\theta^{*}}\cap\mathbb{B}^{c}(\delta)$
For simplicity of the notation, let us introduce the shorthand
notation $t:\,=m_{2}-m_{1}+1$; notat that this integer satisfies
the relations $t=k_{\ell}/4\geq 81$ and $\lambda:\,=\frac{1}{32\sigma^{2}}b^{2}\epsilon^{2}\nu^{2}_{m_{1}}$, where $\nu_{m_{1}}$
denotes the $m_{1}$-th largest eigenvalue of matrix $H$.
Lemma 6.
Given the distribution $\mathbb{Q}$ defined as above, we have
$$\displaystyle\mathbb{E}_{\eta,\eta^{\prime}\sim\mathbb{Q}}\Big{[}\exp\big{(}%
\frac{\langle\eta,\,\eta^{\prime}\rangle}{\sigma^{2}}\big{)}\Big{]}$$
$$\displaystyle\leq\exp\left(-\left(1-\frac{1}{\sqrt{t}}\right)^{2}+\exp\left(%
\frac{2+\lambda}{\sqrt{t}-1}\right)\right).$$
(47)
See Appendix B.4 for the proof of this lemma.
We can now complete the proof of Case 2. If the radius satisfies
$\epsilon^{2}\leq\frac{\sqrt{k_{\ell}}}{4}\sigma^{2}$, as a consequence of
the eigenvalue bound (44), we find that
$$\displaystyle\frac{\lambda}{\sqrt{t}}=\frac{\epsilon^{2}b^{2}\nu^{2}_{m_{1}}}{%
32\sigma^{2}\sqrt{m_{2}-m_{1}+1}}\leq\frac{\sqrt{k_{\ell}}\cdot b^{2}\cdot%
\frac{4a^{2}}{9b^{2}}}{144\sqrt{m_{2}-m_{1}+1}}\leq\frac{1}{18},$$
where the last inequality uses $a<1$ and $m_{2}-m_{1}+1=k_{\ell}/4.$
With control of these quantities in hand, we can bound the testing
error. The right hand side in expression (47) can be
upper bounded as
$$\displaystyle\exp\left(-\left(1-\frac{1}{\sqrt{t}}\right)^{2}+\exp\left(\frac{%
2}{\sqrt{t}-1}+\frac{\sqrt{t}}{\sqrt{t}-1}\frac{\lambda}{\sqrt{t}}\right)%
\right)\leq\exp\left(-\left(1-\frac{1}{9}\right)^{2}+\exp\left(\frac{1}{4}+%
\frac{9}{8}\cdot\frac{1}{18}\right)\right)<2,$$
where we use the fact that $t=k_{\ell}/4\geq 81$. As a
consequence of Lemma 3, we have
$$\displaystyle\inf_{\psi}\operatorname{Err}(\psi;\{0\},\mathcal{E}_{\theta^{*}}%
,c\epsilon)\geq 1-\frac{1}{2}\Big{[}\mathbb{E}_{\eta,\eta^{\prime}}\,\exp(%
\frac{\langle\eta,\,\eta^{\prime}\rangle}{\sigma^{2}})-1\Big{]}^{1/2}\;>\;%
\frac{1}{2}\geq\rho,$$
which completes the proof of Case 2, and hence the theorem.
4.3 Proof of Corollary 1
Our proof of Corollary 1 is based on a particular way of sandwiching the upper and
lower critical radii, which we first introduce.
4.3.1 Sandwiching the upper and lower critical radii
Recall the upper and lower critical dimensions from
equations (9) and (14), respectively.
Suppose that we can find pair a pair of functions $(\hat{k}_{\ell},\hat{k}_{u})$
such that the sandwich relation
$$\displaystyle\hat{k}_{\ell}(\epsilon)\,\leq\,k_{\ell}(\epsilon)\,\leq\,k_{u}(%
\epsilon)\leq\hat{k}_{u}(\epsilon),~{}~{}\text{ for }\epsilon>0.$$
(48)
holds. Under an additional monotonicity property, any such pair can
be used to bound the interval $[\epsilon_{\ell},\epsilon_{u}]$.
Lemma 7.
Consider a pair of functions $\hat{k}_{u}$ and $\hat{k}_{\ell}$ that are each
non-increasing in $\epsilon$, and satisfy the sandwich
relation (48). Then we have $\big{[}\epsilon_{\ell},\;\epsilon_{u}\big{]}\subseteq\big{[}\widehat{\epsilon}%
_{\ell},\;\widehat{\epsilon}_{u}\big{]}$, where
$$\displaystyle\widehat{\epsilon}_{u}:\,=\inf\left\{\epsilon\mid\epsilon\geq%
\frac{8}{\sqrt{\rho}}\sigma^{2}\frac{\sqrt{\hat{k}_{u}(\epsilon)}}{\epsilon}%
\right\}\quad\mbox{and}\quad\widehat{\epsilon}_{\ell}:\,=\sup\left\{\epsilon%
\mid\epsilon\leq\frac{1}{4}\sigma^{2}\frac{\sqrt{\hat{k}_{\ell}(\epsilon)}}{%
\epsilon}\right\}.$$
Proof.
These bounds follow directly by replacing $k_{u}$ by $\hat{k}_{u}$
(respectively $k_{\ell}$ by $\hat{k}_{\ell}$) in the definition of $\widehat{\epsilon}_{u}$
(respectively $\widehat{\epsilon}_{\ell}$). The non-increasing nature of the functions
$(\hat{k}_{\ell},\hat{k}_{u})$ ensure that the quantities $(\widehat{\epsilon}_{\ell},\widehat{\epsilon}_{u})$ are
well-defined.
∎
4.3.2 Proof of upper bound
We prove the upper bound via an application of Lemma 7
combined with Theorem 1. In particular, we construct
functions333Strictly speaking, these functions also depend on
$\theta^{*}$ and the ellipse $\mathcal{E}$, but we omit this dependence
so as to simplify notation. $(f_{k}^{\ell},f_{k}^{u})$ that sandwich
local Kolmogorov width above and below by
$$\displaystyle f_{k}^{\ell}(\epsilon)~{}\leq~{}\omega_{k}(\mathcal{E}_{\theta^{%
*}}\cap\mathbb{B}(a\epsilon))\leq\omega_{k}(\mathcal{E}_{\theta^{*}}\cap%
\mathbb{B}(\epsilon))~{}\leq f_{k}^{u}(\epsilon)\qquad\mbox{for every integer %
$1\leq k\leq d$.}$$
(49)
Here $a\in(0,1)$ is the constant involved in the statement of
Theorem 2. It is then straightforward to verify
that the functions
$$\displaystyle\hat{k}_{u}(\epsilon):\,=\arg\min_{1\leq k\leq d}\big{\{}f_{k}^{u%
}(\epsilon)\leq\frac{\epsilon}{\sqrt{2}}\big{\}}~{}~{}\text{ and
}~{}~{}\hat{k}_{\ell}(\epsilon):\,=\arg\min_{1\leq k\leq d}\big{\{}f_{k}^{\ell%
}(\epsilon)\leq 3b\epsilon\big{\}}.$$
(50)
satisfy the conditions of Lemma 7, so that it can be
applied.
We begin by upper bounding the localized Kolmogorov
width (8) of $\mathcal{E}\cap\mathbb{B}(\epsilon)$. Let
$\Pi_{k}$ denote the orthogonal projection onto the first $k$
coordinates of the ellipse. We then have
$$\displaystyle\omega^{2}_{k}(\mathcal{E}_{\theta}^{*}\cap\mathbb{B}(\epsilon))%
\leq\max_{\theta\in\mathcal{E},\|\theta-\theta^{*}\|_{2}\leq\epsilon}\|(\theta%
-\theta^{*})-\Pi_{k}(\theta-\theta^{*})\|^{2}_{2}=\max_{\theta\in\mathcal{E},%
\|\theta\|_{2}\leq\epsilon}\sum_{i=k+1}^{d}\theta_{i}^{2}.$$
Since the scalars $\{\mu_{i}\}_{i=1}^{d}$ are ranked in
non-increasing order, for any vector $\theta\in\mathcal{E}$, we have
$$\displaystyle\sum_{i=k+1}^{d}\theta_{i}^{2}=\sum_{i=k+1}^{d}\mu_{i}\frac{%
\theta_{i}^{2}}{\mu_{i}}$$
$$\displaystyle\;\leq\;\mu_{k+1}\sum_{i=k+1}^{d}\frac{\theta_{i}^{2}}{\mu_{i}}\;%
\leq\;\mu_{k+1},$$
where the final inequality follows from the inclusion $\theta\in\mathcal{E}$. Consequently, we have proved that
$$\displaystyle\omega_{k}(\mathcal{E}_{\theta^{*}}\cap\mathbb{B}(\epsilon))$$
$$\displaystyle\leq f_{k}^{u}(\epsilon)\;:\,=\;\underbrace{\min\{\epsilon,\sqrt{%
\mu_{k+1}}}_{=:f_{k}^{u}(\epsilon)}\}.$$
It is easy to check that the function $\epsilon\mapsto\hat{k}_{u}(\epsilon)$ from equation (50) is non-increasing, so
that an application of Lemma 7 yields the claimed upper
bound (20).
4.3.3 Proof of lower bound
We prove the lower bound by a combination of Lemma 7 and
Theorem 2, and in order to do so, we need to lower
bound the localized Kolmogorov width. As previously noted, the
Bernstein width (35) is always a lower bound on the
Kolmogorov width, so that
$$\displaystyle\omega_{k}(\mathcal{E}\cap\mathbb{B}(a\epsilon))\geq b_{k,2}(%
\mathcal{E}\cap\mathbb{B}(a\epsilon))=\underbrace{\min\{a\epsilon,\sqrt{\mu_{k%
+1}}\}}_{=:f_{k}^{\ell}(\epsilon)}$$
where the last inequality follows from the fact that we can always
inscribe a Euclidean ball of radius $\sqrt{\mu_{k+1}}$ centered at
zero inside the ellipse $\mathcal{E}$ truncated to its first
$(k+1)$-coordinates. It is straightforward to verify that the
corresponding $\hat{k}_{\ell}$ that is defined in (50) is
non-increasing in $\epsilon$, so that the the claimed lower
bound (20) follows by applying Lemma 7 with
Theorem 2.
4.4 Proof of Corollary 2
We again separate our proof into two parts, corresponding to the upper
bound and lower bounds. The upper bound is proved via an application
of Lemma 7 and Theorem 1. On the other
hand, we prove the lower bound by using Theorem 2
and Lemma 1 in conjunction with Lemma 7.
4.4.1 Proof of upper bound
Let us introduce the convenient shorthand $\theta^{*}_{-s}:\,=(\theta^{*}_{1},\ldots,\theta^{*}_{s-1},\theta^{*}_{s+1},%
\ldots,\theta^{*}_{d})$. In terms of the
the local minimax testing radius $\epsilon_{\text{{\tiny OPT}}}(\theta^{*};\mathcal{E})$
previously defined in equation (5),
Theorem 1 guarantees that $\epsilon_{\text{{\tiny OPT}}}(\theta^{*})\leq\epsilon_{u}(\theta^{*};\mathcal{%
E})$. Consequently, in order to establish
the upper bound stated in Corollary 2, it suffices to
show that for any vector $\theta^{*}$ with $\theta^{*}_{s}\geq\sqrt{\mu_{s}}-t^{*}_{u}(s,\mathcal{E})$ and $\theta^{*}_{-s}=0$, we have
$\epsilon_{u}(\theta^{*};\mathcal{E})\leq t^{*}_{u}(s,\mathcal{E})$.
In order to do so, it suffices to show that
$$\displaystyle\omega_{m_{u}(\delta,s)}(\mathcal{E}_{\theta^{*}}\cap\mathbb{B}(%
\delta))\leq\frac{\delta}{\sqrt{2}}\qquad\mbox{for each $\delta>0$.}$$
(51)
It then follows that $m_{u}(\delta;s)\geq k_{u}(\delta)$, from
which an application of Lemma 7 implies that
$\epsilon_{u}(s,\mathcal{E})\leq t^{*}_{u}(s,\mathcal{E})$, as
desired.
Accordingly, the remainder of our argument is devoted to establishing
inequality (51). Let $\Pi_{m}$ denote the
orthogonal projection onto the span of the first $m$ standard
basis vectors $\{e_{i}\}_{i=1}^{m}$. By
definition (8), the localized Kolmogorov width is
upper bounded by
$$\displaystyle\omega_{m}(\mathcal{E}_{\theta^{*}}\cap\mathbb{B}(\delta))\leq%
\underbrace{\max_{\theta\in C}\sqrt{\sum_{i=m+1}^{d}\theta_{i}^{2}}}_{=\,:T_{m%
}^{u}}\qquad\mbox{where}\quad C:\,=\big{\{}\theta\in{}^{d}\mid\sum_{i=1}^{d}%
\frac{\theta_{i}^{2}}{\mu_{i}}\leq 1,~{}~{}(\theta_{s}-\theta^{*}_{s})^{2}+%
\sum_{i\neq s}\theta_{i}^{2}\leq\delta^{2}\big{\}},$$
where we have used the fact that $\theta^{*}_{-s}=0.$
Let us introduce the shorthand $m\backslash s:\,=\{1,2,\ldots,s-1,s+1,\ldots,m\}$. Observe that setting $\theta_{j}\neq 0$ for any $j\in m\backslash s$ only makes the constraints defining
$C$ more difficult to satisfy, and does not improve the objective
function defining $T_{m}^{u}$. Therefore, we can reduce the problem to
$$\displaystyle\max_{\theta_{s},\{\theta_{i}\}_{i=m+1}^{d}}\sum_{i=m+1}^{d}%
\theta_{i}^{2},\quad\mbox{such that}\quad\sum_{i=m+1}^{d}\frac{\theta^{2}_{i}}%
{\mu_{i}}\leq 1\text{ and
}(\theta_{s}-\theta^{*}_{s})^{2}+\sum_{i=m+1}^{d}\theta^{2}_{i}\leq\delta^{2}.$$
Since the scalars $\{\mu_{i}\}_{i=m+1}^{d}$ are ranked in
non-increasing order, any optimal solution must satisfy $\theta_{i}=0$
for all $i\in\{m+2,\ldots,d\}$. Putting pieces together,
the optimal solution is defined by a pair $(\theta_{s},\theta_{m+1})$ that satisfy the relations
$$\displaystyle(\theta_{s}-\theta^{*}_{s})^{2}+\theta_{m+1}^{2}$$
$$\displaystyle=\delta^{2}~{}~{}\text{ and }~{}~{}\frac{\theta^{2}_{s}}{\mu_{s}}%
+\frac{\theta^{2}_{m+1}}{\mu_{m+1}}=1.$$
Solving these equations yields
$$\displaystyle\theta_{s}=\frac{\theta^{*}_{s}}{1-t}-\sqrt{\frac{\delta^{2}-\mu_%
{m+1}}{1-t}+\frac{t(\theta^{*}_{s})^{2}}{(1-t)^{2}}},\quad\mbox{and}\quad%
\theta_{m+1}^{2}=\delta^{2}-\left(\sqrt{\frac{\delta^{2}-\mu_{m+1}}{1-t}+\frac%
{t(\theta^{*}_{s})^{2}}{(1-t)^{2}}}-\frac{t\theta^{*}_{s}}{1-t}\right)^{2},$$
(52)
where we define $t:\,=\mu_{m+1}/\mu_{s}.$ We introduce the
following lemma.
Lemma 8.
For any $\delta\geq t^{*}_{u}(s,\mathcal{E})$ and $\sqrt{\mu_{s}}-t^{*}_{u}(s,\mathcal{E})\leq\theta^{*}_{s}\leq\sqrt{\mu_{s}}$, we have
$\theta_{m+1}^{2}\leq\frac{\delta^{2}}{2}$.
See Appendix C for the proof of this result.
Therefore, the localized Kolmogorov width is at most
$\omega^{2}_{m}(\mathcal{E}_{\theta^{*}}\cap\mathbb{B}(\delta))\leq(T_{m}^{u})^%
{2}=\theta_{m+1}^{2}\leq\frac{\delta^{2}}{2}$, which
completes the proof of the upper bound stated in
Corollary 2.
4.4.2 Proof of lower bound
In terms of the the local minimax testing radius $\epsilon_{\text{{\tiny OPT}}}(\theta^{*};\mathcal{E})$ previously defined in
equation (5), Lemma 1
guarantees that $\epsilon_{\text{{\tiny OPT}}}(\theta^{*})\geq\epsilon_{B}(\theta^{*};\mathcal{%
E})$. Therefore, in order to establish the lower bound stated in
Corollary 2, it is sufficient for us to show that for
any vector $\theta^{*}$ with $\theta^{*}_{s}\leq\sqrt{\mu_{s}}-t^{*}_{\ell}(s,\mathcal{E})$ and $\theta^{*}_{-s}=0$, we have
$\epsilon_{B}(\theta^{*};\mathcal{E})\geq t^{*}_{\ell}(s,\mathcal{E})$.
Suppose that we can show that
$$\displaystyle mb^{2}_{m-1,\infty}(\mathcal{E}_{\theta^{*}})\geq\delta^{2},$$
(53)
where we have introduced the shorthand $m:\,=m_{\ell}(\delta;s)$. It then follows that then $k_{B}(\delta)\geq m$, from which
an application of Lemma 7 guarantees that
$\epsilon_{B}(\theta^{*};\mathcal{E})\geq t^{*}_{\ell}(s,\mathcal{E})$.
Here we apply Lemma 7 with respect to $k_{B}$ and $\epsilon_{B}$
replacing quantities $k_{\ell}$ and $\epsilon_{\ell}$ in the statement
which holds by the same argument of the original proof.
The remainder of our argument is to devoted to proving
inequality (53). Consider the set
$$\displaystyle\mathcal{H}$$
$$\displaystyle:\,=\left\{\theta\mid\theta_{s}=\theta^{*}_{s},~{}\theta_{m+1},%
\ldots,\theta_{d}=0,~{}\theta_{i}\in[-\frac{\delta}{\sqrt{|\mathcal{M}|}},%
\frac{\delta}{\sqrt{|\mathcal{M}|}}],~{}i\in\mathcal{M}\right\},~{}~{}\mathcal%
{M}:\,=\{1\leq i\leq m,i\neq s\}.$$
We claim that the set $\mathcal{H}$ is contained within the ellipse
$\mathcal{E}$. In order to see this, first note that $\mu_{i}$ are ranked in
a non-increasing order, so we have $\frac{\delta^{2}}{|\mathcal{M}|}\sum_{i\neq s}^{m}\frac{1}{\mu_{i}}\leq\delta^%
{2}/\mu_{m}$. It implies
$$\displaystyle\|\theta\|_{\mathcal{E}}^{2}=\frac{(\theta^{*}_{s})^{2}}{\mu_{s}}%
+\sum_{i\neq s}^{m}\frac{\delta^{2}}{\mu_{i}|\mathcal{M}|}\leq\frac{(\theta^{*%
}_{s})^{2}}{\mu_{s}}+\frac{\delta^{2}}{\mu_{m}}=1-\frac{2w}{\sqrt{\mu_{s}}}+%
\frac{w^{2}}{\mu_{s}}+\frac{\delta^{2}}{\mu_{m}},$$
(54)
where we plugged in $\theta^{*}_{s}=\sqrt{\mu_{s}}-w$ with $w\geq t^{*}_{\ell}(s,\mathcal{E}).$ On one hand, inequality $w\leq\sqrt{\mu_{s}}$
guarantees that $\frac{w^{2}}{\mu_{s}}-\frac{w}{\sqrt{\mu_{s}}}<0.$ On the
other hand, inequalities $\mu_{k}^{2}\geq\delta^{2}\mu_{s}$ and $\delta\leq t^{*}_{\ell}(s,\mathcal{E})$ yield that $\frac{\delta^{2}}{\mu_{k}}\leq\frac{\delta}{\sqrt{\mu_{s}}}\leq\frac{w}{\sqrt{%
\mu_{s}}}$. Combining
these two facts, we have $\|\theta\|_{\mathcal{E}}^{2}\leq 1$ for every $\theta\in\mathcal{H}$, namely set $\mathcal{H}\subset\mathcal{E}$.
It means that we are able to inscribe a $|\mathcal{M}|-1$-dimensional
$\ell_{\infty}$ ball with radius $\delta/\sqrt{|\mathcal{M}|}$ into
the ellipse $\mathcal{E}$, namely $mb^{2}_{m-1,\infty}(\mathcal{E}_{\theta^{*}})\geq\delta^{2}$. Putting pieces
together, we finish the proof of the lower bound of
Corollary 2.
5 Discussion
In this paper, we have studied the local geometry of compound testing
problems in ellipses. Our main contribution was to show that it is
possible to obtain a sharp characterization of the localized minimax
testing radius at a given vector $\theta^{*}$ in terms of an equation
involving the Kolmogorov width of a particular set. This set involves
the geometry of the ellipse, but is localized to the vector
$\theta^{*}$ via intersection with a Euclidean ball. This form of
localization and the resulting fixed point equation is similar to
localization techniques that are used in proving optimal rates for
estimation (e.g., [33]), and our work demonstrates that
similar concepts are also useful in establishing optimal testing
rates. We anticipate that this perspective will be useful in studying
adaptivity and local geometry in other testing problems.
Acknowledgments
This work was partially supported by Office of Naval Research grant
DOD ONR-N00014, and National Science Foundation grant NSF-DMS-1612948
to MJW.
Appendix A Proof of Lemma 1
In order to simplify notation, we use $k$ as a shorthand for $k_{B}$
throughout this proof. Our goal is to prove that if $\epsilon^{2}\leq\frac{\sqrt{k}\sigma^{2}}{4}$, then any test has testing error at least
$1/2$. Let $e_{i}\in{}^{d}$ denote the standard basis vector
with the $i$-th coordinate equal to one. The definition of $k$
guarantees that we can construct a $2^{k}$-sized collection of
perturbation vectors of the form
$$\displaystyle\theta_{b}=\theta^{*}+\frac{\epsilon}{\sqrt{k}}\sum_{i=1}^{k}b_{i%
}e_{i},~{}~{}~{}b\in\{-1,+1\}^{k}$$
(55)
such that each $\theta_{b}$ lies on the boundary of the $\ell_{\infty}$
ball around $\theta^{*}$ with radius $\frac{\epsilon}{\sqrt{k}}$.
This last property ensures that each $\theta_{b}$ belongs to the ellipse
$\mathcal{E}$. At the same time, we have $\|\theta_{b}-\theta^{*}\|^{2}=k(\frac{\epsilon}{\sqrt{k}})^{2}=\epsilon^{2}$,
valid for all Boolean vectors $b\in\{-1,+1\}^{k}$, so that each
perturbation vector $\theta_{b}$ indexes a distribution in the
alternative hypothesis class $\mathcal{H}_{1}$.
By Lemma 3, we can lower bound the testing error
as
$$\displaystyle\inf_{\psi}\operatorname{Err}(\psi;\{\theta^{*}\},\mathcal{E},%
\epsilon)=\inf_{\psi}\operatorname{Err}(\psi;\{0\},\mathcal{E}_{\theta^{*}},%
\epsilon)\geq 1-\frac{1}{2}\sqrt{\mathbb{E}_{\eta,\eta^{\prime}\sim\mathbb{Q}}%
\,\exp(\frac{\langle\eta,\,\eta^{\prime}\rangle}{\sigma^{2}})-1}.$$
Let $\mathbb{Q}$ denote the uniform distribution over the set $\{\theta_{b}-\theta^{*},b\in\{-1,+1\}^{k}\}$. Introducing the shorthand
$N:\,=2^{k}$, we have
$$\displaystyle\mathbb{E}_{\eta,\eta^{\prime}\sim\mathbb{Q}}\,\exp(\frac{\langle%
\eta,\,\eta^{\prime}\rangle}{\sigma^{2}})\;=\;\frac{1}{N^{2}}\sum_{b,b^{\prime%
}}\exp(\frac{(\theta_{b}-\theta^{*})^{T}(\theta_{b^{\prime}}-\theta^{*})}{%
\sigma^{2}})$$
$$\displaystyle=\frac{1}{N}\sum_{b}\exp(\frac{(\theta_{b}-\theta^{*})^{T}(%
\epsilon\text{1}_{d}/\sqrt{k})}{\sigma^{2}})$$
(56)
$$\displaystyle=2^{-k}\sum_{i=0}^{k}\binom{k}{i}\exp(\frac{\epsilon^{2}(k-2i)}{k%
\sigma^{2}})$$
$$\displaystyle=\Big{(}\frac{\exp(\epsilon^{2}/k\sigma^{2})+\exp(-\epsilon^{2}/k%
\sigma^{2})}{2}\Big{)}^{k},$$
where we have used the symmetry of the problem. It can be verified
via elementary calculation that $\frac{\exp(x)+\exp(-x)}{2}\leq 1+x^{2}$ for all $|x|\leq 1/2$. Setting $x=\frac{\epsilon^{2}}{k\sigma^{2}}$, we have $x\leq\frac{1}{4\sqrt{k}}<1/2$, and hence
$$\displaystyle\frac{1}{N^{2}}\sum_{b,b^{\prime}}\exp(\frac{(\theta_{b}-\theta^{%
*})^{T}(\theta_{b^{\prime}}-\theta^{*})}{\sigma^{2}})\leq(1+\frac{\epsilon^{4}%
}{k^{2}\sigma^{4}})^{k}\leq\exp(\frac{\epsilon^{4}}{k\sigma^{4}}),$$
where in the last step we used the standard bound $1+x\leq e^{x}$.
Therefore, the testing error is lower bounded by $1-\frac{1}{2}\sqrt{e^{\epsilon^{4}/k\sigma^{4}}-1}$. Thus, whenever $\epsilon^{2}\leq\sqrt{k}\sigma^{2}/4$, the testing error is lower bounded as
$\inf_{\psi}\operatorname{Err}(\psi;\{\theta^{*}\},\mathcal{E},\delta)\geq 1-%
\frac{1}{2}\sqrt{e^{1/16}-1}\geq\frac{1}{2}$, which completes the
proof of Lemma 1.
Appendix B Proofs for Theorem 2
In this appendix, we collect the proofs of various auxiliary results involved
in the proof of Theorem 2.
B.1 Proof of Lemma 2
Note that the inequality $b_{k,2}(\mathcal{E}_{\theta^{*}}\cap\mathbb{B}(a\epsilon))\leq\omega_{k}(%
\mathcal{E}_{\theta^{*}}\cap\mathbb{B}(a\epsilon))$ follows as an immediate consequence of the relation
between widths. It remains to prove inequalities (i)-(iii).
Proof of inequalities (i) and (iii):
Recalling the definition of the Bernstein width, we claim that
$\omega_{k}((\mathcal{E}\cap\mathbb{B}(a\epsilon)))=b_{k,2}(\mathcal{E}\cap%
\mathbb{B}(a\epsilon))=\min\{a\epsilon,\sqrt{\mu_{k+1}}\}$. In order
to prove this claim, it suffices to show that
$$\displaystyle\omega_{k}(\mathcal{E}\cap\mathbb{B}(a\epsilon))\stackrel{{%
\scriptstyle(i)}}{{\leq}}\min\{a\epsilon,\sqrt{\mu_{k+1}}\},\quad\mbox{and}%
\quad b_{k,2}(\mathcal{E}\cap\mathbb{B}(a\epsilon))\stackrel{{\scriptstyle(ii)%
}}{{\geq}}\min\{a\epsilon,\sqrt{\mu_{k+1}}\}.$$
Inequality (i) follows easily by direct calculation after taking the
$k$-dimensional projection in the definition of Kolmogorov width to be
projecting to the span of $\{e_{1},\ldots,e_{k}\}$. To show the second
part, note that any vector $v\in{}^{d}$ with $\|v\|_{2}\leq\sqrt{\mu_{k+1}}$ and $v_{i}=0$ for all $i=k+2,\ldots,d$
satisfies
$$\displaystyle\|v\|_{\mathcal{E}}^{2}=\sum_{i=1}^{d}\frac{v_{i}^{2}}{\mu_{i}}=%
\sum_{i=1}^{k+1}\frac{v_{i}^{2}}{\mu_{i}}\,\stackrel{{\scriptstyle(a)}}{{\leq}%
}\,\frac{1}{\mu_{k+1}}\sum_{i=1}^{k+1}v_{i}^{2}\stackrel{{\scriptstyle(b)}}{{%
\leq}}1,$$
where inequality (a) follows from the non-increasing ordering of the
sequence $\{\mu_{i}\}_{i=1}^{d}$ and inequality (b) follows from
the structure of vector $v.$ Consequently, the ellipse $\mathcal{E}$
contains a $k+1$-dimensional $\ell_{2}$ ball centered at $0$ of radius
$\sqrt{\mu_{k+1}}$. Therefore we have $b_{k,2}(\mathcal{E}\cap\mathbb{B}(a\epsilon))\geq\min\{a\epsilon,\sqrt{\mu_{k+%
1}}\}$. We
complete the proof of inequality (iii).
Consider every vector $v\in\mathcal{E}$ that lies in the
$k+1$-dimensional $\ell_{2}$ ball that specified above. Since
$\|\theta^{*}\|_{\mathcal{E}}\leq 1/2$, we have $\|2\theta^{*}\|_{\mathcal{E}}\leq 1$ which implies
that $2\theta^{*}\in\mathcal{E}.$ By the convexity of $\mathcal{E}$, we have $\theta^{*}+v/2\in\mathcal{E}$. It means that $b_{k,2}(\mathcal{E}_{\theta^{*}}\cap\mathbb{B}(a\epsilon))\geq\min\{a\epsilon,%
\frac{1}{2}\sqrt{\mu_{k+1}}\}$, which completes the proof of
inequality (i).
Proof of inequality (ii):
Again consider projecting to the span of $\{e_{1},\ldots,e_{k}\}$,
therefore the Kolmogorov width can be controlled as
$$\displaystyle\omega_{k}^{2}(\mathcal{E}_{\theta^{*}}\cap\mathbb{B}(a\epsilon))%
\leq(a\epsilon)^{2}\wedge\max\Big{\{}\sum_{i=k+1}^{d}\Delta_{i}^{2}~{}\mid~{}%
\|\theta^{*}+\Delta\|_{\mathcal{E}}\leq 1,~{}\|\Delta\|_{2}\leq a\epsilon\Big{%
\}}.$$
By the triangle inequality, we obtain $\|\Delta\|_{\mathcal{E}}\leq 1+\|\theta^{*}\|_{\mathcal{E}}\leq 3/2$ where the last inequality uses $\|\theta^{*}\|_{\mathcal{E}}\leq 1/2.$ Letting $\Delta_{k+1}=\sqrt{\mu_{k+1}}$ and $\Delta_{i}=0$ for
$i\neq k+1$, we have $\omega_{k}(\mathcal{E}_{\theta^{*}}\cap\mathbb{B}(a\epsilon))\leq\min\{a%
\epsilon,3/2\sqrt{\mu_{k+1}}\}$ which
thus proves the inequality (ii).
B.2 Proof of 4
We break the proof of Lemma 4 into two parts. In the
first part, we construct the vector $\theta^{\dagger}$ and the collection
$\{u_{i}\}_{i=1}^{k_{\ell}}$, and show that the properties (a)–(d) hold.
In the second part, we show that matrix $H$ satisfies the
eigenvalue property.
B.2.1 Part I
Recall that the ellipse norm is defined as $\|\theta\|_{\mathcal{E}}^{2}=\sum_{i=1}^{d}\frac{\theta_{i}^{2}}{\mu_{i}}$, so that $\theta\in\mathcal{E}$ is equivalent to $\|\theta\|_{\mathcal{E}}\leq 1$. Recall that
$M$ denotes a diagonal matrix with diagonal entries
$(1/\mu_{1},\ldots,1/\mu_{d})$.
Constructing $\theta^{\dagger}$:
Let us first define a vector $\theta^{\dagger}\in\mathcal{E}$ that satisfies the
assumptions in Lemma 4. Define the function $\psi:(0,\infty)\rightarrow(0,\infty)$ via
$$\displaystyle\psi(r)$$
$$\displaystyle:\,=r^{2}\|M(\mathbb{I}_{d}+rM)^{-1}\theta^{*}\|_{2}^{2}=r^{2}%
\sum_{i=1}^{d}\frac{1}{(1+r/\mu_{i})^{2}}\frac{(\theta^{*}_{i})^{2}}{\mu_{i}^{%
2}}=\sum_{i=1}^{d}\frac{r^{2}}{(\mu_{i}+r)^{2}}(\theta^{*}_{i})^{2}.$$
Note that $\psi$ is a continuous and non-decreasing function on
$[0,\infty)$ such that $\psi(0)=0$ and $\lim_{r\rightarrow+\infty}\psi(r)=\|\theta^{*}\|$. Since $a\in(0,1)$, there must
exists some $r>0$ such that $\psi(r)=a\epsilon$. Given this
choice of $r$, we then define $\theta^{\dagger}:\,=(\mathbb{I}_{d}+rM)^{-1}\theta^{*}$. Since $r>0$, our definition ensures
that $\theta^{\dagger}\in\mathcal{E}$: more precisely, we have
$$\displaystyle\|\theta^{\dagger}\|_{\mathcal{E}}^{2}=\sum_{i=1}^{d}\frac{1}{(1+%
r/\mu_{i})^{2}}\frac{(\theta^{*}_{i})^{2}}{\mu_{i}}<\|\theta^{*}\|_{\mathcal{E%
}}^{2}\leq 1.$$
Now we are ready to construct orthogonal unit vectors
$u_{1},\ldots,u_{k_{\ell}}$ such that proper perturbations of $\theta^{\dagger}$
towards linear combinations of those directions still lie in the set
$\mathcal{E}.$ We do this in a sequential way.
Constructing the vector $u_{1}$.
Defining the vector $v_{1}:\,=\theta^{*}-\theta^{\dagger}$, the definition
of $\theta^{\dagger}$ guarantees that $v_{1}$ is parallel to both $\theta^{\dagger}-\theta^{*}$ and $M\theta^{\dagger}$, so that condition 1 of the lemma
holds. Note also that $\|\theta^{\dagger}-v_{1}\|_{\mathcal{E}}^{2}=\|\theta^{\dagger}+v_{1}\|_{%
\mathcal{E}}^{2}-4(\theta^{\dagger})^{T}Mv_{1}\leq 1$ because $\theta^{\dagger}+v_{1}=\theta^{*}\in\mathcal{E}$ and $(\theta^{\dagger})^{T}Mv_{1}=r\|M\theta^{\dagger}\|_{2}^{2}\geq 0$. These properties guarantee the
inclusion $\theta^{\dagger}\pm v_{1}\in\mathcal{E}$, whence by convexity, the
ellipse $\mathcal{E}$ contains the line segment connecting the two points
$\theta^{\dagger}\pm v_{1}$. We let $u_{1}$ be the normalized version of
$v_{1}$—namely, $u_{1}:\,=v_{1}/\|v_{1}\|_{2}$.
Constructing $u_{2}$.
Without loss of generality, we can assume that $k_{\ell}\geq 2$. By
definition of $k_{\ell}$, there exists a vector $\Delta\in\mathcal{E}_{\theta^{*}}\cap\mathbb{B}(a\epsilon)$ satisfying $\|\Delta-\Pi_{v_{1}}(\Delta)\|_{2}\geq 3b\epsilon$. Accordingly, we may define
$$\displaystyle\Delta_{2}$$
$$\displaystyle:\,=\arg\max_{\begin{subarray}{c}\Delta\in\mathcal{E}_{\theta^{*}%
}\cap\mathbb{B}(a\epsilon):\\
\|\Delta-\Pi_{v_{1}}(\Delta)\|_{2}\geq 3b\epsilon\end{subarray}}\|\Delta-\Pi_{%
v_{1}}(\Delta)\|_{\mathcal{E}}.$$
(57)
With this choice, we set $v_{2}:\,=\Delta_{2}-\Pi_{v_{1}}(\Delta_{2})$
and $u_{2}:\,=v_{2}/\|v_{2}\|_{2}$. Note that these choices ensure that
$\|v_{2}\|_{2}\geq 3b\epsilon$ and $v_{2}\perp v_{1}$.
In order to complete the proof, it suffices to show that $\theta^{\dagger}\pm\frac{1}{3}v_{2}\in\mathcal{E}$. Indeed, when this inclusion holds,
then we have $\theta^{\dagger}\pm b\epsilon\begin{bmatrix}u_{2}&\cdots u_{k_{\ell}}\end{%
bmatrix}H_{\cdot,1}=\theta^{\dagger}\pm b\epsilon u_{2}\in\mathcal{E}.$ In order to show that the required
inclusion holds, we begin by noting that since $\mathcal{E}$ is a convex
set containing $\theta^{\dagger}-v_{1}$ and $\theta^{*}+\Delta_{2}$, it
contains the segment connecting these two points. In particular, we
have
$$\displaystyle\theta^{\dagger}+\frac{\|(\theta^{\dagger}-v_{1})-\theta^{\dagger%
}\|_{2}}{\|(\theta^{\dagger}-v_{1})-(\theta^{*}+\Pi_{v_{1}}(\Delta_{2}))\|_{2}%
}v_{2}=\theta^{\dagger}+\frac{\|v_{1}\|_{2}}{\|-2v_{1}-\Pi_{v_{1}}(\Delta_{2})%
\|_{2}}v_{2}\in\mathcal{E},$$
a result that follows from the proportional segments
theorem [13], when we consider a line
passing through $\theta^{\dagger}$ that is parallel to $v_{2}$. See also
Figure 3 for an illustration.
Since $\|\Delta_{2}\|_{2}\leq a\epsilon\leq\|v_{1}\|_{2}$ and $\|\Pi_{v_{1}}(\Delta_{2})\|_{2}\leq\|\Delta_{2}\|_{2}$, we thus have $\|-2v_{1}-\Pi_{v_{1}}(\Delta_{2})\|_{2}\leq 3\|v_{1}\|_{2}$ and it implies that $\theta^{\dagger}+\frac{1}{3}v_{2}\in\mathcal{E}$.
Finally, using the fact that $\theta^{\dagger}+\frac{1}{3}v_{2}\in\mathcal{E}$
and $v_{2}^{T}M\theta^{\dagger}=0$, we have
$$\displaystyle\|\theta^{\dagger}-\frac{1}{3}v_{2}\|_{\mathcal{E}}^{2}=\|\theta^%
{\dagger}+\frac{1}{3}v_{2}\|_{\mathcal{E}}^{2}-\frac{2}{3}v_{2}^{T}M\theta^{%
\dagger}\leq 1,$$
which completes the proof.
Constructing the remaining $u_{i}$.
We sequentially construct pairs $(v_{i},v^{\prime}_{i})$ for $i=3,\ldots,k_{\ell}$ as
follows and $u_{i}$ is a just a scaled version of $v_{i}$.
Given $v_{1},\ldots,v_{i-1}$ for $i\leq k_{\ell}$, by definition of $k_{\ell}$,
there still exists
some $\Delta_{i}\in\mathcal{E}_{\theta^{*}}\cap\mathbb{B}(a\epsilon)$ such
that $\|\Delta_{i}-\Pi_{v_{1},\ldots,v_{i-1}}(\Delta_{i})\|_{2}\geq 3b\epsilon$. We then define
$$\displaystyle v_{i}:\,=\Delta_{i}-\Pi_{v_{1},\ldots,v_{i-1}}(\Delta_{i}),\quad%
\mbox{and}\quad v^{\prime}_{i}:\,=\Delta_{i}-\Pi_{v_{1}}(\Delta_{i}),$$
which ensures that $\|v_{i}\|_{2}\geq 3b\epsilon$ and
both $v_{i}$ and $v_{i}^{\prime}$ are perpendicular to $v_{1}.$
Finally, we set $u_{i}:\,=v_{i}/\|v_{i}\|_{2}$.
We now claim that
$$\displaystyle\theta^{\dagger}\pm\frac{1}{3}v_{i}^{\prime}\in\mathcal{E}.$$
(58)
Taking this claim as given for the moment, let us first
establish property (c) and (d).
Observe that the vectors
vectors $v_{1},\ldots,v_{i}$ are orthogonal, so we can write $v^{\prime}_{i}$ as the
linear combination of $v_{2},\ldots,v_{i}$. For $i\geq 3$, there exist
scalars $r_{i,j}$ such that
$$\displaystyle v^{\prime}_{i}=h_{i,2}v_{2}+\cdots+h_{i,i-1}v_{i-1}+v_{i},$$
and by orthogonality of those $v_{i}$, we have
$$\displaystyle\|v^{\prime}_{i}-v_{i}\|_{2}^{2}=h_{i,1}^{2}\|v_{2}\|_{2}^{2}+%
\cdots+h_{i,i-1}^{2}\|v_{i-1}\|_{2}^{2}.$$
For $i\geq 3$, since $(v^{\prime}_{i}-v_{i})\perp v_{i}$, it is further guaranteed that
$$\displaystyle\|v^{\prime}_{i}-v_{i}\|_{2}^{2}$$
$$\displaystyle=\|v^{\prime}_{i}\|_{2}^{2}-\|v_{i}\|_{2}^{2}\leq(a^{2}-9b^{2})%
\epsilon^{2}$$
where the last inequality follows from fact $v^{\prime}_{i}\in\mathbb{B}(a\epsilon)$ and the definition of fact that $\|v_{i}\|_{2}\geq 3b\epsilon$. Putting the pieces together, we have
$$\displaystyle h_{i,1}^{2}+\cdots+h_{i,i-1}^{2}\leq\frac{(a^{2}-9b^{2})\epsilon%
^{2}}{9b^{2}\epsilon^{2}}=\frac{a^{2}-9b^{2}}{9b^{2}}.$$
This proves property property (c) where $\|H_{i,\cdot}\|_{2}^{2}\leq 1+(a^{2}-9b^{2})/9b^{2}$. Thus by relation (58), for $i\geq 3$
we have shown
$$\displaystyle\theta^{\dagger}\pm b\epsilon x_{i-1}=\theta^{\dagger}\pm b%
\epsilon(h_{i,2}u_{2}+\cdots+h_{i,i-1}u_{i-1}+u_{i})\in\mathcal{E},$$
which completes the proof of property (d).
It is only left to prove relation (58). To see this
fact, first note $\|v_{i}\|_{2}\leq\|v^{\prime}_{i}\|_{2}$ due to Pythagoras
theorem. Also, by the maximality of $v_{2}$ in inequality
(57), note that $\|v^{\prime}_{i}\|_{\mathcal{E}}\leq\|v_{2}\|_{\mathcal{E}}$. Thus,
$$\displaystyle\|\theta^{\dagger}+\frac{1}{3}v^{\prime}_{i}\|_{\mathcal{E}}^{2}=%
\|\theta^{\dagger}\|_{\mathcal{E}}^{2}+\frac{1}{9}\|v^{\prime}_{i}\|_{\mathcal%
{E}}^{2}\leq\|\theta^{\dagger}\|_{\mathcal{E}}^{2}+\frac{1}{9}\|v_{2}\|_{%
\mathcal{E}}^{2}=\|\theta^{\dagger}+\frac{1}{3}v_{2}\|_{\mathcal{E}}^{2}\leq 1,$$
where we have used the fact that $v^{\prime}_{i}$ and $v_{2}$ are perpendicular to
$v_{1}=rM\theta^{\dagger}$ to ignore the cross terms, as well as the
fact that $\theta^{\dagger}+\frac{1}{3}v_{2}\in\mathcal{E}$. Thus $\theta^{\dagger}+\frac{1}{3}v^{\prime}_{i}\in\mathcal{E}$. We can also show $\theta^{\dagger}-\frac{1}{3}v^{\prime}_{i}\in\mathcal{E}$ similarly as before
since $\theta^{\dagger}Mv_{i}^{\prime}=0.$
B.2.2 Part II
We deal with inequalities (i) and (ii) separately.
For any $s\in[k_{\ell}-1]$, we have
$$\displaystyle s\nu_{s}^{2}\leq\sum_{i=1}^{k_{\ell}-1}\nu_{i}^{2}=\mbox{trace}(%
H^{T}H)\leq\sum_{i=1}^{k_{\ell}-1}\|H_{i,\cdot}\|_{2}^{2}\leq(k_{\ell}-1)\frac%
{a^{2}}{9b^{2}},$$
where the last inequality uses property (c).
It implies that $s\nu_{s}^{2}\leq(k_{\ell}-1)\frac{a^{2}}{9b^{2}}$ which proves inequality (i).
For any $t\in[k_{\ell}-2]$,
we claim that for any $t$-dimensional subspace $W\subseteq{}^{m}$, there exists some $\ell$, such that standard basis vector $e_{\ell}$ satisfies that
$\|e_{\ell}-\Pi_{W}(e_{\ell})\|_{2}^{2}\geq 1-\frac{t}{m}$. In order
to prove this claim, we take an orthonormal basis $z_{1},\ldots,z_{t}$ of
$W$ and extend it to an orthonormal basis $z_{1},\ldots,z_{m}$ for
${}^{m}$. We then have
$$\displaystyle\sum_{i=1}^{m}\|e_{i}-\Pi_{W}(e_{i})\|_{2}^{2}=\sum_{i=1}^{m}\sum%
_{j=t+1}^{m}\langle e_{i},z_{j}\rangle^{2}=\sum_{j=t+1}^{m}\sum_{i=1}^{m}%
\langle e_{i},z_{j}\rangle^{2}=m-t,$$
where the last equality is due to the fact that $e_{i},1\leq i\leq m$
forms a standard basis of ${}^{m}$, and $z_{j}$ is a unit vector. So the claim
holds by the pigeonhole principle.
Now let $H^{T}=\sum_{i=1}^{k_{\ell}-1}\sigma_{i}u_{i}v_{i}^{T}$ with
$\sigma_{1}\geq\cdots\geq\sigma_{k_{\ell}-1}$ be the SVD of $H^{T}$.
Take $W$ be the span of the first $t$ singular vectors
$v_{1},\ldots,v_{t}$. By taking dimension $m=k_{\ell}-1$, the above claim implies there
exists some $e_{\ell}$ such that $\|e_{\ell}-\Pi_{W}(e_{\ell})\|_{2}^{2}\geq 1-\frac{t}{k_{\ell}-1}$. Let $v:\,=(h_{\ell,2},\ldots,h_{\ell,\ell-1},0,\ldots,0)$ so that the $\ell-1$
row of $H^{T}$ can be written as $H^{T}_{\cdot,\ell-1}=e_{\ell}^{T}+v$. Note by definition of matrix $H$, we have $\|v\|_{2}^{2}\leq a^{2}-9b^{2}$; thus
$$\displaystyle\|v-\Pi_{W}(v)\|_{2}\leq\|v\|_{2}\leq\sqrt{a^{2}-9b^{2}}/3b.$$
Applying the triangle inequality yields
$$\displaystyle\|H_{\cdot,\ell-1}-\Pi_{W}(H_{\cdot,\ell-1})\|_{2}\geq\|e_{\ell}-%
\Pi_{W}(e_{\ell})\|_{2}-\|v-\Pi_{W}(v)\|_{2}\geq 1-\frac{t}{k_{\ell}-1}-\frac{%
\sqrt{a^{2}-9b^{2}}}{3b}.$$
(59)
Define the matrix $\tilde{H}:\,=\sum_{i=1}^{t}\sigma_{i}u_{i}v_{i}^{T}$, note that the rows of $\tilde{H}$ are the projections of
the rows of $H^{T}$ onto $W$, and therefore, the definition
of the operator norm implies that
$$\displaystyle\|H^{T}-\tilde{H}\|_{op}\geq\|H_{\cdot,\ell-1}-\Pi_{W}(H_{\cdot,%
\ell-1})\|_{2}.$$
It is easy to check that $\|H^{T}-\tilde{H}\|_{op}=\sigma_{t+1}$. Combining with inequality (59), we have
$$\displaystyle\sigma_{t+1}\geq 1-\frac{t}{k_{\ell}-1}-\frac{\sqrt{a^{2}-9b^{2}}%
}{3b},$$
which completes the proof of inequality (ii).
Combining our results from Parts I and II concludes the
proof of Lemma 4.
B.3 Proof of Lemma 5
We break our proof into two parts, corresponding to the two claims in
the lemma.
Proof of inequality (i):
This inequality is proved via a probabilistic argument. Recall that
the ellipse norm is defined as $\|x\|_{\mathcal{E}}^{2}=\sum_{i=1}^{d}\frac{x_{i}^{2}}{\mu_{i}}$, and that we write $x\in\mathcal{E}$ to mean to
$\|x\|_{\mathcal{E}}\leq 1$. We use $M$ to denote the diagonal matrix
with entries $(1/\mu_{1},\ldots,1/\mu_{d})$.
Let us define auxiliary vector
$$\displaystyle\tilde{\theta}^{S}:\,=\theta^{\dagger}+\frac{b\epsilon}{2\sqrt{2s%
}}XVz^{S},$$
(60)
where matrix $X:\,=UH$ is defined in Lemma 4
and matrix $H$ has eigen-decomposition $Q\Sigma V^{T}$.
We claim that the following two facts:
(a)
There exists a sign vector $z^{S}\in\{-1,0,1\}^{k_{\ell}-1}$
such that $\tilde{\theta}^{S}\in\mathcal{E}$.
(b)
Defining $v_{1}:\,=\theta^{*}-\theta^{\dagger}$, we then have
$\theta^{*}+v_{1}\in\mathcal{E}$.
Taking these two facts as given for now, we can prove inequality (i).
Noticing that $\theta^{\dagger}+v_{1}=\theta^{*}\in\mathcal{E}$ and $\theta^{\dagger}+2v_{1}=\theta^{*}+v_{1}\in\mathcal{E}$, by convexity of set $\mathcal{E}$,
if $\theta^{\dagger}+u\in\mathcal{E}$ for some vector $u$ then
$$\displaystyle\theta^{*}+\frac{1}{2}u=\theta^{\dagger}+v_{1}+\frac{1}{2}u=\frac%
{1}{2}(\theta^{\dagger}+u)+\frac{1}{2}(\theta^{\dagger}+2v_{1})\in\mathcal{E}.$$
Letting $u:\,=b\epsilon\frac{1}{\sqrt{s}}XVz^{S}$, then
$\theta^{S}:\,=\theta^{*}+\frac{1}{2}b\epsilon\frac{1}{\sqrt{s}}XVz^{S}=\theta^%
{*}+\frac{1}{2}u$. Then we have $\theta^{S}\in\mathcal{E}$.
It remains to establish claims (a) and (b) stated above.
Proof of claim (a):
Let $z^{S}\in\{-1,+1\}^{|S|}$ be a vector of i.i.d. Rademacher random
variables. Expanding the square yields
$$\displaystyle\|\tilde{\theta}^{S}\|_{\mathcal{E}}^{2}$$
$$\displaystyle=(\theta^{\dagger})^{T}M\theta^{\dagger}+\frac{b\epsilon}{\sqrt{2%
s}}(\theta^{\dagger})^{T}MXVz^{S}+\frac{b^{2}\epsilon^{2}}{8s}(XVz^{S})^{T}%
MXVz^{S}$$
$$\displaystyle=(\theta^{\dagger})^{T}M\theta^{\dagger}+b^{2}\epsilon^{2}\frac{1%
}{8s}(XVz^{S})^{T}MXVz^{S},$$
where the last inequality is due to the fact that $(\theta^{\dagger})^{T}MX=0.$ Let us consider the expectation $\mathbb{E}\|\tilde{\theta}^{S}\|_{\mathcal{E}}^{2}$. Note
that
$$\displaystyle\mathbb{E}(XVz^{S})^{T}MXVz^{S}$$
$$\displaystyle=\mathbb{E}~{}\mbox{trace}[(XVz^{S})^{T}MXVz^{S}]\;=\;\mathbb{E}~%
{}\mbox{trace}[V^{T}X^{T}MXVz^{S}(z^{S})^{T}],$$
where the last inequality uses the property of the trace function. By
linearity, we can switch the order of the expectation and trace
function, and doing so yields
$$\displaystyle\mathbb{E}~{}\mbox{trace}[V^{T}X^{T}MXVz^{S}(z^{S})^{T}]=\mbox{%
trace}[\underbrace{V^{T}X^{T}MXV}_{:\,=A}\mathbb{E}[z^{S}(z^{S})^{T}]]=\sum_{i%
\in S}A_{i}.$$
We claim $\sum_{i\in S}A_{i}\leq 8s\max_{i}\{x_{i}^{T}Mx_{i}\}$.
In order to show this, first note that
$$\displaystyle\mbox{trace}[A]=\mbox{trace}[V^{T}X^{T}MXV]=\mbox{trace}[X^{T}MX]%
=\sum_{i=1}^{k_{\ell}-1}x_{i}^{T}Mx_{i},$$
using the fact that $VV^{T}=\mathbb{I}$.
Since $S$ is the subset of $F:\,=\{m_{1},m_{1}+1,\ldots,m_{2}\}\cap\mathcal{M}(7(k_{\ell}-1)/8)$, so for $t:\,=\frac{7(k_{\ell}-1)}{8}$, we have
$$\displaystyle((k_{\ell}-1)-t)A_{(t)}\leq\sum_{i=t}^{k_{\ell}-1}A_{(i)}\leq%
\mbox{trace}[A]\leq(k_{\ell}-1)\max_{i}\{x_{i}^{T}Mx_{i}\}.$$
It implies that $A_{(t)}\leq 8\max_{i}\{x_{i}^{T}Mx_{i}\}$ which further verifies our claim.
Putting the pieces together, we have
$$\displaystyle\mathbb{E}\|\tilde{\theta}^{S}\|_{\mathcal{E}}^{2}$$
$$\displaystyle=(\theta^{\dagger})^{T}M\theta^{\dagger}+b^{2}\epsilon^{2}\frac{1%
}{8s}\mathbb{E}(XVz^{S})^{T}MXVz^{S}$$
$$\displaystyle\stackrel{{\scriptstyle(i)}}{{\leq}}(\theta^{\dagger})^{T}M\theta%
^{\dagger}+b^{2}\epsilon^{2}\frac{1}{8s}8s\max_{i}\{x_{i}^{T}Mx_{i}\}\stackrel%
{{\scriptstyle(ii)}}{{=}}\|\theta^{\dagger}+b\epsilon x_{j}\|_{\mathcal{E}}^{2%
}\stackrel{{\scriptstyle(iii)}}{{\leq}}1,$$
where step (i) uses claim on the $A_{i}$s, step (ii) uses fact that
$\theta^{\dagger}Mx_{j}=0$ and
step (iii) uses the fact that $\theta^{\dagger}+b\epsilon x_{j}\in\mathcal{E}$. Since the average squared ellipse norm is at most one,
there must exist at least one $\tilde{\theta}^{S}$ with
$\|\tilde{\theta}^{S}\|_{\mathcal{E}}^{2}\leq 1$, as claimed.
Proof of claim (b):
Since $v_{1}=\theta^{*}-\theta^{\dagger}$, the results of
Lemma 4 guarantee that
$$\displaystyle\|v_{1}\|_{\mathcal{E}}^{2}=\sum_{i=1}^{d}\left(\frac{r(\epsilon)%
}{r(\epsilon)+\mu_{i}}\right)^{2}(\theta^{*}_{i})^{2}$$
$$\displaystyle=\sum_{i=1}^{d}\frac{r(\epsilon)^{2}\mu_{i}}{(r(\epsilon)+\mu_{i}%
)^{2}}\frac{(\theta^{*}_{i})^{2}}{\mu_{i}}$$
$$\displaystyle\leq\|\theta^{*}\|_{\mathcal{E}}^{2}\;\max_{i=1,\ldots,d}\left\{%
\frac{r(\epsilon)^{2}\mu_{i}}{(r(\epsilon)+\mu_{i})^{2}}\right\}.$$
Since $(r(\epsilon)+\mu_{i})^{2}\geq 4r(\epsilon)\mu_{i}$, we have
$\|v_{1}\|_{\mathcal{E}}\leq\frac{\sqrt{r(\epsilon)}}{2}\|\theta^{*}\|_{%
\mathcal{E}}$. Triangle
inequality guarantees that
$$\displaystyle\|\theta^{*}+v_{1}\|_{\mathcal{E}}\leq\|\theta^{*}\|_{\mathcal{E}%
}+\|v_{1}\|_{\mathcal{E}}\leq\left(1+\frac{\sqrt{r(\epsilon)}}{2}\right)\|%
\theta^{*}\|_{\mathcal{E}}\leq 1,$$
where the last inequality uses assumption that
$r(\epsilon)=\Phi(\epsilon)<(\|\theta^{*}\|_{\mathcal{E}}^{-1}-1)^{2}$
and $\|\theta^{*}\|_{\mathcal{E}}\leq 1.$
Proof of inequality (ii):
Letting $\tilde{\Delta}^{S}:\,=\tilde{\theta}^{S}-\theta^{\dagger}$ and
by construction $\Delta^{S}=\frac{1}{2}\tilde{\Delta}^{S}.$
By definition, $\|\tilde{\Delta}\|_{2}^{2}=\frac{1}{8s}b^{2}\epsilon^{2}\|U\Sigma Vz^{S}\|_{2}%
^{2}.$ Substituting matrix $H$ as its
decomposition $Q\Sigma V^{T}$ yields
$$\displaystyle\|\tilde{\theta}^{S}-\theta^{\dagger}\|_{2}^{2}$$
$$\displaystyle=\frac{1}{8s}b^{2}\epsilon^{2}\|UQ\Sigma z^{S}\|_{2}^{2}=\frac{1}%
{8s}b^{2}\epsilon^{2}\|\Sigma z^{S}\|_{2}^{2},$$
(61)
where the last inequality uses the orthogonality of matrices $U,Q$.
Recall that set $S$ is a subset of $\{m_{1},\ldots,m_{2}\}$. Using
eigenvalue bound (44)(ii) from part (e) of
Lemma 4, we have $\|\Sigma z^{S}\|_{2}^{2}\geq(1-\frac{m_{2}}{t}-\frac{\sqrt{a^{2}-9b^{2}}}{3b})%
^{2}\|z^{S}\|_{2}^{2}$. Combining with equality (61) yields
inequality (ii).
Finally, putting the two parts together completes the proof
of Lemma 5.
B.4 Proof of Lemma 6
Let us denote the cardinality of $\mathcal{S}$ as $N:\,=\binom{m_{2}-m_{1}+1}{s}.$ Then the expectation can be written as
$$\displaystyle\mathbb{E}\exp\big{(}\frac{\langle\eta,\,\eta^{\prime}\rangle}{%
\sigma^{2}}\big{)}=\frac{1}{N^{2}}\sum_{S,S^{\prime}\in\mathcal{S}}\exp\big{(}%
\frac{1}{\sigma^{2}}\langle\Delta^{S},\,\Delta^{S^{\prime}}\rangle\big{)}.$$
From the definition (45) of $\theta^{S}$, we have
$$\displaystyle\langle\Delta^{S},\,\Delta^{S^{\prime}}\rangle=\frac{1}{32}b^{2}%
\epsilon^{2}\frac{1}{s}(z^{S})^{T}V^{T}H^{T}U^{T}UHVz^{S^{\prime}}=\frac{1}{32%
}b^{2}\epsilon^{2}\frac{1}{s}(z^{S})^{T}\Sigma^{2}z^{S^{\prime}},$$
where we used the relation $H^{T}H=V\Sigma^{2}V^{T}$, and the
orthogonality of the matrices $V$ and $U$. In order to further
control the right hand side, note that
$$\displaystyle(z^{S})^{T}\Sigma^{2}z^{S^{\prime}}=\sum_{i\in S\cap S^{\prime}}%
\nu_{i}^{2}z^{S}_{i}z^{S^{\prime}}_{i}\leq\nu^{2}_{m_{1}}\sum_{i\in S\cap S^{%
\prime}}|z^{S}_{i}z^{S^{\prime}}_{i}|=\nu^{2}_{m_{1}}|S\cap S^{\prime}|.$$
Combining these three inequalities above yields
$$\displaystyle\mathbb{E}\exp\big{(}\frac{\langle\eta,\,\eta^{\prime}\rangle}{%
\sigma^{2}}\big{)}\leq\frac{1}{N^{2}}\sum_{S,S^{\prime}\in\mathcal{S}}\exp\big%
{(}\frac{1}{32\sigma^{2}}b^{2}\epsilon^{2}\nu^{2}_{m_{1}}\frac{|S\cap S^{%
\prime}|}{s}\big{)}.$$
Let us now further compute the right hand side above. Recall that we
denote $t:\,=m_{2}-m_{1}+1$. Note that intersection cardinality
$|S\cap S^{\prime}|$ takes values in $\{0,1,\ldots,s\}$. Given every set
$S$ and integer $i\in\{0,1,\ldots,s\}$, the number of $S^{\prime}$ such
that $|S\cap S^{\prime}|=i$ equals to ${s\choose i}{t-s\choose s-i}$.
Consequently, if we let $\lambda:\,=\frac{1}{32\sigma^{2}}b^{2}\epsilon^{2}\nu^{2}_{m_{1}}$, we obtain
$$\displaystyle\mathbb{E}\exp\big{(}\frac{\langle\eta,\,\eta^{\prime}\rangle}{%
\sigma^{2}}\big{)}={t\choose s}^{-1}\sum_{i=0}^{s}{s\choose i}{t-s\choose s-i}%
e^{\lambda i/s}=\sum_{i=0}^{s}\frac{G_{i}z^{i}}{i!},$$
(62)
where $z:\,=e^{\lambda/s}$ and $G_{i}:\,=\frac{(s!(t-s)!)^{2}}{((s-i)!)^{2}t!(t-2s+i)!}$.
Let us set integer $s:\,=\lfloor\sqrt{t}\rfloor$. We claim
quantity $G_{i}$ satisfies the following bound
$$\displaystyle G_{i}\leq\exp\Big{(}-(1-\frac{1}{\sqrt{t}})^{2}+\frac{2i}{\sqrt{%
t}-1}\Big{)}\qquad\mbox{for all $i\in\{0,1,\ldots,s\}$.}$$
(63)
Taking expression (63) as given for now and plugging
into inequality (62), we have
$$\displaystyle\mathbb{E}\exp(\frac{\langle\eta,\,\eta^{\prime}\rangle}{\sigma^{%
2}})\leq\exp\Big{(}-(1-\frac{1}{\sqrt{t}})^{2}\Big{)}\sum_{i=0}^{s}\frac{(z%
\exp(\frac{2}{\sqrt{t}-1}))^{i}}{i!}$$
$$\displaystyle\,\stackrel{{\scriptstyle(a)}}{{\leq}}\,\exp\Big{(}-(1-\frac{1}{%
\sqrt{t}})^{2}\Big{)}\exp\left(z\exp(\frac{2}{\sqrt{t}-1})\right)$$
$$\displaystyle\,\stackrel{{\scriptstyle(b)}}{{\leq}}\,\exp\left(-\left(1-\frac{%
1}{\sqrt{t}}\right)^{2}+\exp\left(\frac{2+\lambda}{\sqrt{t}-1}\right)\right),$$
where step (a) follows from the standard power series expansion $e^{x}=\sum_{i=0}^{\infty}\frac{x^{i}}{i!}$ and step (b) follows by $z=e^{\lambda/s}$ and $s=\lfloor\sqrt{t}\rfloor>\sqrt{t}-1$. We have thus established
inequality (47).
It only remains to check inequality (63) for $G_{i}$.
Using the fact that $1-x\leq e^{-x}$, we have
$$\displaystyle G_{0}=\frac{((t-s)!)^{2}}{t!(t-2s)!}$$
$$\displaystyle=(1-\frac{s}{t})(1-\frac{s}{t-1})\cdots(1-\frac{s}{t-s+1})\leq%
\exp(-s\sum_{i=1}^{s}\frac{1}{t-s+i}).$$
(64)
Recalling that $s:\,=\lfloor\sqrt{t}\rfloor$, then we can
bound the sum in expression (64) as
$$\displaystyle s\sum_{i=1}^{s}\frac{1}{t-s+i}\geq s\sum_{i=1}^{s}\frac{1}{t}=%
\frac{s^{2}}{t}\geq(1-\frac{1}{\sqrt{t}})^{2},$$
which, when combined with inequality (64), implies that
$G_{0}\leq\exp(-(1-\frac{1}{\sqrt{t}})^{2})$.
Moreover, direct calculations yield that we have
$$\displaystyle\frac{G_{i}}{G_{i-1}}=\frac{(s-i+1)^{2}}{t-2s+i}\;\leq\;\frac{G_{%
1}}{G_{0}}\quad\mbox{for all $i=1,\ldots,s$,}$$
(65)
where the last inequality follows from the fact that
$\frac{(s-i+1)^{2}}{t-2s+i}$ is non-increasing with index $i$.
Therefore, recalling that $s=\lfloor\sqrt{t}\rfloor$, we
have
$$\displaystyle\frac{G_{i}}{G_{i-1}}\leq\frac{G_{1}}{G_{0}}\;\leq\;\frac{t}{t-2%
\sqrt{t}+1}=(1+\frac{1}{\sqrt{t}-1})^{2}\leq\exp(\frac{2}{\sqrt{t}-1}),$$
where the last inequality follows from $1+x\leq e^{x}.$ Putting pieces
together validates bound (63) thus finishing the proof
of inequality (47).
Appendix C Proof of Lemma 8
Letting $\Delta:\,=\sqrt{\frac{\delta^{2}-\mu_{m+1}}{1-t}+\frac{t(\theta^{*}_{s})^{2}}{%
(1-t)^{2}}}-\frac{t\theta^{*}_{s}}{1-t}$, we then have
$\theta_{m+1}^{2}=\delta^{2}-\Delta^{2}$. Direct calculations yield
$$\displaystyle\Delta=\frac{\frac{\delta^{2}-\mu_{m+1}}{1-t}+\frac{t(\theta^{*}_%
{s})^{2}}{(1-t)^{2}}-\frac{t^{2}(\theta^{*}_{s})^{2}}{(1-t)^{2}}}{\sqrt{\frac{%
\delta^{2}-\mu_{m+1}}{1-t}+\frac{t(\theta^{*}_{s})^{2}}{(1-t)^{2}}}+\frac{t%
\theta^{*}_{s}}{1-t}}=\frac{\frac{\delta^{2}-\mu_{m+1}}{1-t}+\frac{t(\theta^{*%
}_{s})^{2}}{1-t}}{\sqrt{\frac{\delta^{2}-\mu_{m+1}}{1-t}+\frac{t(\theta^{*}_{s%
})^{2}}{(1-t)^{2}}}+\frac{t\theta^{*}_{s}}{1-t}}.$$
Re-organizing the terms yields
$$\displaystyle\Delta$$
$$\displaystyle=\frac{\delta^{2}-\mu_{m+1}+t(\theta^{*}_{s})^{2}}{t\theta^{*}_{s%
}+\sqrt{(\delta^{2}-\mu_{m+1})(1-t)+t(\theta^{*}_{s})^{2}}}=\delta\Big{(}\frac%
{1+\frac{t(\theta^{*}_{s})^{2}-\mu_{m+1}}{\delta^{2}}}{\frac{t\theta^{*}_{s}}{%
\delta}+\sqrt{1-t+\frac{t\mu_{m+1}}{\delta^{2}}+\frac{t(\theta^{*}_{s})^{2}-%
\mu_{m+1}}{\delta^{2}}}}\Big{)}.$$
Now let us first analyze the denominator. Since $t$ is defined as
$\frac{\mu_{m+1}}{\mu_{s}}$, we obtain
$\frac{t\mu_{m+1}}{\delta^{2}}=\frac{\mu^{2}_{m+1}}{\delta^{2}\mu_{s}}$; by $\theta^{*}_{s}\leq\sqrt{\mu_{s}}$,
we obtain that $\frac{t\theta^{*}_{s}}{\delta}\leq\frac{t\sqrt{\mu_{s}}}{\delta}\leq\frac{\mu_%
{m+1}}{\delta\sqrt{\mu_{s}}}$. Plugging into the expression of $\Delta$ gives
$$\displaystyle\frac{\Delta}{\delta}\geq\frac{1+\frac{t(\theta^{*}_{s})^{2}-\mu_%
{m+1}}{\delta^{2}}}{\frac{\mu_{m+1}}{\delta\sqrt{\mu_{s}}}+\sqrt{1+\frac{\mu^{%
2}_{m+1}}{\delta^{2}\mu_{s}}+\frac{t(\theta^{*}_{s})^{2}-\mu_{m+1}}{\delta^{2}%
}-t}}.$$
Moreover, since $m$ is chosen as the maximum index that satisfies
$\mu_{k}^{2}\geq\frac{1}{64}\delta^{2}\mu_{s}$, it is guaranteed that
$\mu_{m+1}^{2}\leq\frac{1}{64}\delta^{2}\mu_{s}$ which further implies
that
$$\displaystyle\frac{\Delta}{\delta}\geq\frac{1+\frac{t(\theta^{*}_{s})^{2}-\mu_%
{m+1}}{\delta^{2}}}{\frac{1}{8}+\sqrt{1+\frac{1}{64}+\frac{t(\theta^{*}_{s})^{%
2}-\mu_{m+1}}{\delta^{2}}-t}}.$$
(66)
Now, in order to control $\frac{\Delta}{\delta}$, we only need to control
quantity $\frac{t(\theta^{*}_{s})^{2}-\mu_{m+1}}{\delta^{2}}$. Recall that
for any $\delta>t^{*}_{u}(s,\mathcal{E})$, we only consider those
$\theta^{*}$ satisfying $\theta^{*}_{s}\geq\sqrt{\mu_{s}}-t^{*}_{u}(s,\mathcal{E})$.
Let us write $\theta^{*}_{s}=\sqrt{\mu_{s}}-c\delta$ with $c\in[0,1]$, then
$$\displaystyle\frac{t(\theta^{*}_{s})^{2}-\mu_{m+1}}{\delta^{2}}=t\frac{(\sqrt{%
\mu_{s}}-c\delta)^{2}-\mu_{s}}{\delta^{2}}=\underbrace{-2c\frac{\mu_{m+1}}{%
\delta\sqrt{\mu_{s}}}}_{:\,=x}+c^{2}t.$$
Here $x:\,=-2c\frac{\mu_{m+1}}{\delta\sqrt{\mu_{s}}}\geq-2\frac{\mu_{m+1}}{\delta%
\sqrt{\mu_{s}}}\geq-\frac{1}{4}$ since
$c\leq 1$ and $\mu_{m+1}^{2}\leq\frac{1}{64}\delta^{2}\mu_{s}$. Now
we are ready to substitute this expression into
inequality (66). Therefore it is guaranteed that
$$\displaystyle\frac{\Delta}{\delta}\geq\frac{1+x+c^{2}t}{\frac{1}{8}+\sqrt{%
\frac{65}{64}+x+c^{2}t-t}}\geq\frac{1+x}{\frac{1}{8}+\sqrt{\frac{65}{64}+x}}%
\geq\min_{x\geq-1/4}\Big{\{}\frac{1+x}{\frac{1}{8}+\sqrt{\frac{65}{64}+x}}\Big%
{\}}=\frac{3}{4}.$$
Consequently, we have
$$\displaystyle\theta^{2}_{m+1}=\delta^{2}-\Delta^{2}\leq\delta^{2}-(\frac{3%
\delta}{4})^{2}\leq\frac{1}{2}\delta^{2}.$$
Appendix D Auxiliary results on the critical radii
In this appendix, we collect a number of auxiliary results concerning
the upper and low critical inequalities, as defined in
equations (10) and (15), respectively.
D.1 Existence and uniqueness
First, let us show that there exists a unique and strictly positive
solution to inequalities (10) and (15).
Let us prove this claim for the solution of
inequality (10), since the argument for
inequality (15) is entirely analogous. It suffices to
show that the function $g(t):\,=k_{u}(t,\theta^{*},\mathcal{E})$ is
non-increasing on the positive real line. Since the function $t\mapsto 1/t$ is strictly decreasing on the positive real line, we are
then guaranteed that the function $t\mapsto\sqrt{g(t)}/t=\sqrt{k_{u}(t,\theta^{*},\mathcal{E})}/t$ is strictly decreasing on the
positive real line. This property ensures that
inequality (10) has a unique and strictly positive
solution: the left-side function $t$ is strictly increasing, whereas
the right-side function is strictly decreasing.
It remains to show that $g(t)=k_{u}(t,\theta^{*},\mathcal{E})$ is
non-increasing. Equivalently, we need to show that for any $t>0$,
we have
$$\displaystyle g(ct)\;=\;k_{u}(ct,\theta^{*},\mathcal{E})\geq k_{u}(t,\theta^{*%
},\mathcal{E})\;=\;g(t)\qquad\mbox{for all $c\in(0,1]$.}$$
By linearity of the Kolmogorov width, we have
$$\displaystyle k_{u}(t,\theta^{*},\mathcal{E}):\,=\arg\min_{1\leq k\leq d}\left%
\{\omega_{k}(\mathcal{E}_{\theta^{*}}\cap\mathbb{B}(t))\leq\frac{1}{2}t\right%
\}=\arg\min_{1\leq k\leq d}\left\{\omega_{k}(c(\mathcal{E}_{\theta^{*}}\cap%
\mathbb{B}(t)))\leq\frac{1}{2}c\,t\right\}.$$
(67)
Given any vector $v\in\mathcal{E}_{\theta^{*}}\cap\mathbb{B}(t)$, convexity
ensures that $cv\in\mathcal{E}_{\theta^{*}}\cap\mathbb{B}(ct)$, which is
equivalent to the containment $c\,\big{(}\mathcal{E}_{\theta^{*}}\cap\mathbb{B}(t)\big{)}\subset\mathcal{E}_{%
\theta^{*}}\cap\mathbb{B}(ct)$. This
containment implies that
$$\displaystyle\omega_{k}(c(\mathcal{E}_{\theta^{*}}\cap\mathbb{B}(t)))\leq%
\omega_{k}(\mathcal{E}_{\theta^{*}}\cap\mathbb{B}(ct)).$$
Combined with our earlier inequality (67), we
conclude that $k_{u}(t,\theta^{*},\mathcal{E})\leq k_{u}(ct,\theta^{*},\mathcal{E})$, as desired.
D.2 Well-definedness of the function $\Phi$
In this appendix, we verify that the function $\Phi$ from
equation (16) is well-defined. In order to provide
intuition, Figure 4 provides an illustration of $\Phi$.
We begin with the case when $\delta<\|\theta^{*}\|_{2}/a$. For notation
simplicity, let $r(\delta):\,=\min\Big{\{}r\geq 0\,\mid a^{2}\delta^{2}\leq\sum_{i=1}^{d}\frac{%
r^{2}}{(r+\mu_{i})^{2}}(\theta^{*}_{i})^{2}\Big{\}}.$ Since each $\mu_{i}\geq 0$, function $f:r\to\sum_{i=1}^{d}\frac{r^{2}}{(r+\mu_{i})^{2}}(\theta^{*}_{i})^{2}$ is
non-decreasing in $r$. It is easy to check that
$$\displaystyle\lim_{r\to 0^{+}}f(r)=0,\qquad\text{ and }\lim_{r\to\infty}f(r)=%
\|\theta^{*}\|_{2}^{2}.$$
Then quantity $r(\delta)$ is uniquely defined and positive whenever
$\delta<\|\theta^{*}\|_{2}/a$. Note that as $\delta\to\frac{\|\theta^{*}\|_{2}}{a}$, $a^{2}\delta^{2}\to\|\theta^{*}\|_{2}^{2}$ therefore
$r(\delta)\to\infty.$
It is worth noticing that given any $\theta^{*}$ where $\|\theta^{*}\|_{2}$ does
not depend on $\delta$, $r$ goes to zero when $\delta\to 0$, namely
$\lim_{\delta\to 0^{+}}\Phi(\delta)=0.$
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Possible Coexistence of Antiferromagnetic and Ferromagnetic Spin Fluctuations in the Spin-triplet Superconductor UTe${}_{2}$ Revealed by ${}^{125}$Te NMR under Pressure
Devi V. Ambika
Ames Laboratory, U.S. DOE, Ames, Iowa 50011, USA
Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA
Qing-Ping Ding
Ames Laboratory, U.S. DOE, Ames, Iowa 50011, USA
Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA
Khusboo Rana
Ames Laboratory, U.S. DOE, Ames, Iowa 50011, USA
Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA
Corey E. Frank
NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA
Center for Nanophysics and Advanced Materials, Department of Physics, University of Maryland, College Park, MD 20742, USA
Elizabeth L. Green
National High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32310, USA
Sheng Ran111Present address: Department of Physics, Washington University, St. Louis, MO
NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA
Center for Nanophysics and Advanced Materials, Department of Physics, University of Maryland, College Park, MD 20742, USA
Nicholas P. Butch
NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA
Center for Nanophysics and Advanced Materials, Department of Physics, University of Maryland, College Park, MD 20742, USA
Yuji Furukawa
Ames Laboratory, U.S. DOE, Ames, Iowa 50011, USA
Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA
Abstract
A spin-triplet superconducting state mediated by ferromagnetic (FM) spin fluctuations has been suggested to occur in the newly discovered heavy-fermion superconductor UTe${}_{2}$. However, the recent neutron scattering measurements revealed the presence of antiferromagnetic (AFM) spin fluctuations in UTe${}_{2}$. Here, we report the ${}^{125}$Te nuclear magnetic resonance (NMR) studies of a single-crystal UTe${}_{2}$, suggesting the coexistence of FM and AFM spin fluctuations in UTe${}_{2}$. Owing to the two different Te sites in the compound, we conclude that the FM spin fluctuations are dominant within ladders and the AFM spin fluctuations originate from the inter-ladder magnetic coupling. Although AFM spin fluctuations exist in the system, the FM spin fluctuations in the ladders may play an important role in the appearance of the spin-triplet superconducting state of UTe${}_{2}$.
The interplay between magnetic fluctuations and superconductivity (SC) is one of the central issues in unconventional superconductors such as iron-based superconductors, high $T_{\rm c}$ cuprates, and heavy fermions.
UTe${}_{2}$ is a newly discovered heavy-fermion superconductor with a superconducting transition temperature $T_{\rm c}$ $\sim$ 1.6–2.0 K Ran2019 ; Aoki2019 ; Rosa2021 and was proposed to be located at the end member of U-based ferromagnetic (FM) superconductors.
Distinct from the previously discovered FM superconductors such as UGe${}_{2}$, UCoGe, and URhGe which exhibit long-range FM order Aoki20192 , UTe${}_{2}$ does not show any magnetic order down to 0.25 K Ran2019 ; Aoki2019 ; Ran2019_Nat ; Sundar2019 , making it unique in the family of U-based FM superconductors.
The observation of the nature of unconventional spin-triplet SC in UTe${}_{2}$, such as very anisotropic SC upper critical field $H_{\rm c2}$ exceeding the Pauli limit Ran2019 ; Aoki2019 ; Knebel2019 , multiple SC phases Braithwaite2019 ; Hayes2021 ; Thomas2020 , and time-reversal symmetry breaking Hayes2021 ; Wei2022 , has also sparked a large volume of research activity on the compound Aoki20192 .
Initially, FM spin fluctuations have been understandably considered to play an important role for the triplet paring, as suggested by muon spin relaxation Sundar2019 and nuclear magnetic resonance (NMR) measurements Tokunaga2019 .
On the other hand, recently, antiferromagnetic (AFM) spin fluctuations with the incommensurate wave-vector of $q$ = (0, 0.57, 0) have been detected by neutron scattering (NS) measurements Knafo2021 ; Duan2020 .
Those results indicate that the nature of the magnetic fluctuations in UTe${}_{2}$ are complicated and still under debate.
The nature of the magnetically ordered state as well as the SC phases under pressure is also still an open question.
With the application of pressure, two SC phases appear around 0.25 GPa Thomas2020 .
While the SC phase with lower $T_{\rm c}$ (SC1) is suppressed continuously with pressure, the SC phase with higher $T_{\rm c}$ (SC2) is enhanced and takes a maximum $T_{\rm c}$ $\sim$ 3 K at around 1.2 GPa, and is suppressed rapidly at higher pressures.
Above the critical pressure $p_{\rm c}$ $\sim$ 1.5 GPa, a magnetic phase appears Ran_PRB2020 ; Braithwaite2019 ; Thomas2020 ; Li2021 .
Initially, a FM ordered state was suggested for the pressure-induced magnetically ordered state Ran_PRB2020 ; Braithwaite2019 , however, recent studies proposed an AFM state Thomas2020 ; Li2021 .
NMR is a powerful technique to investigate low-energy spin fluctuations and superconducting properties from a microscopic point of view.
The temperature ($T$) dependence of the nuclear spin-lattice relaxation rate (1/$T_{1}$) reflects the wave vector $q$-summed dynamical susceptibility at nuclear sites.
On the other hand, NMR spectrum measurements, in particular, the Knight shift $K$, give us information on local static magnetic properties.
Furthermore, in the SC state, the temperature dependences of $K$ and 1/$T_{1}$ provide important information about the SC gap structure.
In fact, recent NMR measurements on single crystals revealed strong and slow spin fluctuations in the normal state Tokunaga2019 ; Tokunaga2022 and also provided key experimental results supporting the spin-triplet superconducting state in UTe${}_{2}$ Nagamine2019 ; Nagamine2021 ; Fujibayashi2022 .
In this paper, we have carried out ${}^{125}$Te NMR measurements using a ${}^{125}$Te-enriched single crystal of UTe${}_{2}$ to investigate the evolution of magnetic fluctuations in UTe${}_{2}$ under pressure.
The most striking result obtained here is the observation of the possible coexistence of AFM and FM spin fluctuations in UTe${}_{2}$.
Owing to the two different Te sites in the compound, the FM spin fluctuations are considered to be dominated within the ladders while the AFM spin fluctuations originate from the inter-ladder magnetic couplings.
The observed results are consistent with the recent NS data where intra-ladder and inter-ladder magnetic couplings are ferromagnetic and antiferromagnetic, respectively Knafo2021 ; Duan2020 .
The ${}^{125}$Te-enriched single crystal (3 $\times$ 1 $\times$ 0.2 mm${}^{3}$) of UTe${}_{2}$ with $T_{\rm c}$ = 1.6 K at zero magnetic field ($H$) at ambient pressure was synthesized by chemical vapor transport method using iodine as the transport agent Ran2019 .
NMR measurements of ${}^{125}$Te ($I$ = $\frac{1}{2}$, $\frac{\gamma_{\rm N}}{2\pi}$ = 13.454 MHz/T) nuclei were conducted using a laboratory-built phase-coherent spin-echo pulse spectrometer up to a pressure of 1.57 GPa with a NiCrAl/CuBe piston-cylinder cell using Daphne 7373 as the pressure transmitting medium.
Pressure calibration was accomplished by ${}^{63}$Cu nuclear quadrupole resonance in Cu${}_{2}$O Fukazawa2007 ; Reyes1992 at 77 K.
$T_{\rm c}$ = 2.3 K for 0.52 GPa and 2.8 K for 1.0 GPa at $H$ = 0 were determined by in-situ AC susceptibility measurements using an NMR tank circuit.
The values of $T_{\rm c}$ are consistent with previous papers Ran_PRB2020 ; Thomas2020 .
The ${}^{125}$Te-NMR spectra were obtained by sweeping $H$ at fixed NMR frequencies ($f$) where $H$ was applied parallel to the $b$ axis.
The 1/$T_{\rm 1}$ was measured with a saturation recovery method T1 .
UTe${}_{2}$ crystallizes in a body-centered orthorhombic structure with the $Immm$ space group Hutanu2020 where the U atoms form a two-leg ladder structure with legs along $a$ axis and rung along the $c$ axis as shown in Fig. 1(a) VESTA .
There are two crystallographic inequivalent Te sites occupying $4j$ and $4h$ sites with point symmetries $mm2$ and $m2m$, respectively.
Following the previous paper Tokunaga2019 , these sites are denoted by Te1 and Te2.
Te1 is located inside the distorted tetrahedron formed by the first and second nearest-neighbors four U atoms which are belonging to three ladders.
On the other hand, Te2 is surrounded by the four nearest neighbor U atoms forming a square-like structure within a ladder.
Thus NMR measurements for Te2 mainly pick up the local magnetic properties of each ladder while Te1 NMR provides the local information of magnetic properties related to inter-ladder coupling.
Those differences in the environments for the Te sites are important as discussed below.
Figure 1(b) shows the $H$-swept ${}^{125}$Te-NMR spectra at 30 K under various pressures ($p$ = 0–1.57 GPa) with $H$ parallel to the $b$ axis, where two lines corresponding to the two Te sites are observed.
The units of the horizontal axis are Knight shift $K$ defined by $K$ = (2$\pi$$f$/$\gamma_{\rm N}$ – $H$)/$H$.
The two lines with lower and higher $K$ values have been assigned to Te1 and Te2, respectively Tokunaga2019 .
The temperature and pressure dependences of full width at half maximum (FWHM) of lines for Te1 and Te2 are shown in Fig. 1(c).
At high $T$ above 200 K, the FWHMs are relatively small $\sim$ 0.8 mT for both Te1 and Te2 at ambient pressure, showing a good sample quality.
With decreasing $T$, FWHM increases and shows a broad maximum around 35 K, similar to the $T$ dependence of magnetic susceptibility $\chi$ and also $K$ as shown in Fig. 2(a).
This indicates that the $T$ dependence of FWHM reflects the $T$ dependence of $\chi$.
Similar $T$ dependence of FWHM can be observed with the lowered peak temperatures down to $\sim$20 K for $p$ = 0.52 and 1.0 GPa and $\sim$10 K at 1.57 GPa.
The large enhancements of FWHM at higher $p$ at low $T$ suggest the increase of $\chi$, as actually observed in the $K$ data.
The $T$ dependences of ${}^{125}$Te Knight shift for Te1 [$K$(Te1)] and Te2 [$K$(Te2)] sites determined by the peak positions of the NMR lines are shown in Fig. 2(a) for the measured pressures under $H$ $\parallel$ $b$.
All $K$ values for Te2 are greater than those for Te1 due to the different hyperfine coupling constants ($A_{\rm hf}$).
The $A_{\rm hf}$ values have been reported to be 34.1 and 51.8 kOe/$\mu_{\rm B}$ for Te1 and Te2, respectively, at ambient pressure Tokunaga2019 .
At ambient pressure, $K$s for both Te sites show a similar temperature dependence with a broad maximum around $T_{\rm max}$ $\sim$ 35–40 K, similar to the magnetic susceptibility data Ran2019 ; Aoki2019 ; Li2021 .
With increasing $p$, $T_{\rm max}$ shifts to lower temperatures to $\sim$30 K at $p$ = 0.52 GPa and to $\sim$25 K at $p$ = 1.0 GPa.
These behaviors are consistent with the previously reported $p$-dependence of magnetic susceptibility Li2021 and the recent NMR data Kinjo2022 .
At $p$ = 1.57 GPa, both $K$s keep increasing with decreasing $T$ and level off below $\sim$15 K without showing a clear maximum.
We were able to measure the spectrum down to 5.5 K although the signal intensity becomes weak below 10 K due to the shortening of nuclear spin-spin relaxation time $T_{2}$.
However, we could not observe any signals at 4.2 K.
This could be due to the pressure-induced short-range magnetically ordered state whose onset temperature has been reported to be $\sim$5 K at $p$ = 1.57 GPa Ran_PRB2020 ; Thomas2020 ; Li2021 .
We notice that the $T$ dependences of $K$(Te1) and $K$(Te2) are slightly different below $\sim$30 K, a little bit lower than $T_{\rm max}$ under $p$ $<$ 1.0 GPa.
This can be seen in Fig. 2(b) where the ratios of Knight shifts for the two Te sites, $K$(Te2)/$K$(Te1), are plotted as a function of $T$.
The ratio is nearly 1.48 at higher $T$ above $T_{\rm max}$ at ambient pressure and increases to 1.62 at 1.6 K.
This indicates that the $T$ dependences of $K$ for Te1 and Te2 are scaled above $\sim$30 K, but do not scale below 30 K.
Since the $T$ dependent part of $K$ is proportional to hyperfine coupling constant as $K(T)$ = $A_{\rm hf}$$\chi(T)$/$N_{\rm A}$$\mu_{\rm B}$
where $N_{\rm A}$ is Avogadro’s number and $\mu_{\rm B}$ is Bohr magneton, this indicates that the ratio of the $A_{\rm hf}$ for Te1 and Te2 changes slightly below $\sim$30 K.
To understand how the $A_{\rm hf}$ changes at low temperatures, we plotted $K$ as a function of the magnetic susceptibility.
As can be seen in Fig. 2(c), all the data points for Te2 are on the same straight line above and below 30 K, indicating no change in $A_{\rm hf}$ for this site.
From the slope of the line, $A_{\rm hf}$ is estimated to be 52.0 kOe/$\mu_{\rm B}$ which is very close to 51.8 kOe/$\mu_{\rm B}$ reported previously Tokunaga2019 .
On the other hand, although a clear linear relationship between $K$(Te1) and $\chi(T)$ can be seen above 30 K (shown by the solid red circles), the data points start deviating below the temperature as shown by the open red circles.
From the change in the slopes shown by solid and broken red lines in Fig. 2(c), we found the $A_{\rm hf}$ changes from 34.8 kOe/$\mu_{\rm B}$ above 30 K to 32.2 kOe/$\mu_{\rm B}$ at 1.6 K.
Thus we attribute the small increase in $K$(Te2)/$K$(Te1) below 30 K to the small reduction of $A_{\rm hf}$ for the Te1 site.
Since 30 K is close to the crossover temperature below which Fermi-liquid (FL) behavior 1/$T_{1}T$ = constant is observed as shown in Fig. 3, the results suggest that $A_{\rm hf}$ changes slightly in the FL state in UTe${}_{2}$.
The similar $T$ dependence of the ratios can be seen at 0.52 and 1.0 GPa, but the deviation from the constant values at high temperatures starts at slightly lower temperatures of $\sim$ 20 K for 0.52 GPa and $\sim$15 K for 1.0 GPa, respectively.
This is consistent with $T_{1}$ data where the crossover temperature to a low-temperature FL state decrease with increasing $p$.
It is interesting to point out that the ratios decrease with increasing $p$, indicating that the ratio of $A_{\rm hf}$ changes.
To determine the origin of the changes in the ratios under pressure, one needs $\chi$ data at the same pressure which are not available at present.
Further studies for the $\chi$ measurements are required to elucidate the origin.
At 1.57 GPa, the values of the ratio decrease slightly with $p$, but a clear upturn cannot be observed down to 5.5 K.
This may suggest no crossover to the FL state at 1.57 GPa down to $\sim$5.5 K.
It is also worth mentioning that, since UTe${}_{2}$ does not exhibit SC but has a magnetically ordered ground state under $p$ $>$ 1.5 GPa Thomas2020 , the unconventional SC state only appears in the FL state, as has been pointed out in Ref. Kinjo2022 .
Now we discuss magnetic fluctuations in UTe${}_{2}$ based on the results of $T_{1}$ measurements for both Te sites.
Figures 3(a)-3(d) show the $T$ dependence of 1/$T_{1}T$ at various pressures.
At ambient pressure, 1/$T_{1}T$ for both the Te sites exhibits Curie-Weiss (CW) behavior above $\sim$40 K.
Below $\sim$40 K, due to the hybridization between the localized 5$f$-electron and conduction electron bands, 1/$T_{1}T$ exhibits nearly constant behavior, a characteristics of heavy-fermion states.
The solid lines represents the FL behaviors of 1/$T_{1}T$ = 325 and 250 1/sK for Te1 and Te2, respectively.
These results are consistent with the data reported previously Tokunaga2019 .
The FL-crossover temperature defined as $T_{\rm FL}$ is also usually considered as Kondo temperature.
Under pressure, the crossover temperature decreases down to $T_{\rm FL}$ $\sim$ 20 and $\sim$ 10 K for $p$ = 0.52 and 1.0 GPa, respectively.
At the same time, the values of 1/$T_{1}T$ = constant increase to 260 (390) 1/sK for Te2(Te1) at 0.52 GPa and to 510 (765) 1/sK for Te2(Te1) at 1.0 GPa, suggesting the increase of the density of states at the Fermi energy and/or the enhancement of electron correlations in the FL state.
The CW behavior of 1/$T_{1}T$ above $\sim$40 K is nearly independent of $p$, however, we observed a different $T$ dependence of 1/$T_{1}T$ for the two Te sites at low temperatures.
With decreasing $T$ below $\sim$40 K, 1/$T_{1}T$ for Te1 is enhanced more than 1/$T_{1}T$ for Te2, which is clearly recognized under $p$ = 1.57 GPa.
Even for the case of $p$ = 0.52 and 1.0 GPa, one can see difference in the $T$ dependence of 1/$T_{1}T$ below $\sim$40 K.
The different $T$ dependence between Te1 and Te2 is more clearly seen in the 1/$T_{1}$ vs. $T$ plots in Figs. 3(e)-3(h).
1/$T_{1}$ for Te2 is nearly $T$ independent at high temperatures and shows a FL behavior below $T_{\rm FL}$ $\sim$ 20 K and $\sim$ 10 K for $p$ = 0.52 and 1.0 GPa, respectively.
1/$T_{1}$ for Te1 shows an enhancement where a clear difference in the $T$ dependence can be detected below $\sim$40 K down to $T_{\rm FL}$.
A similar difference in the $T$ dependence of 1/$T_{1}$ between Te1 and Te2 is also observed at $p$ = 1.57 GPa where the ground state is magnetic.
Those results indicate that Te1 and Te2 pick up different magnetic fluctuations.
The CW behavior in 1/$T_{1}T$ has been pointed out to be scaled with the $T$ dependence of the magnetic susceptibility under $H$ parallel to the $a$ axis (the magnetic easy axis), expected for FM spin fluctuations with $q$ = 0 Tokunaga2019 .
The FM spin fluctuations were also suggested from nuclear spin-spin relaxation time $T_{2}$ measurements Tokunaga2019 ; Tokunaga2022 .
Therefore, the further enhancements of 1/$T_{1}T$ at the Te1 site in comparison with those at the Te2 site indicate an additional contribution of magnetic fluctuations with $q$ $\neq$ 0, AFM spin fluctuations, at the Te1 site AFM_Te1 .
These results suggest the existence of AFM and FM spin fluctuations in the paramagnetic state of UTe${}_{2}$.
It is noted that the AFM spin fluctuations develop below around 40 K under pressure, although the fluctuations were not clearly observed at ambient pressure.
As mentioned above, Te2 is surrounded by four U atoms forming a ladder and picks up magnetic fluctuations originating from each ladder.
Thus our results indicate that FM spin fluctuations are dominant within the ladders.
On the other hand, since Te1 is surrounded by four U atoms belonging to three different ladders, it is possible to pick up magnetic fluctuations originating from not only intra-ladder but also inter-ladders magnetic couplings as schematically shown in Fig. 4.
Therefore, the AFM spin fluctuations detected at Te1 are considered to be due to AFM coupling between the ladders in UTe${}_{2}$.
These results are consistent with the NS measurements where the intra-ladder magnetic interactions are ferromagnetic for the leg and the rung directions and the magnetic interactions between the ladders is antiferromagnetic Duan2020 ; Knafo2021 .
It is noted that the NS measurements detect the AFM spin fluctuations at ambient pressure while our NMR data do not show clear AFM spin fluctuation at $p$ = 0.
Although the reason for the difference is not clear at present, it may be possible to explain by taking into consideration the different energy scale between the two experimental techniques because NMR may not detect the magnetic fluctuations if those energy (frequency) are much higher than NMR frequency.
If this were the case, our NMR results might suggest that the fluctuation frequency of the AFM spin fluctuations gets lower with increasing pressure.
Further NS experiments under pressure are required to clarify this.
Finally it is interesting to point out that the energy scale of $\sim$40 K (below which the AFM spin fluctuations develop under $p$) has been detected in other experiments even at ambient $p$ such as the observation of peak in the magnetic excitation spectrum by inelastic NS measurements Butch2022 , the peak $T$ in the magnetic susceptibility, hybridization gap observed in scanning tunneling spectroscopy Jiao2020 and so on.
Therefore, the energy of $\sim$40 K could be considered as one of the characteristic energy scales in determining the physical properties of the heavy-fermion UTe${}_{2}$.
In summary, we performed ${}^{125}$Te NMR measurements on UTe${}_{2}$ under pressure up to 1.57 GPa.
From the different temperature dependence of 1/$T_{1}T$ between the two different Te sites, Te1 and Te2, we suggest the coexistence of AFM and FM spin fluctuations in UTe${}_{2}$.
The FM spin fluctuations are considered to exist inside each ladder while the AFM spin fluctuations are assigned to originate from the inter-ladder magnetic interactions.
We point out that the FM spin fluctuations inside the ladders may play an important role in the appearance of the spin-triplet superconducting state in UTe${}_{2}$, although AFM spin fluctuations start to develop below $\sim$40 K which are clearly observed under pressure.
Further detailed investigations of the relationship between the magnitude of AFM/FM fluctuations and $T_{\rm c}$, the magnetic ordered states, and also about the superconducting properties under pressures are highly called for.
The authors thank A. Sapkota, B. Li, M. Hirata, and R. Yamamoto for helpful discussions. The research was supported by the U.S. Department of Energy (DOE), Office of Basic Energy Sciences, Division of Materials Sciences and Engineering. Ames Laboratory is operated for the U.S. DOE by Iowa State University under Contract No. DE-AC02-07CH11358. The National High Magnetic Field Laboratory is supported by the National Science Foundation through NSF/DMR-1644779 and the State of Florida.
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In addition, if the AFM fluctuations are characterized by the incommensurate wave vector determined by the neutron scattering measurements, the AFM spin fluctuations will not be canceled out at the Te1 site.
(28)
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(29)
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Quark anti-quark expectation value in finite volume
Johan Bijnens
Karim Ghorbani111Speaker
Department of Theoretical Physics, Lund University, Sölvegatan 14A, SE 223 62, Sweden
Abstract
We have computed the quark anti-quark expectation value in finite volume at two loop
in chiral perturbation theory and compare it with a formula obtained in analogy to
the Lüscher formula for pion mass in finite volume. We observe
that due to the small finite size correction at two loop it is not possible to
obtain conclusions on the accuracy of the extended Lüscher formula.
LU TP 06-37
hep-lat/0610083
October 2006
I Introduction
Quantum chromo dynamics (QCD) is a strongly coupled theory at low energy.
Therefore it is difficult to study the physics below the QCD scale of about 1 GeV.
Lattice QCD on the other hand is an attempt to tackle this problem by taking the
Lagrangian and calculating (parts of) the generating functional numerically.
Although these computations with fairly small quark masses are now possible,
at the same time the finite volume effects are also becoming important.
In this work we employ chiral perturbation theory (ChPT) in order to study these
effects analytically. Chiral perturbation theory, in its modern form,
was first proposed by
Weinberg Weinberg and extended in GL1 ; BCE . ChPT is an effective field
theory to describe the strong interactions at low energy. To do the
computation in finite volume as suggested in GL2 one
imposes a periodic boundary condition on fields which result in
the momentum quantization and consequently modification on the quantum
corrections. Since the applicability of ChPT is limited to small momenta, in
finite volume this leads to
$$F_{\pi}L>1\,.$$
(1)
which holds for L larger than about 2 fm.
$F_{\pi}$ is the pion decay constant and L is the linear
size of the boundary. In addition we stay in a limit where the
Compton wavelength of the pion is smaller than the lattice size
which corresponds to
$$m_{\pi}^{2}F_{\pi}^{2}V>>1\,.$$
(2)
This is the so-called p-regime.
This condition ensures that perturbative treatment will work.
Expectation value of the quark condensate per se will be achieved by varying
the generating functional with respect to the scalar external field as follows
$$<\overline{q}T^{a}q>=\frac{1}{Z_{0}}(-i\frac{\delta}{\delta S_{a}})Z_{eff}[s_{%
a}]_{\arrowvert s_{a}=0}.$$
(3)
Where $T^{a}$ stands for the Gell-Mann matrices. We then carry out
the $\langle\overline{q}q\rangle$ at two loop in finite volume within
the framework of chiral perturbation theory. This quantity has already been evaluated at
one loop order in DG . Another way for the computation of the finite volume
effect is the Lüscher approach. In this method one obtains the leading finite size effects
for the pion mass to all order in perturbation theory from the scattering
amplitude Luscher1 ; Luscher2 . It has later been extended to evaluate the volume dependence
of the pion decay constantCH1 .
As the second goal we shall then obtain a formula for the the quark condensate volume
dependence by following the same line of reasoning as in the Lüscher work.
This work was published in Ref. BGH .
II A Lüscher formula for the vacuum condensate
We were inspired by the work of Lüscher, where he showed how the finite size effect on
the masses of spin-less particles are related to the forward elastic scattering amplitude
in infinite volume. This occurs when particles are enclosed in a lattice box
and as considered in this letter with temporal direction of the space-time extended to infinity.
To see explicitly where these effects stem from, one looks at the two-point correlation function
with fields subjected to a periodic boundary condition. The correlator takes on the form
$$<\phi(x)\phi(0>_{L}=\int\frac{dp_{0}}{2\pi}\sum_{\vec{p}}L^{-d}e^{-ip.x}G_{L}^%
{-1}(\vec{p}^{2},m_{0}^{2})$$
(4)
where the inverse propagator reads
$$G_{L}(\vec{p}^{2},m_{0}^{2})=p^{2}-m_{0}-\Sigma_{L}(p^{2})$$
(5)
and $\Sigma_{L}(p^{2})$ denotes the sum of the one-particle irreducible diagrams
to be evaluated in momentum space with discretized momenta and $\vec{p}=\frac{2\pi\vec{n}}{L}$,
with $\vec{n}$ a three dimensional vector of integers.
In practice the desired quantity to compute is the change in $\Sigma(p^{2})$ due to
the finite volume correction and the difficult task in Lüscher’s work was to show that
this consists of all self-energy diagrams with just one propagator in each diagram allowed
to wrap around the whole position space and therefore experiencing the boundary.
Using the Poisson summation formula the modified propagator can be obtained in terms of the
propagator in the infinite volume accompanied by an exponential factor as follows
$$G_{L}(q)=\sum_{\vec{n}}e^{-iq_{i}n_{i}L_{i}}G_{\infty}$$
(6)
where in three dimensions the length squared for a given
$\vec{n}$ reads $k=\vec{n}^{2}=n_{1}^{2}+n_{2}^{2}+n_{2}^{2}$.
For $\vec{n}=0$ it gives back the propagator in infinite volume.
The crucial point comes about in Euclidean space where $q_{0}\to-iq_{0}$
and, as shown below, in our result for $\langle\bar{q}q\rangle$
the exponential factor falls off rapidly
for large $\vec{n}^{2}$. Lüscher in his analysis kept only $\vec{n}^{2}=1$
but summation over all values shall be carried out numerically in our formula
as suggested in CH2 .
Out of three self-energy graphs shown in Ref. Luscher1 , only two are potentially relevant
to the $\langle\bar{q}q\rangle$ as depicted in Fig. 1.
The diagram (a) has however no contribution when we apply
ChPT to the case of parity even operators.
Extending our task to the evaluation of a general operator in the the diagram (b)
we arrive at the following relation
$$\langle O\rangle_{V}-\langle O\rangle_{\infty}=\int\frac{d^{d}q}{(2\pi)^{d}}%
\sum_{\vec{n}\neq 0}e^{-iLq.n}G_{\infty}\langle\phi|O|\phi\rangle\,.$$
(7)
Integration over momentum can be split into parallel and transverse components
with respect to the $\vec{n}$, followed by the continuation
$q_{\parallel}\to q_{\parallel}-is$.
For $s\to\infty$ the integration over this path is negligible.
Unlike the mass case it turns out that the computation of the diagram (b)
for the $\langle\bar{q}q\rangle$ is simpler mainly because the matrix
element $\langle\phi|O|\phi\rangle$ representing the amputated
vertex function has no external momenta in it and in fact it should be
evaluated at zero momentum transfer.
After isolating the poles and picking out the right one we, complete the integration
over $q_{\parallel}$ which finally leads to the following formula
$$\displaystyle\langle O_{V}\rangle-\langle O_{\infty}\rangle=-\langle\phi|O|\phi\rangle$$
(8)
$$\displaystyle\!\!\!\!\times\left(\sum_{\vec{n}\neq 0}\frac{1}{16\pi^{2}}\int_{%
0}^{\infty}\frac{q^{2}dq}{\sqrt{q^{2}+m^{2}}}e^{-\sqrt{\vec{n}^{2}(m^{2}+q^{2}%
)L^{2}}}\right)$$
As the last step we identify the remaining integral as the modified Bessel function
$K_{1}$
$$\langle O\rangle_{V}-\langle O\rangle_{\infty}=-\langle\phi|O|\phi\rangle\sum_%
{k=1,\infty}\frac{x(k)}{16\pi^{2}}\frac{m^{2}}{\sqrt{\zeta(k)}}K_{1}(\zeta(k))$$
(9)
where $\zeta(k)=\sqrt{(}k)mL$ and $x(k)$ is the multiplicity,i.e. $x(k)$
is the number of times the value $k=\vec{n}^{2}$ appears in the sum overt $\vec{n}$.
Deriving this formula also has the advantage that one can estimate the size of the
sigma term by evaluating the volume dependence of the
the quark anti-quark expectation value.
III The vacuum expectation value at two loop
The calculation of the quark condensate at two loop was done in ABP . We go back to
these computation while taking into account the finite size effects.
As mentioned before, this effect comes about as a modification in the
propagator. In Fig. 2 we show all the Feynman diagrams that contribute up to $p^{6}$.
In infinite volume the loop diagrams we deal with, contain the functions
$$\displaystyle A(m^{2})=\frac{i}{(2\pi)^{d}}\int d^{d}q\frac{1}{q^{2}-m^{2}}$$
$$\displaystyle B(m^{2})=\frac{i}{(2\pi)^{d}}\int d^{d}q\frac{1}{(q^{2}-m^{2})^{%
2}}$$
(10)
These integrals are solved in the dimensional regularization scheme
with $\epsilon=(4-d)/2$ .
$$\displaystyle A(m^{2})$$
$$\displaystyle=$$
$$\displaystyle\frac{m^{2}}{16\pi^{2}}\left(\frac{1}{\epsilon}-\gamma_{E}+log(4%
\pi)+1\right)$$
(11)
$$\displaystyle+\bar{A}(m^{2})+\epsilon A^{\epsilon}(m^{2})+o(\epsilon^{2})$$
$$\displaystyle B(m^{2})$$
$$\displaystyle=$$
$$\displaystyle\frac{m^{2}}{16\pi^{2}}\left(\frac{1}{\epsilon}-\gamma_{E}+log(4%
\pi)+1\right)$$
(12)
$$\displaystyle+\bar{B}(m^{2})+\epsilon B^{\epsilon}(m^{2})+o(\epsilon^{2}).$$
At finite volume integration over momenta should be replaced by a summation
over discretized momenta. It is also important to note that the pole structure
does not change its form, since the singularity arises when momenta in the
loop become very large and therefore the pole remains unchanged despite the replacement
of integration with summation. The final result for the integrals
at finite volume becomes
$$\displaystyle\bar{A}(m^{2})$$
$$\displaystyle=$$
$$\displaystyle-\frac{m^{2}}{16\pi^{2}}\log\left(\frac{m^{2}}{\mu^{2}}\right)$$
(13)
$$\displaystyle-\frac{1}{16\pi^{2}}\sum_{k=1,\infty}x(k)\frac{4m^{2}}{\zeta(k)}K%
_{1}(\zeta(k))$$
$$\displaystyle\bar{B}(m^{2})$$
$$\displaystyle=$$
$$\displaystyle-\frac{1}{16\pi^{2}}\log\left(\frac{m^{2}}{\mu^{2}}\right)-\frac{%
1}{16\pi^{2}}$$
(14)
$$\displaystyle+\frac{1}{16\pi^{2}}\sum_{k=1,\infty}x(k)2K_{o}(\zeta(k))$$
with $\zeta(k)=\sqrt{k}mL$.
It is evident from equations (13) and (14) that the simple relation
$\bar{A}(m^{2})=m^{2}(1/16\pi^{2}+\bar{B}(m^{2}))$ is no longer valid in
finite volume computations. After putting all the diagrams together and
applying the dimensional regularization scheme with renormalization
parameter $\mu$ we ensure that the divergent parts cancel and for finite part
we use the expressions in (13) and (14) instead of those in
(11) and (12).
The parts containing $A^{\epsilon}$ and $B^{\epsilon}$ cancel in the final
result for the vacuum expectation value.
We denote the lowest order masses for pions, kaons and etas as $\chi_{\pi}$, $\chi_{K}$ and
$\chi_{\eta}$ respectively. They are given in terms of the strange quark mass, $m_{s}$
and the average of up quark and down quark masses, $\hat{m}$ by
$$\displaystyle\chi_{\pi}=2B_{0}\hat{m}\,,\quad\chi_{K}=B_{0}(m_{s}+\hat{m})$$
$$\displaystyle\chi_{\eta}=2B_{0}(2m_{s}+\hat{m})/3\,.$$
(15)
We also define the following quantities
$$\displaystyle\langle\bar{u}u\rangle$$
$$\displaystyle=$$
$$\displaystyle-B_{0}F_{0}^{2}\left(1+\frac{v_{4}^{(u)}}{F_{0}}+\frac{v_{6}^{(u)%
}}{F_{0}^{2}}\right)\,,$$
$$\displaystyle\langle\bar{s}s\rangle$$
$$\displaystyle=$$
$$\displaystyle-B_{0}F_{0}^{2}\left(1+\frac{v_{4}^{(s)}}{F_{0}}+\frac{v_{6}^{(s)%
}}{F_{0}^{2}}\right)\,.$$
(16)
The analytical expressions for $v_{4}^{(u)}$,$v_{4}^{(s)}$,
$v_{6}^{(u)}$,$v_{6}^{(s)}$
are evaluated in ChPT and can be found in BGH . Numerical results are shown
in Fig. 3 and Fig. 4, where the ratio $R_{q}$ is defined as
$$R_{q}\equiv\frac{\langle\bar{q}q\rangle_{V}-\langle\bar{q}q\rangle_{\infty}}{%
\langle\bar{q}q\rangle_{\infty}}\,.$$
(17)
The $C^{r}_{i}$, the low energy coupling constants (LECs) from diagrams at $p^{6}$ BCE
cancel out in the numerator and we have ignored them in the denominator but the
$L^{r}_{i}$ will be considered as inputs and we use the values from fit 10 ofABT .
As expected the volume size effects are pronounced for small L and small pion mass.
But the full NNLO contribution to the finite volume effect
is much smaller than the total NNLO numerical correction.
In turn this total NNLO correction is smaller than the one
presented in ABP . This is due to the fact that in Ref. ABP
the physical value of pion decay constant $F_{\pi}$ and also physical masses for
mesons are put in the formula.
IV Conclusions
We compared the quark condensate in finite volume obtained in two different ways.
The method referred to as Lüscher’s approach yields the leading term of this quantity
to all orders in perturbation theory while only one propagator experiences the finite
volume. One the other hand, we try to do this computation up to $p^{6}$ rather directly
by letting the modification of all propagators be present at the same time.
The comparison of the direct calculation at one-loop with the extended formula is
trivial but at two-loop level, however, this is not the case.
This allows in principle a check of the accuracy of the extended Lüscher approach.
Unfortunately, because of the very small size of the finite volume effect at $p^{6}$,
no conclusion can be drawn concerning the accuracy level of the extended Lüscher
formula from our work.
Acknowledgements.
One of us, KG, wishes to thank the organizers of the IPM-LHP06 conference for the travel grant and
great hospitality.
This work is supported by the European Union RTN network,
Contract No.
MRTN-CT-2006-035482 (FLAVIAnet) and by
the European Community-Research Infrastructure
Activity Contract No. RII3-CT-2004-506078 (HadronPhysics).
KG acknowledges a fellowship from the Iranian Ministry of Science.
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Symmetry Reduction in Twisted Noncommutative Gravity
with Applications to Cosmology and Black Holes
Thorsten Ohl
Alexander Schenkel
Institut für Theoretische Physik und Astrophysik
Universität Würzburg, Am Hubland, 97074 Würzburg, Germany
e-mail: [email protected]: [email protected]
(October 2008)
Abstract
As a preparation for a mathematically consistent study of the
physics of symmetric spacetimes in a noncommutative setting, we
study symmetry reductions in deformed gravity. We focus on
deformations that are given by a twist of a Lie algebra acting on the
spacetime manifold. We derive conditions on those twists that allow a
given symmetry reduction. A complete classification of admissible
deformations is possible in a class of twists generated by
commuting vector fields. As examples, we explicitly construct the families of
vector fields that generate twists which are compatible with
Friedmann-Robertson-Walker cosmologies and Schwarzschild black
holes, respectively. We find nontrivial isotropic twists of FRW
cosmologies and nontrivial twists that are compatible with all
classical symmetries of black hole solutions.
1 Introduction
The study of noncommutative geometry is an active topic in both
theoretical physics and mathematics. From the mathematical perspective
it is a generalization of classical (commutative) geometry. From the
physics perspective it is suggested by the Gedankenexperiment
of localizing events in spacetime with a Planck scale
resolution [1]. In this
Gedankenexperiment, a sharp localization induces an
uncertainty in the spacetime coordinates, which can naturally be
described by a noncommutative spacetime. Furthermore, noncommutative
geometry and quantum gravity appear to be connected strongly and one
can probably model “low energy” effects of quantum gravity theories
using noncommutative geometry.
There have been many attempts to formulate scalar, gauge and gravity
theories on noncommutative spacetime, in particular using the simplest
example of a Moyal-Weyl spacetime having constant noncommutativity
between space and time coordinates, see
[2, 3] for reviews. Furthermore,
this framework had been applied to phenomenological particle physics
with [4, 5] and without Seiberg-Witten maps (see the
review [6] and references therein),
cosmology [7] and black hole physics (see the
review [8] and references therein).
Our work is based on the approach outlined
in [9, 10, 11], where a
noncommutative gravity theory based on an arbitrary twist deformation
is established. This approach has the advantages of being formulated
using the symmetry principle of deformed diffeomorphisms, being
coordinate independent and applicable to nontrivial topologies.
However, there is also the disadvantage that it does not match
the Seiberg-Witten limit of string theory [12].
Nevertheless, string theory is not the only candidate for a
fundamental theory of quantum gravity. Therefore, the investigation of
deformed gravity remains interesting on its own terms and it could very well
emerge from a fundamental theory of quantum gravity different from
string theory.
The outline of this paper is as follows. In section 2
we review the basics of the formalism of twisted noncommutative
differential geometry. For more details and the proofs we refer to the
original paper [10] and the
review [11]. We will work with a general twist and
do not restrict ourselves to the Moyal-Weyl deformation.
In section 3 we will study symmetry reduction in
theories based on twisted symmetries, such as the twisted
diffeomorphisms in our theory of interest. The reason is that we aim
to investigate which deformations of cosmological and black hole
symmetries are possible. We will derive the conditions that the twist
has to satisfy in order to be compatible with the reduced symmetry.
In section 4 we restrict the twists to the class of
Reshetikhin-Jambor-Sykora twists [13, 14],
that are twists generated by commuting vector fields and are convenient
for practical applications. Within this restricted class of twists we
can classify more explicitly the possible deformations of Lie algebra
symmetries acting on a manifold $\mathcal{M}$.
In section 5 and 6 we apply the
formalism to cosmological symmetries as well as the black hole. We
classify the possible Reshetikhin-Jambor-Sykora deformations of these models and
obtain physically interesting ones. In section 7 we
conclude and give an outlook to possible further investigations. In
particular possible applications to phenomenological cosmology and
black hole physics will be discussed.
2 Basics of Twisted Differential Geometry and Gravity
In order to establish notation, we will give a short summary of the framework of twisted differential geometry and gravity.
More details can be found in [9, 10, 11].
There is a quite general procedure for constructing noncommutative spaces and their corresponding symmetries
by using a twist. For this we require the following ingredients [11]:
1.
a Lie algebra $\mathfrak{g}$
2.
an action of the Lie algebra on the space we want to deform
3.
a twist element $\mathcal{F}$, constructed from the generators of the Lie algebra $\mathfrak{g}$
By a twist element we denote an invertible element of $U\mathfrak{g}\otimes U\mathfrak{g}$, where
$U\mathfrak{g}$ is the universal enveloping algebra of $\mathfrak{g}$. $\mathcal{F}$ has to fulfill some
conditions, which will be specified later. The basic idea in the following is to combine any bilinear map with the inverse twist
and therefore deform these maps. This leads to a mathematically consistent deformed theory covariant under
the deformed transformations. We will show this now for the deformation of diffeomorphisms.
For our purpose we are interested in the Lie algebra of vector fields $\Xi$ on a manifold $\mathcal{M}$.
The transformations induced by $\Xi$ can be seen as infinitesimal diffeomorphisms. A natural action of these
transformations on the algebra of tensor fields $\mathcal{T}:=\bigoplus\limits_{n,m}\bigotimes^{n}\Omega\otimes\bigotimes^{m}\Xi$
is given by the Lie derivative $\mathcal{L}$. $\Omega$ denotes the space of one-forms.
In order to deform this Lie algebra, as well as its action on tensor fields and the tensor fields themselves, we first have
to construct the enveloping algebra $U\Xi$. This is the associative
tensor algebra generated by the elements of $\Xi$ and the unit $1$, modulo the left and right ideals generated by
the elements $[v,w]-vw+wv$. This algebra can be seen as a Hopf algebra by using the following
coproduct $\Delta$, antipode $S$ and counit $\epsilon$ defined on the generators $u\in\Xi$ and $1$ by:
$$\displaystyle\begin{array}[]{ll}\Delta(u)=u\otimes 1+1\otimes u,&\Delta(1)=1%
\otimes 1~{},\\
\epsilon(u)=0,&\epsilon(1)=1~{},\\
S(u)=-u,&S(1)=1~{}.\end{array}$$
(1)
These definitions can be consistently carried over to the whole enveloping algebra demanding $\Delta$ and $\epsilon$
to be algebra homomorphisms and $S$ to be an anti-homomorphism, i.e. for any two elements $\eta,\xi\in U\Xi$, we require
$$\displaystyle\Delta(\eta\xi)=\Delta(\eta)\Delta(\xi)~{},$$
(2a)
$$\displaystyle\epsilon(\eta\xi)=\epsilon(\eta)\epsilon(\xi)~{},$$
(2b)
$$\displaystyle S(\eta\xi)=S(\xi)S(\eta)~{}.$$
(2c)
The action of the enveloping algebra on the tensor fields can be defined by extending the Lie derivative
$$\displaystyle\mathcal{L}_{\eta\xi}(\tau):=\mathcal{L}_{\eta}(\mathcal{L}_{\xi}%
(\tau))~{},~{}\forall\eta,\xi\in U\Xi~{},~{}\tau\in\mathcal{T}~{}.$$
(3)
This action is consistent with the Lie algebra properties, since $\mathcal{L}_{[u,v]}(\tau)=\mathcal{L}_{uv}(\tau)-\mathcal{L}_{vu}(\tau)$
for all $u,v\in\Xi$ by the properties of the Lie derivative.
The extension of the Lie algebra $\Xi$ to the Hopf algebra $(U\Xi,\cdot,\Delta,S,\epsilon)$, where $\cdot$ is the multiplication in $U\Xi$,
can now be used in order to construct deformations of it. For the deformations we restrict ourselves
to twist deformations, which is a wide class of possible deformations. The reason is that for twist deformations the
construction of deformed differential geometry and gravity can be performed explicitly by only using properties of the twist,
see [10]. Other deformations require further investigations.
In order to perform the deformation we require a twist element $\mathcal{F}=f^{\alpha}\otimes f_{\alpha}\in U\Xi\otimes U\Xi$
(the sum over $\alpha$ is understood) fulfilling the following conditions
$$\displaystyle\mathcal{F}_{12}(\Delta\otimes\mathrm{id})\mathcal{F}=\mathcal{F}%
_{23}(\mathrm{id}\otimes\Delta)\mathcal{F}~{},$$
(4a)
$$\displaystyle(\epsilon\otimes\mathrm{id})\mathcal{F}=1=(\mathrm{id}\otimes%
\epsilon)\mathcal{F}~{},$$
(4b)
$$\displaystyle\mathcal{F}=1\otimes 1+\mathcal{O}(\lambda)~{},$$
(4c)
where $\mathcal{F}_{12}:=\mathcal{F}\otimes 1$, $\mathcal{F}_{23}:=1\otimes\mathcal{F}$ and $\lambda$ is the deformation parameter.
The first condition will assure the associativity of the deformed products, the second will assure that deformed
multiplications with unit elements will be trivial and the third condition assures the existence of the undeformed
classical limit $\lambda\to 0$.
Furthermore, we can assume without loss of generality that $f_{\alpha}$ (and also $f^{\alpha}$) are linearly independent for all $\alpha$,
what can be assured by combining linearly dependent $f$.
Note that $\mathcal{F}$ is regarded as formal power series in $\lambda$, such as the deformation itself.
Strict (convergent) deformations will not be regarded here.
The simplest example is the twist on $\mathbb{R}^{n}$ given by
$\mathcal{F}_{\theta}:=\exp{\bigl{(}-\frac{i\lambda}{2}\theta^{\mu\nu}\partial_%
{\mu}\otimes\partial_{\nu}\bigr{)}}$ with $\theta^{\mu\nu}=\mathrm{const.}$
and antisymmetric, leading to the Moyal-Weyl deformation, but there are also more complicated ones.
From a twist, one can construct the twisted triangular Hopf algebra
$(U\Xi_{\mathcal{F}},\cdot,\Delta_{\mathcal{F}},S_{\mathcal{F}},\epsilon_{%
\mathcal{F}})$
with $R$-matrix $R:=\mathcal{F}_{21}\mathcal{F}^{-1}=:R^{\alpha}\otimes R_{\alpha}$, inverse $R^{-1}=:\bar{R}^{\alpha}\otimes\bar{R}_{\alpha}=R_{21}$
and
$$\displaystyle\Delta_{\mathcal{F}}(\xi):=\mathcal{F}\Delta(\xi)\mathcal{F}^{-1}%
~{},\quad\epsilon_{\mathcal{F}}(\xi):=\epsilon(\xi)~{},\quad S_{\mathcal{F}}(%
\xi):=\chi S(\xi)\chi^{-1}~{},$$
(5)
where $\chi:=f^{\alpha}S(f_{\alpha})$, $\chi^{-1}:=S(\bar{f}^{\alpha})\bar{f}_{\alpha}$ and $\bar{f}^{\alpha}\otimes\bar{f}_{\alpha}:=\mathcal{F}^{-1}$.
Furthermore, $\mathcal{F}_{21}:=f_{\alpha}\otimes f^{\alpha}$ and $R_{21}:=R_{\alpha}\otimes R^{\alpha}$. Again, we can assume without loss of generality that all summands of $\mathcal{F}^{-1}$, $R$ and $R^{-1}$ are linearly independent.
However, as explained in [10], it is simpler to use the triangular $\star$-Hopf algebra $\mathcal{H}_{\Xi}^{\star}=(U\Xi_{\star},\star,\Delta_{\star},S_{\star},%
\epsilon_{\star})$,
isomorphic to $(U\Xi_{\mathcal{F}},\cdot,\Delta_{\mathcal{F}},S_{\mathcal{F}},\epsilon_{%
\mathcal{F}})$. The operations
in this algebra on its generators $u,v\in\Xi$ (note that this algebra has the same generators as the classical Hopf algebra) are defined by
$$\displaystyle u\star v:=\bar{f}^{\alpha}(u)\bar{f}_{\alpha}(v)~{},$$
(6a)
$$\displaystyle\Delta_{\star}(u):=u\otimes 1+X_{\bar{R}^{\alpha}}\otimes\bar{R}_%
{\alpha}(u)~{},$$
(6b)
$$\displaystyle\epsilon_{\star}(u):=\epsilon(u)=0~{},$$
(6c)
$$\displaystyle S_{\star}^{-1}(u):=-\bar{R}^{\alpha}(u)\star X_{\bar{R}_{\alpha}%
}~{},$$
(6d)
where for all $\xi\in U\Xi$ we define $X_{\xi}:=\bar{f}^{\alpha}\xi\chi S^{-1}(\bar{f}_{\alpha})$. The action of the twist on the elements of $U\Xi$
is defined by extending the Lie derivative to the adjoint action [10]. Note that $U\Xi=U\Xi_{\star}$ as vector spaces. The $R$-matrix is
given by $R_{\star}:=X_{R^{\alpha}}\otimes X_{R_{\alpha}}$ and is triangular. The coproduct and antipode (6)
is defined consistently on $U\Xi_{\star}$ by using for all $\xi,\eta\in U\Xi_{\star}$ the definitions
$$\displaystyle\Delta_{\star}(\xi\star\eta):=\Delta_{\star}(\xi)\star\Delta_{%
\star}(\eta)~{},\quad~{}S_{\star}(\xi\star\eta):=S_{\star}(\eta)\star S_{\star%
}(\xi)~{}.$$
(7)
The next step is to define the $\star$-Lie algebra of deformed infinitesimal diffeomorphisms. It has been shown [10]
that for the twist deformation case the choice $(\Xi_{\star},[~{},~{}]_{\star})$, where $\Xi_{\star}=\Xi$ as vector spaces and
$$\displaystyle[u,v]_{\star}:=[\bar{f}^{\alpha}(u),\bar{f}_{\alpha}(v)]$$
(8)
is a natural choice for a $\star$-Lie algebra. It fulfills all conditions which are necessary for a sensible $\star$-Lie algebra given by
1.
$\Xi_{\star}\subset U\Xi_{\star}$ is a linear space, which generates $U\Xi_{\star}$
2.
$\Delta_{\star}(\Xi_{\star})\subseteq\Xi_{\star}\otimes 1+U\Xi_{\star}\otimes%
\Xi_{\star}$
3.
$[\Xi_{\star},\Xi_{\star}]_{\star}\subseteq\Xi_{\star}$
The advantage of using the $\star$-Hopf algebra $(U\Xi_{\star},\star,\Delta_{\star},S_{\star},\epsilon_{\star})$
instead of the $\mathcal{F}$-Hopf algebra $(U\Xi_{\mathcal{F}},\cdot,\Delta_{\mathcal{F}},S_{\mathcal{F}},\epsilon_{%
\mathcal{F}})$
is that the $\star$-Lie algebra of vector fields is isomorphic to $\Xi$ as a vector space.
For the $\mathcal{F}$-Hopf algebra this is not the case and the $\mathcal{F}$-Lie algebra consists in general of multidifferential
operators.
The algebra of tensor fields $\mathcal{T}$ is deformed by using the $\star$-tensor product [10]
$$\displaystyle\tau\otimes_{\star}\tau^{\prime}:=\bar{f}^{\alpha}(\tau)\otimes%
\bar{f}_{\alpha}(\tau^{\prime})~{},$$
(9)
where as basic ingredients the deformed algebra of functions $A_{\star}:=(C^{\infty}(M),\star)$ as well as the $A_{\star}$-bimodules of
vector fields $\Xi_{\star}$ and one-forms $\Omega_{\star}$ enter. We call $\mathcal{T}_{\star}$ the deformed algebra of tensor fields. Note that
$\mathcal{T}_{\star}=\mathcal{T}$ as vector spaces.
The action of the deformed infinitesimal diffeomorphisms on $\mathcal{T}_{\star}$ is defined by the $\star$-Lie derivative
$$\displaystyle\mathcal{L}^{\star}_{u}(\tau):=\mathcal{L}_{\bar{f}^{\alpha}(u)}(%
\bar{f}_{\alpha}(\tau))~{},~{}\forall\tau\in\mathcal{T}_{\star}~{},~{}u\in\Xi_%
{\star}~{},$$
(10)
which can be extended to all of $U\Xi_{\star}$ by $\mathcal{L}^{\star}_{\xi\star\eta}(\tau):=\mathcal{L}^{\star}_{\xi}(\mathcal{L%
}^{\star}_{\eta}(\tau))$.
Furthermore, we define the $\star$-pairing $\langle\cdot,\cdot\rangle_{\star}:\Xi_{\star}\otimes_{\mathbb{C}}\Omega_{\star%
}\to A_{\star}$ between
vector fields and one-forms as
$$\displaystyle\langle v,\omega\rangle_{\star}:=\langle\bar{f}^{\alpha}(v),\bar{%
f}_{\alpha}(\omega)\rangle~{},~{}\forall v\in\Xi_{\star},~{}\omega\in\Omega_{%
\star}~{},$$
(11)
where $\langle\cdot,\cdot\rangle$ is the undeformed pairing.
Based on the deformed symmetry principle one can define covariant derivatives, torsion and curvature. This leads to deformed Einstein
equations, see [10], which we do not have to review here, since we do not use them in the following.
3 Symmetry Reduction in Twisted Differential Geometry
Assume that we have constructed a deformed gravity theory based on a twist $\mathcal{F}\in U\Xi\otimes U\Xi$. Like in Einstein
gravity, the physical applications of this theory is strongly dependent on symmetry reduction. In this section we first define
what we mean by symmetry reduction of a theory covariant under a Lie algebraic symmetry (e.g. infinitesimal diffeomorphisms)
and then extend the principles to deformed symmetries and $\star$-Lie algebras.
In undeformed general relativity we often face the fact that the systems we want to describe have certain (approximate) symmetries.
Here we restrict ourselves to Lie group symmetries.
For example in cosmology one usually constrains oneself to fields invariant under certain symmetry groups $G$, like e.g. the euclidian
group $E_{3}$ for flat universes or the $SO(4)$ group for universes with topology $\mathbb{R}\times S_{3}$, where the spatial hypersurfaces
are 3-spheres.
For a non rotating black hole one usually demands the metric to be stationary and spherically symmetric. Practically, one
uses the corresponding Lie algebra $\mathfrak{g}$ of the symmetry group $G$, represents it faithfully on the Lie algebra
of vector fields $\Xi$ on the manifold $\mathcal{M}$ and demands the fields $\tau\in\mathcal{T}$, which occur in the theory,
to be invariant under these transformations, i.e. we demand
$$\displaystyle\mathcal{L}_{v}(\tau)=0~{},~{}\forall v\in\mathfrak{g}~{}.$$
(12)
Since the Lie algebra $\mathfrak{g}$ is a linear space we can choose a basis $\{t_{i}:i=1,\cdots,\mathrm{dim}(\mathfrak{g})\}$
and can equivalently demand
$$\displaystyle\mathcal{L}_{t_{i}}(\tau)=0~{},~{}\forall i=1,2,\cdots,\mathrm{%
dim}(\mathfrak{g})~{}.$$
(13)
The Lie bracket of the generators has to fulfill
$$\displaystyle[t_{i},t_{j}]=f_{ij}^{~{}~{}k}t_{k}~{},$$
(14)
where $f_{ij}^{~{}~{}k}$ are the structure constants.
One can easily show that if we combine two invariant tensors with the tensor product, the resulting tensor is invariant too
because of the trivial coproduct
$$\displaystyle\mathcal{L}_{t_{i}}(\tau\otimes\tau^{\prime})=\mathcal{L}_{t_{i}}%
(\tau)\otimes\tau^{\prime}+\tau\otimes\mathcal{L}_{t_{i}}(\tau^{\prime})~{}.$$
(15)
The same holds true for pairings $\langle v,\omega\rangle$ of invariant objects $v\in\Xi$ and $\omega\in\Omega$.
Furthermore, if a tensor is invariant under infinitesimal transformations, it is also invariant under (at least small) finite transformations,
since they are given by exponentiating the generators. The exponentiated generators are part of the enveloping algebra,
i.e. $\exp(\alpha^{i}t_{i})\in U\mathfrak{g}$, where $\alpha^{i}$ are parameters. For large finite transformations the topology of the
Lie group can play a role, such that the group elements may not simply be given by exponentiating the generators. In the following
we will focus only on small finite transformations in order to avoid topological effects.
We now generalize this to the case of $\star$-Hopf algebras and their corresponding $\star$-Lie algebras. Our plan is
as follows: we start with a suitable definition of a $\star$-Lie subalgebra constructed from the Lie algebra $(\mathfrak{g},[~{},~{}])$.
This definition is guided by conditions, which allow for deformed symmetry reduction using infinitesimal transformations. Then
we complete this $\star$-Lie subalgebra in several steps to a $\star$-enveloping subalgebra, a $\star$-Hopf subalgebra
and a triangular $\star$-Hopf subalgebra. We will always be careful that the dimension of the $\star$-Lie subalgebra remains the same
as the dimension of the corresponding classical Lie algebra. At each step we obtain several restrictions between the twist
and $(\mathfrak{g},[~{},~{}])$.
We start by taking the generators $\{t_{i}\}$ of $\mathfrak{g}\subseteq\Xi$
and representing their deformations in the $\star$-Lie algebra $(\Xi_{\star},[~{},~{}]_{\star})$ as
$$\displaystyle t_{i}^{\star}=t_{i}+\sum\limits_{n=1}^{\infty}\lambda^{n}t_{i}^{%
(n)}~{},$$
(16)
where $\lambda$ is the deformation parameter and $t_{i}^{(n)}\in\Xi_{\star}$.
The span of these deformed generators, together with the $\star$-Lie bracket, should form a $\star$-Lie subalgebra $(\mathfrak{g}_{\star},[~{},~{}]_{\star}):=(\mathrm{span}(t_{i}^{\star}),[~{},~%
{}]_{\star})$. Therefore $(\mathfrak{g}_{\star},[~{},~{}]_{\star})$ has
to obey certain conditions. Natural conditions are
$$\displaystyle[\mathfrak{g_{\star}},\mathfrak{g}_{\star}]_{\star}\subseteq%
\mathfrak{g}_{\star},$$
i.e. $$[t_{i}^{\star},t_{j}^{\star}]_{\star}=f_{ij}^{\star~{}k}t^{\star}_{k}$$ with $$f_{ij}^{\star~{}k}=f_{ij}^{~{}~{}k}+\mathcal{O}(\lambda)$$
(17a)
$$\displaystyle\Delta_{\star}(\mathfrak{g}_{\star})\subseteq\mathfrak{g}_{\star}%
\otimes 1+U\Xi_{\star}\otimes\mathfrak{g}_{\star},$$
which is equivalent to $$\bar{R}_{\alpha}(\mathfrak{g}_{\star})\subseteq\mathfrak{g}_{\star}~{}\forall_%
{\alpha}$$
(17b)
The first condition is a basic feature of a $\star$-Lie algebra.
The second condition implies that if we have two $\mathfrak{g}_{\star}$ invariant tensors $\tau,\tau^{\prime}\in\mathcal{T}_{\star}$, the
$\star$-tensor product of them is invariant as well
$$\displaystyle\mathcal{L}^{\star}_{t_{i}^{\star}}(\tau\otimes_{\star}\tau^{%
\prime})=\mathcal{L}^{\star}_{t_{i}^{\star}}(\tau)\otimes_{\star}\tau^{\prime}%
+\bar{R}^{\alpha}(\tau)\otimes_{\star}\mathcal{L}^{\star}_{\bar{R}_{\alpha}(t_%
{i}^{\star})}(\tau^{\prime})=0~{},$$
(18)
since $\bar{R}_{\alpha}(t_{i}^{\star})\in\mathfrak{g}_{\star}$.
The $\star$-pairings $\langle v,\omega\rangle_{\star}$ of two invariant objects $v\in\Xi_{\star}$ and $\omega\in\Omega_{\star}$ are also invariant
under the $\star$-action of $\mathfrak{g}_{\star}$.
These are important features if one wants to combine invariant objects to e.g. an invariant action.
Furthermore, the conditions are sufficient such that the following consistency relation is fulfilled for
any invariant tensor $\tau\in\mathcal{T}_{\star}$
$$\displaystyle 0=f_{ij}^{\star~{}k}\mathcal{L}^{\star}_{t^{\star}_{k}}(\tau)=%
\mathcal{L}^{\star}_{[t^{\star}_{i},t^{\star}_{j}]}(\tau)=\mathcal{L}^{\star}_%
{t^{\star}_{i}}(\mathcal{L}^{\star}_{t^{\star}_{j}}(\tau))-\mathcal{L}^{\star}%
_{\bar{R}^{\alpha}(t^{\star}_{j})}(\mathcal{L}^{\star}_{\bar{R}_{\alpha}(t^{%
\star}_{i})}(\tau))~{},$$
(19)
since $\bar{R}_{\alpha}(t_{i}^{\star})\in\mathfrak{g}_{\star}$.
Hence by demanding the two conditions (17) for the $\star$-Lie subalgebra
$(\mathrm{span}(t_{i}^{\star}),[~{},~{}]_{\star})$ we can consistently perform symmetry reduction by using deformed
infinitesimal transformations. In the classical limit $\lambda\to 0$ we obtain the classical Lie algebra
$(\mathfrak{g}_{\star},[~{},~{}]_{\star})\stackrel{{\scriptstyle\lambda\to 0}}{%
{\longrightarrow}}(\mathfrak{g},[~{},~{}])$.
Next, we consider the extension of the $\star$-Lie subalgebra $(\mathfrak{g}_{\star},[~{},~{}]_{\star})\subseteq(\Xi_{\star},[~{},~{}]_{\star})$
to the triangular $\star$-Hopf subalgebra $\mathcal{H}_{\mathfrak{g}}^{\star}=(U\mathfrak{g}_{\star},\star,\Delta_{\star}%
,S_{\star},\epsilon_{\star})\subseteq\mathcal{H}_{\Xi}^{\star}$.
This can be seen as extending the infinitesimal transformations to a quantum group. We will divide this path into several steps,
where in every step we have to demand additional restrictions on the twist.
Firstly, we construct the $\star$-tensor algebra generated by the elements of $\mathfrak{g}_{\star}$ and $1$. We take this tensor algebra
modulo the left and right ideals generated by the elements $[u,v]_{\star}-u\star v+\bar{R}^{\alpha}(v)\star\bar{R}_{\alpha}(u)$.
It is necessary that these elements are part of $U\mathfrak{g}_{\star}$, i.e. we require
$$\displaystyle\bar{R}^{\alpha}(\mathfrak{g}_{\star})\star\bar{R}_{\alpha}(%
\mathfrak{g}_{\star})\subseteq U\mathfrak{g}_{\star}~{}.$$
(20)
This leads to the algebra $(U\mathfrak{g}_{\star},\star)$, which is a subalgebra of $(U\Xi_{\star},\star)$.
Secondly, we extend this subalgebra to a $\star$-Hopf subalgebra. Therefore we additionally have to require that
$$\displaystyle\Delta_{\star}(U\mathfrak{g}_{\star})$$
$$\displaystyle\subseteq U\mathfrak{g}_{\star}\otimes U\mathfrak{g}_{\star}~{},$$
(21a)
$$\displaystyle S_{\star}(U\mathfrak{g}_{\star})$$
$$\displaystyle\subseteq U\mathfrak{g}_{\star}~{}.$$
(21b)
Note that we do not demand that $S^{-1}_{\star}$ (defined on $U\Xi_{\star}$)
closes in $U\mathfrak{g}_{\star}$, since this is in general not the case for a nonquasitriangular Hopf algebra and we do not want to
demand quasitriangularity at this stage.
Then the $\star$-Hopf algebra $\mathcal{H}_{\mathfrak{g}}^{\star}$ is a Hopf subalgebra of $\mathcal{H}_{\Xi}^{\star}$.
Thirdly, we additionally demand that there exists an $R$-matrix $R_{\star}\in U\mathfrak{g}_{\star}\otimes U\mathfrak{g}_{\star}$. It is natural
to take the $R$-matrix of the triangular $\star$-Hopf algebra $\mathcal{H}_{\Xi}^{\star}$ defined by $R_{\star}:=X_{R^{\alpha}}\otimes X_{R_{\alpha}}$.
This leads to the restrictions
$$\displaystyle X_{R^{\alpha}},X_{R_{\alpha}}\in U\mathfrak{g}_{\star}~{},~{}%
\forall_{\alpha}.$$
(22)
Since $R_{\star}$ is triangular, i.e. $R_{\star}^{-1}=\bar{R}_{\star}^{\alpha}\otimes\bar{R}_{\star\alpha}=R_{\star 2%
1}=R_{\star\alpha}\otimes R_{\star}^{\alpha}$, we also have
$X_{\bar{R}^{\alpha}},X_{\bar{R}_{\alpha}}\in U\mathfrak{g}_{\star}~{},~{}%
\forall_{\alpha}$.
If these conditions are fulfilled, $\mathcal{H}_{\mathfrak{g}}^{\star}$ is a triangular $\star$-Hopf subalgebra of $\mathcal{H}_{\Xi}^{\star}$
with the same $R$-matrix.
As we have seen, extending the $\star$-Lie subalgebra to a (triangular) $\star$-Hopf subalgebra gives severe restrictions on
the possible deformations, more than just working with the deformed infinitesimal transformations given by a $\star$-Lie subalgebra
or the finite transformations given by the $\star$-enveloping subalgebra $(U\mathfrak{g}_{\star},\star)$.
Now the question arises if we actually require the deformed finite transformations to form a (triangular) $\star$-Hopf algebra in order to
use them for a sensible symmetry reduction. Because $(U\mathfrak{g}_{\star},\star)$ describes deformed finite transformations and we
have the relation
$$\displaystyle\mathcal{L}^{\star}_{U\mathfrak{g}_{\star}\backslash\{1\}}(\tau)=%
\{0\}\Leftrightarrow\mathcal{L}^{\star}_{\mathfrak{g}_{\star}}(\tau)=\{0\}~{},$$
(23)
we can consistently demand tensors to be invariant under $(U\mathfrak{g}_{\star},\star)$, since we
require tensors to be invariant under $(\mathfrak{g}_{\star},[~{},~{}]_{\star})$.
Therefore, a well defined $(U\mathfrak{g}_{\star},\star)$ leads to a structure sufficient for symmetry reduction.
The equivalence (23) can be shown by using linearity
of the $\star$-Lie derivative and the property $\mathcal{L}^{\star}_{\xi\star\eta}(\tau)=\mathcal{L}^{\star}_{\xi}(\mathcal{L}%
^{\star}_{\eta}(\tau))$.
In order to better understand the different restrictions necessary for constructing the $\star$-Lie subalgebra $(\mathfrak{g}_{\star},[~{},~{}]_{\star})$,
the $\star$-enveloping subalgebra and the (triangular) $\star$-Hopf subalgebra $(U\mathfrak{g}_{\star},\star,\Delta_{\star},S_{\star},\epsilon_{\star})$,
we restrict ourselves in the following sections to the class of Reshetikhin-Jambor-Sykora twists [13, 14]. This is a
suitable nontrivial generalization of the Moyal-Weyl product, also containing e.g. $\kappa$ and $q$ deformations when applied to Poincaré symmetry.
4 The Case of Reshetikhin-Jambor-Sykora Twists
Let $\{V_{a}\in\Xi\}$ be an arbitrary set of mutually commuting vector fields,
i.e. $[V_{a},V_{b}]=0~{},~{}\forall_{a,b}$, on an $n$ dimensional manifold $\mathcal{M}$. Then the object
$$\displaystyle\mathcal{F}_{V}:=\exp\bigl{(}-\frac{i\lambda}{2}\theta^{ab}V_{a}%
\otimes V_{b}\bigr{)}\in U\Xi\otimes U\Xi$$
(24)
is a twist element, if $\theta$ is constant and antisymmetric [10, 13, 14].
We call (24) a Reshetikhin-Jambor-Sykora twist. Note that this twist is not restricted to the topology $\mathbb{R}^{n}$ for the manifold
$\mathcal{M}$.
Furthermore, we can restrict ourselves to $\theta$ with maximal rank and an even number of vector fields $V_{a}$,
since we can lower the rank of the Poisson structure afterwards by choosing some of the $V_{a}$ to be zero.
We can therefore without loss of generality use the standard form
$$\displaystyle\theta=\begin{pmatrix}0&1&0&0&\cdots\\
-1&0&0&0&\cdots\\
0&0&0&1&\cdots\\
0&0&-1&0&\cdots\\
\vdots&\vdots&\vdots&\vdots&\ddots\end{pmatrix}$$
(25)
by applying a suitable $GL(n)$ transformation on the $V_{a}$.
This twist element is easy to apply and in particular we have for the inverse and the $R$-matrix
$$\displaystyle\mathcal{F}_{V}^{-1}=\exp\bigl{(}\frac{i\lambda}{2}\theta^{ab}V_{%
a}\otimes V_{b}\bigr{)}~{},\quad~{}R=\mathcal{F}_{V,21}\mathcal{F}_{V}^{-1}=%
\mathcal{F}_{V}^{-2}=\exp\bigl{(}i\lambda\theta^{ab}V_{a}\otimes V_{b}\bigr{)}%
~{}.$$
(26)
Now let $(\mathfrak{g},[~{},~{}])\subseteq(\Xi,[~{},~{}])$ be the Lie algebra of the symmetry we want to deform. We choose a
basis of this Lie algebra $\{t_{i}:i=1,\cdots,\mathrm{dim}(\mathfrak{g})\}$ with $[t_{i},t_{j}]=f_{ij}^{~{}~{}k}t_{k}$.
Next, we discuss the symmetry reduction based on the $\star$-Lie subalgebra, as explained in section 3.
Therefore we make the ansatz (16) for the generators $t_{i}^{\star}$.
Furthermore, we evaluate the two conditions (17) the $t_{i}^{\star}$ have to satisfy.
We start with the coproduct condition (17b), which
is equivalent to $\bar{R}_{\alpha}(t_{i}^{\star})\in\mathrm{span}(t_{i}^{\star}),~{}\forall_{\alpha}$, where $\alpha$ is a multi index. Using the
explicit form of the inverse $R$-matrix (26) we arrive at the conditions
$$\displaystyle[V_{a_{1}},\cdots[V_{a_{n}},t_{i}^{\star}]\cdots]=\mathcal{N}_{a_%
{1}\cdots a_{n}i}^{\star j}t_{j}^{\star}~{},$$
(27)
where $\mathcal{N}_{a_{1}\cdots a_{n}i}^{\star j}:=\mathcal{N}_{a_{1}\cdots a_{n}i}^{%
j}+\sum\limits_{k=1}^{\infty}\lambda^{k}\mathcal{N}_{a_{1}\cdots a_{n}i}^{(k)~%
{}j}$ are constants.
The only independent condition in (27) is given by
$$\displaystyle[V_{a},t_{i}^{\star}]=\mathcal{N}^{\star j}_{ai}t^{\star}_{j}~{},$$
(28)
since it implies all the other ones by linearity.
In particular, the zeroth order in $\lambda$ of (28) yields
$$\displaystyle[V_{a},t_{i}]=\mathcal{N}_{ai}^{j}t_{j}~{}.$$
(29)
This leads to the following
Proposition 1.
Let $(\mathfrak{g},[~{},~{}])\subseteq(\Xi,[~{},~{}])$ be a classical Lie algebra and $(\Xi_{\star},[~{},~{}]_{\star})$ the
$\star$-Lie algebra of vector fields deformed by a Reshetikhin-Jambor-Sykora twist, constructed with vector fields $V_{a}$.
Then for a symmetry reduction respecting the minimal axioms (17), it is necessary that
the following Lie bracket relations hold true
$$\displaystyle[V_{a},\mathfrak{g}]\subseteq\mathfrak{g}~{},\forall_{a}~{}.$$
(30)
In other words, $(\mathrm{span}(t_{i},V_{a}),[~{},~{}])\subseteq(\Xi,[~{},~{}])$ forms a Lie algebra with ideal $\mathfrak{g}$.
Here $t_{i}$ are the generators of $\mathfrak{g}$.
Note that this gives conditions relating the classical Lie algebra $(\mathfrak{g},[~{},~{}])$ with the twist.
Next, we evaluate the $\star$-Lie bracket condition (17a).
Using the explicit form of the inverse twist (26) and (28) we obtain
$$\displaystyle\bar{f}_{\alpha_{(n)}}(t_{i}^{\star})=[V_{a_{1}},\cdots[V_{a_{n}}%
,t_{i}^{\star}]\cdots]=\bigl{(}\mathcal{N}^{\star}_{a_{n}}\cdots\mathcal{N}^{%
\star}_{a_{1}}\bigr{)}_{i}^{j}t_{j}^{\star}=:\mathcal{N}^{\star j}_{\alpha_{(n%
)}i}t_{j}^{\star}~{},$$
(31a)
$$\displaystyle\bar{f}^{\alpha_{(n)}}(t_{i}^{\star})=\Theta^{\beta_{(n)}\alpha_{%
(n)}}\bar{f}_{\beta_{(n)}}(t_{i}^{\star})~{},$$
(31b)
$$\displaystyle\Theta^{\beta_{(n)}\alpha_{(n)}}:=\frac{1}{n!}\left(\frac{i%
\lambda}{2}\right)^{n}\theta^{b_{1}a_{1}}\cdots\theta^{b_{n}a_{n}}~{},$$
(31c)
where $\alpha_{(n)},\beta_{(n)}$ are multi indices. This leads to
$$\displaystyle[t_{i}^{\star},t_{j}^{\star}]_{\star}=\Theta^{\beta_{(n)}\alpha_{%
(n)}}\mathcal{N}^{\star k}_{\beta_{(n)}i}\mathcal{N}^{\star l}_{\alpha_{(n)}j}%
[t_{k}^{\star},t_{l}^{\star}]~{}.$$
(32)
Note that in particular for the choice $t_{i}^{\star}=t_{i},~{}\forall_{i},$ the $\star$-Lie subalgebra closes with structure constants
$$\displaystyle[t_{i},t_{j}]_{\star}=\Theta^{\beta_{(n)}\alpha_{(n)}}\mathcal{N}%
^{k}_{\beta_{(n)}i}\mathcal{N}^{l}_{\alpha_{(n)}j}[t_{k},t_{l}]=\Theta^{\beta_%
{(n)}\alpha_{(n)}}\mathcal{N}^{k}_{\beta_{(n)}i}\mathcal{N}^{l}_{\alpha_{(n)}j%
}f_{kl}^{~{}~{}m}t_{m}=:f_{ij}^{\star~{}m}t_{m}~{},$$
(33)
where we have used the $\mathcal{N}$ defined in (29).
This leads to the following
Proposition 2.
Let $[V_{a},\mathfrak{g}_{\star}]\subseteq\mathfrak{g}_{\star}~{},\forall_{a}$. Then we can always construct a $\star$-Lie subalgebra
$(\mathfrak{g}_{\star},[~{},~{}]_{\star})\subseteq(\Xi_{\star},[~{},~{}]_{\star})$
by choosing the generators as $t_{i}^{\star}=t_{i}$ for all $i$. With this we have $\mathfrak{g}_{\star}=\mathfrak{g}$ as vector spaces
and the structure constants are deformed as
$$\displaystyle f_{ij}^{\star~{}m}=\Theta^{\beta_{(n)}\alpha_{(n)}}\mathcal{N}^{%
k}_{\beta_{(n)}i}\mathcal{N}^{l}_{\alpha_{(n)}j}f_{kl}^{~{}~{}m}~{}.$$
(34)
Since the condition (17b) together with the requirement $t_{i}^{\star}=t_{i}$, for all $i$, automatically fulfills
(17a), we choose $t_{i}^{\star}=t_{i}$, for all $i$, as a
canonical embedding. In general, other possible embeddings
require further constructions to fulfill condition
(17a) and are therefore less natural.
We will discuss possible differences between this and other embeddings later on, when we construct the $\star$-Hopf subalgebra and the
$\star$-Lie derivative action on $\star$-tensor fields.
In addition, we obtain that the necessary condition (20) for extending $\mathfrak{g}_{\star}$ to the $\star$-enveloping
subalgebra $(U\mathfrak{g}_{\star},\star)\subseteq(U\Xi_{\star},\star)$
is automatically fulfilled, since we have $\bar{R}_{\alpha_{(n)}}(\mathfrak{g}_{\star})\subseteq\mathfrak{g}_{\star}$
for all $\alpha_{(n)}$ and additionally
$$\displaystyle\bar{R}^{\alpha_{(n)}}(\mathfrak{g}_{\star})=(-2)^{n}\Theta^{%
\beta_{(n)}\alpha_{(n)}}\bar{R}_{\beta_{(n)}}(\mathfrak{g}_{\star})\subseteq%
\mathfrak{g}_{\star}~{},~{}\forall{\alpha}_{(n)}.$$
(35)
Next, we evaluate the conditions (21), which have to be fulfilled in order to construct the $\star$-Hopf subalgebra
$\mathcal{H}_{\mathfrak{g}}^{\star}\subseteq\mathcal{H}_{\Xi}^{\star}$. For the particular choice of the twist (24)
we obtain the following
Proposition 3.
Let $(U\mathfrak{g}_{\star},\star)\subseteq(U\Xi_{\star},\star)$ be a $\star$-enveloping subalgebra and let the deformation parameter
$\lambda\neq 0$.
Then in order to extend $(U\mathfrak{g}_{\star},\star)$ to the $\star$-Hopf subalgebra $\mathcal{H}_{\mathfrak{g}}^{\star}=(U\mathfrak{g}_{\star},\star,\Delta_{\star}%
,S_{\star},\epsilon_{\star})\subseteq\mathcal{H}_{\Xi}^{\star}$ the
condition
$$\displaystyle V_{a_{1}}\in\mathfrak{g}_{\star}~{},\quad\text{if }~{}[V_{a_{2}}%
,\mathfrak{g}_{\star}]\neq\{0\}$$
(36)
has to hold true for all pairs of indices $(a_{1},a_{2})$ connected by the antisymmetric matrix $\theta$ (25), i.e. $(a_{1},a_{2})\in\big{\{}(1,2),(2,1),(3,4),(4,3),\dots\big{\}}$.
Note that these conditions depend on the embedding $t_{i}^{\star}=t_{i}^{\star}(t_{j})$.
The proof of this proposition is shown in the appendix A.
Finally, if we demand $\mathcal{H}_{\mathfrak{g}}^{\star}$ to be a triangular $\star$-Hopf algebra (22) we obtain the stringent condition
$$\displaystyle V_{a}\in\mathfrak{g}_{\star}~{},~{}\forall_{a}~{}.$$
(37)
This can be shown by using $X_{R_{\alpha}}=R_{\alpha}$ and $V_{a}\star V_{b}=V_{a}V_{b}$, which holds true for the class of Reshetikhin-Jambor-Sykora twists.
As we have seen above, there are much stronger restrictions on the Lie algebra $(\mathfrak{g},[~{},~{}])$
and the twist, if we want to extend the deformed infinitesimal transformations $(\mathfrak{g}_{\star},[~{},~{}]_{\star})$
to the (triangular) $\star$-Hopf subalgebra $\mathcal{H}_{\mathfrak{g}}^{\star}$. In particular this extension restricts the $V_{a}$ themselves,
while for infinitesimal transformations and the finite transformations $(U\mathfrak{g}_{\star},\star)$ only the images of $V_{a}$
acting on $\mathfrak{g}_{\star}$ are important.
Next, we study the $\star$-action of the $\star$-Lie and Hopf algebra on the deformed tensor fields. The $\star$-action of the generators
$t_{i}^{\star}$ on $\tau\in\mathcal{T}_{\star}$ is defined by (10) and simplifies to
$$\displaystyle\mathcal{L}^{\star}_{t_{i}^{\star}}(\tau)=\Theta^{\alpha_{(n)}%
\beta_{(n)}}\mathcal{N}^{\star j}_{\alpha_{(n)}i}~{}\mathcal{L}_{t_{j}^{\star}%
}\bigl{(}\bar{f}_{\beta_{(n)}}(\tau)\bigr{)}~{}.$$
(38)
For invariant tensors, the $\star$-Lie derivative has to vanish to all orders in $\lambda$, since we work with formal power series.
If we now for explicitness take the natural choice $t_{i}^{\star}=t_{i}$ we obtain the following
Proposition 4.
Let $[V_{a},\mathfrak{g}_{\star}]\subseteq\mathfrak{g}_{\star}~{},\forall_{a}$ and $t_{i}^{\star}=t_{i},~{}\forall_{i}$. Then a tensor $\tau\in\mathcal{T}_{\star}$ is $\star$-invariant
under $(\mathfrak{g}_{\star},[~{},~{}]_{\star})$, if and only if it is invariant under the undeformed action of $(\mathfrak{g},[~{},~{}])$, i.e.
$$\displaystyle\mathcal{L}^{\star}_{\mathfrak{g}_{\star}}(\tau)=\{0\}~{}%
\Leftrightarrow~{}\mathcal{L}_{\mathfrak{g}}(\tau)=\{0\}~{}.$$
(39)
Proof.
For the proof we make the ansatz $\tau=\sum\limits_{n=0}^{\infty}\lambda^{n}\tau_{n}$ and investigate $\mathcal{L}^{\star}_{t_{i}}(\tau)$
order by order in $\lambda$, since we work with formal power series. By using (29) to reorder the
Lie derivatives such that $t_{i}$ is moved to the right, it can be shown recursively in powers of $\lambda$ that the proposition
holds true.
∎
Note that for $t_{i}^{\star}\neq t_{i}$ this does not necessarily hold true. We can not make statements for this case, since
we would require a general solution of (32), which we do not have yet. But we mention again that
we consider choosing $t_{i}^{\star}$ different from $t_{i}$ quite unnatural.
This proposition translates to the case of finite symmetry transformations with $t_{i}^{\star}=t_{i}$ because of the properties of
the $\star$-Lie derivative.
The framework developed in this section will now be applied to
cosmology and black holes in order to give some specific examples and
discuss possible physical implications.
5 Application to Cosmology
In this section we will investigate models with symmetry group $E_{3}$ in four spacetime dimensions with topology $\mathbb{R}^{4}$.
These are flat Friedmann-Robertson-Walker (FRW) universes.
The undeformed Lie algebra of this group is generated by the “momenta” $p_{i}$ and “angular momenta” $L_{i}$, $i\in\{1,2,3\}$,
which we can represent in the Lie algebra of vector fields as
$$\displaystyle p_{i}=\partial_{i}\quad,\quad L_{i}=\epsilon_{ijk}x^{j}\partial_%
{k}~{},$$
(40)
where $\epsilon_{ijk}$ is the Levi-Civita symbol.
The undeformed Lie bracket relations are
$$\displaystyle[p_{i},p_{j}]=0~{},\quad[p_{i},L_{j}]=-\epsilon_{ijk}p_{k}~{},%
\quad[L_{i},L_{j}]=-\epsilon_{ijk}L_{k}~{}.$$
(41)
We will work with the natural embedding $t_{i}^{\star}=t_{i}$, and therefore the $\star$-Lie subalgebra is given by
$\mathfrak{g}_{\star}=\mathfrak{e}_{3\star}=\mathfrak{e}_{3}=\mathrm{span}(p_{i%
},L_{i})$.
We can now explicitly evaluate the condition each twist vector field $V_{a}$ has to satisfy given by
$[V_{a},\mathfrak{e}_{3}]\subseteq\mathfrak{e}_{3}$ (cf. proposition 1).
Since the generators are at most linear in the spatial coordinates, $V_{a}$ can be at most quadratic in order to
fulfill this condition. If we make a quadratic ansatz with time dependent coefficients we obtain that each $V_{a}$
has to be of the form
$$\displaystyle V_{a}=V_{a}^{0}(t)\partial_{t}+c_{a}^{i}\partial_{i}+d_{a}^{i}L_%
{i}+f_{a}x^{i}\partial_{i}~{},$$
(42)
where $c_{a}^{i},~{}d_{a}^{i},~{}f_{a}\in\mathbb{R}$ and $V_{a}^{0}(t)\in C^{\infty}(\mathbb{R})$ in order to obtain hermitian deformations.
If all $V_{a}$ have the form (42), the $\star$-Lie algebra closes (cf. proposition 2).
Next, we have to find conditions such that the $V_{a}$ are mutually commuting. A brief calculation shows that the
following conditions have to be fulfilled:
$$\displaystyle d_{a}^{i}d_{b}^{j}\epsilon_{ijk}=0~{},\forall_{k}~{},$$
(43a)
$$\displaystyle c_{a}^{i}d_{b}^{j}\epsilon_{ijk}-c_{b}^{i}d_{a}^{j}\epsilon_{ijk%
}+f_{a}c_{b}^{k}-f_{b}c_{a}^{k}=0~{},\forall_{k}~{},$$
(43b)
$$\displaystyle[V_{a}^{0}(t)\partial_{t},V_{b}^{0}(t)\partial_{t}]=0~{}.$$
(43c)
As a first step, we will now work out all possible deformations of $\mathfrak{e}_{3}$ when twisted with two commuting vector fields.
We will classify the possible solutions. Therefore we divide the solutions into classes depending on the value of $d_{a}^{i}$ and $f_{a}$.
We use as notation for our cosmologies $\mathfrak{C}_{AB}$, where $A\in\{1,2,3\}$ and $B\in\{1,2\}$,
which will become clear later on, when we sum up the results in table 1.
Type $\mathfrak{C}_{11}$ is defined to be vector fields with $d_{1}^{i}=d_{2}^{i}=0$ and $f_{1}=f_{2}=0$, i.e.
$$\displaystyle V_{1(\mathfrak{C}_{11})}=V_{1}^{0}(t)\partial_{t}+c_{1}^{i}%
\partial_{i}~{},\quad V_{2(\mathfrak{C}_{11})}=V_{2}^{0}(t)\partial_{t}+c_{2}^%
{i}\partial_{i}~{}.$$
(44)
These vector fields fulfill the first two conditions (43a) and (43b). The solutions of the third condition
(43c) will be discussed later, since this classification we perform now does not depend on it.
Type $\mathfrak{C}_{21}$ is defined to be vector fields with $d_{1}^{i}=d_{2}^{i}=0$, $f_{1}\neq 0$ and $f_{2}=0$.
The first condition (43a) is trivially fulfilled and the second (43b) is fulfilled,
if and only if $c_{2}^{i}=0,~{}\forall_{i}$, i.e. type $\mathfrak{C}_{21}$ is given by the vector fields
$$\displaystyle\tilde{V}_{1(\mathfrak{C}_{21})}=V_{1}^{0}(t)\partial_{t}+c_{1}^{%
i}\partial_{i}+f_{1}x^{i}\partial_{i}~{},\quad\tilde{V}_{2(\mathfrak{C}_{21})}%
=V_{2}^{0}(t)\partial_{t}~{}.$$
(45)
These vector fields can be simplified to
$$\displaystyle V_{1(\mathfrak{C}_{21})}=c_{1}^{i}\partial_{i}+f_{1}x^{i}%
\partial_{i}~{},\quad V_{2(\mathfrak{C}_{21})}=V_{2}^{0}(t)\partial_{t}~{},$$
(46)
since both lead to the same twist (24).
Solutions with $d_{1}^{i}=d_{2}^{i}=0$, $f_{1}\neq 0$ and $f_{2}\neq 0$ lie in type $\mathfrak{C}_{21}$, since we can perform the twist conserving map
$V_{2}\to V_{2}-\frac{f_{2}}{f_{1}}V_{1}$, which transforms $f_{2}$ to zero. Furthermore $\mathfrak{C}_{31}$ is defined by
$d_{1}^{i}=d_{2}^{i}=0$, $f_{1}=0$ and $f_{2}\neq 0$ and is equivalent to $\mathfrak{C}_{21}$ by interchanging the labels of the vector fields.
Next, we go on to solutions with without loss of generality $\mathbf{d}_{1}\neq 0$ and $\mathbf{d}_{2}=0$ ($\mathbf{d}$ denotes the vector).
Note that this class contains also the class with $\mathbf{d}_{1}\neq 0$ and $\mathbf{d}_{2}\neq 0$.
To see this, we use the first condition (43a) and obtain that $\mathbf{d}_{1}$ and $\mathbf{d}_{2}$ have to be parallel,
i.e. $d^{i}_{2}=\kappa d^{i}_{1}$.
Then we can transform $\mathbf{d}_{2}$ to zero by using the twist conserving map $V_{2}\to V_{2}-\kappa V_{1}$.
Type $\mathfrak{C}_{12}$ is defined to be vector fields with $\mathbf{d}_{1}\neq 0$, $\mathbf{d}_{2}=0$ and $f_{1}=f_{2}=0$.
The first condition (43a) is trivially fulfilled, while the second condition (43b)
requires that $\mathbf{c}_{2}$ is parallel to $\mathbf{d}_{1}$, i.e. we obtain
$$\displaystyle V_{1(\mathfrak{C}_{12})}=V_{1}^{0}(t)\partial_{t}+c_{1}^{i}%
\partial_{i}+d_{1}^{i}L_{i}~{},\quad V_{2(\mathfrak{C}_{12})}=V_{2}^{0}(t)%
\partial_{t}+\kappa~{}d_{1}^{i}\partial_{i}~{},$$
(47)
where $\kappa\in\mathbb{R}$ is a constant.
Type $\mathfrak{C}_{22}$ is defined to be vector fields with $\mathbf{d}_{1}\neq 0$, $\mathbf{d}_{2}=0$, $f_{1}\neq 0$ and $f_{2}=0$.
Solving the second condition (43b) (therefore we have to use that the vectors are real!) we obtain
$$\displaystyle V_{1(\mathfrak{C}_{22})}=c_{1}^{i}\partial_{i}+d_{1}^{i}L_{i}+f_%
{1}x^{i}\partial_{i}~{},\quad V_{2(\mathfrak{C}_{22})}=V_{2}^{0}(t)\partial_{t%
}~{},$$
(48)
where we could set without loss of generality $V_{1}^{0}(t)$ to zero, as in type $\mathfrak{C}_{21}$. Note that $\mathfrak{C}_{21}$ is
contained in $\mathfrak{C}_{22}$ by violating the condition $\mathbf{d}_{1}\neq 0$.
Finally, we come to the last class, type $\mathfrak{C}_{32}$, defined by $\mathbf{d}_{1}\neq 0$, $\mathbf{d}_{2}=0$, $f_{1}=0$ and $f_{2}\neq 0$.
This class contains also the case $\mathbf{d}_{1}\neq 0$, $\mathbf{d}_{2}=0$, $f_{1}\neq 0$ and $f_{2}\neq 0$ by using the twist
conserving map $V_{1}\to V_{1}-\frac{f_{1}}{f_{2}}V_{2}$. The vector fields are given by
$$\displaystyle V_{1(\mathfrak{C}_{32})}=V_{1}^{0}(t)\partial_{t}+\frac{d_{1}^{j%
}c_{2}^{k}\epsilon_{jki}}{f_{2}}\partial_{i}+d_{1}^{i}L_{i}~{},\quad V_{2(%
\mathfrak{C}_{32})}=V_{2}^{0}(t)\partial_{t}+c_{2}^{i}\partial_{i}+f_{2}x^{i}%
\partial_{i}~{}.$$
(49)
Note that type $\mathfrak{C}_{11}$ and $\mathfrak{C}_{12}$ can be extended to a triangular $\star$-Hopf
subalgebra by choosing $V_{1}^{0}(t)=V_{2}^{0}(t)=0$ in each case.
For a better overview we additionally present the the results in table 1, containing all possible two vector field
deformations $\mathfrak{C}_{AB}$ of the Lie algebra of the euclidian group.
From this table the notation $\mathfrak{C}_{AB}$ becomes clear.
Next, we discuss solutions to the third condition (43c) $[V_{1}^{0}(t)\partial_{t},V_{2}^{0}(t)\partial_{t}]=0$.
It is obvious that choosing either
$V_{1}^{0}(t)=0$ or $V_{2}^{0}(t)=0$ and the other one arbitrary is a solution. Additionally, we consider solutions with $V_{1}^{0}(t)\neq 0$ and $V_{2}^{0}(t)\neq 0$.
Therefore there has to be some point $t_{0}\in\mathbb{R}$, such that without loss of generality $V_{1}^{0}(t)$ is unequal zero in some open region
$U\subseteq\mathbb{R}$ around $t_{0}$.
In this region we can perform the diffeomorphism $t\to\tilde{t}(t):=\int\limits_{t_{0}}^{t}dt^{\prime}\frac{1}{V_{1}^{0}(t^{%
\prime})}$
leading to $\tilde{V}_{1}^{0}(\tilde{t})=1$. With this the third condition (43c) becomes
$$\displaystyle 0=[V_{1}^{0}(t)\partial_{t},V_{2}^{0}(t)\partial_{t}]=[\tilde{V}%
_{1}^{0}(\tilde{t})\partial_{\tilde{t}},\tilde{V}_{2}^{0}(\tilde{t})\partial_{%
\tilde{t}}]=\Bigl{(}\partial_{\tilde{t}}\tilde{V}_{2}^{0}(\tilde{t})\Bigr{)}%
\partial_{\tilde{t}}~{}.$$
(50)
This condition is solved if and only if $\tilde{V}_{2}^{0}(\tilde{t})=\mathrm{const.}$ for $t\in U\subseteq\mathbb{R}$.
For the subset of analytical functions $C^{\omega}(\mathbb{R})\subset C^{\infty}(\mathbb{R})$ we can continue this condition
to all $\mathbb{R}$ and obtain the global relation $V_{2}^{0}(t)=\kappa V_{1}^{0}(t)$, with some constant $\kappa\in\mathbb{R}$.
For non analytic, but smooth functions, we can not continue these relations to all $\mathbb{R}$ and therefore only obtain local conditions restricting the functions in the overlap of their supports to be linearly dependent.
In particular non analytic functions with disjoint supports fulfill the condition (43c) trivially.
After characterizing the possible two vector field deformations of $\mathfrak{e}_{3}$ we briefly give a method how to obtain twists
generated by a larger number of vector fields. For this purpose we use the canonical form of $\theta$ (25).
Assume that we want to obtain deformations with e.g. four vector fields. Then of course all vector fields have to be of the form (42).
According to the form of $\theta$ we have two blocks of vector fields $(a,b)=(1,2)$ and $(a,b)=(3,4)$, in which the classification described
above for two vector fields can be performed. This means that all four vector field twists can be obtained by using two types
of two vector field twists.
We label the twist by using a tuple of types, e.g. $(\mathfrak{C}_{11},\mathfrak{C}_{22})$ means that $V_{1},V_{2}$ are of type $\mathfrak{C}_{11}$ and $V_{3},V_{4}$ of type $\mathfrak{C}_{22}$.
But this does only assure that $[V_{a},V_{b}]=0$ for $(a,b)\in\{(1,2),(3,4)\}$ and we have to demand further restrictions in
order to fulfill $[V_{a},V_{b}]=0$ for all $(a,b)$ and that all vector fields give independent contributions
to the twist. In particular twists constructed with linearly dependent vector fields can be reduced to a twist
constructed by a lower number of vector fields.
This method naturally extends to a larger number of vector fields, until we cannot find anymore independent and
mutually commuting vector fields. We will now give two examples for the $\mathfrak{e}_{3}$ case in order to clarify the construction.
As a first example we construct the four vector field twist $(\mathfrak{C}_{11},\mathfrak{C}_{11})$.
In this case all four vector fields commute without imprinting further restrictions. We assume that three of the four vectors
$\mathbf{c}_{a}$ are linearly independent, such that the fourth one, say $\mathbf{c}_{4}$, can be decomposed into the other ones.
If we now choose four linearly independent functions $V_{a}^{0}(t)$ (this means that they are non analytic) leads to a proper four vector field
twist.
As a second simple example we construct the four vector field twist $(\mathfrak{C}_{21},\mathfrak{C}_{21})$.
In order to have commuting vector fields we obtain the condition $c_{3}^{i}=\frac{f_{3}}{f_{1}}c_{1}^{i}$.
We therefore have $V_{3}=\frac{f_{3}}{f_{1}}V_{1}$ and the four vector field twist can be reduced to the two vector field twist
of type $\mathfrak{C}_{21}$ with $\tilde{V}_{1}=V_{1}$ and $\tilde{V}_{2}=V_{2}+\frac{f_{3}}{f_{1}}V_{4}$. This is an example
of an improper four vector field twist.
This method can be applied in order to investigate general combinations of two vector field twists, if one requires them.
Because this construction is straightforward and we do not require these twists for our discussions, we do not present them here.
At the end we calculate the $\star$-commutator of the linear coordinate functions $x^{\mu}\in A_{\star}$ for the various types of models
in first order in the deformation parameter $\lambda$. It is given by
$$\displaystyle c^{\mu\nu}:=[x^{\mu}\stackrel{{\scriptstyle\star}}{{,}}x^{\nu}]:%
=x^{\mu}\star x^{\nu}-x^{\nu}\star x^{\mu}=i\lambda\theta^{ab}V_{a}(x^{\mu})V_%
{b}(x^{\nu})+\mathcal{O}(\lambda^{2})~{}.$$
(51)
The results are given in appendix B and show that these commutators can be at most quadratic in
the spatial coordinates $x^{i}$.
Possible applications of these models will be discussed in the outlook, see section 7.
6 Application to Black Holes
In this section we investigate possible deformations of non rotating black holes.
We will do this in analogy to the cosmological models and therefore do not have to explain every single step.
The undeformed Lie algebra of the symmetry group $\mathbb{R}\times SO(3)$ of a non rotating black hole is generated by
the vector fields
$$\displaystyle p^{0}=\partial_{t}\quad,\quad L_{i}=\epsilon_{ijk}x^{j}\partial_%
{k}~{},$$
(52)
given in cartesian coordinates. We choose $t_{i}^{\star}=t_{i}$ for all $i$
and define $\mathfrak{g}_{\star}=\mathfrak{g}=\mathrm{span}(p^{0},L_{i})$.
It can be shown that each twist vector field $V_{a}$ has to be of the form
$$\displaystyle V_{a}=(c^{0}_{a}(r)+N_{a}^{0}t)\partial_{t}+d_{a}^{i}L_{i}+f_{a}%
(r)x^{i}\partial_{i}$$
(53)
in order to fulfill $[V_{a},\mathfrak{g}]\subseteq\mathfrak{g}$. Here $r=\|\mathbf{x}\|$ is the euclidian
norm of the spatial position vector.
The next task is to construct the two vector field deformations. Therefore we additionally have to
demand $[V_{a},V_{b}]=0,~{}\forall_{a,b}$, leading to the conditions
$$\displaystyle d_{a}^{i}d_{b}^{j}\epsilon_{ijk}=0~{},~{}\forall_{k}~{},$$
(54a)
$$\displaystyle(f_{a}(r)x^{j}\partial_{j}-N_{a}^{0})c^{0}_{b}(r)-(f_{b}(r)x^{j}%
\partial_{j}-N_{b}^{0})c^{0}_{a}(r)=0~{},$$
(54b)
$$\displaystyle f_{a}(r)f_{b}^{\prime}(r)-f_{a}^{\prime}(r)f_{b}(r)=0~{},$$
(54c)
where $f_{a}^{\prime}(r)$ means the derivative of $f_{a}(r)$.
Note that (54c) is a condition similar to (43c), and therefore
has the same type of solutions. Because of this, the functions $f_{1}(r)$ and $f_{2}(r)$ have to be parallel in the overlap
of their supports. From this we can always eliminate locally one $f_{a}(r)$
by a twist conserving map and simplify the investigation of the condition (54b).
At the end, the local solutions have to be glued together.
We choose without loss of generality $f_{1}(r)=0$ for our classification of local solutions.
The solution to (54a) is that the $\mathbf{d}_{a}$ have to be parallel. We use
$$\displaystyle\mathbf{d}_{a}=\kappa_{a}\mathbf{d}$$
(55)
with constants $\kappa_{a}\in\mathbb{R}$ and some arbitrary vector $\mathbf{d}\neq 0$.
We now classify the solutions to (54b) according to $N_{a}^{0}$ and $f_{2}(r)$ and label them by $\mathfrak{B}_{AB}$.
We distinguish between $f_{2}(r)$ being the zero function or not. The result is shown in table 2.
Other choices of parameters can be mapped by a twist conserving map into these classes.
Note that in particular for analytical functions $f_{a}(r)$ the twist conserving map transforming $f_{1}(r)$ to zero
can be performed globally, and with this also the classification of twists given in table 2.
In type $\mathfrak{B}_{32}$ we still have to solve a differential equation for $c_{1}^{0}(r)$ given by
$$\displaystyle c_{1}^{0}(r)=\frac{f_{2}(r)}{N_{2}^{0}}rc_{1}^{0\prime}(r)~{},$$
(56)
for an arbitrary given $f_{2}(r)$.
We will not work out the solutions to this differential equation, since type $\mathfrak{B}_{32}$ is a quite unphysical model,
in which the noncommutativity is increasing linear in time due to $N_{2}^{0}\neq 0$.
Note that $\mathfrak{B}_{11}$ can be extended to a triangular $\star$-Hopf algebra by choosing $c_{a}^{0}(r)=c_{a}^{0}$, for $a\in\{1,2\}$.
In addition, $\mathfrak{B}_{12}$ is a $\star$-Hopf algebra for $\kappa_{1}=\kappa_{2}=0$.
The $\star$-commutators $c^{\mu\nu}=[x^{\mu}\stackrel{{\scriptstyle\star}}{{,}}x^{\nu}]$ of the coordinate functions $x^{\mu}\in A_{\star}$ in order $\lambda^{1}$ for these
models are given in the appendix B. They can be used in order to construct sensible
physical models of a noncommutative black hole.
By using the method explained in the previous section, the two vector field twists can be extended to multiple vector field
twists. Since we do not require these twists in our work and their construction is straightforward, we do not
present them here.
7 Conclusion and Outlook
We have discussed symmetry reduction in noncommutative gravity using
the formalism of twisted noncommutative differential geometry. Our
motivation for these investigations derives from the fact that, for
most physical applications of gravity theories, including cosmology,
symmetry reduction is required due to the complexity of such models,
already in the undeformed case.
In section 3 we have presented a general method for
symmetry reduction in twisted gravity theories. As a result we have
obtained restrictions on the twist, depending on the structure of the
twisted symmetry group. In particular, we find that deforming the
infinitesimal symmetry transformations results in weaker restrictions
than deforming the finite transformations and demanding a quantum
group structure. In section 4 we have applied this
general method to gravity theories twisted by Reshetikhin-Jambor-Sykora twists.
These are twists constructed from commuting vector fields. In this
case we could give explicit conditions, which have to be fulfilled in
order to allow symmetry reduction of a given Lie group.
In sections 5 and 6 we have
investigated admissible deformations of FRW and black hole symmetries
by a Reshetikhin-Jambor-Sykora twist. In this class we have classified all
possible deformations. This lays the foundation for phenomenological
studies of noncommutative cosmology and black hole physics based on
twisted gravity.
In a forthcoming work [15] we will
investigate cosmological implications of twisted FRW models by
studying fluctuations of quantum fields living on twisted FRW
backgrounds. Quantum fields were already introduced in a twisted
framework in [16]. As we see from
proposition 4, the noncommutative backgrounds
are also invariant under the undeformed action of the classical
symmetry. This means that they have the same coordinate
representations with respect to the undeformed basis vectors as the
commutative fields in Einstein gravity. With this we have a construction
principle for noncommutative backgrounds, in their natural basis,
by representing the classical fields in the deformed basis.
A class of models of particular interest is
type $\mathfrak{C}_{22}$ in section 5
(cf. table 1). These twists break classical translation
invariance, but classical rotation invariance can be retained by
tuning $\mathbf{d}_{1}$ and $\mathbf{c}_{1}$ to small values. Furthermore,
the global factor $V_{2}^{0}(t)$ in the exponent of the twist can be used
in order to tune noncommutativity effects depending on time.
Obviously, enforcing a suitable $V_{2}^{0}(t)$ by hand leads to
phenomenologically valid models.
Since there is no natural choice of $V_{2}^{0}(t)$, it is interesting to
investigate the dynamics of $V_{2}^{0}(t)$ in a given field configuration
and study if it leads to a model consistent with cosmological
observations. In this case, the model would be physically
attractive. This will also be subject of future
work [15]. Dynamical noncommutativity has
already been studied in the case of scalar field theories on Minkowski
spacetime [17].
In the case of black hole physics, models of particular interest would
be $\mathfrak{B}_{11}$ with functions $c^{0}_{a}(r)$ decreasing
sufficiently quickly with $r$ and $\mathfrak{B}_{12}$ with $f_{2}(r)$
and $c_{2}^{0}(r)$ decreasing sufficiently quickly with $r$
(cf. table 2). It will again be interesting to
investigate the dynamics of these functions on a given field
configuration. Note that the type $\mathfrak{B}_{12}$
with $\kappa_{1}=\kappa_{2}=0$ is invariant under the classical black hole
symmetries, and therefore particularly interesting for physical
applications. On the other hand, models with nonvanishing $N_{a}^{0}$ are
of little physical interest, because the noncommutativity is growing
linearly in time, which would be unphysical.
Other avenues for future work are the classification of models on
nontrivial topologies (like, e.g., $\mathbb{R}\times S_{3}$ in
cosmology), investigating nontrivial
embeddings $t_{i}^{\star}=t_{i}^{\star}(t_{j})$ and using a wider class of twist
elements.
Acknowledgements
AS thanks Christoph Uhlemann and Julian Adamek for discussions and
comments on this work. This research is supported by Deutsche
Forschungsgemeinschaft through the Research Training Group 1147
Theoretical Astrophysics and Particle Physics.
Appendix A Proof of Proposition 3
In this appendix we show that for Reshetikhin-Jambor-Sykora twists (24) the conditions (21) necessary for
extending the $\star$-enveloping subalgebra $(U\mathfrak{g}_{\star},\star)$ to a $\star$-Hopf subalgebra are
equivalent to the simplified conditions of proposition 3.
The plan is as follows: we use (21a) and show that
it is equivalent to the conditions of proposition 3. In a second step, we show that (21b)
is automatically satisfied if (21a) is fulfilled, and thus does not lead to additional conditions.
We start with (21a) and show the identity
$$\displaystyle\Delta_{\star}(U\mathfrak{g}_{\star})\subseteq U\mathfrak{g}_{%
\star}\otimes U\mathfrak{g}_{\star}~{}\Leftrightarrow~{}\Delta_{\star}(%
\mathfrak{g}_{\star})\subseteq U\mathfrak{g}_{\star}\otimes U\mathfrak{g}_{%
\star}~{}.$$
(57)
The direction $\Rightarrow$ is trivial, since $\mathfrak{g}_{\star}\subset U\mathfrak{g}_{\star}$, and the direction $\Leftarrow$ can be shown
using that $\Delta_{\star}$ is a $\star$-algebra homomorphism and that $U\mathfrak{g}_{\star}$ closes under $\star$-multiplication.
Furthermore, using (6) we obtain
$$\displaystyle\Delta_{\star}(\mathfrak{g}_{\star})\subseteq U\mathfrak{g}_{%
\star}\otimes U\mathfrak{g}_{\star}~{}\Leftrightarrow~{}X_{\bar{R}^{\alpha}}%
\in U\mathfrak{g}_{\star}~{},~{}~{}\text{for all }\alpha\text{ with }\bar{R}_{%
\alpha}(\mathfrak{g}_{\star})\neq\{0\}~{}.$$
(58)
Therefore we have to use that all $\bar{R}^{\alpha}$ are linearly independent, that $X$ is a vector space isomorphism [10]
and that we have $\bar{R}_{\alpha}(\mathfrak{g}_{\star})\subseteq\mathfrak{g}_{\star}$,
due to the minimal axioms (17).
Additionally, we can show that $X_{\bar{R}^{\alpha}}=\bar{f}^{\beta}\bar{R}^{\alpha}\chi S^{-1}(\bar{f}_{\beta%
})=\bar{R}^{\alpha}$. This is done
by applying the explicit form of the twist (24) and using that the $V_{a}$ mutually commute.
Next, we show that
$$\displaystyle\bar{R}^{\alpha}\in U\mathfrak{g}_{\star}~{},~{}~{}\text{for all %
}\alpha\text{ with }\bar{R}_{\alpha}(\mathfrak{g}_{\star})\neq\{0\}~{}%
\Leftrightarrow~{}\theta^{ba}V_{b}\in\mathfrak{g}_{\star}~{},~{}~{}\text{for %
all }a\text{ with }[V_{a},\mathfrak{g}_{\star}]\neq\{0\}~{}.$$
(59)
The direction $\Rightarrow$ is trivial, since the RHS is a special case of the LHS.
The direction $\Leftarrow$ can be shown by using that the $V_{a}$ mutually commute and the explicit expression of the $R$-matrix
(26).
Finally, the RHS of (59) is equivalent to the condition of proposition 3 by
using the canonical form of $\theta$ (25).
Next, we show that (21b) is satisfied, if (21a) is fulfilled.
For this we use that for Reshetikhin-Jambor-Sykora twists we have $\chi=f^{\alpha}S(f_{\alpha})=1$, which leads to the identity
$$\displaystyle S_{\mathcal{F}}(\xi)=\chi S(\xi)\chi^{-1}=S(\xi)=S^{-1}(\xi)=S_{%
\mathcal{F}}^{-1}(\xi)~{},~{}\forall\xi\in U\Xi$$
(60)
for the antipode in the $\mathcal{F}$-Hopf algebra.
This property translates to the $\star$-Hopf algebra, since it is isomorphic to the $\mathcal{F}$-Hopf algebra and we obtain
the following equivalences of (21b)
$$\displaystyle S_{\star}(U\mathfrak{g}_{\star})\subseteq U\mathfrak{g}_{\star}~%
{}\Leftrightarrow~{}S_{\star}(\mathfrak{g}_{\star})\subseteq U\mathfrak{g}_{%
\star}~{}\Leftrightarrow~{}S^{-1}_{\star}(\mathfrak{g}_{\star})\subseteq U%
\mathfrak{g}_{\star}~{}.$$
(61)
For the first equivalence we had to use that $S_{\star}$ is a $\star$-anti homomorphism.
Using the RHS of (59), which is equivalent to (21a), and the definition of $S_{\star}^{-1}$
(6), we obtain
$$\displaystyle S^{-1}_{\star}(\mathfrak{g}_{\star})=-\sum_{n=0}^{\infty}\frac{(%
-i\lambda)^{n}}{n!}\theta^{a_{1}b_{1}}\cdots\theta^{a_{n}b_{n}}[V_{a_{1}},%
\cdots,[V_{a_{n}},\mathfrak{g}_{\star}]\cdots]~{}V_{b_{1}}\cdots V_{b_{n}}\in U%
\mathfrak{g}_{\star}~{},$$
(62)
where we have used $\xi\star V_{a}=\xi V_{b}$ for all $\xi\in U\Xi_{\star}$, since the action of the twist on $V_{a}$ is trivial.
Appendix B $\star$-Commutators of the Coordinate
Functions in FRW and Black Hole Models
In tables 3 and 4, we list the $\star$-commutators among the linear
coordinate functions to order $\lambda^{1}$ in the FRW and black hole
models. In these expressions, $(i\leftrightarrow j)$ denotes the
same term with $i$ and $j$ interchanged.
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On the origin of the radial flow in low energy heavy ion reactions
Regina Nebauer${}^{*+}$ and Jörg Aichelin${}^{*}$
${}^{*}$SUBATECH
Laboratoire de Physique Subatomique et des Technologies Associées
UMR Universitè de Nantes, IN2P3/CNRS, Ecole des Mines de Nantes
4, rue Alfred Kastler
F-44070 Nantes Cedex 03, France.
${}^{+}$ Universität Rostock, Rostock, Germany
Abstract
The average transverse energy of nucleons and intermediate mass fragments
observed in the heavy ion reaction Xe(50A MeV)+Sn shows the same linear increase as a
function of their mass as observed in heavy ion collisions up to the highest energies
available today and fits well into the systematics. At higher energies this
observation has been interpreted as a sign of a strong radial flow in an
otherwise thermalized system. Investigating the reaction with Quantum Molecular
dynamics simulations we find in between 50A MeV and 200A MeV a change in the
reaction mechanism. At 50A MeV the apparent radial flow is merely
caused by an in-plane flow and Coulomb repulsion. The average transverse
fragment energy does not
change in the course of the reaction and is equal to the initial fragment energy
due to the Fermi motion.
At 200A MeV, there are two kinds of fragments: those formed from spectator matter and those
from the center of the reaction. There the transverse energy is caused by the pressure
from the compressed nuclear matter. In both cases we observe a binary event structure,
even in central collisions. This demonstrates as well the non thermal character
of the reaction. The actual process which leads to multifragmentation is rather
complex and is discussed in detail.
I Introduction
It is known since long that for almost all particles observed in heavy ion
reactions between $30A~{}MeV$ and $200~{}A.GeV$
the transverse kinetic energy spectra have a
Maxwell-Boltzmann form, predicted for an emission from an equilibrated source.
However, the apparent temperature of the spectra and hence the average
kinetic energy of the particles is quite different for different hadrons and
fragments and increases with increasing mass and increasing energy. This
observation seemed to exclude an identification of the apparent temperature
with the real temperature of the system.
Recently it has been conjectured [1, 2, 3] that at all energies
between $50A~{}MeV$ and $200A~{}GeV$ the assumption of a strong radial flow can
reconcile the mass dependence of the apparent temperature with
thermodynamics. At relativistic
and ultra-relativistic energies this has been inferred by comparing transverse
pion, kaon and proton spectra [1]. At energies below
$500A~{}MeV$ the lever arm is still larger because one can include the
intermediate
mass fragments (IMF’s) of masses in between 2 and 10 [2, 3], emitted
at mid rapidity, to separate radial flow and temperature. The deviations in forward
and backward direction are usually interpreted as preequilibrium emission.
This observation has renewed the interest in the thermal analysis of heavy ion
reactions in the high as well as in the low energy heavy ion community
[2, 4]. It revealed a very large value for
the radial flow velocity (up to 40% of the speed of light) and a very peculiar
beam energy dependence
[4] which
has been discussed as a possible sign for a
transition from a hadronic phase to a Quark Gluon Plasma phase.
Especially at low beam energies the observed
value of the radial velocity in units of the speed of light of
$.05\leq\beta_{r}/c\leq 0.2$ [2] (the exact value depends on the event
selection) is hard to understand in terms of physical processes because
none of the other observables present evidence that the system is sufficiently
compressed for attributing the radial flow to an equation of state effect.
The very moderate in-plane flow is negative (as in deep inelastic collisions)
and points therefore to an other
origin than compression and subsequent release of the compressional energy,
in contradistinction to higher energies where the in-plane flow is positive.
Nevertheless, the low energy points follow smoothly the above mentioned
systematics.
This radial flow at 50A MeV has been reproduced[5]
by simulations using the Quantum Molecular Dynamics (QMD) [6]
approach. The same is true at higher energies
where QMD simulations have been performed by the FOPI collaboration
[3].
It is the purpose of this article to take advantage of this agreement and to
study the origin of the large increase of the transverse energy with the
mass and hence the origin of the radial flow. Here we concentrate
ourselves on the reaction at 50A MeV and study in detail how the fragments
are produced. The reaction mechanism at this low energy is quite different than
that at higher energies. To demonstrate this we present as well the same study
for a beam energy of 200A MeV.
A detailed study of the radial flow at higher energies, its equation of state
dependence and its physical origin will be the subject of a
forthcoming publication [7].
For our study at $(50A~{}MeV)$
we use simulations of the reaction $Xe+Sn$
which has recently been measured by the INDRA collaboration. The INDRA detector
at GANIL has been constructed to study
multifragmentation and therefore the angular coverage and the
energy thresholds
have been chosen to be better than that of any other $4\pi$ detector
elsewhere. Hence the data taken with this detector are most suitable
to confirm or disprove the theories embedded in the simulation programs.
A detailed comparison of our results with the experimental data will
be published elsewhere [5]. Here we mention that not only the mass
dependence of the average kinetic energy but also the kinetic energy spectra
themselves are in reasonable agreement with experiment.
For details about the QMD approach we refer to reference [6]. In
this program the nucleons are represented by Gaussian wave packets with a
constant width. The time evolution of the centers of these wave packets
is given by Euler Lagrange equations derived from the Lagrangian of the system.
The nucleons have an effective charge of Z/A. The fragment distribution becomes
stable after 240 fm/c [8]. From this point on we employ a Coulomb
trajectory
program for the fragments and nucleons until the Coulomb energy is released.
This second step is necessary at this low energy because a large fraction of
the kinetic energy of the fragments is due to the Coulomb energy.
II The radial flow at beam energies of 50A MeV
A first general idea of the time evolution of the collision can be obtained
from the time evolution of the density of the system.
If the maximal density is reached the nuclei have their maximal overlap, after
that the system
expands and the density decreases. This permits to find the time scale of
the reaction.
The total density is
the sum over all nucleons which are described by Gaussians:
$$\rho(\vec{r},t)\propto\sum_{i=1}^{A}e^{-\frac{(\vec{r}-\vec{r_{i}}(t))^{2}}{2{%
L}}}$$
(1)
The width of the Gaussians is $4L=4.33~{}fm^{2}$ and A is the number of nucleons
present in the system.
In figure 1, left hand side, we plot the time evolution of the
total density in the center of the reaction.
The maximum density is obtained at $\approx 50~{}fm/c$, on
the same time scale the
system expands and reaches at $120~{}fm/c$ a low density phase where the
fragments do not interact anymore.
On the right hand side of the same figure we display the density profile along the
beam (z) axis. Here we can follow the two nuclei. They occupy the same coordinate
space at
$50..60~{}fm/c$. The system expands after $120~{}fm/c$. We find
that this quasi-central ($b=3~{}fm$) collision is semi transparent. Projectile
and target pass through each other without being seriously decelerated.
For $b=0~{}fm$ we get the same result.
That binary character is confirmed by experiment:
In the center of mass system the experiment
shows
a flat
angular distribution ($dN/dcos\theta_{cm}$) between
$60^{\circ}\leq\theta_{cm}\leq 120^{\circ}$ as well as
a constant average kinetic
energy for fragments $Z\geq 3$ even in central collisions.
In forward and backward direction a strongly enhanced cross section
is observed. The INDRA collaboration made use of this observation and
presented their data in two angular bins:
$60^{o}~{}\leq~{}\theta_{CM}~{}\leq~{}120^{o}$ (IMF’s emitted in this angular range are called
mid-rapidity fragments (MRF’s)) and $\theta_{CM}~{}<~{}60^{o},\theta_{CM}~{}>~{}120^{o}$ called
projectile/target like fragments (PTF’s) [2].
How do the transverse fragment momenta reflect this passage through the other nucleus?
This is displayed in
figure 2 for a b = 3 fm reaction. Here we show the time evolution
of the average transverse momentum of
all fragments with $5\leq A\leq 10$ (left) and $A>10$ (right). For the
dotted points the transverse momentum is represented with respect to the beam
direction, for the triangles with respect to the largest eigenvector of the
momentum tensor which is tilted by the flow angle $\theta_{flow}$.
It is the first seminal result of this article that in the rotated system the
average transverse momentum is initially and finally the same. A very similar
result we obtain for b = 1 and b = 5 fm. During the reaction
the particles are accelerated in transverse direction but later they
feel a force into the opposite direction. The origin of this acceleration will
be discussed later.
As squares we display the time evolution of the in-plane
flow ${1\over N_{F}}\sum_{N_{F}}sign(p^{cm}_{z})p_{x}$. $N_{F}$ is the number of
fragments. We see that it remains moderate and is negative.
The average transverse energy of the fragments as a function of the fragment
charge is
displayed in fig. 3. On the left hand side we display the transverse
energy with respect to the beam axis, on the right hand side with respect to
the rotated system. The mean value is presented after the initialization, after
250 fm/c, when the fragment distribution gets stable, and finally after the
mutual Coulomb repulsion has ceased.
It is the second seminal result of this article that the average transverse
fragment energy initially is about 14 MeV and independent of the fragment size.
As Goldhaber
[9] has pointed out many years ago this is expected if the
fragment formation is a fast process. In this case the momentum distribution of
the fragments is a convolution of the momentum distribution of the
entrained nucleons and one expects for the
fragment momentum squared
$$P^{2}=<(\sum_{i}^{A}p_{i})^{2}>={3k_{Fermi}^{2}\over 5}A{N-A\over N-1}$$
where A is the number of nucleons entrained in the fragment and N is the
number of nucleons of the disintegrating nucleus. Hence the fragment
kinetic energy is
$$E_{A}={3E_{Fermi}\over 5}{N-A\over N-1}$$
and as well about 14 MeV in the QMD calculations. (Due to surface effects it
differs from the value of a Fermi gas in a square wall potential).
How it can happen that the fragments pass the reaction zone without
being heated up and are finally observed at midrapidity or in PTF’s
we will investigate in the second part of this article.
The nucleons interact via the potential
$$V=-124\frac{\rho}{\rho_{0}}+70.5(\frac{\rho}{\rho_{0}})^{2}~{}[MeV]$$
(2)
where the density is given by equation 1 and $\rho_{0}$ is the normal
nuclear matter density.
In order to reveal the physics which drives the reaction we display the relative
density of
those nucleons which are finally entrained in MRF’s or PTF’s as a function of time in
the x-z plane
$$\rho^{\tiny MRF/PTF}_{rel}(x,z,t)={\rho_{\tiny MRF/PTF}(x,z,t)\over\rho_{total%
}(x,z,t)}$$
(3)
and superimpose the gradient of the potential in the x-z plane as arrows where
x is the direction of the impact parameter.
For the sake of a clearer display we plot nucleons coming
from the projectile only. We start out with a discussion of the reaction
at zero impact parameter then we continue with $b=3fm$ reactions.
The motion of the nucleons in the potential of a nucleus is a sequence of acceleration
and deceleration. Nucleons on the surface are almost at rest, due to the density (and
thus the potential) gradient they become accelerated towards the center of the nucleus.
They
reach their maximal momentum when they pass
the center of the nucleus, climb up the potential on the
other side and are finally
at rest again when arriving at the surface.
When a heavy ion collision occurs, the position of
the nucleons in the projectile or target determines whether they ”feel”
the heavy ion collision right from the beginning or only when the
high density phase has already passed. We will show that the initial position
of the entrained nucleons decides
as
well whether the fragment is finally observed at mid rapidity or in forward/
backward direction.
In figure 4 we display the motion of the nucleons finally entrained in
PTF’s for a reaction at $b=0~{}fm$. The spatial distribution of those nucleons
is almost identical with
that of all nucleons present in the projectile.
In the first step of the collision the nucleons move away from the target
into the yet unperturbed part of the projectile. When they arrive
at the back end of the projectile they invert the direction of their momenta.
They are then accelerated in longitudinal direction towards the
center of the reaction. When they arrive finally there the high density
zone has disappeared already.
Hence the nucleons pass the center without a larger change of their initial
momentum. The initial correlations [10] among the nucleons which
finally form a fragment survive the reaction because
all potential gradients are small.
Nucleons finally emitted as MRF’s (fig. 5) are strongly
located at the front end of the nuclei. These are the nucleons which
climb the
nuclear potential created by the higher density in the reaction zone and which are
at rest before they are on the top of the potential wall.
Due to their position they are involved in
the collisions between projectile and target nucleons right from the beginning.
Collisions support the
deceleration. (Later collisions are to a large extend Pauli suppressed.)
They escape the barrier
in transverse direction.
As their (longitudinal as well as
transverse)momentum
is quite small, the nucleons stay longer in the
center of the reaction what favours the mixing of projectile and target
nucleons.
When leaving the reaction zone the fragments become decelerated due to the
potential interaction with the rest of the system. This
deceleration balances
the gain in energy due to the prior acceleration in transverse direction,
although the
physics of both processes is rather independent.
The same analysis for PTF’s for $b=3~{}fm$ is displayed in figure 6.
The general reaction mechanisms, even the time scales,
are the same as discussed before. In addition to the scenario at $b=0~{}fm$
asymmetry
effects occur here. Up to $40~{}fm/c$ the scenario is the same as at
zero impact parameter. When the high density phase occurs, the nucleons take the line of
least resistance, i.e. they follow the minimum of the potential on the right hand
side (for the projectile nucleons, the target nucleons take the inverse direction on the
other side). The larger part of the nucleons pass the reaction center when the
potential barrier has disappeared. As already discussed for zero impact
parameter case, the
nucleons emitted as fragments in forward/backward direction pass through the
reaction zone without an important change of their initial momenta.
For the reaction Xe(50A MeV)+Sn we could find two modes for the emission of fragments:
1.
The Fragments are already ”preformed” in the nuclei. The initial correlations
between the nucleons survive as they avoid the high density zone. For a finite impact
parameter a strong in-plane flow is observed. The fragments are quite big and emitted in
forward/backward direction.
2.
Fragments emitted in midrapidity are smaller. They are formed by mixing projectile
and target nucleons. Due to their initially small longitudinal momentum they stay in the
high density zone and emitted in transverse direction. These fragments are rather small as
they are formed via collisions.
III The radial flow at beam energies of 200A MeV
In the reaction at 200 A MeV the velocity of the nucleons due to
internal motion and the velocity of the two colliding nuclei in the center of mass frame
are now significantly different. As a consequence we expect a change of the reaction
mechanism.
That this is indeed the case can already be seen from the time evolution of the
transverse momenta of the fragments, figure 7. The collective directed in-plane
flow has the opposite sign as compared to 50A MeV. This is as well a sign for the
change of the reaction mechanism which develops between 50A MeV and 200A MeV from a
deep inelastic scenario
with a negative flow angle to a bounce off type reaction with a positive flow angle.
Despite of the higher beam
energy the transverse acceleration of the light
fragments is delayed as compared to the reaction at 50A MeV and sets on only
at the maximum overlap of the two nuclei. Then they are
accelerated and remain with a high transverse momentum after the expansion.
The large fragments have a completely different time evolution as compared to
50A MeV. First we see a deceleration in transverse direction of the
fragments. Then, when the highest density is reached this turns into an
acceleration. Finally, as at lower energies, the
fragments are decelerated again during the expansion of the system. Finally they have
a quite low transverse momentum.
To understand this behaviour it is useful to study the initial-final state correlations.
Particularly in momentum space we observe such strong correlations. In figure 8 we
display the initial distribution of the nucleons at t=0 (color level) and superimpose
the relative fraction of nucleons finally emitted in small (lhs) or big (middle, rhs) fragments.
Here we can clearly see that the nucleons finally emitted in heavy fragments have
an initial momentum which points away from the reaction partner. This initial - final -
state correlation, already
allusively present at 50A MeV increases with the mass of the fragment and the beam energy
[10].
For nucleons finally
emitted in small fragments the correlations are less pronounced but with preference of momenta
which point into the reaction zone what favours the mixing of projectile and
target nucleons in the fragments.
Nucleons emitted finally in small fragments are located in the center of the collision
partners and become only accelerated when a higher density is built up in the center of the
reaction. Following the density gradient they develop a radial flow. Finally they
disentangle form the system. This costs energy and therefore the momentum decreases.
We expect a constant acceleration of all fragment nucleons and therefore
a final transverse energy which increases linearly with the fragment mass.
The light fragments show only a small in-plane-flow which is positive in contrast to the
reaction at the lower energy. In contradistinction to 50A MeV we observe
for the light fragments a genuine acceleration in transverse direction independent of the
system in which it is measured. At 200A MeV the density built up in the center of the reaction
is sufficiently high for a noticeable acceleration of the fragments.
We see that especially for the heavy fragments that the initial transverse momentum in
the rotated system is quite different as compared to the beam system due to the initial
correlations in momentum space, discussed above.
For a better understanding of the time evolution of the momenta of the big fragments we
study as at 50A MeV the movement of their constituent nucleons in the nuclear potential,
figure 9. Here we display the gradient of the nuclear potential and the
relative fraction of nucleons emitted in fragments $A>10$.
In addition to their strong initial - final - state correlations in momentum space
the fragments have as well strong correlations in coordinate space. The big fragments
are at t=0 located at the surface of the collisions partners and have
therefore an average density well below the normal nuclear matter density. At the beginning
of the reaction
they are accelerated towards the center of the reaction because the density is
higher there.
This lowers the transverse momentum. When the high density at the center of the reaction is
built up, the acceleration changes sign. Thus we can already at 200A MeV and 3 fm
impact parameter distinguish between participants and spectators.
Now we can draw up two competing processes of fragment formation at 200A MeV:
1.
Large fragments are formed from spectator matter. The entrained nucleons do not pass
the region of high density and therefore the acceleration in transverse direction is moderate.
For the entrained nucleons we observe as well strong initial - final - state correlations
in momentum space. In other words, the more the nucleons avoid the high density zone the larger
is the change that they find themselves finally in big fragments. At 200A MeV the
probability that nucleons passes the high density zone without having a collision is well
reduced with respect to 50A MeV. Therefore any larger correlated part of nuclear matter which
passes the high density becomes destroyed by collisions and cannot form finally a fragment
anymore.
2.
Small correlated parts of nuclear matter can still survive the passage through the high
density zone. That will be different at still higher energies. These
nucleons form small fragments mixing projectile and target
via collisions. The nucleons finally emitted in this fragments are close to
the high density zone and accelerated due to the internal pressure. Their
transverse energy is high and expected to be proportional to their mass (radial flow).
These two competing processes allow to understand
the behaviour of the transverse energy as a function of the
charge (figure 10,lhs). At t=0, the beginning of the reaction, the strong
initial correlations in
momentum space (high transverse momentum for big fragments)
cause a linear increase of the transverse energy with the charge of the
fragment. For small fragments the process 2 dominates, finally the
transverse energy of fragments is large and increases with increasing charge (mass) of the
fragment.
For big fragments the process 1 is dominant, the average transverse
energy is small. In between we observe a transition phase.
On the rhs of the same figure we display as in figure 3 the transverse
energy of the fragments in the rotated systems which is almost identical with the left hand
side. Hence the radial energies are genuine and not artifacts due to a strong in-plane flow as
at 50A MeV.
IV Conclusion
In conclusion we have found that the kinetic energy of the fragments observed in
the heavy ion reaction $Xe+Sn$ at $50A~{}MeV$ reflects the initial Fermi energy of
the entrained nucleons. There is no hint that the system becomes equilibrated. Rather, in
agreement with experiment,
even in central collisions the reaction is semi transparent. We find that the final
longitudinal momentum of the fragments depends strongly on the time point when the
entrained nucleons pass the center of the reaction.
Nucleons which are passing early in the reaction are decelerated
due to collisions with the reaction partner and due to the strong potential gradient.
They are deviated into the transverse direction as their longitudinal momentum is not
high enough to
overcome the potential barrier formed in the reaction center in the phase of
highest density.
Nucleons which arrive later
do not encounter a strong potential gradient anymore and pass the reaction center
freely. Thus, they keep almost their initial momentum. We find that
compressional effects have little influence on the final momentum of the
fragments at this energy and the observed apparent radial flow is
not real. The linear increase of the fragment
kinetic energy with the mass for small fragments
finds its natural explication in terms of the initial Fermi motion,
the Coulomb barrier and a small in-plane flow.
At 200A MeV we found that the reaction mechanism has changed and approaches the
participant-spectator model as far as heavy fragments are concerned. Due to the smaller mean
free path heavy fragments cannot come anymore form the center of the reaction. Their
transverse energy remains therefore moderate and the in-plane flow weak.
Small fragments show a strong radial flow and have passed the center of the reaction
where the forces are much stronger than at 50A MeV.
At higher energies fragments formed from of participants become more and more rare due to
the shorter mean free path and due to stronger force gradients. Hence fragment formation
of spectator matter will dominate. Initial - final - state correlations in momentum space become
stronger and collisions start to contribute in addition to the radial
flow. How these different process influence the final radial flow and its dependence on the
nuclear equation of state is presently under investigation [7].
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Abstract
We consider in this paper a spray constituted of an incompressible
viscous gas and of small droplets which can breakup. This spray is
modeled by the coupling (through a drag force term) of the
incompressible Navier-Stokes equation and of the Vlasov-Boltzmann
equation, together with a fragmentation kernel. We first show at
the formal level that if the droplets are very small after the
breakup, then the solutions of this system converge towards the
solution of a simplified system in which the small droplets produced
by the breakup are treated as part of the fluid. Then, existence of
global weak solutions for this last system is shown to hold, thanks
to the use of the DiPerna-Lions theory for singular transport
equations.
Existence theory for the kinetic-fluid coupling when small droplets are
treated as part of the fluid
S. BENJELLOUN111CMLA, ENS Cachan & CNRS, 61 av du Président Wilson, 94235, Cachan, Cedex, France., L. DESVILLETTES111CMLA, ENS Cachan & CNRS, 61 av du Président Wilson, 94235, Cachan, Cedex, France., AND A. MOUSSA222UPMC Université Paris 06 & CNRS, UMR 7598, LJLL, F-75005, Paris & INRIA Paris-Rocquencourt, Équipe-Projet REO, BP 105, F-78153 Le Chesnay Cedex, France.
1 Introduction
Sprays are complex flows which are constituted of an underlying gas in
which a population of droplets (or dust specks) are dispersed,
cf. [18]. There are various possibilities for modeling
such flows, depending in particular on the volume fraction of the
liquid phase (cf. [9] for example).
We focus here on the case when the volume fraction occupied by the
droplets is small enough to be neglected in the equations (such sprays
are called thin sprays, cf. [18]), so that the
modeling of the liquid phase can be performed by the use of a pdf
(particles density function) which solves Vlasov-Boltzmann
equation (cf. [21, 1]). Denoting $f:=f(t,\mathbf{x},\boldsymbol{\xi},r)\geq 0$ the number density of droplets of radius $r$ which at
time $t$ and point $\mathbf{x}$ have velocity $\boldsymbol{\xi}$, the Vlasov equation
writes
$$\partial_{t}f+{\boldsymbol{\xi}}\cdot\nabla_{\mathbf{x}}f+\nabla_{\boldsymbol{%
\xi}}\cdot(f\mathbf{\Gamma})=Q(f),$$
(1)
where $\mathbf{\Gamma}$ represents the acceleration felt by the
droplets (resulting from the drag force exerted by the gas), and $Q$
is an operator taking into account the complex phenomena happening at
the level of the droplets (collisions, coalescences, breakup).
We also restrict ourselves to the case when the gas is incompressible
and viscous, which is for instance the usual framework when studying
the transport of sprays in the upper airways of the human lungs,
cf. [17].
Accordingly, the gas is modeled by the incompressible Navier-Stokes
equation
$$\displaystyle\nabla_{\mathbf{x}}\cdot\mathbf{u}=0,$$
(2)
$$\displaystyle\rho_{g}\left[\partial_{t}\mathbf{u}+\nabla_{\mathbf{x}}\cdot(%
\mathbf{u}\otimes\mathbf{u})\right]+\nabla_{\mathbf{x}}p-\mu\Delta_{\mathbf{x}%
}\mathbf{u}=\mathbf{F}_{\text{ret}},$$
(3)
where $\rho_{g}$ is the constant density of the gas, $\mathbf{u}:=\mathbf{u}(t,\mathbf{x})\in\mathbb{R}^{3}$ is its velocity, $\mu$ is its constant (dynamic) viscosity,
and $\mathbf{F}_{\text{ret}}$ is the retroaction of the drag force:
$$\mathbf{F}_{\text{ret}}(t,\mathbf{x})=-\int_{0}^{+\infty}\int_{\mathbb{R}^{3}}%
\frac{4}{3}\,\rho_{l}\,r^{3}\,f\,\mathbf{\Gamma}\,\mathrm{d}\boldsymbol{\xi}%
\mathrm{d}r.$$
(4)
Finally, $\rho_{l}$ is the constant density of the liquid (so that the
mass of the droplets of radius $r$ is $\frac{4}{3}\,\rho_{l}\,r^{3}$).
For particles with small Reynolds numbers, the drag force is given by the simple formula (known as “Stokes’ law”)
$$\mathbf{\Gamma}(t,\mathbf{x},\boldsymbol{\xi},r)=-\frac{9}{2}\frac{\mu}{\rho_{%
l}}\frac{\boldsymbol{\xi}-\mathbf{u}(t,\mathbf{x})}{r^{2}},$$
(5)
that we shall systematically use in the sequel.
The modeling of the breakup phenomena is important in the applications
and has led to various models appearing in the literature
(cf. [1]). A typical form of the breakup kernel is
obtained when assuming that the droplets after breakup have the same
velocities as before breakup. The operator writes then
$$\displaystyle Q(f)(t,\mathbf{x},\boldsymbol{\xi},r)=-\nu(\boldsymbol{\xi},r)\,%
f(t,\mathbf{x},\boldsymbol{\xi},r)+\int_{r^{*}>r}b(r^{*},r)\,\nu(\boldsymbol{%
\xi},r^{*})\,f(t,\mathbf{x},\boldsymbol{\xi},r^{*})\,dr^{*},$$
(6)
where $\nu:=\nu(\boldsymbol{\xi},r)\geq 0$ is the fragmentation rate, and $b:=b(r^{*},r)\geq 0$ is related to the probability of ending up with droplets of radius $r$ out of the breakup of droplets of radius $r^{*}$.
We finally obtain the following system
$$\displaystyle\nabla_{\mathbf{x}}\cdot\mathbf{u}=0,$$
(7)
$$\displaystyle\partial_{t}f+\boldsymbol{\xi}\cdot\nabla_{\mathbf{x}}f+\nabla_{%
\boldsymbol{\xi}}\cdot[f\mathbf{\Gamma}]=Q(f),$$
(8)
$$\displaystyle\rho_{g}\,\Big{[}\partial_{t}\mathbf{u}+\nabla_{\mathbf{x}}\cdot(%
\mathbf{u}\otimes\mathbf{u})\Big{]}+\nabla_{\mathbf{x}}p-\mu\Delta_{\mathbf{x}%
}\mathbf{u}=\mathbf{F}_{\text{ret}}.$$
(9)
Because of the dependence w.r.t. $r$ of $\mathbf{\Gamma}$ in eq. (5),
we see that the drag force acting on the small droplets leads to very quick equilibration
of their velocity with the velocity of the gas. It is therefore natural to try
to write a set of equations replacing (4) – (9),
in which the small droplets are considered as part of the gaseous phase.
Indeed, in the context of the numerical simulation of eq. (1) thanks to a particle method, the small droplets which are produced
because of the breakup can lead to a high computational cost.
Once integrated (at the level of the continuous equations) in the gaseous phase, they can be discretized along with the gas
thanks to a finite volume scheme, and their contribution to the computational cost remains in this way reasonable.
In order to perform a mathematical study of the system obtained by such an approximation, we consider the simplest
possible case, namely when the fragmentation rate $\nu$ takes the form $\nu(\boldsymbol{\xi},r):=\tau^{-1}\mathds{1}_{r>r_{2}}$, for some constants $r_{2}>0$ and $\tau>0$, the latter being seen as a characteristic time of fragmentation.
We also assume that the aerosol is bidispersed: only two possible radii $r_{1}>r_{2}$ exist for the droplets, and the result of the breakup of particles of radius $r_{1}$ are particles of radius $r_{2}$. This implies that the density $f$ splits in the following way
$$\displaystyle f(t,\mathbf{x},\boldsymbol{\xi},r)=f_{1}(t,\mathbf{x},%
\boldsymbol{\xi})\,\delta_{r=r_{1}}+f_{2}(t,\mathbf{x},\boldsymbol{\xi})\,%
\delta_{r=r_{2}},$$
and that system (4) – (9) rewrites, after having
normalized all the constants (except $r_{2}$):
$$\displaystyle[left=\empheqlbrace]$$
$$\displaystyle\nabla_{\mathbf{x}}\cdot\mathbf{u}=0,$$
(10)
$$\displaystyle\partial_{t}f_{1}+\boldsymbol{\xi}\cdot\nabla_{\mathbf{x}}f_{1}+%
\nabla_{\boldsymbol{\xi}}\cdot\left[{f_{1}\,(\mathbf{u}-\boldsymbol{\xi})}%
\right]=-{f_{1}},$$
(11)
$$\displaystyle\partial_{t}f_{2}+\boldsymbol{\xi}\cdot\nabla_{\mathbf{x}}f_{2}+%
\nabla_{\boldsymbol{\xi}}\cdot\left[\frac{f_{2}\,(\mathbf{u}-\boldsymbol{\xi})%
}{r_{2}^{2}}\right]=\frac{f_{1}}{r_{2}^{3}},$$
(12)
$$\displaystyle\partial_{t}\mathbf{u}+\nabla_{\mathbf{x}}\cdot(\mathbf{u}\otimes%
\mathbf{u})+\nabla_{\mathbf{x}}p-\Delta_{\mathbf{x}}\mathbf{u}=-\int_{\mathbb{%
R}^{3}}(\mathbf{u}-\boldsymbol{\xi})\,f_{1}\,\mathrm{d}\boldsymbol{\xi}-r_{2}%
\int_{\mathbb{R}^{3}}(\mathbf{u}-\boldsymbol{\xi})\,f_{2}\,\mathrm{d}%
\boldsymbol{\xi}.$$
(13)
As we already explained, we are interested in the asymptotic regime when the particles resulting from breakup are becoming smaller and smaller. In our bidispersed model, this reduces to study the limit $r_{2}\to 0$.
Denoting $\rho=r_{2}^{3}\displaystyle\int_{\mathbb{R}^{3}}f_{2}\,\mathrm{d}\boldsymbol{\xi}$, and noticing that
$$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\bigg{\{}\int_{{\mathbb{T}}^{3}}%
\int_{\mathbb{R}^{3}}f_{1}\,\frac{|\boldsymbol{\xi}|^{2}}{2}\,\mathrm{d}%
\boldsymbol{\xi}\mathrm{d}\mathbf{x}$$
$$\displaystyle+r_{2}^{3}\,\int_{{\mathbb{T}}^{3}}\int_{\mathbb{R}^{3}}f_{2}\,%
\frac{|\boldsymbol{\xi}|^{2}}{2}\,\mathrm{d}\boldsymbol{\xi}\mathrm{d}\mathbf{%
x}+\int_{{\mathbb{T}}^{3}}\frac{|\mathbf{u}|^{2}}{2}\,\mathrm{d}\mathbf{x}%
\bigg{\}}$$
$$\displaystyle+\int_{{\mathbb{T}}^{3}}\int_{\mathbb{R}^{3}}f_{1}\,{|\boldsymbol%
{\xi}-\mathbf{u}|^{2}}\,\mathrm{d}\boldsymbol{\xi}\mathrm{d}\mathbf{x}$$
$$\displaystyle+\frac{1}{r_{2}^{2}}\,\int_{{\mathbb{T}}^{3}}\int_{\mathbb{R}^{3}%
}r_{2}^{3}f_{2}\,|\boldsymbol{\xi}-\mathbf{u}|^{2}\,\mathrm{d}\boldsymbol{\xi}%
\mathrm{d}\mathbf{x}+\int_{{\mathbb{T}}^{3}}|\nabla_{\mathbf{x}}\mathbf{u}|^{2%
}\,\mathrm{d}\mathbf{x}=0,$$
we see that (at the formal level) $r_{2}^{3}\,f_{2}(t,\mathbf{x},\boldsymbol{\xi})\to\rho(t,x)\,\delta_{%
\boldsymbol{\xi}=\mathbf{u}(t,\mathbf{x})}$ when $r_{2}\to 0$.
Integrating eq. (12) against $r_{2}^{3}\,\mathrm{d}\boldsymbol{\xi}$, we end up with
$$\partial_{t}\rho+\nabla_{\mathbf{x}}\cdot[\rho\,\mathbf{u}]=\int_{\mathbb{R}^{%
3}}f_{1}\,\mathrm{d}\boldsymbol{\xi}.$$
(14)
Then, integrating eq. (12) against $r_{2}^{3}\,\boldsymbol{\xi}\,\mathrm{d}\boldsymbol{\xi}$ and adding the result with eq. (13),
we obtain
$$\partial_{t}((1+\rho)\,\mathbf{u})+\nabla_{\mathbf{x}}\cdot((1+\rho)\,\mathbf{%
u}\otimes\mathbf{u})+\nabla_{\mathbf{x}}p-\Delta_{\mathbf{x}}\mathbf{u}=-\int_%
{\mathbb{R}^{3}}(\mathbf{u}-\boldsymbol{\xi})\,f_{1}\,\mathrm{d}\boldsymbol{%
\xi}+\int_{\mathbb{R}^{3}}f_{1}\,\boldsymbol{\xi}\,\mathrm{d}\boldsymbol{\xi}.$$
Combining this last equation with eq. (14) and replacing the notation $f_{1}$ by $f$, we write down the
system that we wish to study (we close it with periodic boundary conditions for the mathematical proof of existence):
$$\displaystyle[left=\empheqlbrace]$$
$$\displaystyle\nabla_{\mathbf{x}}\,\mathbf{u}=0,$$
(15)
$$\displaystyle\partial_{t}\rho+\nabla_{\mathbf{x}}\cdot[\rho\,\mathbf{u}]=\int_%
{\mathbb{R}^{3}}f\,\mathrm{d}\boldsymbol{\xi},$$
(16)
$$\displaystyle\partial_{t}f+\boldsymbol{\xi}\cdot\nabla_{\mathbf{x}}f+\nabla_{%
\boldsymbol{\xi}}\cdot[(\mathbf{u}-\boldsymbol{\xi})\,f]=-f,$$
(17)
$$\displaystyle(1+\rho)\Big{[}\partial_{t}\mathbf{u}+\nabla_{\mathbf{x}}\cdot(%
\mathbf{u}\otimes\mathbf{u})\Big{]}+\nabla_{\mathbf{x}}p-\Delta_{\mathbf{x}}%
\mathbf{u}=2\,\int_{\mathbb{R}^{3}}(\boldsymbol{\xi}-\mathbf{u})\,f\,\mathrm{d%
}\boldsymbol{\xi},$$
(18)
where $\rho:=\rho(t,\mathbf{x})\geq 0$, $\mathbf{u}:=\mathbf{u}(t,\mathbf{x})\in\mathbb{R}^{3}$, $p:=p(t,\mathbf{x})\geq 0$,
$f:=f(t,\mathbf{x},\boldsymbol{\xi})\geq 0$, and $t\geq 0$, $\mathbf{x}\in{\mathbb{T}}^{3}$, $\boldsymbol{\xi}\in\mathbb{R}^{3}$. Let us recall
that $\rho$ represents here the “added density” resulting from the very small particles. This is the reason why, though we normalized the fluid density in (13) by $\rho_{g}=1$, we have in (18) the term $(1+\rho)$ in front of the convective part of the fluid equation. For a more detailed version of the previous computation, see [5].
Our goal is to study the existence theory for this system, completed with the following initial data:
$$\displaystyle f(0,\mathbf{x},\boldsymbol{\xi})$$
$$\displaystyle=f_{\textnormal{in}}(\mathbf{x},\boldsymbol{\xi})\geq 0,\quad%
\mathbf{x}\in{\mathbb{T}}^{3},\quad\boldsymbol{\xi}\in{\mathbb{R}^{3}},$$
(19)
$$\displaystyle\rho(0,\mathbf{x})$$
$$\displaystyle=\rho_{\textnormal{in}}(\mathbf{x})\geq 0,\quad\mathbf{x}\in{%
\mathbb{T}}^{3},$$
(20)
$$\displaystyle\mathbf{u}(0,\mathbf{x})$$
$$\displaystyle=\mathbf{u}_{\textnormal{in}}(\mathbf{x})\in{\mathbb{T}}^{3},%
\quad\mathbf{x}\in{\mathbb{T}}^{3}.$$
(21)
Before stating our result, let us introduce a few notations. If there is no ambiguity on the considered time interval, we simply denote by $\textnormal{L}^{p}_{t}(\textnormal{L}^{q}_{\mathbf{x}})$ and $\textnormal{L}^{p}_{t}(\textnormal{L}^{q}_{\mathbf{x},\boldsymbol{\xi}})$ the spaces $\textnormal{L}^{p}\big{(}[0,T];\textnormal{L}^{q}({\mathbb{T}}^{3})\big{)}$ and $\textnormal{L}^{p}\big{(}[0,T];\textnormal{L}^{q}({\mathbb{T}}^{3}\times%
\mathbb{R}^{3})\big{)}$, for any pair of exponents $(p,q)\in[1,\infty]^{2}$. In particular
when $p=q$ we simply use the notation $\textnormal{L}^{p}$. We adopt the same convention for Sobolev spaces $\textnormal{W}^{1,p}_{\mathbf{x}}$ and $\textnormal{H}^{m}_{\mathbf{x}}$ ($m\in\mathbb{N}$) and denote by $\textnormal{H}^{-m}_{\mathbf{x}}$ (or equivalently $\textnormal{H}^{-m}({\mathbb{T}}^{3})$) the dual of the latter.
When the subscript “ div ” is added to any space,
the corresponding subspace are the divergence-free (in $\mathbf{x}$ only) elements of the ambient space. We denote by $\mathbb{P}$ the Leray projector onto divergence-free vector fields
$$\displaystyle\mathbb{P}=\textbf{Id}-\nabla_{\mathbf{x}}\Delta^{-1}_{\mathbf{x}%
}\textnormal{div}_{\mathbf{x}},$$
easily defined on $\mathscrbf{D}([0,T]\times{\mathbb{T}}^{3})$ thanks to Fourier analysis, extended by duality to $\mathscrbf{D}^{\prime}([0,T]\times{\mathbb{T}}^{3})$.
Finally, if $h$ is a scalar function defined on $\mathbb{R}_{+}\times{\mathbb{T}}^{3}\times\mathbb{R}^{3}$, we define the following moments for $h:=h(t,\mathbf{x},\boldsymbol{\xi})$ (and for $\alpha\in\mathbb{R}$)
$$\displaystyle m_{\alpha}h(t,\mathbf{x}):=\int_{{\mathbb{R}^{3}}}|\boldsymbol{%
\xi}|^{\alpha}\,h(t,\mathbf{x},\boldsymbol{\xi})\,\mathrm{d}\boldsymbol{\xi},%
\quad M_{\alpha}h(t):=\int_{{\mathbb{T}}^{3}}m_{\alpha}(h)(t,\mathbf{x})\,%
\mathrm{d}\mathbf{x},\quad\mathbf{m}_{1}h(t,\mathbf{x}):=\int_{{\mathbb{R}^{3}%
}}\boldsymbol{\xi}h(t,\mathbf{x},\boldsymbol{\xi})\,\mathrm{d}\boldsymbol{\xi}.$$
Thanks to those notations, we are able to write down our main
Theorem:
Theorem 1.1.
Let $T>0$ and assume that
$$f_{\textnormal{in}}\in\textnormal{L}^{\infty},(1+|\boldsymbol{\xi}|^{2})f_{%
\textnormal{in}}\in\textnormal{L}^{1},\rho_{\textnormal{in}}\in\textnormal{L}^%
{\infty},\mathbf{u}_{\textnormal{in}}\in\mathbf{L}_{\textnormal{div}}^{2}.$$
(22)
Then the system (15) – (21)
admits a global weak solution $(\rho\geq 0,\,f\geq 0,\,\mathbf{u}\in\mathbb{R}^{3})$ such that
$$\rho\in\textnormal{L}^{\infty}([0,T],\textnormal{L}^{5/3}({\mathbb{T}}^{3})),f%
\in\textnormal{L}^{\infty}([0,T]\times{\mathbb{T}}^{3}\times{\mathbb{R}^{3}}),%
\mathbf{u}\in\textnormal{L}^{2}([0,T],\textnormal{H}^{1}_{\textnormal{div}}({%
\mathbb{T}}^{3}))\cap\textnormal{L}^{\infty}([0,T],\textnormal{L}^{2}({\mathbb%
{T}}^{3})).$$
(23)
Moreover, the triplet $(\rho,\,f,\,\mathbf{u})$ satisfies the following energy estimate
$$\displaystyle\frac{1}{2}\left\{M_{2}f(t)+\|\sqrt{1+\rho(t)}\,\mathbf{u}(t)\|_{%
\textnormal{L}^{2}({\mathbb{T}}^{3})}^{2}\right\}$$
$$\displaystyle+\int_{0}^{t}\|\nabla_{\mathbf{x}}\mathbf{u}(s)\|_{\textnormal{L}%
^{2}({\mathbb{T}}^{3})}^{2}\mathrm{d}s+\frac{3}{2}\int_{0}^{t}\int_{{\mathbb{T%
}}^{3}\times\mathbb{R}^{3}}|\mathbf{u}-\boldsymbol{\xi}|^{2}f\mathrm{d}%
\boldsymbol{\xi}\mathrm{d}\mathbf{x}\mathrm{d}s$$
$$\displaystyle\leq\frac{1}{2}\left\{M_{2}f_{\text{\bf in}}+\|\sqrt{1+\rho_{%
\text{\bf in}}}\,\mathbf{u}_{\text{\bf in}}\|_{\textnormal{L}^{2}({\mathbb{T}}%
^{3})}^{2}\right\},$$
(24)
and the bound
$$\displaystyle\|f\|_{\textnormal{L}^{\infty}\left([0,\,T]\times{\mathbb{T}}^{3}%
\times{\mathbb{R}^{3}}\right)}\leq e^{2T}\|f_{\text{\bf in}}\|_{\textnormal{L}%
^{\infty}\left({\mathbb{T}}^{3}\times{\mathbb{R}^{3}}\right)}.$$
Remark 1.1.
We explain here the meaning of “ weak solutions ” in the above Theorem. For $p\in[1,\infty]$, when $\mathbf{u}\in\mathbf{L}^{p}$, $f\in\textnormal{L}^{\infty}$ and $\rho\in\textnormal{L}^{p^{\prime}}$,
equations (15)–(16) have a clear meaning in the distributional sense. However the meaning of (18) is not completely obvious, and we rewrite this equation, using both (18) and (17), as
$$\displaystyle\mathbb{P}\left\{\partial_{t}[(1+\rho)\mathbf{u}]+\textnormal{div%
}_{\mathbf{x}}\Big{[}(1+\rho)\mathbf{u}\otimes\mathbf{u}\Big{]}\right\}-\Delta%
_{\mathbf{x}}\mathbf{u}=\mathbb{P}\left\{2\mathbf{m}_{1}f-\mathbf{u}m_{0}f%
\right\}.$$
(25)
For the initial condition, we adopt the following general definition : consider $v_{\textnormal{in}}\in\mathscr{D}^{\prime}({\mathbb{T}}^{3})$ and $w$ in $\textnormal{L}_{\textnormal{loc}}^{1}([0,T];\textnormal{H}^{-m}({\mathbb{T}}^{%
3}))$ for some integer $m\in\mathbb{N}$. A
distribution $v\in\mathscr{D}^{\prime}\big{(}]-\infty,T[\times{\mathbb{T}}^{3}\big{)}$ is a solution of the Cauchy problem
$$\partial_{t}v=w,\qquad v(0)=v_{\textnormal{in}},$$
if the support of $v$ is included in $\mathbb{R}_{+}$ and if
$$\displaystyle\partial_{t}v$$
$$\displaystyle=\widetilde{w}+\delta_{0}\otimes v_{\textnormal{in}}\text{ in }%
\mathscr{D}^{\prime}\big{(}]-\infty,T[\times{\mathbb{T}}^{3}\big{)},$$
where $\widetilde{w}$ is the extension of $w$ by $0$ on $\mathbb{R}_{-}$.
The study of the existence of solutions to coupled (through drag force interaction) fluid-kinetic equations
is now a well-established subject.
As far as viscous equations are concerned for the modeling of the gaseous phase, we would like to quote the works on the Vlasov-Stokes equations in [12], on the Vlasov/incompressible Navier-Stokes equations in [2, 6, 20], on the Vlasov-Fokker-Planck/incompressible Euler equations in [8]
, on the Vlasov-Fokker-Planck/incompressible Navier-Stokes equations in [15], and on the Vlasov-Fokker-Planck/compressible Navier-Stokes equations in [16].
Without any viscosity or Fokker-Planck damping, the study is more difficult, and only smooth local solutions are known to exist, in the compressible setting (cf. [4] and [14]).
Our system (15) – (18) is reminiscent of
the Vlasov/incompressible Navier-Stokes equations with a variable density.
It is known that variable densities in the incompressible Navier-Stokes
equations lead to extra difficulties (w.r.t. constant densities), cf. [7, 13]. Here extra difficulties (w.r.t. [6]) appear in the final passage to the limit (that is, in the stability result for solutions satisfying the natural a priori estimates
of the problem).
They are linked with nonlinearities which are specific of our model, and which necessitate a
careful treatment using refined versions of Lemmas presented in [10].
The equations coming out of the theory of sprays are known to be difficult to approximate in a good way (that is, in a way in which the a priori estimates satisfied by the
equations themselves, such as the energy estimate, are also satisfied or well approximated by the approximating equations). This difficulty appears in this paper, and the approximating scheme which is used is rather complicated and necessitates successive steps, which are
reminiscent of those used in [6].
Our paper is structured as follows:
Section 2 is devoted to the Proof of existence of solutions to an approximated
version of this system. The approximation is
removed in Section 3 and Theorem 1.1
is proven there. Finally, we present in a short Appendix some autonomous results
which are used in the Proof of Theorem 1.1.
2 Existence for a regularized system
In this section, we begin the Proof of Theorem 1.1 by introducing a regularized system.
2.1 Definition of the regularized system
In order to do so,
we introduce a mollifier $\varphi_{\varepsilon}\in\mathscr{C}^{\infty}({\mathbb{T}}^{3})$ for all $\varepsilon>0$.
We also introduce a truncation in this way: we define $\gamma_{\varepsilon}$ a nonnegative element of $\mathscr{D}(\mathbb{R}^{3})$,
whose support lies in $\textnormal{{B}}(0,2/\varepsilon)$, bounded by $1$, and equal to $1$ on the ball $\textnormal{{B}}(0,1/\varepsilon)$.
Our regularized system writes
$$\displaystyle[left={\empheqlbrace}]$$
$$\displaystyle\textnormal{div}_{\mathbf{x}}\,\mathbf{u}_{\varepsilon}=0,$$
(26)
$$\displaystyle\partial_{t}\rho_{\varepsilon}+(\mathbf{u}_{\varepsilon}\star%
\varphi_{\varepsilon})\cdot\nabla_{\mathbf{x}}\,\rho_{\varepsilon}=m_{0}(f_{%
\varepsilon}\gamma_{\varepsilon}),$$
(27)
$$\displaystyle\partial_{t}f_{\varepsilon}+\boldsymbol{\xi}\cdot\nabla_{\mathbf{%
x}}f_{\varepsilon}+\nabla_{\boldsymbol{\xi}}\cdot[(\mathbf{u}_{\varepsilon}%
\star\varphi_{\varepsilon}-\boldsymbol{\xi})f_{\varepsilon}]=-f_{\varepsilon},$$
(28)
$$\displaystyle\mathbb{P}\Big{\{}(1+\rho_{\varepsilon})\big{[}\partial_{t}%
\mathbf{u}_{\varepsilon}+(\mathbf{u}_{\varepsilon}\star\varphi_{\varepsilon})%
\cdot\nabla_{\mathbf{x}}\mathbf{u}_{\varepsilon}\big{]}\Big{\}}-\Delta_{%
\mathbf{x}}\mathbf{u}_{\varepsilon}=2\mathbb{P}\Big{\{}\mathbf{m}_{1}(f_{%
\varepsilon}\gamma_{\varepsilon})-\mathbf{u}_{\varepsilon}m_{0}(f_{\varepsilon%
}\gamma_{\varepsilon})\Big{\}},$$
(29)
with initial conditions $f_{\textnormal{in}}^{\varepsilon}\in\mathscr{D}({\mathbb{T}}^{3}\times\mathbb{%
R}^{3})$, $\rho_{\textnormal{in}}^{\varepsilon}\in\mathscr{C}^{\infty}({\mathbb{T}}^{3})$ and $\mathbf{u}_{\textnormal{in}}^{\varepsilon}\in\mathscrbf{C}^{\infty}_{%
\textnormal{div}}({\mathbb{T}}^{3})$ approximating respectively $f_{\textnormal{in}}$ in all $\textnormal{L}_{\textnormal{loc}}^{p}({\mathbb{T}}^{3}\times\mathbb{R}^{3})$ ($p<\infty$) and in $\textnormal{L}^{1}({\mathbb{T}}^{3}\times\mathbb{R}^{3},(1+|\boldsymbol{\xi}|^%
{2})\,\mathrm{d}\mathbf{x}\,\mathrm{d}\boldsymbol{\xi})$, $\rho_{\textnormal{in}}$ in all $\textnormal{L}^{p}({\mathbb{T}}^{3})$ ($p<\infty$), and $\mathbf{u}_{\textnormal{in}}$ in $\mathbf{L}^{2}({\mathbb{T}}^{3})$.
Next subsection is devoted to the Proof of existence of solutions to this approximated system (that is, for a given
parameter $\varepsilon>0$).
2.2 Existence of solutions for the regularized system
In this Subsection, the parameter $\varepsilon$ is fixed, and we drop out the corresponding indices in order to make
the formulas more readable.
We fix a triplet of initial data $(\boldsymbol{\omega},h,\nu)\in\mathscrbf{C}^{\infty}_{\textnormal{div}}\times(%
\mathscr{C}^{1}\cap\textnormal{L}^{\infty})\times\mathscr{C}^{1}$ (the two last ones being nonnegative). The considered system (in which we do not explicitly write the initial data) is then the following:
$$\displaystyle[left={\empheqlbrace}]$$
$$\displaystyle\textnormal{div}_{\mathbf{x}}\,\mathbf{u}=0,$$
(30)
$$\displaystyle\partial_{t}\rho+(\mathbf{u}\star\varphi)\cdot\nabla_{\mathbf{x}}%
\,\rho=m_{0}(f\gamma),$$
(31)
$$\displaystyle\partial_{t}f+\boldsymbol{\xi}\cdot\nabla_{\mathbf{x}}f+\nabla_{%
\boldsymbol{\xi}}\cdot[(\mathbf{u}\star\varphi-\boldsymbol{\xi})f]=-f,$$
(32)
$$\displaystyle\mathbb{P}\Big{\{}(1+\rho)\big{[}\partial_{t}\mathbf{u}+(\mathbf{%
u}\star\varphi)\cdot\nabla_{\mathbf{x}}\mathbf{u}\big{]}\Big{\}}-\Delta_{%
\mathbf{x}}\mathbf{u}=2\mathbb{P}\Big{\{}\mathbf{m}_{1}(f\gamma)-\mathbf{u}m_{%
0}(f\gamma)\Big{\}}.$$
(33)
We are able to prove the
Proposition 2.1.
For any triplet of initial data $(\boldsymbol{\omega},h,\nu)\in\mathscrbf{C}^{\infty}_{\textnormal{div}}\times(%
\mathscr{C}^{1}\cap\textnormal{L}^{\infty})\times\mathscr{C}^{1}$ (the two last ones being nonnegative), there exists a weak solution (in the sense of Remark 1.1)
to the system (30) –
(33).
Proof of Proposition 2.1:
We shall show existence of a solution to the nonlinear system (30) – (33) by applying Schauder’s fixed point Theorem to the following mapping:
$$\displaystyle\textnormal{S}:\mathscr{C}^{0}_{t}(\mathbf{L}^{2}_{\textnormal{%
div}})$$
$$\displaystyle\longrightarrow\mathscr{C}^{0}_{t}(\mathbf{L}^{2}_{\textnormal{%
div}})$$
$$\displaystyle\mathbf{u}$$
$$\displaystyle\longmapsto\mathbf{u}^{\diamond},$$
where $\mathbf{u}^{\diamond}$ will be built thanks to the following steps :
(i)
We first consider $f^{\diamond}\in\mathscr{C}^{1}$, unique classical solution of
$$\displaystyle\partial_{t}f^{\diamond}+\boldsymbol{\xi}\cdot\nabla_{\mathbf{x}}%
f^{\diamond}+\nabla_{\boldsymbol{\xi}}\cdot[(\mathbf{u}\star\varphi-%
\boldsymbol{\xi})f^{\diamond}]=-f^{\diamond},$$
(34)
with $h$ as initial datum,
(ii)
then $\rho^{\diamond}\in\mathscr{C}^{1}$ is defined as the unique classical solution of
$$\displaystyle\partial_{t}\rho^{\diamond}+(\mathbf{u}\star\varphi)\cdot\nabla_{%
\mathbf{x}}\rho^{\diamond}=m_{0}(f^{\diamond}\gamma),$$
(35)
with $\nu$ as initial datum,
(iii)
and finally $\mathbf{u}^{\diamond}\in\mathscr{C}^{0}_{t}(\mathbf{L}^{2}_{\textnormal{div}})$ is built as the unique divergence-free weak solution of
$$\displaystyle\mathbb{P}\Bigg{\{}(1+\rho^{\diamond})\Big{[}\partial_{t}\mathbf{%
u}^{\diamond}+(\mathbf{u}\star\varphi)\cdot\nabla_{\mathbf{x}}\mathbf{u}^{%
\diamond}\Big{]}\Bigg{\}}-\Delta_{\mathbf{x}}\mathbf{u}^{\diamond}=2\mathbb{P}%
\Big{\{}\mathbf{m}_{1}(f^{\diamond}\gamma)-\mathbf{u}^{\diamond}m_{0}(f^{%
\diamond}\gamma)\Big{\}},$$
(36)
with $\boldsymbol{\omega}$ as initial datum.
In the sequel $\textnormal{P}_{T}(\boldsymbol{\omega},h,\nu)$ will denote any nonnegative function (which also may depend on $\varepsilon$, but as explained before we omit the corresponding index here) that splits into a finite sum of positively homogeneous functions of strictly positive degree of one of the following terms : $\|\boldsymbol{\omega}\|_{\mathbf{H}^{1}}$,$\|h\|_{\infty}$ or $\|\nu\|_{\infty}$. $\textnormal{P}_{T}(\boldsymbol{\omega},h,\nu)$ may change from one line to another,
but will always have the structure that we just described.
In the next paragraph, we show that $f^{\diamond}$, $u^{\diamond}$ and $\rho^{\diamond}$ are well defined.
2.2.1 Existence and uniqueness of $f^{\diamond}$, $\rho^{\diamond}$ and $\mathbf{u}^{\diamond}$
We know that $\mathbf{u}\star\varphi\in\mathscr{C}^{0}_{t}(\mathscrbf{C}^{1}_{\mathbf{x}})$ and $\nabla_{\mathbf{x}}(\mathbf{u}\star\varphi)\in\mathbf{L}^{\infty}$, so that the characteristic curves of (34) and (35)
are globally well-defined. We hence easily obtain the existence and uniqueness of two nonnegative
functions $f^{\diamond}$ and $\rho^{\diamond}$ (classically) solving
(34) and (35). Thanks to the maximum principle, we also know that
$$\displaystyle\|f^{\diamond}\|_{\infty}\leq e^{2T}\|h\|_{\infty},$$
from which we get
$$\displaystyle\|m_{0}(f^{\diamond}\gamma)\|_{\infty}+\|\mathbf{m}_{1}(f^{%
\diamond}\gamma)\|_{\infty}\leq C_{\gamma}e^{2T}\|h\|_{\infty}\leq\textnormal{%
P}_{T}(\boldsymbol{\omega},h,\nu),$$
(37)
and hence
$$\displaystyle\|\rho^{\diamond}\|_{\infty}$$
$$\displaystyle\leq\textnormal{P}_{T}(\boldsymbol{\omega},h,\nu).$$
(38)
We now consider $f^{\diamond}$ and $\rho^{\diamond}$ as given. The existence and uniqueness of $\mathbf{u}^{\diamond}\in\mathbf{H}^{1}_{t,\mathbf{x}}\subset\mathscr{C}_{t}^{0%
}(\mathbf{L}^{2}_{\mathbf{x}})$, divergence-free weak solution of (36) (with $\boldsymbol{\omega}$ for initial data) may be obtained thanks to the usual Galerkin method. Since this type of construction is rather standard, we won’t detail it here, and refer to [5] for a precise treatment of this procedure in our system, or for
instance to [3], Chapter 3, for a more generic approach.
In next subsection, we obtain natural estimates for the quantity $\mathbf{u}^{\diamond}$.
2.2.2 Estimates for $\mathbf{u}^{\diamond}$
Taking $\mathbf{u}^{\diamond}$ as test function in (36) and using (35),
we obtain the following estimate (since $\mathbf{u}\star\varphi$ is divergence free and $f^{\diamond}$ is nonnegative)
$$\displaystyle\frac{1}{2}\,\left\{\int_{{\mathbb{T}}^{3}}(1+\rho^{\diamond}(t,%
\mathbf{x}))|\mathbf{u}^{\diamond}(t,\mathbf{x})|^{2}\mathrm{d}\mathbf{x}\right\}$$
$$\displaystyle+\int_{0}^{t}\int_{{\mathbb{T}}^{3}}|\nabla_{\mathbf{x}}\mathbf{u%
}^{\diamond}(s,\mathbf{x})|^{2}\mathrm{d}\mathbf{x}\,\mathrm{d}s$$
$$\displaystyle\leq-\frac{3}{2}\,\int_{0}^{t}\int_{{\mathbb{T}}^{3}}|\mathbf{u}^%
{\diamond}(s,\mathbf{x})|^{2}m_{0}(f^{\diamond}\gamma)(s,\mathbf{x})\,\mathrm{%
d}\mathbf{x}\,\mathrm{d}s$$
$$\displaystyle+2\int_{0}^{t}\int_{{\mathbb{T}}^{3}}\mathbf{u}^{\diamond}(s,%
\mathbf{x})\cdot\mathbf{m}_{1}(f^{\diamond}\gamma)(s,\mathbf{x})\,\mathrm{d}%
\mathbf{x}\,\mathrm{d}s$$
$$\displaystyle+\frac{1}{2}\,\left\{\int_{{\mathbb{T}}^{3}}(1+\nu(\mathbf{x}))|%
\boldsymbol{\omega}(\mathbf{x})|^{2}\mathrm{d}\mathbf{x}\right\},$$
(39)
so that using $\rho^{\diamond}\geq 0$ and Grönwall’s lemma together with
estimates (37)–(38), we get
$$\displaystyle\|\mathbf{u}^{\diamond}\|_{\textnormal{L}^{\infty}_{t}(\mathbf{L}%
^{2}_{\mathbf{x}})}+\|\mathbf{u}^{\diamond}\|_{\textnormal{L}^{2}_{t}(\mathbf{%
H}^{1}_{\mathbf{x}})}\leq\textnormal{P}_{T}(\boldsymbol{\omega},h,\nu)\exp(%
\textnormal{P}_{T}(\boldsymbol{\omega},h,\nu)).$$
(40)
Taking then $\partial_{t}\mathbf{u}^{\diamond}$ as a test function in (36),
we get (using again $\rho^{\diamond}\geq 0$ and $f^{\diamond}\geq 0$)
$$\displaystyle\|\partial_{t}\mathbf{u}^{\diamond}\|_{2}^{2}+\frac{1}{2}\,\|%
\nabla_{\mathbf{x}}\mathbf{u}^{\diamond}(t)\|_{2}^{2}$$
$$\displaystyle\leq\|(1+\rho^{\diamond})(\mathbf{u}\star\varphi)\|_{\infty}\,\|%
\nabla_{\mathbf{x}}\mathbf{u}^{\diamond}\|_{2}\,\|\partial_{t}\mathbf{u}^{%
\diamond}\|_{2}+2\,\|\mathbf{m}_{1}(f^{\diamond}\gamma)\|_{2}\,\|\partial_{t}%
\mathbf{u}^{\diamond}\|_{2}$$
$$\displaystyle+2\,\|m_{0}(f^{\diamond}\gamma)\|_{\infty}\,\|\mathbf{u}^{%
\diamond}\|_{2}\,\|\partial_{t}\mathbf{u}^{\diamond}\|_{2}+\frac{1}{2}\,\|%
\nabla_{\mathbf{x}}\boldsymbol{\omega}\|_{2}^{2},$$
so that using Young’s inequality, estimate (40) above,
and estimates (37)–(38), we end up with
$$\displaystyle\|\partial_{t}\mathbf{u}^{\diamond}\|_{2}\leq\textnormal{P}_{T}(%
\boldsymbol{\omega},h,\nu)\,\exp(\textnormal{P}_{T}(\boldsymbol{\omega},h,\nu)%
)\,\Big{[}\|\mathbf{u}\|_{\textnormal{L}^{\infty}_{t}(\mathbf{L}^{2}_{\mathbf{%
x}})}+1\Big{]},$$
(41)
thanks to the elementary convolution inequality
$\|\mathbf{u}\star\varphi\|_{\infty}\leq\|\mathbf{u}\|_{\textnormal{L}^{\infty}%
_{t}(\mathbf{L}^{2}_{\mathbf{x}})}\|\varphi\|_{2}$.
Given any triplet $(\boldsymbol{\omega},h,\nu)\in\mathscrbf{C}^{\infty}_{\textnormal{div}}\times%
\mathscr{C}^{1}\cap\textnormal{L}^{\infty}\times\mathscr{C}^{1}$ of initial data, we can then define the mapping
$$\displaystyle\textnormal{S}:\mathscr{C}^{0}_{t}(\mathbf{L}^{2}_{\textnormal{%
div}})$$
$$\displaystyle\longrightarrow\mathscr{C}^{0}_{t}(\mathbf{L}^{2}_{\textnormal{%
div}})$$
$$\displaystyle\mathbf{u}$$
$$\displaystyle\longmapsto\mathbf{u}^{\diamond}.$$
In next paragraph, we use Schauder’s fixed point Theorem for this mapping.
2.2.3 Application of Schauder’s Theorem
According to the estimates of the previous paragraph, we see that
for any $\mathbf{u}\in\mathscr{C}^{0}_{t}(\mathbf{L}^{2}_{\textnormal{div}})$,
$$\displaystyle\|\textnormal{S}(\mathbf{u})\|_{\textnormal{L}^{\infty}_{t}(%
\mathbf{L}^{2}_{\mathbf{x}})}\leq\textnormal{P}_{T}(\boldsymbol{\omega},h,\nu)%
\exp(\textnormal{P}_{T}(\boldsymbol{\omega},h,\nu)):=R.$$
Hence, S sends $\textnormal{B}_{R}$ (the closed ball of radius $R$ of the normed space $\mathscr{C}^{0}_{t}(\mathbf{L}^{2}_{\textnormal{div}})$) to itself.
Thanks to estimates (40) and (41), a bounded subset of $\mathscr{C}^{0}_{t}(\mathbf{L}^{2}_{\textnormal{div}})$ is sent by S to a bounded subset of $\mathbf{H}^{1}_{t,\mathbf{x}}$, and Lemma 4.3 of the Appendix allows us to conclude that $\textnormal{S}(\textnormal{B}_{R})$ is relatively compact in $\mathscr{C}^{0}_{t}(\mathbf{L}^{2}_{\textnormal{div}})$.
It remains to study the continuity of S.
Notice first that for any $\mathbf{u}\in\mathscr{C}^{0}_{t}(\mathbf{L}^{2}_{\textnormal{div}})$, any weak bounded solution of (34) or (35) is a renormalized solution in the DiPerna-Lions sense introduced in [10]: the absorption coefficient ($=1$) of the first equation and the right-hand side of second one are bounded and one easily checks that the vector fields $(t,\mathbf{x},\boldsymbol{\xi})\mapsto(\boldsymbol{\xi},\mathbf{u}\star\varphi%
(t,\mathbf{x})-\boldsymbol{\xi})$ and $(t,\mathbf{x})\mapsto\mathbf{u}\star\varphi(t,\mathbf{x})$ satisfy all the desired assumptions given in [10].
Now if $(\mathbf{u}_{n})_{n}$ converges to $\mathbf{u}$ in $\mathscr{C}^{0}_{t}(\mathbf{L}^{2}_{\textnormal{div}})$, we know from the previous step that $(\textnormal{S}(\mathbf{u}_{n}))_{n\in\mathbb{N}}$ has a converging subsequence. Showing that the whole sequence converges to $\textnormal{S}(\mathbf{u})$ reduces hence to prove that it has only one accumulation point, namely $\textnormal{S}(\mathbf{u})$. Assume therefore that $\mathbf{v}\in\mathscr{C}^{0}_{t}(\mathbf{L}^{2}_{\textnormal{div}})$ is an accumulation point of $(\textnormal{S}(\mathbf{u}_{n}))_{n\in\mathbb{N}}$ and denote by $\sigma$ the corresponding extraction. Since $(\mathbf{u}_{\sigma(n)})_{n}$ is still bounded, using (40)–(41) and adding a subsequence if necessary,
we may assume that $\mathbf{v}$ is also the limit of $(\textnormal{S}(\mathbf{u}_{n}))_{n\in\mathbb{N}}$ in $\mathbf{H}^{1}_{t,\mathbf{x}}$, for the weak topology. On the other hand, since $(\mathbf{u}_{\sigma(n)})_{n}$ still converges to $\mathbf{u}$ in $\mathscr{C}^{0}_{t}(\mathbf{L}^{2}_{\textnormal{div}})$, DiPerna-Lions stability result of [10] ensures that $(f_{\sigma(n)})_{n}$ and $(\rho_{\sigma(n)})_{n}$ both converge in $\textnormal{L}^{p}_{\textnormal{loc}}$
(for all $p<\infty$) to $f$ and $\rho$, the corresponding (unique) weak and bounded solutions of the associated equations (with initial data $h$ and $\nu$ and vector field defined by $\mathbf{u}$). At this point, we have enough (strong) convergences to ensure that all the nonlinear terms of the fluid equation indeed
converge to the expected limit : $\mathbf{v}\in\mathbf{H}^{1}_{t,\mathbf{x}}$ is hence a weak solution of (36) with $\boldsymbol{\omega}$ as initial datum. We already mentioned the uniqueness of solutions for this equation, so that we eventually get $\mathbf{v}=\mathbf{u}^{\diamond}=\textnormal{S}(\mathbf{u})$.
S is hence a continuous map from a closed convex nonempty
set of a normed space to itself, and it has a relatively compact range.
Thanks to Schauder’s fixed point Theorem (see [11] for instance), S has a fixed point which is a
solution to (30) – (33) with initial data $(\boldsymbol{\omega},h,\nu)$.
This concludes the Proof of Proposition 2.1. $\square$
Reintroducing the parameter $\varepsilon>0$, we have obtained the existence (for any $T>0$, $\varepsilon>0$) of
a divergence-free vector field $\mathbf{u}_{\varepsilon}\in\textnormal{H}^{1}_{t,\mathbf{x}}$, and two nonnegative functions $f_{\varepsilon}\in\mathscr{C}^{1}_{t,\mathbf{x},\boldsymbol{\xi}}$ and $\rho_{\varepsilon}\in\mathscr{C}^{1}_{t,\mathbf{x}}$ solutions of system (26) – (29).
Note that since $\mathbf{u}_{\varepsilon}\star\varphi_{\varepsilon}\in\textnormal{L}^{\infty}_{%
t,\mathbf{x}}$ and $f_{\textnormal{in}}^{\varepsilon}$ is compactly supported, one can
show without difficulty that $f^{\varepsilon}(t,\cdot,\cdot)$ remains compactly supported for a.e. $t$ (with a support depending on $\varepsilon$).
In next section, we pass to the limit when $\varepsilon\to 0$ in the functions $\mathbf{u}_{\varepsilon}$, $f_{\varepsilon}$ and $\rho_{\varepsilon}$, and show that their limit is a (weak) solution of (15) – (18).
3 Passage to the limit and Proof of the main Theorem
We keep on proving Theorem 1.1 in this Section. We start with a local result of existence:
Proposition 3.1.
Under the assumptions on the initial data of Theorem 1.1, there exists
an interval $J_{\textnormal{in}}:=[0,t_{\star}]$ on which the system (15) – (18) admits a weak
solution $(\rho,\,f,\,\mathbf{u})$ (cf. Remark 1.1 for a definition of weak solutions).
Furthermore $\rho$ and $f$ are nonnegative, and
$$\displaystyle\rho$$
$$\displaystyle\in\textnormal{L}^{\infty}(J_{\textnormal{in}};\textnormal{L}^{5/%
3}({\mathbb{T}}^{3})),$$
$$\displaystyle f$$
$$\displaystyle\in\textnormal{L}^{\infty}(J_{\textnormal{in}}\times{\mathbb{T}}^%
{3}\times{\mathbb{R}^{3}}),$$
$$\displaystyle\mathbf{u}$$
$$\displaystyle\in\textnormal{L}^{2}(J_{\textnormal{in}};\textnormal{H}^{1}({%
\mathbb{T}}^{3}))\cap\textnormal{L}^{\infty}(J_{\textnormal{in}};\textnormal{L%
}^{2}({\mathbb{T}}^{3})).$$
Proof of Proposition 3.1 :
We begin with the Proof of bounds for the
solution of (26) – (29), which do not depend on $\varepsilon$.
In the sequel $\textnormal{P}_{T}(\mathbf{u}_{\textnormal{in}},\rho_{\textnormal{in}},f_{%
\textnormal{in}})$ [or more simply $\textnormal{P}_{T}^{\textnormal{in}}$] will denote a polynomial function (with positive coefficients) of the quantity $\|\sqrt{1+\rho_{\textnormal{in}}}\,\mathbf{u}_{\textnormal{in}}\|_{2}+\|f_{%
\textnormal{in}}\|_{\infty}+M_{2}f_{\textnormal{in}}$. Note
that $\textnormal{P}_{T}^{\textnormal{in}}$ will always be independent of $\varepsilon$, but may depend on $T$.
3.1 Uniform bounds with respect to $\varepsilon$
Let us first recall a classical Lemma linking velocity moments, the proof of which may be found in [6] (Lemma 1, Section 3.3):
Lemma 3.1.
Let $\gamma>0$ and $h$ be a nonnegative element of $\textnormal{L}^{\infty}([0,T]\times{\mathbb{T}}^{3}\times\mathbb{R}^{3})$, such that $m_{\gamma}h(t,\mathbf{x})<+\infty$ for a.e. $(t,\mathbf{x})$. The following estimate holds for $0\leq\alpha<\gamma$:
$$\displaystyle m_{\alpha}h(t,\mathbf{x})\leq\left(\frac{4}{3}\pi\|h(t,\mathbf{x%
},\cdot)\|_{\textnormal{L}^{\infty}(\mathbb{R}^{3}_{\boldsymbol{\xi}})}+1%
\right)m_{\gamma}h(t,\mathbf{x})^{\frac{\alpha+3}{\gamma+3}}.$$
Thanks to the maximum principle in eq. (28), we observe that
$$\displaystyle\|f_{\varepsilon}\|_{\infty}$$
$$\displaystyle\leq\|f_{\textnormal{in}}\|_{\infty}\,e^{2T},$$
(42)
from which, thanks to Lemma 3.1, we deduce
$$\displaystyle\left\|{m_{0}f_{\varepsilon}(t)}\right\|_{5/3}$$
$$\displaystyle\leq\textnormal{P}_{T}^{\textnormal{in}}\,M_{2}f_{\varepsilon}(t)%
^{3/5},$$
(43)
$$\displaystyle\left\|{\mathbf{m}_{1}f_{\varepsilon}(t)}\right\|_{5/4}$$
$$\displaystyle\leq\textnormal{P}_{T}^{\textnormal{in}}\,M_{2}f_{\varepsilon}(t)%
^{4/5}.$$
(44)
Since $f_{\varepsilon}$ is compactly supported and solves eq. (28) strongly, one gets by
multiplying this equation by $|\boldsymbol{\xi}|^{2}$ and integrating in $\mathbf{x},\boldsymbol{\xi}$:
$$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}M_{2}f_{\varepsilon}+3M_{2}f_{%
\varepsilon}=2\int_{{\mathbb{T}}^{3}}(\mathbf{u}_{\varepsilon}\star\varphi_{%
\varepsilon})\cdot\mathbf{m}_{1}(f_{\varepsilon})\,\mathrm{d}\mathbf{x}.$$
(45)
Using the previous estimates, we hence have
$$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}M_{2}f_{\varepsilon}+3M_{2}f_{\varepsilon}$$
$$\displaystyle\leq\left\|{\mathbf{u}_{\varepsilon}(t)}\right\|_{5}\,\stackrel{{%
\scriptstyle=1}}{{\overbrace{\left\|{\varphi_{\varepsilon}}\right\|_{1}}}}\,%
\left\|{\mathbf{m}_{1}f_{\varepsilon}(t)}\right\|_{5/4}$$
$$\displaystyle\leq\textnormal{P}_{T}^{\textnormal{in}}\,\left\|{\mathbf{u}_{%
\varepsilon}(t)}\right\|_{5}\,M_{2}f_{\varepsilon}(t)^{4/5},$$
from which we easily deduce
$$\displaystyle M_{2}f_{\varepsilon}(t)\leq\left\{M_{2}f_{\textnormal{in}}^{%
\varepsilon}+\textnormal{P}_{T}^{\textnormal{in}}\int_{0}^{t}\left\|{\mathbf{u%
}_{\varepsilon}(s)}\right\|_{5}\mathrm{d}s\right\}^{5}.$$
Using the Sobolev injection $\mathbf{H}^{1}({\mathbb{T}}^{3})\hookrightarrow\mathbf{L}^{5}({\mathbb{T}}^{3})$, one gets
$$\displaystyle M_{2}f_{\varepsilon}(t)\leq\textnormal{P}_{T}^{\textnormal{in}}%
\left\{1+\int_{0}^{t}\left\|{\mathbf{u}_{\varepsilon}(s)}\right\|_{2}\mathrm{d%
}s+\int_{0}^{t}\left\|{\nabla_{\mathbf{x}}\mathbf{u}_{\varepsilon}(s)}\right\|%
_{2}\mathrm{d}s\right\}^{5}.$$
(46)
Using the previous estimate with (43) – (44), we obtain
$$\displaystyle\left\|{m_{0}f_{\varepsilon}(t)}\right\|_{5/3}$$
$$\displaystyle\leq\textnormal{P}_{T}^{\textnormal{in}}\left\{1+\int_{0}^{t}%
\left\|{\mathbf{u}_{\varepsilon}(s)}\right\|_{2}\mathrm{d}s+\int_{0}^{t}\left%
\|{\nabla_{\mathbf{x}}\mathbf{u}_{\varepsilon}(s)}\right\|_{2}\mathrm{d}s%
\right\}^{3},$$
(47)
$$\displaystyle\left\|{\mathbf{m}_{1}f_{\varepsilon}(t)}\right\|_{5/4}$$
$$\displaystyle\leq\textnormal{P}_{T}^{\textnormal{in}}\left\{1+\int_{0}^{t}%
\left\|{\mathbf{u}_{\varepsilon}(s)}\right\|_{2}\mathrm{d}s+\int_{0}^{t}\left%
\|{\nabla_{\mathbf{x}}\mathbf{u}_{\varepsilon}(s)}\right\|_{2}\mathrm{d}s%
\right\}^{4}.$$
(48)
The energy estimate (39) is satisfied with $\nu:=\rho_{\textnormal{in}}^{\varepsilon}$, $\boldsymbol{\omega}:=\mathbf{u}_{\textnormal{in}}^{\varepsilon}$, and rewrites
$$\displaystyle\frac{1}{2}\left\{\int_{{\mathbb{T}}^{3}}(1+\rho_{\varepsilon}(t,%
\mathbf{x}))|\mathbf{u}_{\varepsilon}(t,\mathbf{x})|^{2}\mathrm{d}\mathbf{x}\right\}$$
$$\displaystyle+\int_{0}^{t}\int_{{\mathbb{T}}^{3}}|\nabla_{\mathbf{x}}\mathbf{u%
}_{\varepsilon}(s,\mathbf{x})|^{2}\,\mathrm{d}\mathbf{x}\,\mathrm{d}s$$
$$\displaystyle\leq-\frac{3}{2}\int_{0}^{t}\int_{{\mathbb{T}}^{3}}|\mathbf{u}_{%
\varepsilon}(s,\mathbf{x})|^{2}m_{0}(f_{\varepsilon}\gamma)(s,\mathbf{x})\,%
\mathrm{d}\mathbf{x}\,\mathrm{d}s$$
$$\displaystyle+2\int_{0}^{t}\int_{{\mathbb{T}}^{3}}\mathbf{u}_{\varepsilon}(s,%
\mathbf{x})\cdot\mathbf{m}_{1}(f_{\varepsilon}\gamma)(s,\mathbf{x})\,\mathrm{d%
}\mathbf{x}\,\mathrm{d}s$$
$$\displaystyle+\frac{1}{2}\left\{\int_{{\mathbb{T}}^{3}}(1+\rho_{\textnormal{in%
}}^{\varepsilon}(\mathbf{x}))\,|\mathbf{u}_{\textnormal{in}}^{\varepsilon}(%
\mathbf{x})|^{2}\,\mathrm{d}\mathbf{x}\right\},$$
so that (since $\rho_{\varepsilon}$ is nonnegative) we obtain
$$\displaystyle\frac{1}{2}\,\|\mathbf{u}_{\varepsilon}(t)\|_{2}^{2}+\int_{0}^{t}%
\|\nabla_{\mathbf{x}}\mathbf{u}_{\varepsilon}(s)\|_{2}^{2}\,\mathrm{d}s$$
$$\displaystyle\leq\frac{1}{2}\|\sqrt{1+\rho_{\textnormal{in}}^{\varepsilon}}\,%
\mathbf{u}_{\textnormal{in}}^{\varepsilon}\|_{2}^{2}+2\int_{0}^{t}\|\mathbf{u}%
_{\varepsilon}\cdot\mathbf{m}_{1}f_{\varepsilon}(s)\|_{1}\,\mathrm{d}s$$
$$\displaystyle\leq\textnormal{P}_{T}^{\textnormal{in}}+2\int_{0}^{t}\|\mathbf{u%
}_{\varepsilon}\cdot\mathbf{m}_{1}f_{\varepsilon}(s)\|_{1}\,\mathrm{d}s.$$
Using Hölder and Young inequalities, we get
$$\displaystyle\frac{1}{2}\|\mathbf{u}_{\varepsilon}(t)\|_{2}^{2}+\frac{1}{2}%
\int_{0}^{t}\|\nabla_{\mathbf{x}}\mathbf{u}_{\varepsilon}(s)\|_{2}^{2}\,%
\mathrm{d}s$$
$$\displaystyle\leq\textnormal{P}_{T}^{\textnormal{in}}+\textnormal{P}_{T}^{%
\textnormal{in}}\int_{0}^{t}\|\mathbf{u}_{\varepsilon}(s)\|_{2}^{2}\,\mathrm{d%
}s+\textnormal{P}_{T}^{\textnormal{in}}\int_{0}^{t}\|\mathbf{m}_{1}f_{%
\varepsilon}(s)\|_{5/4}^{2}\,\mathrm{d}s.$$
We then obtain thanks to (48)
$$\displaystyle\|\mathbf{u}_{\varepsilon}(t)\|_{2}^{2}+\int_{0}^{t}\|\nabla_{%
\mathbf{x}}\mathbf{u}_{\varepsilon}(s)\|_{2}^{2}\,\mathrm{d}s$$
$$\displaystyle\leq\textnormal{P}_{T}^{\textnormal{in}}+\textnormal{P}_{T}^{%
\textnormal{in}}\int_{0}^{t}\left\{\|\mathbf{u}_{\varepsilon}(s)\|_{2}^{2}+%
\int_{0}^{s}\left\|{\nabla\mathbf{u}_{\varepsilon}(\sigma)}\right\|_{2}^{2}\,%
\mathrm{d}\sigma\right\}^{4}\mathrm{d}s.$$
Denote by $\tilde{z}$ the maximal solution of the Cauchy problem
$$z^{\prime}(t)=\textnormal{P}_{T}^{\textnormal{in}}\,z(t)^{4},\qquad z(0)=%
\textnormal{P}_{T}^{\textnormal{in}},$$
defined on some maximal interval $I_{\textnormal{in}}$.
Then if $J_{\textnormal{in}}$ is the closure of $I_{\textnormal{in}}/2$, using the nonlinear Grönwall Lemma 4.1, we get for $t\in J_{\textnormal{in}}$:
$$\displaystyle\left\|{\mathbf{u}_{\varepsilon}(t)}\right\|_{2}^{2}+\int_{0}^{t}%
\left\|{\nabla\mathbf{u}_{\varepsilon}(s)}\right\|_{2}^{2}\mathrm{d}s\leq\|%
\tilde{z}\|_{\textnormal{L}^{\infty}}:=\textnormal{K}_{T}^{\textnormal{in}}.$$
Remark 3.1.
Notice that thanks to Lemma 4.2 (changing $\textnormal{P}_{T}^{\textnormal{in}}$ if necessary) we always have $|J_{\textnormal{in}}|\geq 1/\textnormal{P}_{T}^{\textnormal{in}}$.
We deduce from the previous local estimate that $(\mathbf{u}_{\varepsilon})_{\varepsilon}$ is bounded in $\textnormal{L}^{\infty}\big{(}J_{\textnormal{in}};\textnormal{L}^{2}({\mathbb{%
T}}^{3})\big{)}\cap\textnormal{L}^{2}\big{(}J_{\textnormal{in}};\textnormal{H}%
^{1}({\mathbb{T}}^{3})\big{)}$. Using (46), we hence get the boundedness of $(M_{2}f_{\varepsilon})_{\varepsilon}$ in $\textnormal{L}^{\infty}(J_{\textnormal{in}})$ and then,
with (43) and (44), we see that $(m_{0}f_{\varepsilon})_{\varepsilon}$ and $(\mathbf{m}_{1}f_{\varepsilon})_{\varepsilon}$ are respectively bounded in $\textnormal{L}^{\infty}\big{(}J_{\textnormal{in}};\textnormal{L}^{5/3}({%
\mathbb{T}}^{3})\big{)}$ and $\textnormal{L}^{\infty}\big{(}J_{\textnormal{in}};\textnormal{L}^{5/4}({%
\mathbb{T}}^{3})\big{)}$.
Thanks to a classical transport estimate, using (28), we see that $(\rho_{\varepsilon})_{\varepsilon}$ is bounded in $\textnormal{L}^{\infty}\big{(}J_{\textnormal{in}};\textnormal{L}^{5/3}({%
\mathbb{T}}^{3})\big{)}$.
From (46), we deduce that for all $\varepsilon>0$ and $t\in J_{\textnormal{in}}$,
$$\displaystyle M_{2}{f_{\varepsilon}}(t)\leq\textnormal{K}_{T}^{\textnormal{in}}.$$
(49)
3.2 Compactness properties
Recalling Proposition 2.1 of existence of a solution to the regularized problem (for a given $\varepsilon$), we deduce
from the previous (uniform in $\varepsilon$) bounds
the existence of $\rho\in\textnormal{L}^{\infty}(J_{\textnormal{in}};\textnormal{L}^{5/3}({%
\mathbb{T}}^{3}))$, $f\in\textnormal{L}^{\infty}(J_{\textnormal{in}};\textnormal{L}^{\infty}({%
\mathbb{T}}^{3}\times\mathbb{R}^{3}))$ such that,
up to a subsequence,
$$\displaystyle(\rho_{\varepsilon})_{\varepsilon}$$
$$\displaystyle\operatorname*{\rightharpoonup}_{\varepsilon\rightarrow 0}\rho%
\text{ in }\textnormal{L}^{\infty}(J_{\textnormal{in}};\textnormal{L}^{5/3}({%
\mathbb{T}}^{3}))-\star,$$
$$\displaystyle(f_{\varepsilon})_{\varepsilon}$$
$$\displaystyle\operatorname*{\rightharpoonup}_{\varepsilon\rightarrow 0}f\text{%
in }\textnormal{L}^{\infty}(J_{\textnormal{in}};\textnormal{L}^{\infty}({%
\mathbb{T}}^{3}\times\mathbb{R}^{3}))-\star.$$
Using the boundedness of $(M_{2}f_{\varepsilon})_{\varepsilon}$ in $\textnormal{L}^{\infty}(J_{\textnormal{in}})$ and adding a subsequence if necessary, one manages to also show that
$$\displaystyle(m_{0}f_{\varepsilon})_{\varepsilon}\text{ and }(m_{0}(f_{%
\varepsilon}\gamma_{\varepsilon}))_{\varepsilon}$$
$$\displaystyle\operatorname*{\rightharpoonup}_{\varepsilon\rightarrow 0}m_{0}f%
\text{ in }\textnormal{L}^{\infty}(J_{\textnormal{in}};\textnormal{L}^{5/3}({%
\mathbb{T}}^{3}))-\star,$$
$$\displaystyle(\mathbf{m}_{1}f_{\varepsilon})_{\varepsilon}\text{ and }(\mathbf%
{m}_{1}(f_{\varepsilon}\gamma_{\varepsilon}))_{\varepsilon}$$
$$\displaystyle\operatorname*{\rightharpoonup}_{\varepsilon\rightarrow 0}\mathbf%
{m}_{1}f\text{ in }\textnormal{L}^{\infty}(J_{\textnormal{in}};\mathbf{L}^{5/4%
}({\mathbb{T}}^{3}))-\star.$$
Finally, using the bounds on $(\mathbf{u}_{\varepsilon})_{\varepsilon}$ and the Sobolev injection $\mathbf{H}^{1}({\mathbb{T}}^{3})\hookrightarrow\mathbf{L}^{6}({\mathbb{T}}^{3})$, we get
the existence of $\mathbf{u}\in\textnormal{L}^{\infty}(J_{\textnormal{in}};\mathbf{L}^{2}({%
\mathbb{T}}^{3}))\cap\textnormal{L}^{2}(J_{\textnormal{in}};\mathbf{H}_{%
\textnormal{div}}^{1}({\mathbb{T}}^{3}))$ such that
$$\displaystyle(\mathbf{u}_{\varepsilon})_{\varepsilon}\operatorname*{%
\rightharpoonup}_{\varepsilon\rightarrow 0}\mathbf{u}$$
$$\displaystyle\text{ in }\textnormal{L}^{2}\big{(}J_{\textnormal{in}};\mathbf{H%
}^{1}({\mathbb{T}}^{3})\big{)},$$
$$\displaystyle(\mathbf{u}_{\varepsilon})_{\varepsilon}\operatorname*{%
\rightharpoonup}_{\varepsilon\rightarrow 0}\mathbf{u}$$
$$\displaystyle\text{ in }\textnormal{L}^{2}\big{(}J_{\textnormal{in}};\mathbf{L%
}^{6}({\mathbb{T}}^{3})\big{)},$$
$$\displaystyle(\mathbf{u}_{\varepsilon})_{\varepsilon}\operatorname*{%
\rightharpoonup}_{\varepsilon\rightarrow 0}\mathbf{u}$$
$$\displaystyle\text{ in }\textnormal{L}^{\infty}\big{(}J_{\textnormal{in}};%
\mathbf{L}^{2}({\mathbb{T}}^{3})\big{)}-\star,$$
and of course $(\mathbf{u}_{\varepsilon}\star\varphi_{\varepsilon})_{\varepsilon}$ converges also towards $\mathbf{u}$ for the same topologies.
3.3 Weak convergence of nonlinear terms
In view of the previous weak convergences, it only remains to check that the nonlinear terms
converge to the expected limits, so that the weak limit triplet $(f,\rho,\mathbf{u})$ will indeed be a solution of our system.
Notice that since all the bounds and weak extractions are performed on the local interval $J_{\textnormal{in}}$, the constructed solution will only be local in time. In the last subsection, we shall explain how to extend it. Until then, through all the current subsection, the index ${}_{t}$ will refer to the intervall $J_{\textnormal{in}}$ in the notation “$\textnormal{L}_{t}^{p}(\textnormal{E})$”.
Notice that the system can be written
$$\displaystyle\textnormal{div}_{\mathbf{x}}\mathbf{u}_{\varepsilon}=0,$$
(50)
$$\displaystyle\partial_{t}f_{\varepsilon}+\textnormal{div}_{\mathbf{x},%
\boldsymbol{\xi}}(\mathbf{a}_{\varepsilon}f_{\varepsilon})-2\,f_{\varepsilon}=0,$$
(51)
$$\displaystyle\partial_{t}\rho_{\varepsilon}+\textnormal{div}_{\mathbf{x}}(\rho%
_{\varepsilon}(\mathbf{u}_{\varepsilon}\star\varphi_{\varepsilon}))=m_{0}(f_{%
\varepsilon}\gamma_{\varepsilon}),$$
(52)
$$\displaystyle\mathbb{P}\Big{\{}\partial_{t}[(1+\rho_{\varepsilon})\mathbf{u}_{%
\varepsilon}]+\textnormal{div}_{\mathbf{x}}\big{[}(1+\rho_{\varepsilon})(%
\mathbf{u}_{\varepsilon}\star\varphi_{\varepsilon})\otimes\mathbf{u}_{%
\varepsilon}\big{]}\Big{\}}-\Delta_{\mathbf{x}}\mathbf{u}_{\varepsilon}=%
\mathbb{P}\Big{\{}2\mathbf{m}_{1}(f_{\varepsilon}\gamma_{\varepsilon})-\mathbf%
{u}_{\varepsilon}m_{0}(f_{\varepsilon}\gamma_{\varepsilon})\Big{\}},$$
(53)
where $\mathbf{a}_{\varepsilon}(t,\mathbf{x},\boldsymbol{\xi}):=(\boldsymbol{\xi},[%
\mathbf{u}_{\varepsilon}(t)\star\varphi_{\varepsilon}](\mathbf{x}))$.
In order to handle the nonlinear terms, we shall use several times the Proposition 4.1 of the Appendix.
We denote by $\mathscr{M}_{s}$ the vector space of bounded measures on $J_{\textnormal{in}}\times{\mathbb{T}}^{3}$.
All the coming facts and their proofs are true up to some (finite number of) extractions that we don’t mention in the sequel.
Fact 1 :
The products $(\rho_{\varepsilon}\,(\mathbf{u}_{\varepsilon}\star\varphi_{\varepsilon}))_{\varepsilon}$ and $(\rho_{\varepsilon}\,\mathbf{u}_{\varepsilon})_{\varepsilon}$ both tend to $\rho\,\mathbf{u}$ in $\mathscrbf{M}_{s}$.
Proof of Fact 1: Using Proposition 4.1 of the Appendix, these two terms are handled in the same way: the velocity term (convoluted by $\varphi_{\varepsilon}$ or not) plays the role of $(a_{\varepsilon})_{\varepsilon}$, and is bounded in $\textnormal{L}^{2}_{t}(\textnormal{W}^{1,2}_{\mathbf{x}})$, whereas $(\rho_{\varepsilon})_{\varepsilon}$ plays the role of $(b_{\varepsilon})_{\varepsilon}$ and is bounded in $\textnormal{L}^{\infty}_{t}(\textnormal{L}^{5/3}_{\mathbf{x}})$ ($(5/3)^{\prime}=5/2<2^{\star}=6$). Thanks to (52) and the previous bounds (see subsection 3.1), $(\partial_{t}\rho_{\varepsilon})_{\varepsilon}$ is bounded in $\textnormal{L}^{2}_{t}(\textnormal{H}^{-m}_{\mathbf{x}})$, (where $m$ is taken large enough). We hence have
$$\displaystyle(\rho_{\varepsilon}\,(\mathbf{u}_{\varepsilon}\star\varphi_{%
\varepsilon}))_{\varepsilon}$$
$$\displaystyle\operatorname*{\rightharpoonup}_{\varepsilon\rightarrow 0}\rho\,%
\mathbf{u},\text{ in }\mathscrbf{M}_{s}-\text{w}\star,$$
$$\displaystyle(\rho_{\varepsilon}\,\mathbf{u}_{\varepsilon})_{\varepsilon}$$
$$\displaystyle\operatorname*{\rightharpoonup}_{\varepsilon\rightarrow 0}\rho\,%
\mathbf{u},\text{ in }\mathscr{M}_{s}-\text{w}\star.$$
Fact 2 :
$\displaystyle\big{\langle}(\rho_{\varepsilon}+1)\mathbf{u}_{\varepsilon},%
\mathbf{u}_{\varepsilon}\big{\rangle}_{\mathbf{L}^{2}_{t,\mathbf{x}}}%
\operatorname*{\longrightarrow}_{\varepsilon\rightarrow 0}\big{\langle}(\rho+1%
)\mathbf{u},\mathbf{u}\big{\rangle}_{\mathbf{L}^{2}_{t,\mathbf{x}}}$.
Proof of Fact 2: We wish to prove that
$$\displaystyle\operatorname*{\lim}_{\varepsilon\rightarrow 0}\int_{J_{%
\textnormal{in}}}\int_{{\mathbb{T}}^{3}}(\rho_{\varepsilon}+1)\,|\mathbf{u}_{%
\varepsilon}|^{2}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t=\int_{J_{\textnormal{in}}%
}\int_{{\mathbb{T}}^{3}}(\rho+1)\,|\mathbf{u}|^{2}\,\mathrm{d}\mathbf{x}\,%
\mathrm{d}t.$$
(54)
First write, since $\mathbf{u}_{\varepsilon}$ is divergence-free,
$$\displaystyle\int_{J_{\textnormal{in}}}\int_{{\mathbb{T}}^{3}}(\rho_{%
\varepsilon}+1)\,|\mathbf{u}_{\varepsilon}|^{2}\,\mathrm{d}\mathbf{x}\,\mathrm%
{d}t$$
$$\displaystyle=\int_{J_{\textnormal{in}}}\int_{{\mathbb{T}}^{3}}(\rho_{%
\varepsilon}+1)\,\mathbf{u}_{\varepsilon}\cdot\mathbf{u}_{\varepsilon}\,%
\mathrm{d}\mathbf{x}\,\mathrm{d}t$$
$$\displaystyle=\int_{J_{\textnormal{in}}}\int_{{\mathbb{T}}^{3}}\mathbb{P}\big{%
[}(\rho_{\varepsilon}+1)\mathbf{u}_{\varepsilon}\big{]}\cdot\mathbf{u}_{%
\varepsilon}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t.$$
Harmonic analysis and singular integral theory allow to show that $\mathbb{P}$ is bounded from $\textnormal{L}^{q}_{t}(\textnormal{L}^{p}_{\mathbf{x}})$ to itself, for all $p\in]1,\infty[$ and $q\in[1,\infty]$ (see for instance [19]). Since the strong continuity of an operator implies its weak sequential continuity, we see that $\mathbb{P}$ is sequentially continuous from $\textnormal{L}^{2}_{t}(\textnormal{L}^{30/23}_{\mathbf{x}})$ to itself, equipped with the weak topology. But we have seen in Subsection 3.1 that $(\mathbf{u}_{\varepsilon})_{\varepsilon}$ is bounded in $\textnormal{L}^{2}_{t}(\mathbf{H}^{1}_{\mathbf{x}})$, which is embedded in $\textnormal{L}^{2}_{t}(\mathbf{L}^{6}_{\mathbf{x}})$ by Sobolev injection, and $(\rho_{\varepsilon})_{\varepsilon}$ is bounded in $\textnormal{L}^{\infty}_{t}(\textnormal{L}^{5/3}_{\mathbf{x}})$ so that by Hölder inequality, $((\rho_{\varepsilon}+1)\mathbf{u}_{\varepsilon})_{\varepsilon}$ is bounded in $\textnormal{L}^{2}_{t}(\mathbf{L}^{30/23}_{\mathbf{x}})$ and hence (up to a subsequence) $\Big{(}\mathbb{P}\big{[}(\rho_{\varepsilon}+1)\,\mathbf{u}_{\varepsilon}\big{]%
}\Big{)}_{\varepsilon}$ converges weakly to $\mathbb{P}\big{[}(\rho+1)\,\mathbf{u}\big{]}$ in $\textnormal{L}^{2}_{t}(\textnormal{L}^{30/23}_{\mathbf{x}})$. We now can use Proposition 4.2 of the Appendix, with $q=2$ and $r=\infty$. Indeed,
–
$(\mathbf{u}_{\varepsilon})_{\varepsilon}$ is bounded in $\textnormal{L}^{2}_{t}(\textnormal{W}^{1,2}_{\mathbf{x}})\cap\textnormal{L}^{%
\infty}_{t}(\textnormal{L}^{2}_{\mathbf{x}})$,
–
since $\partial_{t}\mathbb{P}=\mathbb{P}\partial_{t}$, (53) implies that $\Big{(}\partial_{t}\mathbb{P}\big{[}(\rho_{\varepsilon}+1)\mathbf{u}_{%
\varepsilon}\big{]}\Big{)}_{\varepsilon}$ is bounded in $\textnormal{L}^{1}_{t}(\textnormal{H}_{\mathbf{x}}^{-m})$ for $m$ large enough,
–
$(30/23)^{\prime}=30/7<2^{\star}=6$.
Hence the product $\Big{\{}\mathbb{P}\big{[}(\rho_{\varepsilon}+1)\,\mathbf{u}_{\varepsilon}\big{%
]}\cdot\mathbf{u}_{\varepsilon}\Big{\}}_{\varepsilon}$ converges to $\mathbb{P}\big{[}(\rho+1)\,\mathbf{u}\big{]}\cdot\mathbf{u}$ in $\mathscrbf{M}_{s}-\star$. In particular, using
$\mathds{1}_{J_{\textnormal{in}}\times{\mathbb{T}}^{3}}$ as a test function, we get Fact 2.
Fact 3
$(\mathbf{u}_{\varepsilon})_{\varepsilon}$ and $(\mathbf{u}_{\varepsilon}\star\varphi_{\varepsilon})_{\varepsilon}$ both strongly converge to $\mathbf{u}$ in $\mathbf{L}^{2}_{t,\mathbf{x}}$.
Proof of Fact 3: We estimate
$$\displaystyle\int_{J_{\textnormal{in}}}\int_{{\mathbb{T}}^{3}}|\mathbf{u}_{%
\varepsilon}-\mathbf{u}|^{2}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t$$
$$\displaystyle\leq\int_{J_{\textnormal{in}}}\int_{{\mathbb{T}}^{3}}(1+\rho_{%
\varepsilon})\,|\mathbf{u}_{\varepsilon}-\mathbf{u}|^{2}\,\mathrm{d}\mathbf{x}%
\,\mathrm{d}t$$
$$\displaystyle=\int_{J_{\textnormal{in}}}\int_{{\mathbb{T}}^{3}}(1+\rho_{%
\varepsilon})\,|\mathbf{u}_{\varepsilon}|^{2}\,\mathrm{d}\mathbf{x}\,\mathrm{d%
}t+\int_{J_{\textnormal{in}}}\int_{{\mathbb{T}}^{3}}(1+\rho_{\varepsilon})\,|%
\mathbf{u}|^{2}\,\mathrm{d}\mathbf{x}\,\mathrm{d}t$$
$$\displaystyle-2\int_{J_{\textnormal{in}}}\int_{{\mathbb{T}}^{3}}(1+\rho_{%
\varepsilon})\,\mathbf{u}_{\varepsilon}\cdot\mathbf{u}\,\mathrm{d}\mathbf{x}\,%
\mathrm{d}t.$$
The first term of the second line converges to the same expression, but without $\varepsilon$: this
is exactly Fact 2 proven above. We have the same behavior for the second term of this line:
$(\rho_{\varepsilon})_{\varepsilon}$ converges weakly to $\rho$ in $\textnormal{L}^{\infty}_{t}(\textnormal{L}^{5/3}_{\mathbf{x}})-\star$ and $|\mathbf{u}|^{2}\in\textnormal{L}^{1}_{t}(\textnormal{L}^{3}_{\mathbf{x}})$. Eventually, for the third and last term, we use the (already proven) convergence of $[(1+\rho_{\varepsilon})\,\mathbf{u}_{\varepsilon}]_{\varepsilon}$ towards $(1+\rho)\,\mathbf{u}$ in $\textnormal{L}^{2}_{t}(\mathbf{L}^{30/23}_{\mathbf{x}})$,
and the embedding $\mathbf{u}\in\textnormal{L}^{2}_{t}(\mathbf{L}^{6}_{\mathbf{x}})%
\hookrightarrow\textnormal{L}^{2}_{t}(\mathbf{L}^{30/7}_{\mathbf{x}})$.
Strong convergence of $(\mathbf{u}_{\varepsilon})_{\varepsilon}$ is then directly transfered to $(\mathbf{u}_{\varepsilon}\star\varphi_{\varepsilon})_{\varepsilon}$.
Fact 4
$(\mathbf{u}_{\varepsilon})_{\varepsilon}$ converges strongly to $\mathbf{u}$ in all $\textnormal{L}^{c}_{t}(\mathbf{L}^{d}_{\mathbf{x}})$, for $c<2$ and $d<6$.
Proof of Fact 4: Thanks to Fact 3, we have strong convergence of $(\mathbf{u}_{\varepsilon})_{\varepsilon}$,
and hence almost everywhere convergence. Since this family is bounded $\textnormal{L}^{2}_{t}(\mathbf{L}^{6}_{\mathbf{x}})$,
we get the desired convergences.
Fact 5
$(f_{\varepsilon}\,(\mathbf{u}_{\varepsilon}\star\varphi_{\varepsilon}))_{\varepsilon}$ and
$(\mathbf{u}_{\varepsilon}\,m_{0}(f_{\varepsilon}\gamma_{\varepsilon}))_{\varepsilon}$ converges weakly in $\mathbf{L}^{1}$ to respectively $f\,\mathbf{u}$ and $\mathbf{u}\,m_{0}(f)$.
Proof of Fact 5: The family $(f_{\varepsilon})_{\varepsilon}$ converges weakly to $f$ in $\textnormal{L}^{\infty}_{t}(\textnormal{L}^{\infty}_{\mathbf{x},\boldsymbol{%
\xi}})-\star$, which with the strong convergence (Fact 3) of $(\mathbf{u}_{\varepsilon}\star\varphi_{\varepsilon})_{\varepsilon}$ in $\mathbf{L}^{2}_{t,\mathbf{x}}\hookrightarrow\mathbf{L}^{1}_{t,\mathbf{x}}$, ensures the weak $\mathbf{L}^{1}_{t,\mathbf{x},\boldsymbol{\xi}}$ convergence. Similarly, $(m_{0}(f_{\varepsilon}\,\gamma_{\varepsilon}))_{\varepsilon}$ converges weakly to $m_{0}f$ in $\textnormal{L}^{\infty}_{t}(\textnormal{L}^{5/3}_{\mathbf{x}})-\star$, and Fact 4 ensures for instance that $(\mathbf{u}_{\varepsilon})_{\varepsilon}$ converges strongly in $\textnormal{L}^{1}_{t}(\mathbf{L}^{5/2}_{\mathbf{x}})$.
Fact 6
$\big{[}(1+\rho_{\varepsilon})\,(\mathbf{u}_{\varepsilon}\star\varphi_{%
\varepsilon})\otimes\mathbf{u}_{\varepsilon}\big{]}_{\varepsilon}$ converges weakly in $\mathbf{L}^{1}_{t,\mathbf{x}}$ to $(1+\rho)\,\mathbf{u}\otimes\mathbf{u}$.
Proof of Fact 6: Once again, it’s a “ weak $\times$ strong ” type of convergence here. First notice
$$\displaystyle\frac{2}{11}=\frac{\theta}{6}+\frac{1-\theta}{2},$$
where $\theta:=21/22$. Hence, by Hölder’s inequality, we get the following interpolation result: $\textnormal{L}^{\infty}_{t}(\textnormal{L}^{2}_{\mathbf{x}})\cap\textnormal{L}%
^{2}_{t}(\textnormal{L}^{6}_{\mathbf{x}})\hookrightarrow\textnormal{L}^{2/%
\theta}_{t}(\textnormal{L}^{11/2}_{\mathbf{x}})$. This implies that both $(\mathbf{u}_{\varepsilon})_{\varepsilon}$ and $(\mathbf{u}_{\varepsilon}\star\varphi_{\varepsilon})_{\varepsilon}$ are bounded in $\textnormal{L}^{2/\theta}_{t}(\textnormal{L}^{11/2}_{\mathbf{x}})$, and the product $[(\mathbf{u}_{\varepsilon}\star\varphi_{\varepsilon})\otimes\mathbf{u}_{%
\varepsilon}]_{\varepsilon}$ (which converges almost everywhere to $\mathbf{u}\otimes\mathbf{u}$, see Fact 4) is hence bounded in $\textnormal{L}^{1/\theta}_{t}(\textnormal{L}^{11/4}_{\mathbf{x}})$. Since $\theta<1$ and $11/4>5/2$,
we get the strong convergence of this product to $\mathbf{u}\otimes\mathbf{u}$, in $\textnormal{L}^{1}_{t}(\textnormal{L}^{5/2}_{\mathbf{x}})$, and we already know that $(\rho_{\varepsilon})_{\varepsilon}$ converges weakly to $\rho$ in $\textnormal{L}^{\infty}_{t}(\textnormal{L}^{5/3}_{\mathbf{x}})-\star$.
Using all the weak convergences above, we end up with a local in time solution of our system, which
concludes the Proof of Proposition 3.1.
$\square$.
Next subsection is devoted to the prolongation of the local solutions defined above in solutions
defined on $[0,T]$.
3.4 Energy estimate and global existence
In order to prove the existence of global solutions to our system,
a standard strategy consists in reproducing the previous step at time $t_{\star}-\varepsilon$, obtain another local solution and paste it with the previous one, and so on and so forth.
For such a strategy to succeed, one must ensure that the sequence of local times of existence does not decrease too quickly. But, as noticed in Remark 3.1, the local time of existence is bounded
below by $1/\textnormal{P}_{T}^{\textnormal{in}}$, which is (by definition) a non-increasing function of $\|\sqrt{1+\rho_{\textnormal{in}}}\mathbf{u}_{\textnormal{in}}\|_{2}+\|f_{%
\textnormal{in}}\|_{\infty}+M_{2}f_{\textnormal{in}}$.
We already know (by maximum principle) that $\|f\|_{\infty}\leq e^{2t_{\star}}f_{\textnormal{in}}$, and this bound ensures
that all possible extension will always satisfy $\|f\|_{\infty}\leq e^{2T}f_{\textnormal{in}}$, since the sum of all local times of existence does not exceed $T$ (well if it does, we’re done !). It is hence sufficient to prove that, for almost all $t\in J_{\textnormal{in}}$,
$$\displaystyle M_{2}f(t)+\|\sqrt{1+\rho(t)}\,\mathbf{u}(t)\|_{2}\leq M_{2}f_{%
\textnormal{in}}+\|\sqrt{1+\rho_{\textnormal{in}}}\,\mathbf{u}_{\textnormal{in%
}}\|_{2}.$$
(55)
Indeed, such an estimate would propagate for each local solution and we may hence bound from below
all the corresponding times of existence, which means that our strategy would end in a finite number of steps. In fact (55) is a straightforward consequence of the following energy estimate:
Proposition 3.2.
The solution built in Proposition 3.1 satisfies for almost all $t\in J_{\textnormal{in}}$,
$$\displaystyle\frac{1}{2}\left\{M_{2}f(t)+\|\sqrt{1+\rho(t)}\,\mathbf{u}(t)\|_{%
\textnormal{L}^{2}({\mathbb{T}}^{3})}^{2}\right\}$$
$$\displaystyle+\int_{0}^{t}\|\nabla_{\mathbf{x}}\mathbf{u}(s)\|_{\textnormal{L}%
^{2}({\mathbb{T}}^{3})}^{2}\mathrm{d}s+\frac{3}{2}\int_{0}^{t}\int_{{\mathbb{T%
}}^{3}\times\mathbb{R}^{3}}|\mathbf{u}-\boldsymbol{\xi}|^{2}\,f\,\mathrm{d}%
\boldsymbol{\xi}\,\mathrm{d}\mathbf{x}\,\mathrm{d}s$$
$$\displaystyle\leq\frac{1}{2}\left\{M_{2}f_{\text{\bf in}}+\|\sqrt{1+\rho_{%
\text{\bf in}}}\,\mathbf{u}_{\text{\bf in}}\|_{\textnormal{L}^{2}({\mathbb{T}}%
^{3})}^{2}\right\}.$$
(56)
Proof of Proposition 3.2:
We first use inequality (39) for solutions of the regularized system,
and add the integral in time of (45)$\times\frac{1}{2}$ to get
$$\displaystyle\frac{1}{2}\left\{M_{2}f_{\varepsilon}(t)+\|\sqrt{1+\rho(t)}\,%
\mathbf{u}_{\varepsilon}(t)\|_{\textnormal{L}^{2}({\mathbb{T}}^{3})}^{2}\right\}$$
$$\displaystyle+\int_{0}^{t}\|\nabla_{\mathbf{x}}\mathbf{u}_{\varepsilon}(s)\|_{%
\textnormal{L}^{2}({\mathbb{T}}^{3})}^{2}\mathrm{d}s$$
$$\displaystyle+\frac{3}{2}\int_{0}^{t}\int_{{\mathbb{T}}^{3}\times\mathbb{R}^{3%
}}|\mathbf{u}_{\varepsilon}-\boldsymbol{\xi}|^{2}\,f_{\varepsilon}\,\mathrm{d}%
\boldsymbol{\xi}\,\mathrm{d}\mathbf{x}\,\mathrm{d}s$$
$$\displaystyle\leq\frac{1}{2}\,\left\{M_{2}f_{\text{\bf in}}+\|\sqrt{1+\rho_{%
\text{\bf in}}}\,\mathbf{u}_{\text{\bf in}}\|_{\textnormal{L}^{2}({\mathbb{T}}%
^{3})}^{2}\right\}+R_{\varepsilon}(t),$$
where
$$\displaystyle R_{\varepsilon}(t)=\frac{3}{2}\stackrel{{\scriptstyle R_{%
\varepsilon}^{1}(t)}}{{\overbrace{\int_{0}^{t}\int_{{\mathbb{T}}^{3}\times%
\mathbb{R}^{3}}f_{\varepsilon}\,|\mathbf{u}_{\varepsilon}|^{2}\,(1-\gamma_{%
\varepsilon}(\boldsymbol{\xi}))\,\mathrm{d}\boldsymbol{\xi}\,\mathrm{d}\mathbf%
{x}\,\mathrm{d}s}}}$$
$$\displaystyle+2\stackrel{{\scriptstyle R_{\varepsilon}^{2}(t)}}{{\overbrace{%
\int_{0}^{t}\int_{{\mathbb{T}}^{3}\times\mathbb{R}^{3}}f_{\varepsilon}\,%
\boldsymbol{\xi}\cdot\mathbf{u}_{\varepsilon}\,(\gamma_{\varepsilon}(%
\boldsymbol{\xi})-1)\,\mathrm{d}\boldsymbol{\xi}\,\mathrm{d}\mathbf{x}\,%
\mathrm{d}s}}}$$
$$\displaystyle+\stackrel{{\scriptstyle R_{\varepsilon}^{3}(t)}}{{\overbrace{%
\int_{0}^{t}\int_{{\mathbb{T}}^{3}\times\mathbb{R}^{3}}f_{\varepsilon}\,%
\boldsymbol{\xi}\cdot(\mathbf{u}_{\varepsilon}\star\varphi_{\varepsilon}-%
\mathbf{u}_{\varepsilon})\,\mathrm{d}\boldsymbol{\xi}\,\mathrm{d}\mathbf{x}\,%
\mathrm{d}s}}}.$$
Using Lemma 3.1, the maximum principle, and bound (49), we notice that for $\alpha>0$ small enough, $(m_{\alpha}f_{\varepsilon})_{\varepsilon}$ is bounded in $\textnormal{L}^{\infty}_{t}(\textnormal{L}^{3/2}_{\mathbf{x}})$. Now recall that $\gamma_{\varepsilon}$ is chosen so that $|1-\gamma_{\varepsilon}|(\boldsymbol{\xi})\leq\mathds{1}_{|\boldsymbol{\xi}|%
\gtrsim 1/\varepsilon}$, hence
$$\displaystyle R_{\varepsilon}^{1}(t)\lesssim\varepsilon^{\alpha}\|m_{\alpha}f_%
{\varepsilon}\|_{\textnormal{L}^{\infty}_{t}(\textnormal{L}^{3/2}_{\mathbf{x}}%
)}\|\mathbf{u}_{\varepsilon}\|_{\textnormal{L}^{2}_{t}(\textnormal{L}^{6}_{%
\mathbf{x}})}^{2},$$
which goes to $0$ with $\varepsilon$ since $(\mathbf{u}_{\varepsilon})_{\varepsilon}$ is bounded in $\textnormal{L}^{2}_{t}(\mathbf{H}^{1}_{\mathbf{x}})\hookrightarrow\textnormal{%
L}^{2}_{t}(\mathbf{L}^{6}_{\mathbf{x}})$. This shows that $(R_{\varepsilon}^{1})_{\varepsilon}\displaystyle\operatorname*{\longrightarrow%
}_{\varepsilon\rightarrow 0}0$ uniformly on $J_{\textnormal{in}}$, and a similar Proof applies for $(R_{\varepsilon}^{2})_{\varepsilon}$. Then, we have
$$\displaystyle R_{\varepsilon}^{3}(t)=\int_{0}^{t}\int_{{\mathbb{T}}^{3}\times%
\mathbb{R}^{3}}(f_{\varepsilon}\boldsymbol{\xi}\cdot\mathbf{v}_{\varepsilon})%
\,\mathrm{d}\boldsymbol{\xi}\,\mathrm{d}\mathbf{x}\,\mathrm{d}s,$$
where $(\mathbf{v}_{\varepsilon})_{\varepsilon}$ is bounded in $\textnormal{L}^{2}_{t}(\mathbf{L}^{6}_{\mathbf{x}})$ and converges to $0$ in $\textnormal{L}^{2}_{t,\mathbf{x}}\hookrightarrow\textnormal{L}^{1}_{t,\mathbf{%
x}}$. For values of $\boldsymbol{\xi}$ satisfying $|\boldsymbol{\xi}|\leq\displaystyle(\|\mathbf{v}_{\varepsilon}\|_{1}+%
\varepsilon)^{-1/2}$, we simply use the maximum principle for $(f_{\varepsilon})_{\varepsilon}$ to see that their contribution goes to $0$ with $\varepsilon$ (uniformly in time), and for the large values of $\boldsymbol{\xi}$ satisfying the opposite inequality, we handle them as we did for $R_{\varepsilon}^{1}$ and $R_{\varepsilon}^{2}$, using the $\textnormal{L}^{2}_{t}(\mathbf{L}^{6}_{\mathbf{x}})$ bound.
At this stage, we proved that $(R_{\varepsilon})_{\varepsilon}$ converges to $0$ uniformly in time. Let us treat the other terms of inequality (56). We first use the classical estimates of weak convergence to get
$$\displaystyle\int_{0}^{t}\|\nabla_{\mathbf{x}}\mathbf{u}(s)\|_{\textnormal{L}^%
{2}({\mathbb{T}}^{3})}^{2}\mathrm{d}s$$
$$\displaystyle\leq\operatorname*{\underline{\lim}}_{\varepsilon\rightarrow 0}%
\hskip 2.845276pt\int_{0}^{t}\|\nabla_{\mathbf{x}}\mathbf{u}_{\varepsilon}\|_{%
\textnormal{L}^{2}({\mathbb{T}}^{3})}^{2}\mathrm{d}s,$$
$$\displaystyle M_{2}f(t)$$
$$\displaystyle\leq\operatorname*{\underline{\lim}}_{\varepsilon\rightarrow 0}%
\hskip 2.845276ptM_{2}f_{\varepsilon}(t).$$
(57)
Furthermore, adapting the proof of estimate (54) used in Fact $2$,
we obtain for almost all times $t\in J_{\textnormal{in}}$,
$$\displaystyle\|\sqrt{1+\rho_{\varepsilon}(t)}\,\mathbf{u}_{\varepsilon}(t)\|_{%
\textnormal{L}^{2}({\mathbb{T}}^{3})}^{2}\operatorname*{\rightarrow}_{%
\varepsilon\rightarrow 0}\|\sqrt{1+\rho(t)}\,\mathbf{u}(t)\|_{\textnormal{L}^{%
2}({\mathbb{T}}^{3})}^{2},$$
so that it only remains to prove
$$\displaystyle\int_{0}^{t}\int_{{\mathbb{T}}^{3}\times\mathbb{R}^{3}}|\mathbf{u%
}-\boldsymbol{\xi}|^{2}f\mathrm{d}\boldsymbol{\xi}\mathrm{d}\mathbf{x}\mathrm{%
d}s$$
$$\displaystyle\leq\operatorname*{\underline{\lim}}_{\varepsilon\rightarrow 0}%
\int_{0}^{t}\int_{{\mathbb{T}}^{3}\times\mathbb{R}^{3}}|\mathbf{u}_{%
\varepsilon}-\boldsymbol{\xi}|^{2}f_{\varepsilon}\mathrm{d}\boldsymbol{\xi}%
\mathrm{d}\mathbf{x}\mathrm{d}s,$$
(58)
in order to conclude the Proof of Proposition 3.2. We first write
$$\displaystyle\int_{0}^{t}\int_{{\mathbb{T}}^{3}\times\mathbb{R}^{3}}|\mathbf{u%
}_{\varepsilon}-\boldsymbol{\xi}|^{2}f_{\varepsilon}\mathrm{d}\boldsymbol{\xi}%
\mathrm{d}\mathbf{x}\mathrm{d}s=\int_{0}^{t}\int_{{\mathbb{T}}^{3}\times%
\mathbb{R}^{3}}|\mathbf{u}_{\varepsilon}|^{2}f_{\varepsilon}\mathrm{d}%
\boldsymbol{\xi}\mathrm{d}\mathbf{x}\mathrm{d}s+\int_{0}^{t}M_{2}f_{%
\varepsilon}(s)\mathrm{d}s-2\int_{0}^{t}\int_{{\mathbb{T}}^{3}\times\mathbb{R}%
^{3}}\boldsymbol{\xi}\cdot\mathbf{u}_{\varepsilon}f_{\varepsilon}\mathrm{d}%
\boldsymbol{\xi}\mathrm{d}\mathbf{x}\mathrm{d}s.$$
(59)
Since $(\mathbf{u}_{\varepsilon})_{\varepsilon}$ converges strongly in $\mathbf{L}^{2}_{t,\mathbf{x}}$ and $(f_{\varepsilon})_{\varepsilon}$ converges weakly in $\textnormal{L}^{\infty}_{t,\mathbf{x},\boldsymbol{\xi}}-\star$, we have by Fatou’s Lemma
$$\displaystyle\int_{0}^{t}\int_{{\mathbb{T}}^{3}\times\mathbb{R}^{3}}|\mathbf{u%
}|^{2}f\mathds{1}_{|\boldsymbol{\xi}|\leq n}\mathrm{d}\boldsymbol{\xi}\,%
\mathrm{d}\mathbf{x}\,\mathrm{d}s\leq\operatorname*{\underline{\lim}}_{%
\varepsilon\rightarrow 0}\int_{0}^{t}\int_{{\mathbb{T}}^{3}\times\mathbb{R}^{3%
}}|\mathbf{u}_{\varepsilon}|^{2}f_{\varepsilon}\mathrm{d}\boldsymbol{\xi}\,%
\mathrm{d}\mathbf{x}\,\mathrm{d}s,$$
hence by monotone convergence,
$$\displaystyle\int_{0}^{t}\int_{{\mathbb{T}}^{3}\times\mathbb{R}^{3}}|\mathbf{u%
}|^{2}f\,\mathrm{d}\boldsymbol{\xi}\,\mathrm{d}\mathbf{x}\,\mathrm{d}s\leq%
\operatorname*{\underline{\lim}}_{\varepsilon\rightarrow 0}\int_{0}^{t}\int_{{%
\mathbb{T}}^{3}\times\mathbb{R}^{3}}|\mathbf{u}_{\varepsilon}|^{2}f_{%
\varepsilon}\,\mathrm{d}\boldsymbol{\xi}\,\mathrm{d}\mathbf{x}\,\mathrm{d}s.$$
(60)
Using another time Fatou’s Lemma with estimate (57) shows that
$$\displaystyle\int_{0}^{t}M_{2}f(s)\mathrm{d}s\leq\operatorname*{\underline{%
\lim}}_{\varepsilon\rightarrow 0}\int_{0}^{t}M_{2}f_{\varepsilon}(s)\mathrm{d}s.$$
(61)
On the other hand, since $v_{\varepsilon}:=\mathbf{u}_{\varepsilon}-\mathbf{u}$ is bounded in $\textnormal{L}^{2}_{t}(\mathbf{L}^{6}_{\mathbf{x}})$ and converges to $0$ in $\textnormal{L}^{1}_{t,\mathbf{x}}$, we have as before (see the study of $R_{\varepsilon}^{3}$)
$$\displaystyle\int_{0}^{t}\int_{{\mathbb{T}}^{3}\times\mathbb{R}^{3}}f_{%
\varepsilon}\,\boldsymbol{\xi}\cdot(\mathbf{u}_{\varepsilon}-\mathbf{u})\,%
\mathrm{d}\boldsymbol{\xi}\,\mathrm{d}\mathbf{x}\,\mathrm{d}s\operatorname*{%
\longrightarrow}_{\varepsilon\rightarrow 0}0,$$
and handling large velocities as we did for $R_{\varepsilon}^{1}$ and $R_{\varepsilon}^{2}$, we can show that
$$\displaystyle\int_{{\mathbb{T}}^{3}\times\mathbb{R}^{3}}(f\boldsymbol{\xi}%
\cdot\mathbf{u})\mathds{1}_{|\boldsymbol{\xi}|>n}\mathrm{d}\boldsymbol{\xi}\,%
\mathrm{d}\mathbf{x}\,\mathrm{d}s+\sup_{\varepsilon>0}\int_{0}^{t}\int_{{%
\mathbb{T}}^{3}\times\mathbb{R}^{3}}(f_{\varepsilon}\boldsymbol{\xi}\cdot%
\mathbf{u})\mathds{1}_{|\boldsymbol{\xi}|>n}\mathrm{d}\boldsymbol{\xi}\,%
\mathrm{d}\mathbf{x}\,\mathrm{d}s\operatorname*{\longrightarrow}_{n\rightarrow%
+\infty}0.$$
When $n\in\mathbb{N}$ is fixed, we have by weak $\textnormal{L}^{\infty}_{t,\mathbf{x},\boldsymbol{\xi}}-\star$ convergence of $(f_{\varepsilon})_{\varepsilon}$,
$$\displaystyle\int_{0}^{t}\int_{{\mathbb{T}}^{3}\times\mathbb{R}^{3}}(f_{%
\varepsilon}\,\boldsymbol{\xi}\cdot\mathbf{u})\mathds{1}_{|\boldsymbol{\xi}|%
\leq n}\,\mathrm{d}\boldsymbol{\xi}\,\mathrm{d}\mathbf{x}\,\mathrm{d}s%
\operatorname*{\longrightarrow}_{n\rightarrow+\infty}\int_{0}^{t}\int_{{%
\mathbb{T}}^{3}\times\mathbb{R}^{3}}(f_{\varepsilon}\,\boldsymbol{\xi}\cdot%
\mathbf{u})\,\mathrm{d}\boldsymbol{\xi}\,\mathrm{d}\mathbf{x}\,\mathrm{d}s,$$
so that the three last convergences imply together
$$\displaystyle\int_{0}^{t}\int_{{\mathbb{T}}^{3}\times\mathbb{R}^{3}}(f_{%
\varepsilon}\,\boldsymbol{\xi}\cdot\mathbf{u}_{\varepsilon})\,\mathrm{d}%
\boldsymbol{\xi}\,\mathrm{d}\mathbf{x}\,\mathrm{d}s\operatorname*{%
\longrightarrow}_{n\rightarrow+\infty}\int_{0}^{t}\int_{{\mathbb{T}}^{3}\times%
\mathbb{R}^{3}}(f\,\boldsymbol{\xi}\cdot\mathbf{u})\,\mathrm{d}\boldsymbol{\xi%
}\,\mathrm{d}\mathbf{x}\,\mathrm{d}s.$$
(62)
Using (60), (61) and (62) in (59),
we get (58) and this ends the Proof of Proposition 3.2.
$\square$
This also concludes the Proof of Theorem 1.1, thanks to the strategy of prolongation of local solutions explained at the beginning of the Subsection.
$\square$
4 Appendix
We present in the Appendix a few auxiliary Lemmas which are used in various parts of this work. We start
with a standard variant of Grönwall’s Lemma, that we recall for the sake of completeness.
4.1 A variant of Grönwall’s Lemma
Lemma 4.1.
Let $f\in\mathscr{C}^{1}(\mathbb{R}_{+})$ be a convex non-decreasing function and $\alpha\in\mathbb{R}$. Consider $(z,[0,t_{\star}[)$ the maximal solution of $z^{\prime}=f(z)$, $z(0)=\alpha$ on $\mathbb{R}_{+}$.
Let $a:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ be a continuous function such that
$$\displaystyle a(t)\leq\alpha+\int_{0}^{t}f(a(s))\,\mathrm{d}s.$$
Then, for all $t\in[0,t_{\star}[$,
$$\displaystyle a(t)\leq z(t).$$
Proof of Lemma 4.1:
The usual (linear) Grönwall lemma shows that if $u\in\mathscr{C}^{1}(\mathbb{R}_{+})$ satisfies $\displaystyle u(t)\leq\int_{0}^{t}b(s)u(s)\mathrm{d}s$ on $[0,t_{\star}[$, then $u$ is nonpositive on this interval.
We have here by convexity (on $[0,t_{\star}[$)
$$\displaystyle u(t):=a(t)-z(t)\leq\int_{0}^{t}[f(a(s))-f(z(s))]\,\mathrm{d}s%
\leq\int_{0}^{t}f^{\prime}(a(s))\,u(s)\,\mathrm{d}s,$$
and since $f$ is nondecreasing, $u\leq 0$ on $[0,t_{\star}[$.
$\square$
As a consequence, we get the
Lemma 4.2.
Let $A,\gamma>0$. The maximal solution of the Cauchy problem $z^{\prime}=A\,z^{1+\gamma}$, $z(0)=A$ is defined at least on $[0,(\gamma\,A^{\gamma+1})^{-1}[$.
Proof of Lemma 4.2
Indeed : $\displaystyle t\mapsto(A^{-\gamma}-A\,\gamma\,t)^{-1/\gamma}$ is well defined for $t<(\gamma\,A^{\gamma+1})^{-1}$, and solves the Cauchy problem.
$\square$
In next Subsection, we recall for the sake of completeness the following classical compactness Lemma:
4.2 A compactness lemma
Lemma 4.3.
The injection $\textnormal{H}^{1}_{t,\mathbf{x}}\hookrightarrow\mathscr{C}^{0}_{t}(%
\textnormal{L}^{2}_{\mathbf{x}})$ is compact.
Proof of Lemma 4.3:
Let $(f_{n})_{n}$ be a bounded sequence of $\textnormal{H}^{1}_{t,\mathbf{x}}$.
Step 1: $A=\{t\in[0,T]\,:\,\left\|{f_{n}(t)}\right\|_{\textnormal{H}^{1}_{\mathbf{x}}}%
\rightarrow+\infty\}$ is a null set with respect to the Lebesgue measure $\mu$. Indeed, if it were not, then for any $R>0$, we would have
$$\displaystyle R\,\mu(A)\leq\int_{A}\operatorname*{\underline{\lim}}_{n%
\rightarrow\infty}\left\|{f_{n}(t)}\right\|^{2}_{\textnormal{H}^{1}_{\mathbf{x%
}}}(t)\,\mathrm{d}t.$$
Fatou’s lemma would then imply
$$\displaystyle R\,\mu(A)\leq\operatorname*{\underline{\lim}}_{n\rightarrow%
\infty}\int_{A}\left\|{f_{n}(t)}\right\|^{2}_{\textnormal{H}^{1}_{\mathbf{x}}}%
\leq\operatorname*{\underline{\lim}}_{n\rightarrow\infty}\left\|{f_{n}}\right%
\|^{2}_{\textnormal{H}^{1}_{t,\mathbf{x}}},$$
which is impossible since the right-hand side is finite. By definition of $A$, for all $t$ in $[0,T]\backslash A$, the sequence $(\left\|{f_{n}(t)}\right\|_{\textnormal{H}^{1}_{\mathbf{x}}})_{n}$ has a bounded subsequence. Since $\mu(A)=0$, we can find a countable subset $[0,T]\backslash A$ that is dense in $[0,T]$. Let us denote this subset $\textnormal{B}=(t_{p})_{p\in\mathbb{N}}$. We can then extract a (diagonal) subsequence of $(f_{n})_{n}$ (sill denoted $(f_{n})_{n}$) such that for all $p\in\mathbb{N}$, the sequence $(f_{n}(t_{p}))_{n\in\mathbb{N}}$ is bounded in ${\textnormal{H}^{1}_{\mathbf{x}}}$.
Step 2: Since $(\partial_{t}f_{n})_{n}$ is bounded in $\textnormal{L}^{2}_{t,\mathbf{x}}$, $(f_{n})_{n}$ is uniformly equicontinuous w.r.t. $t$, with values in $\textnormal{L}_{\mathbf{x}}^{2}$.
Step 3: Since by Rellich’s Theorem, the injection $\textnormal{H}^{1}_{\mathbf{x}}\hookrightarrow\textnormal{L}^{2}_{\mathbf{x}}$ is compact, we can extract (again, diagonally) a subsequence (still denoted) $(f_{n})_{n}$ such that for all $p$, the sequence $(f_{n}(t_{p}))_{n}$ converges in $\textnormal{L}^{2}_{\mathbf{x}}$ to some element $f(t_{p})\in\textnormal{L}^{2}_{\mathbf{x}}$.
Step 4: $f$ is uniformly continuous on B, with values in $\textnormal{L}^{2}_{\mathbf{x}}$. Indeed, for any $t$, $s$ in B,
$$\displaystyle\left\|{f(t)-f(s)}\right\|_{\textnormal{L}^{2}_{\mathbf{x}}}\leq%
\left\|{f(t)-f_{n}(t)}\right\|_{\textnormal{L}^{2}_{\mathbf{x}}}+\left\|{f_{n}%
(t)-f_{n}(s)}\right\|_{\textnormal{L}^{2}_{\mathbf{x}}}+\left\|{f_{n}(s)-f(s)}%
\right\|_{\textnormal{L}^{2}_{\mathbf{x}}}.$$
The central term of the right-hand side goes to $0$ with $|t-s|$, independently of $n$ because of the uniform equicontinuity obtained in Step 2. When $t$ and $s$ are fixed elements of B, the two other terms of the right-hand side vanish when $n\rightarrow\infty$ because of the extraction of Step 3.
Step 5:
$f$ admits a unique continuous extension to $[0,T]$, which is uniformly continuous on this interval, with values in $\textnormal{L}^{2}_{\mathbf{x}}$.
Step 6: $f$ is actually the limit of $(f_{n})_{n}$ in $\mathscr{C}^{0}_{t}(\textnormal{L}^{2}_{\mathbf{x}})$. Indeed, for $\varepsilon>0$, choose $\delta$ a corresponding modulus of equicontinuity for $f$ and of uniform equicontinuity for $(f_{n})_{n}$. Then pick (B is dense in the compact set $[0,T]$) a finite number $t_{1},\dots,t_{N}$ of elements of B such
that $[0,T]\subseteq\cup_{i=1}^{N}]t_{i}-\delta,t_{i}+\delta[$. Eventually, for any $\sigma\in[0,T]$, if $\sigma\in]t_{i}-\delta,t_{i}+\delta[$, then
$$\displaystyle\left\|{f(\sigma)-f_{n}(\sigma)}\right\|_{\textnormal{L}^{2}_{%
\mathbf{x}}}$$
$$\displaystyle\leq\left\|{f(\sigma)-f(t_{i})}\right\|_{\textnormal{L}^{2}_{%
\mathbf{x}}}+\left\|{f(t_{i})-f_{n}(t_{i})}\right\|_{\textnormal{L}^{2}_{%
\mathbf{x}}}+\left\|{f_{n}(t_{i})-f_{n}(\sigma)}\right\|_{\textnormal{L}^{2}_{%
\mathbf{x}}}$$
$$\displaystyle\leq 2\varepsilon+\left\|{f(t_{i})-f_{n}(t_{i})}\right\|_{%
\textnormal{L}^{2}_{\mathbf{x}}},$$
so that $\left\|{f(\sigma)-f_{n}(\sigma)}\right\|_{\textnormal{L}^{2}_{\mathbf{x}}}\leq
3\varepsilon$ for $n$ large enough. This ends the proof of Lemma 4.3.
$\square$
4.3 Weak convergence of a product
We present here a result based on the method used in [10].
Proposition 4.1.
Let $q\in[1,\infty]$ and $p\in[1,3[$. Consider two families $(a_{\varepsilon})_{\varepsilon}$ and $(b_{\varepsilon})_{\varepsilon}$ respectively in $\textnormal{L}^{q}_{t}(\textnormal{W}^{1,p}_{\mathbf{x}})$ and $\textnormal{L}^{q^{\prime}}_{t}(\textnormal{L}^{s^{\prime}}_{\mathbf{x}})$, with $s<p^{\star}:=\frac{3p}{3-p}$. Assume the weak convergences
$$\displaystyle(a_{\varepsilon})_{\varepsilon}$$
$$\displaystyle\operatorname*{\rightharpoonup}_{\varepsilon\rightarrow 0}a,\text%
{ in}\quad\textnormal{L}^{q}_{t}(\textnormal{W}^{1,p}_{\mathbf{x}})-w\star,$$
$$\displaystyle(b_{\varepsilon})_{\varepsilon}$$
$$\displaystyle\operatorname*{\rightharpoonup}_{\varepsilon\rightarrow 0}b,\text%
{ in}\quad\textnormal{L}^{q^{\prime}}_{t}(\textnormal{L}^{s^{\prime}}_{\mathbf%
{x}})-w\star.$$
If $(\partial_{t}b_{\varepsilon})_{\varepsilon}$ is bounded in $\textnormal{L}^{q^{\prime}}_{t}(\textnormal{H}^{-m}_{\mathbf{x}})$ for some $m\in\mathbb{Z}$, then, up to a subsequence, we have the weak convergence
$$\displaystyle(a_{\varepsilon}\,b_{\varepsilon})_{\varepsilon}\operatorname*{%
\rightharpoonup}_{\varepsilon\rightarrow 0}a\,b\text{ in the sense of measures},$$
i.e. with test functions in $\mathscr{C}^{0}_{t,\mathbf{x}}$.
Proof of Proposition 4.1 :
Let us first notice that for all $\varepsilon>0$ $a_{\varepsilon}\,b_{\varepsilon}\in\textnormal{L}^{1}_{t,\mathbf{x}}$ (Sobolev injection) and that the sequences $(a_{\varepsilon})_{\varepsilon}$ and $(b_{\varepsilon})_{\varepsilon}$ are bounded in the spaces in which they converge weakly.
Step 1.
We have
$$\displaystyle a\,(b\star\varphi_{\eta})\operatorname*{\longrightarrow}_{\eta%
\rightarrow 0}a\,b,\text{ in }\textnormal{L}^{1}_{t,\mathbf{x}}\text{ strong }.$$
Step 2.
Since $(\partial_{t}b_{\varepsilon})_{\varepsilon}$ is bounded in $\textnormal{L}^{q^{\prime}}_{t}(\textnormal{H}^{-m}_{\mathbf{x}})$, $(b_{\varepsilon}\star\varphi_{\eta})_{\varepsilon}$ is bounded in $\textnormal{W}^{1,q^{\prime}}_{t}(\textnormal{W}^{1,p^{\prime}}_{\mathbf{x}})$ so that (thanks to Rellich’s Theorem), for all fixed $\eta$, $(b_{\varepsilon}\star\varphi_{\eta})_{\varepsilon}$ admits a (strongly) converging subsequence in $\textnormal{L}^{q^{\prime}}_{t}(\textnormal{L}^{p^{\prime}}_{\mathbf{x}})$, the limit being necessarily $b\star\varphi_{\eta}$ (this is due to the
uniqueness of the weak$-\star$ limit). In fact, we can choose (but we don’t write it explicitly) a common (diagonal) extraction for all $\eta$ after discretization ($\eta:=1/k$). Since $(a_{\varepsilon})_{\varepsilon}$ converges weakly in $\textnormal{L}^{q}_{t}(\textnormal{L}^{p}_{\mathbf{x}})$, we eventually get, for all fixed $\eta$,
$$\displaystyle(a_{\varepsilon}\,(b_{\varepsilon}\star\varphi_{\eta}))_{%
\varepsilon}\operatorname*{\rightharpoonup}_{\varepsilon\rightarrow 0}a\,(b%
\star\varphi_{\eta})\text{ in }\textnormal{L}^{1}_{t,\mathbf{x}}\text{ weak.}$$
Step 3.
We shall use the following “commutator Lemma”, the Proof of which is rather close to the usual Friedrichs Lemma (which is a key element of [10]).
Lemma 4.4.
Under the assumptions111The assumption on $(\partial_{t}b_{\varepsilon})_{\varepsilon}$ is obviously useless here. of Proposition 4.1, if $(\varphi_{\eta})_{\eta}$ is a sequence of even
mollifiers, then the commutator (convolution in $\mathbf{x}$ only)
$$\displaystyle S_{\varepsilon,\eta}$$
$$\displaystyle:=a_{\varepsilon}\,(b_{\varepsilon}\star\varphi_{\eta})-(a_{%
\varepsilon}\,b_{\varepsilon})\star\varphi_{\eta}$$
goes to $0$ in $\textnormal{L}^{1}_{t,\mathbf{x}}$ as $\eta\rightarrow 0$, uniformly in $\varepsilon$.
Proof of Lemma 4.4:
First recall the following standard fact : since $(a_{\varepsilon})_{\varepsilon}$ is bounded in $\textnormal{L}^{q}_{t}(\textnormal{W}^{1,p}_{\mathbf{x}})$ and $s<p^{\star}$, the sequence $(\tau_{\mathbf{h}}a_{\varepsilon}-a_{\varepsilon})_{\varepsilon}$ tends to $0$ in $\textnormal{L}^{q}_{t}(\textnormal{L}^{s}_{\mathbf{x}})$ as $\mathbf{h}\rightarrow 0$, uniformly in $\varepsilon$. We now write the following equality for the commutator
$$\displaystyle S_{\varepsilon,\eta}(t,\mathbf{x})=\int_{\textnormal{B}_{\eta}}%
\Big{[}a_{\varepsilon}(t,\mathbf{x})-a_{\varepsilon}(t,\mathbf{x}-\mathbf{y})%
\Big{]}\,b_{\varepsilon}(t,\mathbf{x}-\mathbf{y})\,\varphi_{\eta}(\mathbf{y})%
\,\mathrm{d}\mathbf{y},$$
whence thanks to Fubini’s Theorem,
$$\displaystyle\|S_{\varepsilon,\eta}\|_{1}\leq\|b_{\varepsilon}\|_{\textnormal{%
L}^{q^{\prime}}_{t}(\textnormal{L}^{s^{\prime}}_{\mathbf{x}})}\int_{%
\textnormal{B}_{\eta}}|\varphi_{\eta}(\mathbf{y})|\,\|\tau_{\mathbf{y}}a_{%
\varepsilon}-a_{\varepsilon}\|_{\textnormal{L}^{q}_{t}(\textnormal{L}^{s}_{%
\mathbf{x}})}\,\mathrm{d}\mathbf{y},$$
which yields the desired uniform convergence, and concludes the Proof
of Lemma 4.4. $\square$
Step 4.
We have
$$\displaystyle(a_{\varepsilon}b_{\varepsilon})\star\varphi_{\eta}-a_{%
\varepsilon}b_{\varepsilon}\operatorname*{\longrightarrow}_{\eta%
\rightharpoonup 0}0\text{ in the sense of measures,}$$
uniformly in $\varepsilon$ (with a fixed continuous test function). Indeed, if $\theta\in\mathscr{C}_{t,\mathbf{x}}^{0}$, since $\varphi_{\eta}$ is even, we know that
$$\displaystyle\langle(a_{\varepsilon}\,b_{\varepsilon})\star\varphi_{\eta}-a_{%
\varepsilon}\,b_{\varepsilon},\theta\rangle=\langle a_{\varepsilon}\,b_{%
\varepsilon},\theta\star\varphi_{\eta}-\theta\rangle,$$
and the right-hand side tends to $0$ with the desired uniformity because $(a_{\varepsilon}\,b_{\varepsilon})_{\varepsilon}$ is bounded in $\textnormal{L}^{1}_{t,\mathbf{x}}$, and $(\theta\star\varphi_{\eta}-\theta)_{\eta}$ goes to $0$ in $\textnormal{L}^{\infty}_{t,\mathbf{x}}$ ($\theta$ is uniformly continuous).
Step 5.
Write
$$\displaystyle a\,b-a_{\varepsilon}\,b_{\varepsilon}$$
$$\displaystyle=a\,b-a\,(b\star\varphi_{\eta})$$
$$\displaystyle+a\,(b\star\varphi_{\eta})-a_{\varepsilon}\,(b_{\varepsilon}\star%
\varphi_{\eta})$$
$$\displaystyle+a_{\varepsilon}\,(b_{\varepsilon}\star\varphi_{\eta})-(a_{%
\varepsilon}\,b_{\varepsilon})\star\varphi_{\eta}$$
$$\displaystyle+(a_{\varepsilon}\,b_{\varepsilon})\star\varphi_{\eta}-a_{%
\varepsilon}\,b_{\varepsilon}.$$
Fix $\theta\in\mathscr{C}^{0}_{t,\mathbf{x}}$. In the right-hand side, line number $i\in\{1,2,3,4\}$ corresponds to the Step $i$ proven previously. We choose first $\eta$ to handle (uniformly in $\varepsilon$) all the lines of the right-hand side, except the second one. Then, we
choose the appropriate $\varepsilon$ to handle the second line, thanks to Step 2.
This concludes the Proof of Proposition 4.1.
$\square$
Proposition 4.2.
In Proposition 4.1, when
$q<\infty$, the same conclusion holds assuming
only a bound in $\textnormal{L}^{1}_{t}(\textnormal{H}^{-m}_{\mathbf{x}})$ (instead of $\textnormal{L}^{q^{\prime}}_{t}(\textnormal{H}^{-m}_{\mathbf{x}})$) for $(\partial_{t}b_{\varepsilon})_{\varepsilon}$ if, in addition, we assume the convergence of $(a_{\varepsilon})_{\varepsilon}$ to $a$ in $\textnormal{L}^{r}_{t}(\textnormal{L}^{p}_{\mathbf{x}})$ weak$-\star$,
for some $r>q$.
Proof of Proposition 4.2:
The Proof is identical to the Proof of Proposition 4.1, except for the second step (the only one using the bound on $(\partial_{t}b_{\varepsilon})_{\varepsilon}$). For this step, we use a (diagonal) extraction such that,
for all fixed $\eta$, $(b_{\varepsilon}\star\varphi_{\eta})_{\varepsilon}$ converges strongly (and almost everywhere) to $b\star\varphi_{\eta}$ (but only) in $\textnormal{L}^{1}_{t}(\textnormal{L}^{p^{\prime}}_{\mathbf{x}})$. Since $(b_{\varepsilon})_{\varepsilon}$ (and hence $(b_{\varepsilon}\star\varphi_{\eta})_{\varepsilon}$) is bounded in $\textnormal{L}^{q^{\prime}}_{t}(\textnormal{L}^{p^{\prime}}_{\mathbf{x}})$, with $q^{\prime}>r^{\prime}\geq 1$, we see that $(b_{\varepsilon}\star\varphi_{\eta})_{\varepsilon}$ converges to $b\star\varphi_{\eta}$ strongly in $\textnormal{L}^{r^{\prime}}_{t}(\textnormal{L}^{p^{\prime}}_{\mathbf{x}})$, and the added assumption of weak-$\star$ convergence for $(a_{\varepsilon})_{\varepsilon}$ allows hence to get
$$\displaystyle(a_{\varepsilon}\,(b_{\varepsilon}\star\varphi_{\eta}))_{%
\varepsilon}\operatorname*{\rightharpoonup}_{\varepsilon\rightarrow 0}a\,(b%
\star\varphi_{\eta})\text{ in $\textnormal{L}^{1}_{t,\mathbf{x}}$ weak,}$$
which ends the Proof of Proposition 4.2.
$\square$
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Revisiting the Issues On Netflow Sample and Export Performance
Hamed Haddadi, Raul Landa, Miguel Rio
Department of Electronic and Electrical Engineering
University College London
United Kingdom
Email: hamed,mrio,[email protected]
Saleem Bhatti
School of Computer Science
University of St. Andrews
United Kingdom
Email: [email protected]
Abstract
The high volume of packets and packet rates of traffic on some router links makes it exceedingly difficult for routers to examine every packet in order to keep detailed statistics about the traffic which is traversing the router. Sampling is commonly applied on routers in order to limit the load incurred by the collection of information that the router has to undertake when evaluating flow information for monitoring purposes. The sampling process in nearly all cases is a deterministic process of choosing 1 in every $N$ packets on a per-interface basis, and then forming the flow statistics based on the collected sampled statistics. Even though this sampling may not be significant for some statistics, such as packet rate, others can be severely distorted. However, it is important to consider the sampling techniques and their relative accuracy when applied to different traffic patterns.
The main disadvantage of sampling is the loss of accuracy in the collected trace when compared to the original traffic stream. To date there has not been a detailed analysis of the impact of sampling at a router in various traffic profiles and flow criteria. In this paper, we assess the performance of the sampling process as used in NetFlow in detail, and we discuss some techniques for the compensation of loss of monitoring detail.
I Introduction
Packet sampling is an integral part of passive network measurement on today’s Internet. The high traffic volumes on backbone networks and the pressure on routers has resulted in the need to control the consumption of resources in the measurement infrastructure. This has resulted in the definition and use of estimated statistics by routers, generated based on sampling packets in each direction of each port on the routers. The aims of this paper is to analyse the effects of the sampling process as operated by NetFlow, the dominant standard on today’s routers.
There are three constraints on a core router which lead to the use packet sampling: the size of the record buffer, the CPU speed and the record look-up time. In [6], it is noted that in order to manage and analyse the performance of a network, it is enough to look at the basic statistical measures and summary statistics such as average range, variance, and standard deviation. However, in this paper we analyse both analytically and practically the accuracy of the inference of original characteristics from the sampled stream when higher order statistics are used.
This paper focuses on the inference of original network traffic characteristics for flows from a sampled set of packets and examines how the sampling process can affect the quality of the results. In this context, a flow is identified specifically, as the tuple of the following five key fields: Source IP address, Destination IP address, Source port number, Destination port number, Layer 4 protocol type.
I-A NetFlow memory constraints
A router at the core of an internet link is carrying a large number of flows at any given time. this pressure on the router entails the use of strict rules in order to export the statistics and keep the router memory buffer and CPU resources available to deal with changes in traffic patterns by avoiding the handling of large tables of flow records. Rules for expiring NetFlow cache entries include:
•
Flows which have been idle for a specified time are expired and removed from the cache (15 seconds is default)
•
Long lived flows are expired and removed from the cache (30 minutes is default)
•
As the cache becomes full a number of heuristics are applied to aggressively age groups of flows simultaneously
•
TCP connections which have reached the end of byte stream (FIN) or which have been reset (RST) will be expired
I-B Sampling basics
Distributions studies have been done extensively in literature. In brief conclusion, internet traffic is believed to have Heavy-tailed distribution, self-similar nature, Long Range Dependence [2]. Sampling has the following effects on the flows:
•
It is easy to miss short flows [14]
•
Mis-ranking on high flows [4]
•
Sparse flow creation[14]
Packet sampling:
The inversion methods are of little to no use in practice for low sampling probability $q$, such as $q=0.01$ (1 packet in 100) or smaller, and become much worse as $q$ becomes smaller still. For example, on the Abilene network, 50% sampling was needed to detect the top flow correctly [4].
Flow sampling:
Preserves flows intact and the sampling is done on the flow records. In practice, any attempt to gather flow statistics involves classifying individual packets into flows. All packet meta-data has to be organised into flows before sampling can take place. This involves more CPU load and more memory if one uses the traditional hash table approach with one entry per flow. New flow classification techniques, such as bitmap algorithms, could be applied but there is no practical usage in this manner currently.
II Variation of Higher Order statistics
In this section we look at a more detailed analysis of the effect of sampling as performed by netflow on higher order statistics of the packet and flow size distributions. For the analysis of packet sampling application is used by NetFlow, we emulated the NetFlow operation on a 1 hour OC-48 trace, collected from the CAIDA link on $24^{th}$ of April 2003. This data set is available from the public repository at CAIDA [7]. The trace comprises of 84579462 packets with anonymised source and destination IP addresses. An important factor to rememberer in this work is the fact that the memory constraint on the router has been relaxed in generating the flows from the sampled stream. This means that there maybe more than tens of thousands of flow keys present at the memory at a given time, while in NetFlow, the export mechanism empties the buffer list regularly which can have a more severe impact on the resultant distribution of flow rates and statistics333The processing of the data was done using tools which are made available to the public by the authors..
II-A Effects of the short time-out imposed by memory constraints
Table I illustrates the data rates $d(t)$ per interval of measurement. Inverted data rates, by dividing $d(t)$ by the sampling probability $q$, are shown as $dn(t)$.
As observed in table I, the mean does not have a great variation, possibly because distributions of packet sizes within single flows do not exhibit high variability. The standard deviation of the estimated data rate is higher than the corresponding standard deviation for the unsampled data stream. In the absence of any additional knowledge about the higher level protocol, or the nature of the session level activity, in the unsampled data stream, each flow can be thought of as having packets of varying sizes that are more or less independent from one another. Thus, the whole traffic profile results from the addition of many independent random variables which, by the central limit theorem, tend to balance among themselves to produce a more predictable, homogeneous traffic aggregate. However, simple inversion eliminates this multiplicity of randomly distributed values by introducing a very strong correlation effect, whereby the size of all the packets in a reconstructed flow depend on the size of a very small set of sampled packets. This eliminates the possibility for balancing and thus increases the variability of the resulting stream, i.e. its standard deviation.
However, the skewness and kurtosis do change. Skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. Roughly speaking, a distribution has positive skew (right-skewed) if the right (higher value) tail is longer and negative skew (left-skewed) if the left (lower value) tail is longer (confusing the two is a common error). Skewness, the third standardised moment, is written as $\gamma_{1}$ and defined as:
$\gamma_{1}=\frac{\mu_{3}}{\sigma^{3}}$
where $\mu_{3}$ is the third moment about the mean and $\sigma$ is the standard deviation.
Kurtosis is more commonly defined as the fourth cumulant divided by the square of the variance of the probability distribution,
$\gamma_{2}=\frac{\kappa_{4}}{\kappa_{2}^{2}}=\frac{\mu_{4}}{\sigma^{4}}-3$
which is known as excess kurtosis. The ”minus 3” at the end of this formula is often explained as a correction to make the kurtosis of the normal distribution equal to zero. The skewness is a sort of measure of the asymmetry of the distribution function. The kurtosis measures the flatness of the distribution function compared to what would be expected from a Gaussian distribution.
Table II illustrates the packet rates $p(t)$ per interval of measurement. Inverted packet rates, by dividing $p(t)$ by the sampling probability $q$, are shown as $pn(t)$. The distributions before and after sampling are extremely close, and thus their difference tends to exaggerate those small difference that they do have. That is the reason of the enormous skewness and kurtosis that are observed. The skewness of the reconstructed stream is smaller than that of the unsampled stream this means that the reconstructed distribution is more symmetric, that is , it tends to diverge in a more homogeneous manner around the mean. Additionally, it is positive, meaning that in both cases the distribution tends to have longer tails towards large packets rather than towards short packets, concentrating its bulk on the smaller packets. If we concede that small flows (flows consisting of a small number of packets) tend to contain small packets, then it is clear that this smaller packets will be underrepresented and the distribution will shift its weight towards bigger packets (members of bigger flows). Thus, it will become more symmetric and hence less skewed.
The Kurtosis decreases in all of the considered examples. This means that the reconstructed streams are more homogeneous and less prone to outliers when compared with the original traces. Thus, more of the variance in the original traces in packet size can be attributed to infrequent packets that have inordinately big packets that were missed in the sampling process, and thus the variance in the reconstructed stream consists more of homogeneous differences and not large outliers. However, both the reconstructed and unsampled streams are leptokurtic and thus tend to have long, heavy tails.
II-B The two-sample KS test
The two-sample KS test is one of the most useful and general non-parametric methods for comparing two samples, as it is sensitive to differences in both location and shape of the empirical cumulative distribution functions of the two samples. A CDF was calculated for the number of packets per flow and the number of octets per flow for each of the 120 sampling intervals of 30 seconds each, both for the sampled/inverted and unsampled streams. Then, a Two-Sample Kolmogorov-Smirnov Test with $5\%$ significance level was performed between the 120 unsampled and the 120 sampled & inverted distributions. In every case the distributions before and after sampling and inversion were found to be significantly different, and thus it is very clear that the sampling and inversion
process significantly distorts the actual flow behaviour of the network.
III Practical Implications of Sampling
The effects of sampling on network traffic statistics can be measured from different perspectives. In this section we will cover the theories behind the sampling strategy and use some real data captures from CAIDA in an emulation approach to demonstrate the performance constraints of systematic sampling.
III-A Inversion errors on sampled statistics
The great advantage of sampling is the fact that the first order statistics do not show much variation when the sampling is done at consistent intervals and from a large pool of data. This enables the network monitoring to use the sampled statistics to form a relatively good measure of the aggregate measure of network performance. Figure 1 displays the data rates $d(t)$, in number of bytes seen per 30 second interval, on the one hour trace. The inverted data $d(t)$ is also shown with diamond notation, showing the statistics gathered after the sampled data is multiplied by the sampling rate. The black dots display the relative error per interval, $e(t)=\frac{d(t)-dn(t)}{d(t)}$.
Figure 2 displays the packet rates $p(t)$, the number of packets per 30 second interval, versus the sampled and inverted packet rates $pn(t)$. In this figure, it can be observed that the inversion does a very good job at nearly all times and the relative error is negligible. This is a characteristics of systematic sampling and is due to the central limit theorem.
It can be readily seen that the recovery of packet rates by simple inversion is much better than the recovery of data rates. This is because sampling one in a thousand packets deterministically can be trivially inverted by multiplying by the sampling rate (1000): we focus on packet level measurement, as opposed to a flow level measurement. If the whole traffic flow is collapsed into a single link, then if we sample one packet out every thousand and then multiply that by the sampling rate, we will get the total number of packets in that time window. We believe that the small differences that we can see in Figure 2 are due to the fact that at the end of the window some packets are lost (because their ‘representative’ was not sampled) or overcounted (a ‘representative’ for 1000 packets was sampled but the time interval finished before they had passed). We believe these errors happen between measurement windows in time, i.e. they are window-edge effects.
The inversion property described above does not hold for measuring the number of bytes in a sampling interval. Simple inversion essentially assumes that all packets in a given flow are the same size, and of course this assumption is incorrect. It is to be expected that the greater the standard deviation of packet size over an individual flow, the more inaccurate the recovery by simple inversion will be regarding the number of bytes per measurement interval. Figures 3 and 4 displays the standard error rate on data rate and packet rate recovery respectively, in different measurement intervals.
III-B Flow size and packet size distributions
Figure 5, displays the CDF of packet size distribution in all the flows formed from the sampled and unsampled streams. The little variation in the packet size distribution conforms to the findings of the previous section where it was discussed that the packet sampling has low impact on the packet size distribution.
Figure 6:1 shows the effect that the distribution of packet lengths can have on the distribution of flow lengths when periodic packet sampling is applied. As flows reconstructed from a sampled packet stream are predominantly formed by just one packet, their length distribution follows that of single packets (Figure 5). That is the reason for the sharp jump near 1500 octets, as this characteristic originates from the maximum frame size in ethernet networks.
From Figure 6:2 , it can be readily seen that, in the sampled stream, more than 90 percent of flows consist of a single packet, whereas in the unsampled case a much grater diversity in flow lengths exists for small flows. This is due to the fact that simple packet-based deterministic sampling under-represents short flows, and those short flows that are indeed detected by the procedure after sampling usually consist of a single packet. Thus, short flows are either lost or recovered as single packet flows, and long flows have their lengths reduced.
IV Conclusion
In this paper we have reviewed the effects of sampling and flow record creation, as done by NetFlow, on the traffic statistics which are reported by such a process. It is inevitable that systematic sampling can no longer provide a realistic picture of the traffic profile present on internet links. The emergence of applications such as video on-demand, file sharing, streaming applications and even on-line data processing packages prevents the routers from reporting an optimal measure of the traffic traversing them. In the inversion process, it is a mistake to assume that the inversion of statistics by multiplication by the sampling rate is an indicate of even the first order statistics such as packet rates.
An extension to this work and the inversion problem entails the use of more detailed statistics such as port numbers and TCP flags in order to be able to infer the original characteristics from the probability distribution functions of such variables. This will enable a more detailed recovery of original packet and data rates for different applications. The inference of such probabilities, plus use of methods such as Bayesian inference, would enable a forecasting method which would enable the inversion of the sampled stream in near real time.
In a related work, we will be looking at alternative flow synthesis schemes, looking at techniques replacing the NetFlow, such as use of hashing techniques using Bloom filters. The use of a light weight flow indexing system will allow for a larger number of flows to be present at the router, possibly increasing the memory constraints and allowing for a higher sampling rate, which will in turn lead to more accurate inversion.
V Related Work
There has been a great deal of worked done on analysis of sampling process and inversion problem. Choi et al. have explored the sampling error and measurement overhead of NetFlow in [11] though they have not looked at inversion process.
In [3], the authors have compared the Netflow reports with those obtained from SNMP statistics and packet level traces, but without using the sampling feature of NetFlow which is perhaps the dominant version in use nowadays. Estan et al. [5] have proposed a novel method of adapting the sampling rate at a NetFlow router in order to keep the memory resources at a constant level. This is done by upgrading the router firmware, which can be compromised by an attacker injecting varying traffic volume in order to take down the router. Also this work has not considered the flow length statistics which are the primary focus of our work.
Hohn et al. [10] have proposed a flow sampling model which can be used in an offline analysis of flow records formed from an unsampled packet stream. In this model the statistics of the original stream are recovered to a great extent. However the intensive computing and memory resources needed in this process prevents the implementation of such a scheme on highspeed routers. They prove it impossible to accurately recover statistics from a packet sampled stream, but based on the assumption of packets being independent and identically distributed
Roughan at [12] has looked at statistical processes of active measurement using Poisson and uniform sampling and has compared the theoretical performance of the two methods. Papagiannaki et al. at [8] have discussed the effect of sampling on tiny flows when looking at generation of traffic matrices.
Authors at [15] have been looking at anomaly detection using flow statistics, but without sampling. In [17] and [16], authors have looked at inferring the numbers and lengths of flows of original traffic that evaded sampling altogether. They have looked at inversion via multiplication.
Acknowledgements
The authors would like to acknowledge CAIDA[7] for providing the trace files. This work is conducted under the MASTS (EPSRC grant GR/T10503) and the 46PaQ project (EPSRC grant GR/S93707).
References
[1]
NetFlow Services Solutions Guide available at
http://www.cisco.com/univercd/cc/td/doc/cisintwk/intsolns/netflsol/nfwhite.htm
[2]
Will Leland, Murad Taqqu, Walter Willinger, and Daniel Wilson, On the Self-Similar Nature of Ethernet Traffic (Extended Version), IEEE/ACM Transactions on Networking, Vol. 2, No. 1, pp. 1-15, February 1994.
[3]
Sommer, R. and Feldmann, A. 2002. NetFlow: information loss or win?. In Proceedings of the 2nd ACM SIGCOMM Workshop on internet Measurment (Marseille, France, November 06 - 08, 2002). IMW ’02. ACM Press, New York, NY, 173-174. DOI= http://doi.acm.org/10.1145/637201.637226
[4]
Barakat, C., Iannaccone, G., and Diot, C. 2005. Ranking flows from sampled traffic. In Proceedings of the 2005 ACM Conference on Emerging Network Experiment and Technology (Toulouse, France, October 24 - 27, 2005). CoNEXT’05. ACM Press, New York, NY, 188-199.
[5]
Estan, C., Keys, K., Moore, D., and Varghese, G. 2004. Building a better NetFlow. SIGCOMM Comput. Commun. Rev. 34, 4 (Aug. 2004), 245-256. http://doi.acm.org/10.1145/1030194.1015495
[6]
Performance and Fault Management (Cisco Press Core Series) by Paul L Della Maggiora (Author), James M. Thompson (Author), Robert L. Pavone Jr. (Author), Kent J. Phelps (Author), Christopher E. Elliott (Editor), Publisher: Cisco press; 1ST edition, ISBN: 1578701805
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CAIDA, the Cooperative Association for Internet Data Analysis: http://www.caida.org
[8]
K. Papagiannaki, N. Taft, and A. Lakhina. A Distributed Approach to Measure IP Traffic Matrices. In ACM Internet Measurement Conference, Taormina, Italy, October, 2004.
[9]
IETF PSAMP working Group: http://www.ietf.org/html.charters/psamp-charter.html
[10]
N. Hohn and D. Veitch. Inverting Sampled Traffic. In Internet Measurement Conference, 2003. http://citeseer.ist.psu.edu/hohn03inverting.html
[11]
Choi, B. and Bhattacharyya, S. 2005. Observations on Cisco sampled NetFlow. SIGMETRICS Perform. Eval. Rev. 33, 3 (Dec. 2005), 18-23. DOI= http://doi.acm.org/10.1145/1111572.1111579
[12]
A Comparison of Poisson and Uniform Sampling for Active Measurements, Matthew Roughan, accepted to appear in IEEE JSAC.
[13]
InMon Corporation (2004). sFlow accuracy and billing. Available at www.inmon.com/pdf/sFlowBilling.pdf
[14]
Sampling for Passive Internet Measurement: A Review, N.G. Duffield, Statistical Science,Vol. 19, No. 3, 472-498, 2004.
[15]
Brauckhoff, D., Tellenbach, B., Wagner, A., May, M., and Lakhina, A. 2006. Impact of packet sampling on anomaly detection metrics. In Proceedings of the 6th ACM SIGCOMM on internet Measurement (Rio de Janeriro, Brazil, October 25 - 27, 2006). IMC ’06. ACM Press, New York, NY, 159-164. DOI= http://doi.acm.org/10.1145/1177080.1177101
[16]
N. Duffield, C. Lund, and M. Thorup. Properties and Prediction of Flow Statistics from Sampled Packet Streams. In Proc. ACM SIGCOMM IMW’02, Marseille, France, Nov. 2002.
[17]
N. Duffield, C. Lund, and M. Thorup. Estimating Flow Distributions from Sampled Flow Statistics. In Proc. ACM SIGCOMM’03, Karlsruhe,
Germany, Aug. 2003. |
Suppression of excited-state effects in lattice determination of
nucleon electromagnetic form factors
${}^{a}$, S. Capitani${}^{a}$, D. Djukanovic${}^{b}$,
J. Hua${}^{a}$, B. Jäger${}^{c}$, P. Junnarkar${}^{a,b}$, H.B. Meyer${}^{a,b}$,
T.D. Rae${}^{a}$, H. Wittig${}^{a,b}$
${}^{a}$ PRISMA Cluster of Excellence and Institut für Kernphysik,
University of Mainz, 55099 Mainz, Germany
${}^{b}$ Helmholtz Institut Mainz,
University of Mainz, 55099 Mainz, Germany
${}^{c}$ Department of Physics, College of Science,
Swansea University, Swansea, SA2 8PP, UK
E-mail:
Abstract:
We study the ability of a variety of fitting techniques
to extract the ground state matrix elements of the vector current
from ratios of nucleon three- and two-point functions that contain
contaminations from excited states. Extending our high-statistics
study of nucleon form factors, we are able to demonstrate that the
treatment of excited-state contributions in conjunction with
approaching the physical pion mass has a significant impact on the
$Q^{2}$-dependence of the form factors.
1 Introduction
The persistent failure of the vast majority of lattice measurements of nucleon form factors to replicate the experimental value for the proton charge radius is one of the biggest current puzzles in lattice QCD. Possible explanations include the use of unphysically heavy pions, finite-volume effects and excited-state contributions (or indeed a combination of some or all of these). Our purpose here is to study how best to suppress excited-state effects on the isovector electromagnetic form factors of the nucleon in order to better understand their impact on a possible resolution of the puzzle.
The $eN$ scattering cross section is usually parameterized in terms of the Sachs form factors $G_{E}$, $G_{M}$, whereas the matrix element of the vector current between nucleon states has a natural decomposition in terms of the Dirac and Pauli form factors
$F_{1}$ and $F_{2}$,
$$\langle N(p^{\prime},s^{\prime})|V_{\mu}|N(p,s)\rangle=\overline{u}(p^{\prime}%
,s^{\prime})\left[\gamma_{\mu}F_{1}+i\frac{\sigma_{\mu\nu}q_{\nu}}{2m_{N}}F_{2%
}\right]u(p,s)\,.$$
(1)
The relationship between the two sets of form factors is given by
$$G_{E}(Q^{2})=F_{1}(Q^{2})-\frac{Q^{2}}{4m_{N}^{2}}F_{2}(Q^{2}),~{}~{}~{}G_{M}(%
Q^{2})=F_{1}(Q^{2})+F_{2}(Q^{2}).$$
(2)
In order to measure the nucleon form factors on the lattice, we compute the two- and three-point functions of nucleon interpolating operators $N^{\alpha}$
$$C_{2}(\mathbf{q},t_{s})=\sum_{\mathbf{x}}\textrm{e}^{i\mathbf{q}\cdot\mathbf{x%
}}\Gamma_{\alpha\beta}\langle\overline{N}^{\beta}(t_{s},\mathbf{x})N^{\alpha}(%
0)\rangle,~{}~{}~{}~{}~{}~{}~{}C_{3,\mathcal{O}}(\mathbf{q},t,t_{s})=\sum_{%
\mathbf{x},\mathbf{y}}\textrm{e}^{i\mathbf{q}\cdot\mathbf{y}}\Gamma_{\alpha%
\beta}\langle\overline{N}^{\beta}(t_{s},\mathbf{x})\mathcal{O}(t,\mathbf{y})N^%
{\alpha}(0)\rangle,$$
(3)
where $\Gamma=\frac{1}{2}(1+\gamma_{0})(1+i\gamma_{5}\gamma_{3})$ is a polarization matrix, from whose ratios
$$R_{V_{\mu}}(\mathbf{q},t,t_{s})=\frac{C_{3,V_{\mu}}(\mathbf{q},t,t_{s})}{C_{2}%
(\mathbf{0},t_{s})}\sqrt{\frac{C_{2}(\mathbf{q},t_{s}-t)C_{2}(\mathbf{0},t)C_{%
2}(\mathbf{0},t_{s})}{C_{2}(\mathbf{0},t_{s}-t)C_{2}(\mathbf{q},t)C_{2}(%
\mathbf{q},t_{s})}}$$
(4)
we can extract the Sachs form factors as
[1]
$$\displaystyle{\rm Re}\left[R_{V_{0}}(\mathbf{q},t,t_{s})\right]$$
$$\displaystyle=\sqrt{\frac{m_{N}+E_{\mathbf{q}}}{2E_{\mathbf{q}}}}G^{\rm eff}_{%
E}(Q^{2},t,t_{s})$$
(5)
$$\displaystyle{\rm Re}\left[R_{V_{i}}(\mathbf{q},t,t_{s})\right]_{i=1,2}$$
$$\displaystyle=\epsilon_{ij}q_{j}\frac{1}{\sqrt{2E_{\mathbf{q}}(m_{N}+E_{%
\mathbf{q}})}}G^{\rm eff}_{M}(Q^{2},t,t_{s}).$$
(6)
Similar relations hold for the axial and induced pseudoscalar form factors
[2, 3].
2 Excited-State Contributions
For $Q^{2}\not=0$, the excited states contributing to the incoming and outgoing nucleon channels are different in our kinematics (where the final nucleon is at rest), and ignoring finite-size effects we can parameterize the leading excited-state contaminations by
$$G^{\rm eff}_{X}(Q^{2},t,t_{s})=G_{X}(Q^{2})+c_{X,1}(Q^{2})\textrm{e}^{-\Delta t%
}+c_{X,2}(Q^{2})\textrm{e}^{-\Delta^{\prime}(t_{s}-t)}+\ldots$$
(7)
Since our aim is the extraction of the ground-state form factor $G_{X}(Q^{2})$, we need to find means to identify and remove, or else to suppress in some other way, the leading excited-state contributions. For this end we have considered three different methods
[4, 5]:
•
The well-known plateau method is based on the assumption that excited-state contributions in (7) are sufficiently suppressed, and proceeds by looking for a region in $t$ where the signal for $G^{\rm eff}_{X}(Q^{2},t,t_{s})$ for given $Q^{2}$ and $t_{s}$ has reached a plateau, which is then fitted to a constant identified as $G_{X}(Q^{2})$. While well-tried and simple, this method is beset with a number of problems, the most prominent of which is that in order for the plateau to be a good estimate for $G_{X}(Q^{2})$, the excited-state contributions from both sides need to have decayed by at least one or two e-foldings; in order to achieve this, large values for $t_{s}$ are required. Unfortunately, the signal-to-noise ratio becomes rather poor as $t_{s}$ is increased, rendering this difficult to achieve with present statistics. As we observe a systematic trend in the plateau values as a function of $t_{s}$ even for relatively large distances $t_{s}\sim 1.4$ fm (cf. fig. 1), the practicability of this method is seriously in doubt.
•
The by-now also well-established summation method
[6, 2]
proceeds by forming summed ratios
$$S_{X}(Q^{2},t_{s})=\sum_{t=0}^{t_{s}}G^{\rm eff}_{X}(Q^{2},t,t_{s})\rightarrow
c%
+t_{s}\left\{G_{X}(Q^{2})+\mathcal{O}\left(\textrm{e}^{-\Delta t_{s}}\right)\right\}$$
and fitting their $t_{s}$-dependence to a straight line, from the slope of which the desired ground-state form factor $G_{X}(Q^{2})$ can be extracted. The great advantage of the summation method is that it leads to a parametrically reduced excited-state contamination, since any $t$-dependence has been eliminated by summing over $t$, and $\Delta t_{s}\gg\Delta t$ by construction. The summation method’s major disadvantage is a significant increase in statistical errors when compared to the plateau method. One could therefore argue that the reduction in (hard to control) systematic error is paid for by a corresponding increase in (expensive to reduce) statistical error.
•
To have a further handle on the precise behaviour of the excited-state contributions, we have also considered two-state fits
[4, 7],
i.e. we have explicitly fit $G^{\rm eff}_{X}(Q^{2},t,t_{s})$ to leading excited-state contributions of the form (7) with $\Delta=m_{\pi}$, $\Delta^{\prime}=2m_{\pi}$, both
as a function of $t$, $t_{s}-t$ at each $t_{s}$ separately, and
as a function of $t_{s}$, $t$ at all $t_{s}$ simultaneously.
The advantage of these fits is that they are able to fully remove the leading excited-state contamination contained in the signal and project onto the ground state without any need to further increase the range of $t$ and $t_{s}$ under consideration. Their disadvantage lies in that they are somewhat dependent on our assumptions about the excited states (in particular the energy gap), making it somewhat difficult to assess the overall trustworthiness of the fitted results. In light of this latter disadvantage, we do not currently expect to quote final results from the two-state fits, but only to use them as a control to verify that excited-state contaminations do not seriously affect our final answers from the other methods.
In fig. 2, we compare the pion-mass dependence of the value of the electric Sachs form factor evaluated at $Q^{2}=0.1$ GeV${}^{2}$, $G_{E}(0.1~{}\textrm{GeV}^{2})$, as determined using the different methods. For the plateau method, there appears to be no appreciable pion-mass dependence, and the chiral extrapolation does not approach the experimental value as the pion mass is reduced. The summation method shows a slight improvement in so far as a chiral trend is visible, but an extrapolation would still miss a experimental value by a significant margin (although the discrepancy is less significant than for the plateau method, a large portion of this is due to the larger errors associated with the summation method). By contrast, the simultaneous two-state fits yield results that show a clear chiral trend and point towards the experimental value in the physical limit.
By parameterizing each Sachs form factor as a dipole
$$G_{X}(Q^{2})=G_{X}(0)\left(1+\frac{Q^{2}}{M_{X}^{2}}\right)^{-2}$$
(8)
where $G_{E}(0)=1$, $G_{M}(0)=\mu$, we can extract the charge radii and anomalous magnetic moment $\kappa$ via
$$\frac{1}{M_{E}^{2}}=\frac{r_{E}^{2}}{12}=\frac{r_{1}^{2}}{12}+\frac{\kappa}{8m%
_{N}^{2}}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\frac{1}{M_{M}^{2}}=\frac{r_{M}^{2}}{12%
}=\frac{r_{1}^{2}+\kappa r_{2}^{2}}{12(1+\kappa)}$$
(9)
where $\kappa=\mu-1$.
Fig. 3 shows the data for $G_{E}$ and $G_{M}$ together with the dipole fits. The pion-mass dependence of the results obtained for the radii and the magnetic moment from the plateau and summation methods, respectively, is shown in fig. 4.
3 A Possible Scenario
In order to explain why we observe such a different chiral behaviour for the plateau and summation methods on the one hand, and the two-state fits on the other hand, it helps to consider a scenario in which we assume the two-state fits to be a true description of the data. Under this assumption, we can reconstruct the summed ratios that we would obtain at various value of $t_{s}$, and compare the slope obtained from fitting a straight line to the reconstructed values with the “true” value of the form factor.
An example of such a reconstruction on one of our ensembles is shown in fig. 5. In the left panel, we show the reconstructed ratios for the values of $t_{s}$ considered here as blue dots, with the straight-line fit to those reconstructed points shown in red. Also shown as a blue line is the functional form of the summed ratios as a function of $t_{s}$ over a much larger range of source-sink separations. It can be seen that the slope of the red line does not agree well with that of the blue curve as $t_{s}$ grows. In the right panel, we have added the asymptotic linear behaviour of the summed ratio (the dashed blue line). As can be seen from the inset, while the straight-line fit to the reconstructed ratios is indistinguishable from the true functional form at low values of $t_{s}$, it is quite different from the correct asymptotic value.
This behaviour is to be contrasted with the one observed for $g_{A}$
[2],
where the summation method has been found to be very effective in extracting the asymptotic behaviour. A crucial difference between the two cases is that for the form factors at $Q^{2}\not=0$, the gap appearing in the leading exponential corrections to the summed ratio is only $m_{\pi}$, rather than $2m_{\pi}$ as for the $Q^{2}=0$ case. This makes the approach to the asymptotic linear behaviour much slower, and (at least at first sight) seems to require twice as large a source-sink sparation $t_{s}$ in order to obtain the same reduction in systematic error.
4 Conclusions
A systematic trend in the value of $G_{E}$ obtained from the plateau method persists to $t_{s}\sim 1.4$ fm, and even with the summation method, $G_{E}$ systematically comes out too high; considering only the largest values of $t_{s}$ in the summation method brings the results closer to experiment, albeit at the expense of very large statistical errors.
The impact of excited states on the results of the plateau, and to a lesser extent also the summation method, appears to increase even further as the chiral limit is approached.
A possible reason for these observations is indicated by two-state fits: Given the appearance of the small gap $m_{\pi}$ for $Q^{2}\not=0$, the approach to the plateau is very slow, and the summed ratios still receive sizeable corrections to their asymptotic linear behaviour.
Acknowledgments:
We are grateful to our colleagues within the Coordinated Lattice Simulations
(CLS) initiative for sharing ensembles.
This work was granted access to the HPC resources of the Gauss Center for
Supercomputing at Forschungzentrum Jülich, Germany, made available within
the Distributed European Computing Initiative by the PRACE-2IP, receiving
funding from the European Community’s Seventh Framework Programme
(FP7/2007-2013) under grant agreement RI-283493 (project PRA039).
We are grateful for computer time allocated to project HMZ21 on the
JUQUEEN BG/Q computer at NIC, Jülich.
This work was supported by the DFG via SFB 1044
and grant HA 4470/3-1. We thank Dalibor Djukanovic and Christian Seiwerth
for technical support.
References
[1]
B. Jäger et al., PoS(LATTICE 2013)272
[arXiv:1311.5804].
[2]
S. Capitani et al., Phys. Rev. D 86 (2012) 074502
[arXiv:1205.0180].
[3]
P. Junnarkar et al., PoS(LATTICE 2014)150.
[4]
S. Capitani et al., PoS(LATTICE 2012)177
[arXiv:1211.1282].
[5]
S. Capitani et al., PoS(Confinement X)321.
[6]
L. Maiani et al., Nucl. Phys. B 293 (1987) 420.
[7]
J. Green et al., PoS(LATTICE 2011)157
[arXiv:1111.0255].
[8]
J. J. Kelly,
Phys. Rev. C 70 (2004) 068202. |
Spin-polarized currents driven by spin-dependent surface screening
Piotr Graczyk
[email protected]
Institute of Molecular Physics, Polish Academy of Sciences, M. Smoluchowskiego 17, 60-179 Poznan, Poland
Maciej Krawczyk
[email protected]
Faculty of Physics, Adam Mickiewicz University in Poznan, Umultowska 85, 61-614 Poznan, Poland
Abstract
We have examined spin polarization of the electron current in a ferromagnetic metal induced by the spin-dependent surface screening at the dielectric-ferromagnetic metal (D-FM) interface. In an applied ac voltage, the dynamic band splitting driven by the changes in the screening charge at the D-FM interface develops spin accumulation. The resultant spin accumulation gradient produces a time-dependent spin current. We have derived the rate of the spin accumulation dependent on the rate of the screening charge density accumulation within the Stoner band model. The spin-charge dynamics in the system is then modeled by a set of diffusive equations with the contributions from spin-dependent surface screening and spin-dependent conductivity. We show for MgO-Cu-Co-MgO system that the spin-dependent screening in thin Co film produces spin accumulation 7 times higher than that obtained by the spin-dependent conductivity in thick Co films. We propose an experimental approach to validate our numerical predictions and to distinguish between spin accumulation induced by the spin-dependent conductivity and by the spin-dependent surface screening.
The phenomenon of spin-dependent surface screening (SS) leads to the charge-mediated magnetoelectric effect (ME) which is the key mechanism for the electric control of magnetic properties in ultrathin multiferroic heterostructures Burton2012 ; Vaz2012 . The effect has been described first theoretically by Zhang in 1999 Zhang1999 and then confirmed by ab initio calculations Duan2008 ; Niranjan2009 ; Rondinelli2008 . It has been shown experimentally that SS may be utilized to change the magnetization, anisotropy or coercivity of ultrathin ferromagnetic film Molegraaf2009 ; Weisheit2007 ; Maruyama2009 ; Endo2010 ; Shiota2009 ; Shiota2011 ; Seki2011 ; Chiba2011 , to induce magnetic ordering transition Burton2009 ; Vaz2010 or even induce surface magnetism in nonmagnetic materials Sun2010 . In multiferroic tunnel junctions SS provides significant contribution to the tunneling resistance Zhuravlev2010 ; Yin2015 and opens possibilities to manipulate the spin polarization of tunneling electrons Pantel2012 ; Sen2019 . Owing to the surface magnetic anisotropy that is controlled by SS, spin waves may be excited parametrically or resonantly with the ac voltage Rana2017 ; Verba2014 ; Verba2017 ; Chen2017 ; Nozaki2012 .
Spin-dependent surface screening changes the absolute value of magnetization at the dielectric-ferromagnetic metal (D-FM) interface, in contrast to, studied more extensively, strain-driven ME effects Vaz2012 ; Graczyk2015 ; Graczyk2016 ; Graczyk2016a , which rotate the magnetization direction. Although strain-driven ME remains significant for relatively thick layers, it suffers from ageing effects, clamping and integration problems with current technology.
In this letter we study the generation of spin-polarized currents by spin-dependent surface screening effect in D-FM-D system subjected to the ac voltage. The change of the voltage results in the change of the screening charges at the insulator-ferromagnet interface. Since the screening is spin-dependent, this leads to the appearance of nonequilibrium spin density at the interface and then formed spin-density gradient generates diffusive spin current. We discuss this effect within the Stoner band model and derive the term that describes SS in drift-diffusive formalism. With this model we perform numerical simulations and find the spin current with the contributions from spin-dependent conductivity (SC) and spin-dependent surface screening (SS) for MgO-Cu-Co-MgO system. Finally, we propose the strategy to distinguish experimentally between that two current polarization mechanisms.
Considered systems are shown in Fig. 1. The multilayer consists of two 2-nm-thick MgO films separated by Co film of different thicknesses. In system A the Co layer is separated from MgO by 2-nm-thick nonmagnetic metal (Cu) at both interfaces while in the system B it is separated from MgO only from the top. Therefore, in the system A there is no D-FM interface, while in the system B it is present. Consequently, screening charges will appear in the ferromagnet only at the bottom interface in the system B when the voltage is applied to the multilayer structure through bottom and top electrodes (e2 and e1, respectively). It is assumed that charge and spin transport occur only perpendicular to the layers, along $x$ axis.
The coupled charge-spin dynamics driven by the prescribed ac voltage is considered on the base of the diffusive model Valet1993 ; Zhu2008 . The free charge density current is described in the metal layers by the equation:
$$J_{f}=\sigma E-D\frac{\partial n_{f}}{\partial x}+\beta D\frac{e}{\mu_{B}}%
\frac{\partial s}{\partial x},$$
(1)
which includes contributions from electron drift driven by electric field $E$, diffusion driven by the gradient of free charge density $n_{f}$ and diffusive spin polarisation (SC), respectively. Here, $\sigma$ is conductivity, $D$ is diffusion constant, $\beta$ is spin asymmetry coefficient, $\mu_{B}$ is Bohr magneton and $e$ is charge of electron. The displacement current is described by the equation:
$$J_{b}=\epsilon_{0}\epsilon_{r}\frac{\partial E}{\partial t},$$
(2)
where $\epsilon_{0}\epsilon_{r}$ is permittivity of the material. The conservation of free ($i=f$) and bound ($i=b$) charge density $n_{i}$ is described by continuity equation:
$$\frac{\partial n_{i}}{\partial t}=-\frac{\partial J_{i}}{\partial x}.$$
(3)
Electric potential $V$ is given by the Gauss Law:
$$\begin{split}\displaystyle\epsilon_{0}\frac{\partial^{2}V}{\partial x^{2}}&%
\displaystyle=n_{f}+n_{b},\\
\displaystyle E&\displaystyle=-\frac{\partial V}{\partial x},\end{split}$$
(4)
with the boundary conditions at the electrodes:
$$\begin{split}\displaystyle V|_{x=e1}&\displaystyle=U=U_{0}\sin(2\pi f_{0}t),\\
\displaystyle V|_{x=e2}&\displaystyle=0,\end{split}$$
(5)
where $f_{0}$ is the frequency of the oscillating voltage $U$.
The spin current $J_{s}$ in the ferromagnetic material is modeled by the equation Zhu2008 :
$$J_{s}=-D\frac{\partial s}{\partial x}-\beta\frac{\mu_{B}}{e}\left(\sigma E-D%
\frac{\partial n_{f}}{\partial x}\right),$$
(6)
which describes spin current driven by gradient of spin accumulation $s$ (diffusion) and spin-conductivity term.
The continuity equation for the spin accumulation $s$ is:
$$\frac{\partial s}{\partial t}=-\frac{\partial J_{s}}{\partial x}-\frac{s}{T_{1%
}}+f_{s},$$
(7)
which describes the rate of change of $s$ due to gradient of the spin current, the spin-flip relaxation with the characteristic time $T_{1}$. $f_{s}$ is an additional source from spin-dependent charge screening. The formula for $f_{s}$ is derived below. We use in the following paragraphs $n\equiv n_{f}$.
We will consider the voltage-driven magnetization and spin density change in cobalt within the Stoner model Stoner1938 . Due to the exchange splitting of the band d for the majority and minority spins and partially filled minority subband (Fig. 2a) there exists equilibrium spin density (magnetization) in the bulk of the ferromagnet which is equal to the difference between charge densities of the spin-up ($\uparrow$) and spin-down ($\downarrow$) subbands, $m=\mu_{B}(n^{\uparrow}-n^{\downarrow})/e$. The voltage applied to the system shifts the potential of the bands at the D-FM interfaces (Fig. 2b) leading to the accumulation of the screening charge density $n=\delta n^{\uparrow}_{0}+\delta n^{\downarrow}_{0}$ where $\delta n_{0}^{\sigma}$ is accumulated screening charge in the subband $\sigma$. Since the density of states are different between subbands at the Fermi level ($\Delta\rho=\rho^{\uparrow}-\rho^{\downarrow}<0$ for Co), then screening charge changes the value of the magnetization at the interface by $\delta m_{0}=\mu_{B}(\delta n^{\uparrow}_{0}-\delta n^{\downarrow}_{0})/e$ Duan2008 .
Another important feature of the screening in the ferromagnet is that the changes in the band occupancy influence the exchange splitting Niranjan2009 , i.e. the potential is spin-dependent Zhang1999 ; Zhuravlev2010 :
$$V^{\sigma}=V_{c}-\frac{I}{e}(\delta n^{\sigma}-\delta n^{-\sigma}),$$
(8)
where $I$ is exchange constant. Spin-dependent potential (8) means that the subbands shift relative to each other by $V_{s}=V^{\uparrow}-V^{\downarrow}=-2I\delta m$. (Fig. 2c). From Eqs. (5) and (11) in Ref. Zhang1999 we get the relation between $V_{s}$ and screening charge $n=\delta n^{\uparrow}+\delta n^{\downarrow}$ in the linearized Thomas-Fermi model:
$$V_{s}=-\frac{2I\Delta\rho}{\rho+4I\rho^{\uparrow}\rho^{\downarrow}}n,$$
(9)
where $\rho=\rho^{\uparrow}+\rho^{\downarrow}$. This shift has important consequences. Since $\rho^{\uparrow}\neq\rho^{\downarrow}$, the shift changes the equilibrium charge level with respect to the Fermi level by $E_{F}-s_{0}$ (Fig. 2d) where $s_{0}=-e(V^{\uparrow}\rho^{\uparrow}+V^{\downarrow}\rho^{\downarrow})/\rho$ is spin equilibrium level. This leads to the changes in equilibrium charge densities $\delta n^{\sigma}\neq\delta n_{0}^{\sigma}$ and potential $V_{c}$. Therefore, the surface magnetization value changes by $\delta m\neq\delta m_{0}$. Moreover, the screening length in ferromagnet $\lambda_{F}$ deviates from the conventional Thomas-Fermi value $\lambda=\left(\epsilon_{0}/e^{2}\rho\right)^{1/2}$ and thus the capacitance is reduced Zhang1999 . We can implement this effect to our numerical model where the screening length is defined as $\lambda=\left(\epsilon_{0}D/\sigma\right)^{1/2}$ by dividing drift term and multiplying diffusion term in Eq. (1) by $\gamma_{n}=\lambda_{F}/\lambda\approx 1.1$ for Co. We found that modification of the screening length does not influence the effect of the spin-polarized current generation which is considered here so it will be neglected in further part.
Since screening charge and thus $V_{s}$ (Eq. (9)) is time-dependent under the ac volatage, the change of the shift between subbands $\Delta V_{s}$ is the source of nonequilibrium spin density $\Delta s\propto\Delta V_{s}$. From Fig. 2d we see that with the respect to the $s_{0}$:
$$\begin{split}\displaystyle\Delta s&\displaystyle=\frac{\mu_{B}}{e}(\Delta n^{%
\uparrow}-\Delta n^{\downarrow})\\
&\displaystyle=\frac{\mu_{B}}{e}(-eV^{\uparrow}-s_{0})\rho^{\uparrow}-\frac{%
\mu_{B}}{e}(-eV^{\downarrow}-s_{0})\rho^{\downarrow}\\
&\displaystyle=-2\mu_{B}\Delta V_{s}\rho^{\uparrow}\rho^{\downarrow}/\rho e.%
\end{split}$$
(10)
Taking time derivatives we get from (9) and (10) the source term $f_{s}$ to be:
$$f_{s}=\frac{4I\rho^{\uparrow}\rho^{\downarrow}}{\rho}\frac{\Delta\rho}{\rho+4I%
\rho^{\uparrow}\rho^{\downarrow}}\frac{\mu_{B}}{e}\frac{\partial n}{\partial t%
}=\gamma_{s}\frac{\mu_{B}}{e}\frac{\partial n}{\partial t}.$$
(11)
In equilibrium the final change of magnetization $\delta m=\mu_{B}(\delta n^{\uparrow}-\delta n^{\downarrow})/e$ is a consequence of uniform potential shift of the bands and their relative exchange splitting (Fig. 2e).
Equations (1)-(7) with source term (11) are solved numerically by finite element method in Comsol Multiphysics with time-varying voltage applied to the electrodes with amplitude $U_{0}=8$ V and frequency $f_{0}=200$ MHz. The relative permittivity is $\epsilon_{r}=10$ for MgO and $\epsilon_{r}=1$ for metals. We assume the conductivity of metals $\sigma=1.2\cdot 10^{7}$ S/m and the diffusion constant $D=4\cdot 10^{-3}$ m${}^{2}$/s. The spin asymmetry coefficient for Co is $\beta=0.5$ Zhu2008 , the spin-flip relaxation time is $T_{1}=0.9$ ps Zhu2008 and the magnetoelectric coefficient calculated from Eq. (11) and parameters given in Ref. Zhang1999 is $\gamma_{s}=-0.25$. We assume also the magnetization of Co film lies in-plane of the film. Below we analize the distribution of amplitudes of spin accumulation $s$ and spin current $J_{s}$ between MgO layers with the contributions from SC and SS.
Figure 3 shows the calculated spatial distribution of the spin accumulation and the spin current (for the system B and thickness of Co film $L_{\text{F}}$=51 nm) at the moment when they reach maximum values, i.e., when $V$ crosses zero value and charge current flows in the +$x$ direction. We separated the contributions which come from SC and SS. For the current driven by the spin-dependent conductivity, since $\beta>0$ then $\tau_{-}>\tau_{+}$, i.e., the relaxation time for spin down is bigger than for spin up and there is more spin down electrons flowing than spin up electrons. Consequently, $J_{s}$ has negative sign (Fig. 3b, blue line) and its maximum is at the center of Cu/Co bilayer. Within the distance proportional to the spin-diffusion length ($L_{sf}=$60 nm for Co) from the ferromagnet surfaces there builds up a negative spin accumulation gradient (Fig. 3a, blue line) that stands against SC term (compare Eq. (6)) Zayets2012 . In the Cu layer there is weak exponential decay of $s$ from the Cu-Co interface.
For the current driven by the spin-dependent surface screening, since $\gamma_{s}<0$, the outflow of negative charges from the F-D interface generates negative spin accumulation (Fig. 3a, red line). The resulting spin accumulation gradient induces spin current in the direction +$x$ (Fig. 3b, red line). The maximum value of $J_{s}$ is close to the D-FM interface.
We have shown that the spin current driven by SS is a consequence of the spin accumulation gradient build-up. This is in contrast to the SC-polarized current, where spin accumulation gradient is a result of this current and stands against it.
The effects from SS and SC sum up to give resultant spin accumulation and spin current (black lines in Fig. 3). In Co ($\beta>0$, $\gamma_{s}<0$) they add up destructively and for Co thickness of 51 nm discussed above, that contributions balance out at the Cu layer as shown at the left-hand side of dash-dotted line in Fig. 3b.
It is possible to distinguish between SC and SS generated spin currents comparing samples where SS would be absent (system A) and samples where SS would be present (system B). Moreover, spin-dependent conductivity is a phenomenon that acts in a bulk while spin-dependent surface screening is an interface effect. Thus, in thick ferromagnetic layer SC shall dominate while SS should manifest itself in thin ferromagnetic layers.
We consider the values of spin accumulation and spin current in the top Cu layer and we change the thickness of Co layer. The black lines in Fig. 4 show $s$ and $J_{s}$ calculated for the system A. Since there is no D-FM interface, the spin polarization of the current is a result of spin-dependent conductivity. The flow of spin-polarized charges builds up a spin accumulation as discussed above. For a thin Co layer ($L_{\text{F}}<2L_{\text{sf}}$) the spin accumulation counteracts the spin current significantly. For the thickness larger than $2L_{\text{sf}}$ the influence of the interfacial accumulation loses its significance and the value of spin-polarized current reaches maximum value of 105 A/s in Cu (and the value of $J_{s}\approx-\beta J_{f}=7000$ A/s in the bulk of Co layer). The values of $s$ and $J_{s}$ saturate for thick Co films (see insets in Fig. 4).
The red lines in Fig. 4 show $s$ and $J_{s}$ for system B. Because of D-FM interface is present in the system, there are contributions both from SC and SS to the spin transport. Spin-dependent conductivity dominates for $L_{\text{F}}>L_{\text{sf}}$. If the distance from the D-FM interface to Cu layer is less than $2L_{\text{sf}}$, the currents induced by spin-dependent surface screening come into picture and the values of $s$ and $J_{s}$ deviates from those for system A. At $L_{\text{F}}=51$ nm SS balances out SC contribution in Cu and for $L_{\text{F}}<51$ nm spin current driven by SS dominates in the Cu layer. The maximum values obtained for $L_{F}=$ 2 nm are $s=$ 650 mA/m and $J_{s}=$730 A/s which are 7 times higher than those obtained for thick Co films by spin-dependent conductivity.
The dependences of $s$ and $J_{s}$ on $L_{\text{F}}$ in the systems considered above give us simple tool to distinguish between SS- and SC- driven currents. If SS contribution is absent in the system then $s\rightarrow 0$ and $J_{s}\rightarrow 0$ as $L_{\text{F}}\rightarrow 0$ for $L_{\text{F}}<L_{\text{sf}}$. If SS contribution is nonzero, $s$ and $J_{s}$ tend to nonzero values as $L_{\text{F}}\rightarrow 0$ for $L_{\text{F}}<L_{\text{sf}}$. It is rather crucial to choose ferromagnet with relatively long spin-diffusion length like Co, since the effect is hard to resolve for $L_{\text{F}}>L_{\text{sf}}$.
In summary, the mechanism of spin current generation by the spin-dependent surface screening effect has been described within the Stoner model. The mechanism has been implemented to the drift-diffusion model. Then the specific dielectric-ferromagnetic heterostructures have been considered numerically to show the properties of SS-driven spin currents. We found the spin accumulation and the spin polarization of the current driven by spin-dependent surface screening is 7 times higher for 2-nm-thick Co than those obtained by spin-dependent conductivity for thick Co layers. Since the strength of the effect is dependent on the density of the screening charge at the D-FM interface, it may be enhanced with the use of dielectrics of high permittivity. Therefore, it is interesting to consider the influence of SS-driven spin currents onto magnetization dynamics, in particular, the possibility to decrease spin wave damping through currents induced by that mechanism. The proposed systems and the method of extraction of SS-driven spin accumulation is suitable to the X-Ray magnetic circular dichroism experiment Kukreja2015 ; Li2016 , where the measurement of spin accumulation dynamics is element-specific and thus may be constrained to the signal from Cu layer and directly compared to the dependences shown in Fig. 4.
Acknowledgements.
The study has received financial support from the National Science Centre of Poland under grant 2018/28/C/ST3/00052.
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Emotional Musical Prosody: Validated Vocal Dataset for Human Robot Interaction
Richard Savery
Georgia Tech Center for Music Technology
11email: [email protected]
Lisa Zahray
Georgia Tech Center for Music Technology
11email: [email protected]
Gil Weinberg
This material is based upon work supported by the National Science Foundation under Grant No. 1925178
Georgia Tech Center for Music Technology
11email: [email protected]
Abstract
Human collaboration with robotics is dependant on the development of a relationship between human and robot, without which performance and utilization can decrease. Emotion and personality conveyance has been shown to enhance robotic collaborations, with improved human-robot relationships and increased trust. One under-explored way for an artificial agent to convey emotions is through non-linguistic musical prosody. In this work we present a new 4.2 hour dataset of improvised emotional vocal phrases based on the Geneva Emotion Wheel. This dataset has been validated through extensive listening tests and shows promising preliminary results for use in generative systems.
Keywords:robotics, emotion, prosody, music
1 Introduction
As the use of robotics and artificial agents continue to expand, there is a growing need for better forms of communication between human and computer. While speech based mediums are common in home assistants and robots, we contend that for many human-agent collaborations, semantic meaning is not always required. One alternate to speech communication is non-linguistic emotional musical prosody where a robot communicates through musical phrases. Using musical phrases can avoid the challenges of uncanny valley, allowing robot interaction to have it’s own human-inspired form of communication [\citeauthoryearSavery, Rose\BCBL \BBA WeinbergSavery \BOthers.\APACyear2019\APACexlab\BCnt2]. In past work we have successfully shown the capabilities of prosody in robotic systems through simulations of a potential generative system. This includes increased trust in social robots [\citeauthoryearSavery, Rose\BCBL \BBA WeinbergSavery \BOthers.\APACyear2019\APACexlab\BCnt1] and industrial robots [\citeauthoryearSavery, Zahray\BCBL \BBA WeinbergSavery \BOthers.\APACyear2020]. In this paper we present the collection and validation of a emotional prosody dataset that can be used to improve future human robot interactions.
2 Custom Dataset
Our custom dataset with 4.2 hours of audio was created with the singer Mary Carter. After a pilot study collecting data, we decided to use the Geneva Emotion Wheel (GEW) [\citeauthoryearSacharin, Schlegel\BCBL \BBA SchererSacharin \BOthers.\APACyear2012], which is a circular model, containing 20 emotions with position corresponding to the circumplex model (see Figure 2).
One of the primary advantages of the GEW is that it includes 20 different emotions, but these emotions can also be reduced to four separate classes which align with a quadrant from the circumplex model. GEW also includes most of the Eckman’s basic emotions - fear, anger, disgust, sadness, happiness - only leaving out surprise. The ability to use different models of emotion allows for significant future use cases of the dataset [\citeauthoryearSavery \BBA WeinbergSavery \BBA Weinberg\APACyear2020].
This dataset only has one musician, and therefore only captures one person’s perspective on musical emotion. While the dataset can make no claim to represent all emotion and does not create a generalized emotion model, we believe using one person has advantages. By having only one vocalist generative systems have the possibility of recreating one person’s emotional style, instead of incorrectly aggregating multiple peoples to remove distinctive individual and stylistic features.
2.1 Process and Data
Carter was paid $500 and recorded the samples over a week long period at her home studio, using a template we created in Logic while maintaining the same microphone positioning. For the samples we requested phrases to be between 1 and 20 seconds, and to spend about 15 minutes on each emotion, allowing jumping between any order of the emotions. We allowed deletion of a phrase if she felt after singing it did not fit the emotion. The final recorded dataset includes 2441 phrases equalling 4.22 hours of data with an average of 122 for each emotion. Samples from the dataset can be heard online.111www.richardsavery.com/prosodycvae
2.2 Dataset Validation
To validate the dataset, we performed a study with 45 participants from Prolific and Mechanical Turk, paying each $3. Each question in the survey asked the participant to listen to a phrase and select a location on the wheel corresponding to the emotion and intensity they believed the phrase was trying to convey. Phrases fell under two categories of “best” and “all”, with each participant seeing 60 total phrases selected at random. The “best” category consisted of 5 phrases for each emotion that were hand-selected as best representing that emotion, ensuring an even distribution of phrase lengths in each emotion set. The “all” category consisted of a phrase sampled from all phrases in the dataset for that emotion, with a new phrase randomly selected for each participant. Rose plots of the validation results that combine the “best” and “all” categories can be seen in the appendix, separated into each Geneva Wheel quadrant.
2.3 Dataset to Midi
We converted each phrase’s audio into a midi representation to use as training data. We first ran the monophonic pitch detection algorithm CREPE [\citeauthoryearKim, Salamon, Li\BCBL \BBA BelloKim \BOthers.\APACyear2018] on each phrase, which outputs a frequency and a confidence value for a pitch being present every 0.01 seconds. As the phrases included breaths and silence, it was necessary to filter out pitches detected with low confidence. We applied a threshold followed by a median filter to the confidence values, and then forced each detected pitch region to be at least 0.04 seconds long. We then converted the frequencies to midi pitches. We found the most common pitch deviation for each phrase using a histogram of deviations, shifting the midi pitches by this deviation to tune each phrase.
3 Conclusion
We have so far analyzed the dataset by training a Vanilla VAE (without emotion labels) on the pitch alone. Figure 2 shows the latent space reduced from 100 dimensions through t-distributed stochastic neighbor embedding (t-SNE) [\citeauthoryearHinton \BBA RoweisHinton \BBA Roweis\APACyear2003]. This demonstrates that the latent space is able to separate by emotion, without any knowledge of the emotional labels. We then used a conditional VAE (conditioned on emotion labels) to generate phrases with playback through a separate audio sampler. These phrases were validated following the format of our dataset evaluation, and the results are shown in the appendix.
In the future we will run extensive studies with generated prosody applied to human-robot interactions. This will take place between varying group sizes from one human and robot, to groups of humans and robots with different embedded personalities. We expect for emotional musical prosody to enable many future collaborations between human and robot.
References
[\citeauthoryearHinton \BBA RoweisHinton \BBA Roweis\APACyear2003]
\APACinsertmetastarhinton2003stochastic{APACrefauthors}Hinton, G\BPBIE.\BCBT \BBA Roweis, S\BPBIT.
\APACrefYearMonthDay2003.
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\BIn \APACrefbtitleAdvances in neural information processing systems
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\BBOQ\APACrefatitleCREPE: A convolutional representation for pitch
estimation Crepe: A convolutional representation for pitch
estimation.\BBCQ
\BIn \APACrefbtitle2018 IEEE International Conference on Acoustics, Speech
and Signal Processing (ICASSP) 2018 ieee international conference on
acoustics, speech and signal processing (icassp) (\BPGS 161–165).
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\APACinsertmetastarsacharin2012geneva{APACrefauthors}Sacharin, V., Schlegel, K.\BCBL \BBA Scherer, K.
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\BBOQ\APACrefatitleGeneva Emotion Wheel rating study (Report). Geneva,
Switzerland: University of Geneva Geneva emotion wheel rating study
(report). geneva, switzerland: University of geneva.\BBCQ
\APACjournalVolNumPagesSwiss Center for Affective Sciences.
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4 Appendix |
Dispersive Approach to Chiral Perturbation Theory
M. Zdráhal${}^{a,b,}[email protected], J. Novotný${}^{a,}[email protected]
${}^{a}$Institute of Particle and Nuclear Physics, Faculty of Mathematics and Physics,
Charles University, V Holešovičkách 2, CZ-180
00 Prague 8, Czech Republic
${}^{b}$Faculty of Physics, University of Vienna,
Boltzmanngasse 5, A-1090 Vienna, Austria
Abstract
We generalise the reconstruction theorem of Stern, Sazdjian,
and Fuchs based on the dispersion relations to the case of the
($2\rightarrow 2$) scattering of all the pseudoscalar octet mesons
($\pi,K,\eta$). We formulate it in a general way and include also a discussion of the assumptions of the theorem.
It is used to obtain the amplitudes of all such processes in the isospin
limit to the one-loop order (and can be straightforwardly extended
to two loops) independently on the particular power-counting scheme
of the chiral perturbation theory in question. The results in this
general form are presented.
1 Introduction
In low-energy QCD the chiral perturbation theory ($\chi\textrm{PT}$,
cf. [1]) has gained great prominence as a tool for
description interactions of the lightest (pseudoscalar) mesons,
$\pi$, $K$ and $\eta$. Within this framework these mesons are
understood as Goldstone bosons of the spontaneous breaking of the
chiral symmetry $SU(3)_{L}\times SU(3)_{R}$ (appearing if all three
lightest quarks were massless) down to $SU(3)_{V}$, gaining the masses
through the explicit breaking of the symmetry due to the nonzero
quark masses, whereas all the other hadrons (with mass at least of order
$\Lambda\sim 1\,\mathrm{GeV}$) are included only effectively. Using
the symmetry properties and the basic properties like analyticity,
unitarity, and the crossing symmetry, the Lagrangian of this effective
theory is constructed. It contains an infinite number of terms.
Nevertheless, the Weinberg power-counting scheme assigns to a given
diagram of this theory (and term of the Lagrangian) its importance
by means of the chiral dimension $O\!\left(p^{D}\right)$ (see [2]), and so
for computations to a given order in low energies it is sufficient
to use only a finite number of them.
However, we can try to find the amplitudes of a process using the
required properties directly with no need of the Lagrangian to be
explicitly written (with all the advantages and disadvantages that this
involves), which is the main goal of this paper. We extend the
reconstruction theorem given by Stern, Sazdjian, and Fuchs in
[3] and widely used in [4]. Their work was originally
motivated by the problem333The standard power counting is
based on the presumption that the value of $B_{0}$ is of order $\Lambda$.
So there also appeared approaches extending the $\chi\textrm{PT}$ to the case
including the possibility of small $B_{0}$ too, which have a lesser
predictive power connected with larger number of LEC - namely the
generalised $\chi\textrm{PT}$ [5]. of possible smallness of
the scalar condensate $B_{0}$ as the power-counting independent way
to compute the amplitude of $\pi\pi$ scattering in the 2 flavour
case. Even though the problem is not yet satisfactorily
closed444The only direct answer given by experiment is still
the confirmation of the standard power counting in the 2 flavour
case by the $K_{l4}$ measurement at BNL [6]., the method is
interesting also in itself. The extension of this method to the 3
flavour case of the $\pi K$ amplitude has been given by
Ananthanarayan and Büttiker [7]. We have
worked out a generalisation to the processes of all the pseudo-Goldstone bosons (pGB) in the
three flavour case.
In contrast to these previous papers, we have extracted from the
theorem the entire information about the isospin structure and
assumed just the needed crossing symmetry. The isospin properties
are used afterwards separately for particular processes.
Furthermore, the theorem is formulated formally generally by
pointing out all the properties essential for the theorem. This
should simplify the discussion of validity of the theorem and its
eventual further extensions to more complicated cases. In the simple
case of processes from the pure strong $\chi\textrm{PT}$ (conserving the strong
isospin), we have discussed the assumptions and showed their
reasonability and concluded with the construction of their
amplitudes to two loops (with results explicitly written just to the
one loop) in a way independent of the particular power-counting scheme
in use. In this aspect our work is thus also an extension of the work of Osborn [8]
to higher orders.
The plan of the paper is as follows. In Sec. 2 we establish
our notation; the fundamental content of the paper is Sec. 3, where
we reformulate the reconstruction theorem with its assumptions and
give the proof of it. In Sec. 4 we use the theorem (and Appendix A) to compute the amplitudes describing all the 2 pGB
scattering processes to the $O\!\left(p^{4}\right)$ chiral order (explicitly written in Appendix B). Finally, in Sec. 5 we provide a further
discussion of the assumptions and discussion of the results.
Appendix A summarises all the needed amplitudes to describe the
4-pGB processes and lists a general parametrisation of them to the
$O\!\left(p^{2}\right)$ order. Appendix B gives the results of the application of
the theorem to computation of amplitudes to the $O\!\left(p^{4}\right)$ order.
In Appendix C we give relations between our parameters and
renormalised LEC of the standard $\chi\textrm{PT}$ [9] in
a particular renormalisation scheme [10]. The
main points of the analysis of validity of dispersion relations and
the regions of analyticity of amplitudes that are used in the
theorem are briefly summarised in Appendix D.
2 Notation
We consider a scattering process of two pGB of the type
$$A(p_{A})B(p_{B})\rightarrow C(p_{C})D(p_{D})$$
(1)
and write its amplitude according to555This process depends
on two independent kinematical quantities only. It is convenient to
choose the total energy squared $s$ and the angle $\theta$ between
the momenta $p_{A}$ and $p_{C}$ in the center of mass system (CMS) or
two of three Mandelstam variables $s,t,u$. In order to express the crossing and Bose symmetry properties
in a simple way in what follows we keep writing the dependence on all three Mandelstam variables explicitly.
$$\left\langle C(p_{C})D(p_{D})|_{f}A(p_{A})B(p_{B})\right\rangle_{i}=\delta_{if%
}+i(2\pi)^{4}\delta^{(4)}(p_{C}+p_{D}-p_{A}-p_{B})A(s,t,u).$$
(2)
In the relations where we want to distinguish between more processes,
we write explicitly $A_{i\rightarrow f}(s,t,u)$ or $A_{AB\rightarrow CD}(s,t,u)$.
Later on, we will use the crossing symmetry, so let us define the
amplitude of the crossed processes. The amplitude of the direct
process will be denoted by $S(s,t,u)$, i.e. $A_{AB\rightarrow CD}(s,t,u)=S(s,t,u)$. The amplitude of the $T$-crossed channel will be
$A_{A\overline{C}\rightarrow\overline{B}D}(s,t,u)=f_{T}T(s,t,u)$,
where the phase factor $f_{T}$ is defined so that $T(s,t,u)$ fulfils
the crossing relation $S(s,t,u)=T(t,s,u)$ and similarly for the
$U$-crossed channel.
The transition from the CMS variables $s$ and $\cos\theta$ to the
Mandelstam variables $s,t,u$ for the process (1) is
possible using the relation
$$\cos\theta=\frac{s(t-u)+\Delta_{\scriptscriptstyle AB}\Delta_{%
\scriptscriptstyle CD}}{\lambda_{AB}^{1/2}(s)\lambda_{CD}^{1/2}(s)}\,,$$
(3)
where $\Delta_{\scriptscriptstyle ij}$ is a difference of the masses squared $\Delta_{\scriptscriptstyle ij}=m_{i}^{2}-m_{j}^{2}$ and
$$\lambda_{ij}(x)=\lambda(x,m_{i}^{2},m_{j}^{2})=x^{2}+m_{i}^{4}+m_{j}^{4}-2x(m_%
{i}^{2}+m_{j}^{2})-2m_{i}^{2}m_{j}^{2}$$
(4)
refers to the Källen’s quadratic form (also called the triangle
function).
The sum of squared masses of all the particles appearing in the
process is denoted as $\mathcal{M}$, i.e.
$$s+t+u=m_{A}^{2}+m_{B}^{2}+m_{C}^{2}+m_{D}^{2}=\mathcal{M}.$$
(5)
We will use the partial wave decomposition of the amplitudes in the form
$$A_{i\rightarrow f}(s,\cos\theta)=32\pi\sum_{l}A_{l}^{i\rightarrow f}(s)(2l+1)P%
_{l}(\cos\theta),$$
(6)
where $P_{l}$ are Legendre polynomials.
From the unitarity of S-matrix (and real analyticity of the physical
amplitude as well as $T$-invariance) there follows the unitarity
relations. Assuming that the only relevant intermediate states are those containing two
particles $\alpha$ and $\beta$ with masses $m_{\alpha}$ and $m_{\beta}$, its form for the partial wave of
amplitudes reads
$$\operatorname{Im}A_{l}^{i\rightarrow f}(s)=\sum_{\alpha,\beta}\frac{2}{S_{F}}%
\,\frac{\lambda^{1/2}_{\alpha\beta}(s)}{s}A_{l}^{i\rightarrow(\alpha,\beta)}(s%
)\left[A_{l}^{f\rightarrow(\alpha,\beta)}(s)\right]^{*}\theta(s-(m_{\alpha}+m_%
{\beta})^{2}).$$
(7)
Here, $S_{F}$ denotes the symmetry factor ($S_{F}=1$ if the two intermediate
states are distinguishable and $S_{F}=2$ if they are not).
Finally, we use the definition of the (signs of) meson fields given in the matrix
$$\begin{split}\displaystyle\phi(x)=\lambda_{a}\phi_{a}(x)&\displaystyle=\begin{%
pmatrix}\pi^{0}+\frac{1}{\sqrt{3}}\eta&-\sqrt{2}\pi^{+}&-\sqrt{2}K^{+}\\
\sqrt{2}\pi^{-}&-\pi^{0}+\frac{1}{\sqrt{3}}\eta&-\sqrt{2}K^{0}\\
\sqrt{2}K^{-}&-\sqrt{2}\overline{K^{0}}&-\frac{2}{\sqrt{3}}\eta\end{pmatrix}.%
\end{split}$$
(8)
The meson states are chosen so that the matrix element of mesons
gets one minus sign for each charged particle in the final state.
3 Reconstruction theorem
3.1 Statement of the theorem
In this section we prove the following theorem: The amplitude
$S(s,t;u)$ of a given process $A+B\rightarrow C+D$ fulfilling all
the conditions from the next subsection can be reconstructed to (and excluding)
$O\!\left(p^{8}\right)$ order just from the knowledge of imaginary parts of $s$ and
$p$ partial waves of all crossed amplitudes and some polynomial
$$\begin{split}\displaystyle S(s,t;u)&\displaystyle=R_{4}(s,t;u)+\Phi_{0}(s)+%
\left[s(t-u)+\Delta_{\scriptscriptstyle AB}\Delta_{\scriptscriptstyle CD}%
\right]\Phi_{1}(s)+\Psi_{0}(t)\\
&\displaystyle\quad+\left[t(s-u)+\Delta_{\scriptscriptstyle AC}\Delta_{%
\scriptscriptstyle BD}\right]\Psi_{1}(t)+\Omega_{0}(u)+\left[u(t-s)+\Delta_{%
\scriptscriptstyle AD}\Delta_{\scriptscriptstyle BC}\right]\Omega_{1}(u)+O\!%
\left(p^{8}\right),\end{split}$$
(9)
where $R_{4}(s,t;u)$ is a third-order polynomial in Mandelstam
variables (obeying exactly the same symmetries as the original
amplitude $S(s,t;u)$) and
$$\displaystyle\Phi_{0}(s)$$
$$\displaystyle=32s^{3}\int_{\Sigma}^{\Lambda^{2}}\frac{dx}{x^{3}}\frac{%
\operatorname{Im}S_{0}(x)}{x-s}\,,$$
(10)
$$\displaystyle\Phi_{1}(s)$$
$$\displaystyle=96s^{3}\int_{\Sigma}^{\Lambda^{2}}\frac{dx}{x^{3}}\frac{1}{x-s}%
\operatorname{Im}\frac{S_{1}(x)}{\lambda_{AB}^{1/2}(x)\lambda_{CD}^{1/2}(x)}\,,$$
(11)
$$\displaystyle\Psi_{0}(t)$$
$$\displaystyle=32t^{3}\int_{\tau}^{\Lambda^{2}}\frac{dx}{x^{3}}\frac{%
\operatorname{Im}T_{0}(x)}{x-t}\,,$$
(12)
$$\displaystyle\Psi_{1}(t)$$
$$\displaystyle=96t^{3}\int_{\tau}^{\Lambda^{2}}\frac{dx}{x^{3}}\frac{1}{x-t}%
\operatorname{Im}\frac{T_{1}(x)}{\lambda_{AC}^{1/2}(x)\lambda_{BD}^{1/2}(x)}\,,$$
(13)
$$\displaystyle\Omega_{0}(u)$$
$$\displaystyle=32u^{3}\int_{\Upsilon}^{\Lambda^{2}}\frac{dx}{x^{3}}\frac{%
\operatorname{Im}U_{0}(x)}{x-u}\,,$$
(14)
$$\displaystyle\Omega_{1}(u)$$
$$\displaystyle=96u^{3}\int_{\Upsilon}^{\Lambda^{2}}\frac{dx}{x^{3}}\frac{1}{x-u%
}\operatorname{Im}\frac{U_{1}(x)}{\lambda_{AD}^{1/2}(x)\lambda_{BC}^{1/2}(x)}\,,$$
(15)
$S_{0}$ and $S_{1}$ are $s$ and $p$ partial wave of the process in
question, similarly $T_{0}$ and $T_{1}$ and $U_{0}$ and $U_{1}$ the partial
waves of the $T$ and $U$ crossed processes. $\Sigma,\tau,\Upsilon$
are the minima of squared invariant mass of all possible
intermediate states in the $S$, $T$ and $U$ channels (see (18)).
Finally, $\Lambda$ is the scale from the assumptions.
3.2 Assumptions of the theorem
We use this theorem in the (strong part of) chiral perturbation
theory, but for better understanding of it, we are formulating all
the assumptions in the general way and then discussing their
validity in $\chi\textrm{PT}$ separately. The assumptions of the theorem are:
1.
There exists a threshold $\Lambda$ up to which we can regard the
theory under consideration to be complete; this means, for example, that
under this threshold there appears no particle of other types than
those which are already explicitly included in the theory and all the
influence of such extra particles is already taken into account effectively
or negligible under the threshold. In chiral perturbation theory the
threshold $\Lambda$ is the threshold of production of non-pGB
states, i.e. $\Lambda\sim 1\,\mathrm{GeV}$.
2.
There exists a well-behaved expansion of the amplitude in
powers of $p/\Lambda$. In $\chi\textrm{PT}$ it is enough to have any
chiral expansion, not necessarily the standard Weinberg one. As is
common in $\chi\textrm{PT}$, instead of $O\!\left((p/\Lambda)^{n}\right)$ we write just $O\!\left(p^{n}\right)$ for
short.
3.
We can write a three times666As discussed later on, the exact number of subtractions is not so relevant. subtracted dispersion relation for
the amplitude $S(s,t;u)$ in the complex $s$ plane for fixed value of
$u$ (in the form (16)). This is the essential
assumption of the proof - the theorem stands or falls by it. This assumption is also connected with the validity of the
partial wave expansion (for the sake of simplicity, we assume
that this expansion can be also analytically continued below the
physical threshold where needed). The legitimacy of these
assumptions in the case of $\chi\textrm{PT}$ is discussed in
Appendix D.
4.
We further assume that the amplitude considered as a function of a single Mandelstam variable (with the other fixed at some appropriate value) is analytic in some unempty open region. The regions where this and the previous assumptions are valid are discussed (and plotted) in Appendix D.
5.
The crossing relations given in the form of the previous section are
valid, i.e. $S(s,t,u)=T(t,s,u)=U(u,t,s)$. That is a widely accepted assumption, whose validity for a
general $2\rightarrow 2$ process has been proven from axiomatic theory by Bross, Epstein, and Glaser [11].
6.
The absorptive parts of the $l\geq 2$ partial amplitudes are suppressed to
the $O\!\left(p^{8}\right)$ order - thanks to that we can (up to this order) deal just
with the first two partial waves in the theorem. In our case that
follows from the (pseudo-)Goldstone boson character of the particles
under consideration.
In $\chi\textrm{PT}$ the amplitudes behave dominantly as $O\!\left(p^{2}\right)$ and do not
contain any bound state poles. The unitarity relation
(7) then implies that its imaginary part behaves as
$O\!\left(p^{4}\right)$, i.e. the amplitudes are dominantly real. According to the
analyticity of the amplitude, its leading $O\!\left(p^{2}\right)$ part should be a
polynomial in Mandelstam variables. Moreover, it has to be a
polynomial of the first order; otherwise its coefficient would grow
up as mass of the pGB went to zero, and that would contradict the
finiteness of the S-matrix in the chiral limit with the external
momenta fixed777We will see that the finiteness of the
S-matrix in the chiral limit deserves to be regarded as one of the
full-valued assumptions by itself.. However, the first-order
polynomial could not contribute to $l\geq 2$ partial waves, thus these
partial waves should behave in the chiral limit as $O\!\left(p^{4}\right)$, thereby
again using the unitarity relation, the imaginary parts of these partial
waves are at least of the $O\!\left(p^{8}\right)$ order.
3.3 Proof of the theorem
One of our assumptions is that for the amplitude we can write the
three times subtracted dispersion relation in the form:
$$\displaystyle S(s,t;u)=P_{3}(s,t;u)+\frac{s^{3}}{\pi}\int_{\Sigma}^{\infty}%
\frac{dx}{x^{3}}\frac{\operatorname{Im}S(x,\mathcal{M}-x-u;u)}{x-s}\\
\displaystyle+\frac{t^{3}}{\pi}\int_{\tau}^{\infty}\frac{dx}{x^{3}}\frac{%
\operatorname{Im}S(\mathcal{M}-x-u,x;u)}{x-t}\,,$$
(16)
where $P_{n}(s,t;u)$ is a polynomial of the $(n-1)$-th order in $s$
and $t$.888The coefficients of this polynomial depend on $u$. Thanks to the relation for the sum of the
Mandelstam variables, such polynomials could be written in the form
$P_{n}(s,t;u)=\alpha(u)+\beta(u)(s-t)+\gamma(u)(s-t)^{2}+\dots+\omega(u)(s-t)^{%
n-1}.$
(17)
$\Sigma$ is the minimum of the squared invariant mass of all the
possible intermediate states $(\alpha,\beta)$ in the S
channel (or rather the lowest mass
of the state with the same quantum numbers as the in and
out state in the S-channel)999In this relation we anticipate again that we will
consider the problems under conditions implying relevance of the
two-particle intermediate states only.,
$$\Sigma=\min_{(\alpha,\beta)}(m_{\alpha}+m_{\beta})^{2},$$
(18)
and analogically $\tau$ for the $T$ channel (later on we will also need
the same for the $U$ channel, what will be denoted by $\Upsilon$).
Thanks to the crossing symmetry we can replace the $S(s,t;u)$
amplitude in the second integral with the $T(t,s;u)$. Further, since
we assume that we know the complete theory only up to some threshold
$\Lambda$, we have to split the dispersion integral into two parts -
the low-energy $(x\leq\Lambda^{2})$ and the high-energy
$(x\geq\Lambda^{2})$ part. We consider the region where
$s\ll\Lambda^{2}$, i.e. in the high-energy part, $s\ll x$, we
have101010The fact that the remainder $\sum_{n\geq 4}s^{n}R_{n}(u)$
is of order at least $O\!\left(p^{8}\right)$ follows again from the finiteness of the
amplitude in the chiral limit (the importance of this assumption
stems from here) and the crossing symmetry.
$$\frac{s^{3}}{\pi}\int_{\Lambda^{2}}^{\infty}\frac{dx}{x^{3}}\left.\frac{%
\operatorname{Im}S(x,\mathcal{M}-x-u;u)}{x-s}\right|_{s<<x}=s^{3}H_{\Lambda}(u%
)+O\!\left(p^{8}\right),$$
(19)
where
$$H_{\Lambda}(u)=\frac{1}{\pi}\int_{\Lambda^{2}}^{\infty}\frac{dx}{x^{3}}\frac{%
\operatorname{Im}S(x,\mathcal{M}-x-u;u)}{x}$$
(20)
is a function of $u$ only. We can proceed similarly for the high-energy part of
the integration of $T(s,t;u)$. Hence, these high-energy parts can be
to the order $O\!\left(p^{8}\right)$ added to the subtraction polynomial now thereby
of the third order111111The consequence of that is also that we
can formally extend the validity of the original theory even beyond
threshold $\Lambda$ and compute the dispersion integral up to
infinity, pretending that the original theory is really complete
(naturally with the appropriate modification of the third-order
polynomial). Similar arguments would appear even if we introduced
the cutoff $\Lambda$ by other reasons, e.g. by reason of numerical
integration.. Afterwards, the dispersion relation is of the form
$$\displaystyle S(s,t;u)=P_{4}(s,t;u)+\frac{s^{3}}{\pi}\int_{\Sigma}^{\Lambda^{2%
}}\frac{dx}{x^{3}}\frac{\operatorname{Im}S(x,\mathcal{M}-x-u;u)}{x-s}\\
\displaystyle+\frac{t^{3}}{\pi}\int_{\tau}^{\Lambda^{2}}\frac{dx}{x^{3}}\frac{%
\operatorname{Im}T(x,\mathcal{M}-x-u;u)}{x-t}+O\!\left(p^{8}\right).$$
(21)
We decompose the (imaginary parts of) amplitudes into the partial
waves
$$\begin{split}\displaystyle S(s,t;u)&\displaystyle=32\pi\left(S_{0}(s)+3\cos%
\theta S_{1}(s)+S_{l\geq 2}(s,t;u)\right)\\
&\displaystyle=32\pi\left(S_{0}(s)+3\frac{s(t-u)+\Delta_{\scriptscriptstyle AB%
}\Delta_{\scriptscriptstyle CD}}{\lambda_{AB}^{1/2}(s)\lambda_{CD}^{1/2}(s)}S_%
{1}(s)+S_{l\geq 2}(s,t;u)\right).\end{split}$$
(22)
The terms $S_{l\geq 2}(s,t;u)$ incorporate contributions of all the
higher $(l\geq 2)$ partial waves and are suppressed to $O\!\left(p^{8}\right)$ according
to our assumptions.
From the integrals of the P partial waves, we can extract functions
depending just on $u$ and include them in the polynomial, e.g.
$$\displaystyle\int_{\Sigma}^{\Lambda^{2}}\frac{dx}{x^{3}}\frac{x(s+t-x-u)}{x-s}%
\operatorname{Im}\frac{S_{1}(x)}{\lambda_{AB}^{1/2}(x)\lambda_{CD}^{1/2}(x)}\\
\displaystyle=\int_{\Sigma}^{\Lambda^{2}}\frac{dx}{x^{3}}\frac{x(t-u)}{x-s}%
\operatorname{Im}\frac{S_{1}(x)}{\lambda_{AB}^{1/2}(x)\lambda_{CD}^{1/2}(x)}-%
\int_{\Sigma}^{\Lambda^{2}}\frac{dx}{x^{2}}\operatorname{Im}\frac{S_{1}(x)}{%
\lambda_{AB}^{1/2}(x)\lambda_{CD}^{1/2}(x)}.$$
(23)
Finally, we use this rearrangement of the fractions
$$\frac{x(t-u)}{x-s}=\frac{s(t-u)}{x-s}+t-u,$$
(24)
where the second term gives after integration a contribution in the
form $t-u$ times a function of $u$ only and therefore as a
polynomial in $t$ can also be included into the subtraction
polynomial $P_{4}$.
After that, the amplitude reads
$$\displaystyle S(s,t;u)=P_{4}(s,t;u)+\Phi_{0}(s)+[s(t-u)+\Delta_{%
\scriptscriptstyle AB}\Delta_{\scriptscriptstyle CD}]\Phi_{1}(s)\\
\displaystyle+\Psi_{0}(t)+[t(s-u)+\Delta_{\scriptscriptstyle AC}\Delta_{%
\scriptscriptstyle BD}]\Psi_{1}(t)+O\!\left(p^{8}\right),$$
(25)
where $\Phi_{0}(s),\Phi_{1}(s),\Psi_{0}(t)$, and $\Psi_{1}(t)$ are given by (10) – (13).
In order to implement the $s\leftrightarrow u$ crossing symmetry, we
add and subtract the following polynomial in $s$ and $t$ (with
coefficients depending on $u$)
$$\Omega_{0}(u)+[u(t-s)+\Delta_{\scriptscriptstyle AD}\Delta_{\scriptscriptstyle
BC%
}]\Omega_{1}(u),$$
(26)
with $\Omega_{0}(u)$ and $\Omega_{1}(u)$ given by (14) and
(15) and we include the subtraction of this polynomial in
the polynomial $P_{4}(s,t;u)$ and thereby get the amplitude in the
symmetric form (9).
Until now, we have proved the whole theorem except for the
$u$-dependence of the polynomial $P_{4}(s,t;u)$. However, we could write similar
relations also for the amplitudes $T(s,t;u)$ and $U(s,t;u)$ with the
polynomials in $(s,u)$ and the coefficients depending on variable
$t$. Thanks to the crossing symmetry and to the symmetric form of
the rest of the formula, these polynomials should obey the same
crossing symmetry. From this and from the analyticity of the
amplitudes in a neighborhood of $u$ for some fixed $s$ in our
kinematic region, we can conclude that $P_{4}(s,t;u)$ are
polynomials in respect to all Mandelstam variables.
4 Application of the theorem to the 4-meson processes to the $O\!\left(p^{4}\right)$ order
As we have already outlined, in the case of $2\rightarrow 2$
scattering processes of pseudoscalar octet mesons, we can use the
theorem and the unitarity relation to a selfconsistent construction
of all those amplitudes to (and excluding) $O\!\left(p^{8}\right)$ order. We need just a
parametrisation of $O\!\left(p^{2}\right)$ amplitudes as its input.
In the following we will consider the strong isospin conservation
limit121212Recall that in this case the validity of the required
dispersion relations can be proven directly from the axiomatic theory as discussed in Appendix D., where
Ward identities imply that there are just 7 independent processes.
From them the amplitudes of all the possible physical processes can
be determined as summarised in Appendix A. Since we want to keep the
independence on the specific power-counting scheme, we have
parameterised the $O\!\left(p^{2}\right)$ amplitudes in the most general form obeying
their symmetries (crossing, Bose, and isospin) - given also in Appendix A. Inserting them into the
unitarity relations (7) (assuming that we have
amplitudes of all the needed intermediate processes), we get the
imaginary parts to $O\!\left(p^{4}\right)$ order, which can be installed into the
theorem and we receive thereby the amplitudes in that order. With
the same procedure (though more technically involved), we can proceed
with the second iteration and obtain thereby the amplitudes to (and excluding)
$O\!\left(p^{8}\right)$ order. For the sake of simplicity, in this paper we confine
ourselves to the first iteration and thus to the results to $O\!\left(p^{4}\right)$
order.
It remains to show that we already have the amplitudes of all the
relevant intermediate processes, i.e. the self-consistency of
this procedure. We are concerned with the processes to (and excluding) $O\!\left(p^{8}\right)$
chiral order and (in accordance with the first assumption of the
theorem) in the region bellow the appearance of all non-Goldstone
particles and thereby away from poles and cuts of such intermediate
states. Therefore, their effect is included just in the polynomial
of the reconstruction theorem and in the higher orders. The
intermediate states with an odd number of Goldstone bosons are
forbidden by the even intrinsic parity of the effective Lagrangian -
because the effect of the axial anomaly enters at the $O\!\left(p^{4}\right)$ order
(Wess, Zumino and Witten [12]), such intermediate states
induce the contribution of the order at least $O\!\left(p^{8}\right)$. Furthermore, the contribution
of the states with more than two Goldstone bosons is also suppressed
to $O\!\left(p^{8}\right)$ order since the $n$-Goldstone-boson invariant phase space
scales like $p^{2n-4}$ and amplitudes with an arbitrary number of
external Goldstone boson legs behave dominantly as $O\!\left(p^{2}\right)$.
Consequently, the contribution of such ($n>2$) intermediate states
to the imaginary part of the amplitude is according to the unitarity
relation at least $O\!\left(p^{8}\right)$. Thus, we can consider the
two-Goldstone-boson intermediate states only.
Further simplification in the $O\!\left(p^{4}\right)$ order computation appears. Using
again the arguments of finiteness of S-matrix in the chiral limit
with the external momenta fixed, the coefficients of the polynomial
in the theorem should be at least $O\!\left(p^{0}\right)$, and so the polynomial in the
$O\!\left(p^{4}\right)$ amplitude is maximally of the second order in Mandelstam
variables.
4.1 The application schematically
The $O\!\left(p^{2}\right)$ amplitudes can be written (cf. with (17), and Appendix A with use of (5)) in the form131313Let us remind that in the $O\!\left(p^{2}\right)$ case $\alpha(s)=\alpha_{0}+\alpha_{1}s$, i.e. it is just the first order polynomial in $s$.
$$A=\frac{1}{F_{\pi}^{2}}\left(\alpha(s)+\beta(t-u)\right).$$
(27)
The decomposition into the partial waves can be obtained using (3) and (22)
$$\displaystyle A_{0}$$
$$\displaystyle=\frac{1}{32\pi}\frac{1}{F_{\pi}^{2}}\left(\alpha(s)-\beta\frac{%
\Delta_{\scriptscriptstyle AB}\Delta_{\scriptscriptstyle CD}}{s}\right),$$
(28)
$$\displaystyle A_{1}$$
$$\displaystyle=\frac{1}{32\pi}\frac{1}{3F_{\pi}^{2}}\beta\frac{\lambda^{1/2}_{%
AB}(s)\lambda^{1/2}_{CD}(s)}{s}\,.$$
(29)
The (right) discontinuity of our function $S_{0}$ and $S_{1}$ of the theorem, given by the unitarity relation,
is therefore in the $O\!\left(p^{4}\right)$ case very simple:
$$\displaystyle\begin{split}\displaystyle\operatorname{Im}S_{0}(s)&\displaystyle%
=\sum_{\begin{subarray}{c}\scriptscriptstyle k-\text{interm.}\\
\scriptscriptstyle\text{states}\end{subarray}}\frac{2}{S_{F_{k}}}\frac{\lambda%
_{k}^{1/2}(s)}{s}\left(\frac{1}{32\pi}\right)^{2}\frac{1}{F_{\pi}^{4}}\left[%
\left(\alpha_{ik}(s)-\beta_{ik}\frac{\Delta_{\scriptscriptstyle AB}\Delta_{k}}%
{s}\right)\left(\alpha_{kf}(s)-\beta_{kf}\frac{\Delta_{k}\Delta_{%
\scriptscriptstyle CD}}{s}\right)\right]\\
&\displaystyle\qquad\qquad\qquad\times\theta(s-s^{t}_{k}),\end{split}$$
(30)
$$\displaystyle\operatorname{Im}S_{1}(s)$$
$$\displaystyle=\sum_{\begin{subarray}{c}\scriptscriptstyle k-\text{interm.}\\
\scriptscriptstyle\text{states}\end{subarray}}\frac{2}{S_{F_{k}}}\frac{\lambda%
_{k}^{1/2}(s)}{s}\left(\frac{1}{32\pi}\right)^{2}\frac{1}{9F_{\pi}^{4}}\bigg{[%
}\beta_{ik}\frac{\lambda^{1/2}_{AB}(s)\lambda^{1/2}_{k}(s)}{s}\beta_{kf}\frac{%
\lambda^{1/2}_{k}(s)\lambda^{1/2}_{CD}(s)}{s}\bigg{]}\theta(s-s^{t}_{k}),$$
(31)
where the sums go over all the possible (two-pGB) intermediate states $k$ with its symmetry factor $S_{F_{k}}$ and all the objects with lower index $k$ are the quantities relating to this intermediate state, similarly indices $i$ and $f$ refer to initial and final state. Finally $s^{t}_{k}$ is a threshold of this intermediate state.
Thanks to the behaviour of square root of the triangle function, $\lambda^{1/2}(s)\rightarrow s$ for $s\rightarrow\infty$ and the fact that $\alpha(s)$ is the polynomial maximally of the first order in $s$, we know that not all the terms (in the brackets) need all the subtractions to give finite integrals (10) – (15) (e.g. the parts with negative power of $s$ in the polynomial in the brackets do not even need any subtraction). Any subtraction redundant in this aspect then gives just a polynomial, which can be included into the polynomial $R_{4}(s,t,t)$ of the theorem (9) and into the contributions of higher orders. Furthermore, if the integrals rising on both sides of rearrangement (24) have a good mathematical sense, we can pull out the polynomial in front of the integral again with possible change of the polynomial $R_{4}(s,t,t)$. Therefore, applying the reconstruction theorem, we can write $\Phi_{0}$ and $\Phi_{1}$ in the minimal141414With the minimal number of subtraction needed to get a finite nonpolynomial part, i.e. here with one or two subtractions. form (except for the polynomial included to $R_{4}(s,t,t)$):
$$\displaystyle\Phi_{0}(s)=\frac{1}{16\pi^{2}}\frac{1}{F_{\pi}^{4}}\sum_{k}\frac%
{1}{S_{F_{k}}}\Bigg{(}\alpha_{ik}(s)\alpha_{kf}(s)s\int_{s^{t}_{k}}^{\infty}%
\frac{dx}{x}\frac{1}{x-s}\frac{\lambda_{k}^{1/2}(x)}{x}\\
\displaystyle-\big{(}\alpha_{kf}(s)\beta_{ik}\Delta_{\scriptscriptstyle AB}+%
\alpha_{ik}(s)\beta_{kf}\Delta_{\scriptscriptstyle CD}\big{)}\Delta_{k}\int_{s%
^{t}_{k}}^{\infty}\frac{dx}{x}\frac{1}{x-s}\frac{\lambda_{k}^{1/2}(x)}{x}\\
\displaystyle+\beta_{ik}\beta_{kf}\Delta_{k}^{2}\Delta_{\scriptscriptstyle AB}%
\Delta_{\scriptscriptstyle CD}\int_{s_{k}^{t}}^{\infty}\frac{dx}{x^{2}}\frac{1%
}{x-s}\frac{\lambda_{k}^{1/2}(x)}{x}\Bigg{)}.$$
(32)
Similarly for the P-wave function $\Phi_{1}$ (in addition we have used $\lambda_{k}(x)=x^{2}-2x\Sigma_{k}+\Delta_{k}^{2}$, where $\Sigma_{k}$ is the sum of the second powers of masses of the two particles in the intermediate state $k$):
$$\Phi_{1}(s)=\frac{1}{16\pi^{2}}\frac{1}{3F_{\pi}^{4}}\sum_{k}\frac{1}{S_{F_{k}%
}}\beta_{ik}\beta_{kf}\Bigg{(}\big{(}s-2\Sigma_{k}\big{)}\int_{s^{t}_{k}}^{%
\infty}\frac{dx}{x}\frac{1}{x-s}\frac{\lambda_{k}^{1/2}(x)}{x}+\Delta_{k}^{2}%
\int_{s^{t}_{k}}^{\infty}\frac{dx}{x^{2}}\frac{1}{x-s}\frac{\lambda_{k}^{1/2}(%
x)}{x}\Bigg{)}.$$
(33)
In the integrals we recognize the once and twice subtracted dispersion integrals from Appendix B ((68) and (69)), i.e.
$$\displaystyle\begin{split}\displaystyle\Phi_{0}(s)&\displaystyle=\frac{1}{F_{%
\pi}^{4}}\sum_{k}\frac{1}{S_{F_{k}}}\Bigg{(}\beta_{ik}\beta_{kf}\frac{\Delta_{%
k}^{2}}{s^{2}}\Delta_{\scriptscriptstyle AB}\Delta_{\scriptscriptstyle CD}%
\overline{\overline{J}}_{k}(s)\\
&\displaystyle\qquad+\bigg{(}\alpha_{ik}(s)\alpha_{kf}(s)-(\alpha_{kf}(s)\beta%
_{ik}\Delta_{\scriptscriptstyle AB}+\alpha_{ik}(s)\beta_{kf}\Delta_{%
\scriptscriptstyle CD})\frac{\Delta_{\scriptscriptstyle k}}{s}\bigg{)}%
\overline{J}_{k}(s)\Bigg{)},\end{split}$$
(34)
$$\displaystyle\Phi_{1}(s)$$
$$\displaystyle=\frac{1}{3F_{\pi}^{4}}\sum_{k}\frac{1}{S_{F_{k}}}\beta_{ik}\beta%
_{kf}\bigg{(}\bigg{(}1-2\frac{\Sigma_{k}}{s}\bigg{)}\overline{J}_{k}(s)+\frac{%
\Delta_{k}^{2}}{s^{2}}\overline{\overline{J}}_{k}(s)\bigg{)}.$$
(35)
Using the change of the polynomial for the last time (in the replacement of $\frac{\overline{\overline{J}}(s)}{s}$ with $\frac{\overline{J}(s)}{s}$), we have the result:
$$\begin{split}&\displaystyle\Phi_{0}(s)+[s(t-u)+\Delta_{\scriptscriptstyle AB}%
\Delta_{\scriptscriptstyle CD}]\Phi_{1}(s)\\
&\displaystyle\quad=\frac{1}{3F_{\pi}^{4}}\sum_{k}\frac{1}{S_{F_{k}}}\Bigg{(}4%
\Delta_{\scriptscriptstyle AB}\Delta_{\scriptscriptstyle CD}\beta_{ik}\beta_{%
kf}\Delta_{k}^{2}\frac{\overline{\overline{J}}_{k}(s)}{s^{2}}\\
&\displaystyle\qquad+\bigg{(}\beta_{ik}\beta_{kf}\big{(}(t-u)\Delta_{k}^{2}-2%
\Delta_{\scriptscriptstyle AB}\Delta_{\scriptscriptstyle CD}\Sigma_{k}\big{)}-%
3(\alpha_{kf}(s)\beta_{ik}\Delta_{\scriptscriptstyle AB}+\alpha_{ik}(s)\beta_{%
kf}\Delta_{\scriptscriptstyle CD})\Delta_{k}\bigg{)}\frac{\overline{J}_{k}(s)}%
{s}\\
&\displaystyle\qquad+\bigg{(}3\alpha_{ik}(s)\alpha_{kf}(s)+\beta_{ik}\beta_{kf%
}\big{(}(t-u)(s-2\Sigma_{k})+\Delta_{\scriptscriptstyle AB}\Delta_{%
\scriptscriptstyle CD}\big{)}\bigg{)}\overline{J}_{k}(s)\Bigg{)}.\end{split}$$
(36)
We get similar results also for the $T$- and $U$-crossed parts151515If we carry out the same extraction of polynomial also for them, we get again these results by the change $s\leftrightarrow t$ respectively $s\leftrightarrow u$ and appropriate permutation in ABCD and then the symmetries of the polynomial of theorem remain the same.. Final results of this procedure for all the amplitudes are given in Appendix B.
5 Summary and discussions
In this paper we have worked out a generalisation of the dispersive
analysis from [3] to the processes involving all the light
octet mesons. We have formulated the reconstruction theorem for
their amplitudes free of the particular isospin structure of the
processes (which is used afterwards for the individual processes
separately) and have provided (an outline of) its proof. We have
also attempted to point out exactly the properties of the theory
which are needed for its proof mainly due to better understanding of
it as well as of its further generalisation on more complicated
cases (as below). In our particular case we have succeeded in
justifying most of them practically from the first principles of
axiomatic quantum field theory - at least in the regions from Fig. 1.
The amplitudes of 4-octet-meson processes (on condition of the
strong isospin conservation) form a selfconsistent system with
respect to the theorem and the unitarity relations, thanks to the fact that
we can simply construct these amplitudes to (and excluding161616Since we also want to include the power countings where the contributions of odd chiral orders appear (like generalised chiral power counting), we write instead of valid to $O\!\left(p^{6}\right)$ order (what will be case if odd orders do not appear) or to $O\!\left(p^{7}\right)$ order (where the odd orders are included) just valid to and excluding $O\!\left(p^{8}\right)$ order to avoid misunderstanding. We should also remind that in these odd chiral order power countings, the one-loop result is not $O\!\left(p^{4}\right)$ but $O\!\left(p^{5}\right)$ – we can get such a result by simple modification of the results presented keeping the $\beta_{\pi\eta}$ and $\gamma_{KK}$ parameters also in the intermediate processes.) $O\!\left(p^{8}\right)$ order (in this
work explicitly computed only to $O\!\left(p^{4}\right)$ order) independently on the
specific power-counting model in use. The amplitudes for the
particular model can be obtained with the specific choice of the parameters. For example, we can compare the nonpolynomial part of our results (with the standard choice of $O\!\left(p^{2}\right)$ constants from Appendix A) with that part of the standard chiral perturbation results (given by [10]) -- we see an agreement except for the different sign convention171717When
comparing, one should pay attention to their caption of the 4-kaon
process ${\overline{K^{0}}}K^{0}\rightarrow K^{+}K^{-}$, which gives us the false
illusion about the value of the Mandelstam variables, whereas the
right value would be better evoked by the caption ${\overline{K^{0}}}K^{0}\rightarrow K^{-}K^{+}$.. We can also use their result to get a relation between our $O\!\left(p^{4}\right)$ polynomial parameters and low-energy constants (LECs)
of the standard chiral perturbation theory in the particular renormalisation scheme used therein just by comparison of the polynomial parts, as is discussed and given in Appendix C.181818If we used the $O\!\left(p^{4}\right)$ parameters that has been obtained thereby also in the unitarity part, we would be already counting a part of the $O\!\left(p^{6}\right)$ corrections. It is worth mentioning that by subtracting the center of the Mandelstam triangle in the second-order polynomial part of our results, the parameters of the first-order polynomial, i.e. $\alpha$’s, $\beta$’s and $\gamma$’s, do not depend in the subtraction scheme used by [10] in $O\!\left(p^{4}\right)$ order on the LECs of operators with four derivatives from $\mathcal{L}_{4}$ in [9], i.e. $L_{1}^{r}$, $L_{2}^{r}$, and $L_{3}$.
Some of the computed amplitudes ($KK\rightarrow$ anything, processes
involving $\eta$’s) have not been computed in such a general form
yet. However, this pioneerness is connected with one problem of the
assumptions of the theorem. We have assumed the existence of a
threshold separating the processes containing just those 8 mesons
from the processes in which particles of other types (e.g. resonances) appear, and then we focus on a kinematical range far
below that threshold. However, in the case of the processes with the
exception of $\pi\pi\rightarrow\pi\pi$ and $\pi K\rightarrow\pi K$,
this range is very narrow or does not even exist (e.g. in the
process $KK\rightarrow KK$ with a kinematical threshold of
$990\,\mathrm{MeV}$ there can appear the $\rho$ resonance with its
mass of $770\,\mathrm{MeV}$). The significance of our result thereby
decreases. Nevertheless, beside the elegance of our construction due
to its intrinsic self-consistency, our results can become useful for
a check of complicated models or simulations, where one can separate
(or completely turn off) the effects of these resonances. (The
advantage is that even in the most general case in (to and excluding) $O\!\left(p^{8}\right)$ order, there
is a small number ($=47$) of parameters, coupling different processes
together.)
To that end it would be naturally preferable to perform also the
second iteration and get thereby the whole information achievable
from the reconstruction theorem. As has been already mentioned, the
completion of the second iteration is conceptually straightforward; however, there appear certain technical complications which are connected with the computation of partial waves of
amplitudes, where we need to compute integrals of the type $\int t^{n}\overline{J}(t)dt$ and then to “dispersively integrate” them. Because of the appearance of the three different masses, the results are much more complicated than in the 2 flavour $\pi\pi$ case (with only one mass) and so will be presented with all the technical details in a separate paper.
The careful reader has surely noticed that in the relation
(21) or (11) and the other similar (in
opposition to the verbal formulation of the theorem), we have
emphasised that the required imaginary part is not of the partial
wave but of the partial wave divided by
$\lambda_{AB}^{1/2}\lambda_{CD}^{1/2}$, i.e. in principle in the higher order there
could appear a problem under the physical threshold of the
amplitudes where $\lambda_{AB}^{1/2}\lambda_{CD}^{1/2}$ could be
imaginary. There presents itself a question whether we can obtain
this quantity also from the unitarity relations. But we should
remind that the origin of the relation (7) where the
imaginary part of the partial wave times the relevant Legendre
polynomial also occurs. So the only question is whether we can
believe the analytical continuation of the partial wave
decomposition and of the unitarity relation under the physical
threshold. If we do so, then the unitarity relation gives exactly
what we need to the theorem, i.e. what we call the imaginary part of
partial wave for simplicity.
The further important question is of the number of subtractions in
the dispersive relations. Jin and Martin [13] have shown
that thanks to the Froissart bound [14] it suffices to
consider just two subtractions. However, the Froissart bound deals
with the complete theory (full QCD), and in an effective theory one
does not know (or more exactly one does not deal with) what is above
$\Lambda$, and so the situation can occur that more than two subtractions are needed. In other words Froissart tells us two subtractions are
sufficient if we supply the dispersive integral above $\Lambda$ with
the complete theory. How to deal with this unknown, we have
already seen near the relation (19) – for $n$
subtractions we get a series of infinite order beginning with
the $n$-th power of Mandelstam variable. Fortunately, we have shown
that the terms with more than fourth power are at least of $O\!\left(p^{8}\right)$
order. And so, if we assume the finiteness of the S-matrix in the
chiral limit (and the other assumptions used there), we end up with
the third-order polynomial and the remainder of $O\!\left(p^{8}\right)$ order no matter
with how many subtraction we have begun.
From the possible extensions of the theorem we find especially
interesting two of them, the extension to the analysis of the meson
formfactors and the computation of amplitudes of $K\rightarrow 3\pi$
decays (with appearance of cusps). The latter of them is the subject
we are recently working on [15, 16].
Acknowledgment
We are grateful to K. Kampf for careful reading of the manuscript and helpful suggestions. We also thank to M. Knecht for inspiring discussions.
The work was supported by the Center for Particle Physics, Project
No. LC527 of the Ministry of Education of the Czech Republic and by the EU-RTN Programme, Contract No. MRTN–CT-2006-035482, “Flavianet.”
M.Z. would like also gratefully acknowledge an Early Stage
Researcher position supported by the EU-RTN Programme, Contract No. MRTN–CT-2006-035482, “Flavianet.”
Appendix A $O\!\left(p^{2}\right)$ amplitudes and the symmetry properties of amplitudes of the physical mesons
From the isospin Ward identities it follows that all the physical amplitudes could be expressed in terms of the 7 independent amplitudes. In our convention:
1.
$\eta\eta\rightarrow\eta\eta$
$$A(\eta\eta\rightarrow\eta\eta)=A_{\eta\eta}(s,t;u).$$
(37)
2.
$\pi\eta\rightarrow\pi\eta$
$$\displaystyle A(\pi^{\pm}\eta\rightarrow\pi^{\pm}\eta)$$
$$\displaystyle=A_{\pi\eta}(s,t;u),$$
(38)
$$\displaystyle A(\pi^{0}\eta\rightarrow\pi^{0}\eta)$$
$$\displaystyle=A_{\pi\eta}(s,t;u).$$
(39)
3.
$\pi\pi\rightarrow\pi\pi$
$$\displaystyle A(\pi^{+}\pi^{-}\rightarrow\pi^{0}\pi^{0})$$
$$\displaystyle=-A_{\pi\pi}(s,t;u),$$
(40)
$$\displaystyle A(\pi^{+}\pi^{-}\rightarrow\pi^{+}\pi^{-})$$
$$\displaystyle=A_{\pi\pi}(s,t;u)+A_{\pi\pi}(t,s;u),$$
(41)
$$\displaystyle A(\pi^{0}\pi^{0}\rightarrow\pi^{0}\pi^{0})$$
$$\displaystyle=A_{\pi\pi}(s,t;u)+A_{\pi\pi}(t,s;u)+A_{\pi\pi}(u,t;s).$$
(42)
4.
$KK\rightarrow\eta\eta$
$$A({\overline{K^{0}}}K^{0}\rightarrow\eta\eta)=-A(K^{-}K^{+}\rightarrow\eta\eta%
)=A_{\eta K}(s,t;u).$$
(43)
5.
$KK\rightarrow\pi\eta$
$$\displaystyle A(K^{-}K^{+}\rightarrow\pi^{0}\eta)$$
$$\displaystyle=-A_{\eta\pi K}(s,t;u),$$
(44)
$$\displaystyle A({\overline{K^{0}}}K^{0}\rightarrow\pi^{0}\eta)$$
$$\displaystyle=-A_{\eta\pi K}(s,t;u),$$
(45)
$$\displaystyle A(K^{-}K^{0}\rightarrow\pi^{-}\eta)$$
$$\displaystyle=-\sqrt{2}A_{\eta\pi K}(s,t;u),$$
(46)
$$\displaystyle A({\overline{K^{0}}}K^{+}\rightarrow\pi^{+}\eta)$$
$$\displaystyle=-\sqrt{2}A_{\eta\pi K}(s,t;u).$$
(47)
6.
$\pi\pi\rightarrow KK$
$$\displaystyle A(\pi^{0}\pi^{0}\rightarrow{\overline{K^{0}}}K^{0})$$
$$\displaystyle=-A(\pi^{0}\pi^{0}\rightarrow K^{-}K^{+})=A_{\pi K}(s,t;u),$$
(48)
$$\displaystyle A(\pi^{-}\pi^{0}\rightarrow K^{-}K^{0})$$
$$\displaystyle=\sqrt{2}B_{\pi K}(s,t;u),$$
(49)
$$\displaystyle A(\pi^{+}\pi^{0}\rightarrow{\overline{K^{0}}}K^{+})$$
$$\displaystyle=-\sqrt{2}B_{\pi K}(s,t;u),$$
(50)
$$\displaystyle A(\pi^{-}\pi^{+}\rightarrow K^{-}K^{+})$$
$$\displaystyle=A_{\pi K}(s,t;u)+B_{\pi K}(s,t;u),$$
(51)
$$\displaystyle A(\pi^{-}\pi^{+}\rightarrow{\overline{K^{0}}}K^{0})$$
$$\displaystyle=-A_{\pi K}(s,t;u)+B_{\pi K}(s,t;u).$$
(52)
For these processes there are two important amplitudes – the symmetric one $A_{\pi K}(s,t,u)$ and the antisymmetric one $B_{\pi K}(s,t,u)$. In fact, we can consider the only independent amplitude as (51) and then extract these amplitudes as the symmetric and the antisymmetric part of it in respect to the exchange of $t$ and $u$.
7.
$KK\rightarrow KK$
$$\displaystyle A(K^{-}K^{+}\rightarrow{\overline{K^{0}}}K^{0})$$
$$\displaystyle=-A_{KK}(s,t;u),$$
(53)
$$\displaystyle A(K^{-}K^{+}\rightarrow K^{-}K^{+})$$
$$\displaystyle=A({\overline{K^{0}}}K^{0}\rightarrow{\overline{K^{0}}}K^{0})=A_{%
KK}(s,t;u)+A_{KK}(t,s;u).$$
(54)
The most general form of the $O\!\left(p^{2}\right)$ ($O\!\left(p^{3}\right)$) amplitudes
The $O\!\left(p^{2}\right)$ amplitudes could be constructed as the most general
invariant amplitudes satisfying the crossing, Bose and isospin symmetries and using the
fact that they should be polynomials of the first order in the
Mandelstam variables obeying (5)191919In fact, it is the form of the $O\!\left(p^{3}\right)$
amplitudes, because this should hold also for them.. This form is
independent of the power counting used:
$$A_{\eta\eta}=-\frac{M_{\pi}^{2}}{3F_{\pi}^{2}}\alpha_{\eta\eta}\left(1-\frac{4%
M_{\eta}^{2}}{M_{\pi}^{2}}\right),$$
(55)
$$A_{\pi\eta}=\frac{1}{3F_{\pi}^{2}}\left(\beta_{\pi\eta}(3t-2M_{\eta}^{2}-2M_{%
\pi}^{2})+\alpha_{\pi\eta}M_{\pi}^{2}\right),$$
(56)
$$A_{\pi\pi}=\frac{1}{3F_{\pi}^{2}}\left(\beta_{\pi\pi}\left(3s-4M_{\pi}^{2}%
\right)+\alpha_{\pi\pi}M_{\pi}^{2}\right),$$
(57)
$$A_{\eta K}=\frac{1}{4F_{\pi}^{2}}\left[\beta_{\eta K}(3s-2M_{K}^{2}-2M_{\eta}^%
{2})+\alpha_{\eta K}\left(2M_{\eta}^{2}-\frac{2}{3}M_{K}^{2}\right)\right],$$
(58)
$$A_{\eta\pi K}=\frac{1}{4\sqrt{3}F_{\pi}^{2}}\left[\beta_{\eta\pi K}\left(3s-2M%
_{K}^{2}-M_{\pi}^{2}-M_{\eta}^{2}\right)-\left(2M_{K}^{2}-M_{\pi}^{2}-M_{\eta}%
^{2}+\alpha_{\eta\pi K}M_{\pi}^{2}\right)\right],$$
(59)
$$A_{\pi K}=\frac{1}{12F_{\pi}^{2}}\left[\beta_{\pi K}\left(3s-2M_{K}^{2}-2M_{%
\pi}^{2}\right)+2(M_{K}-M_{\pi})^{2}+4\alpha_{\pi K}M_{\pi}M_{K}\right],$$
(60)
$$B_{\pi K}=\frac{1}{4F_{\pi}^{2}}\gamma_{\pi K}(t-u),$$
(61)
$$A_{KK}=\frac{1}{6F_{\pi}^{2}}\left(\beta_{KK}\left(4M_{K}^{2}-3u\right)+3%
\gamma_{KK}(s-t)+2\alpha_{KK}M_{K}^{2}\right).$$
(62)
After a deeper analysis one can show that in $O\!\left(p^{2}\right)$ order the amplitude
$A_{\pi\eta}$ is $t$-independent, so the constant $\beta_{\pi\eta}$
is of the $O\!\left(p^{3}\right)$ order and similarly for $\gamma_{KK}$, which is
also equal to zero in the $O\!\left(p^{2}\right)$ case in an arbitrary power counting.
So, generally
$$\beta_{\pi\eta}\sim O\!\left(p^{1}\right)\qquad\text{and}\qquad\gamma_{KK}\sim
O%
\!\left(p^{1}\right)$$
(63)
The particular power countings differ in the particular values of
the (13+2) constants in those relations. For example, the standard
power counting gives (to $O\!\left(p^{3}\right)$):
$$\displaystyle\alpha_{\eta\eta}^{st}=\alpha_{\pi\eta}^{st}=\alpha_{\pi\pi}^{st}%
=\alpha_{\eta K}^{st}=\alpha_{\pi K}^{st}=\alpha_{KK}^{st}=1,$$
(64)
$$\displaystyle\beta_{\pi\pi}^{st}=\beta_{\eta K}^{st}=\beta_{\eta\pi K}^{st}=%
\beta_{\pi K}^{st}=\gamma_{\pi K}^{st}=\beta_{KK}^{st}=1,$$
(65)
$$\displaystyle\alpha_{\eta\pi K}^{st}=0,$$
(66)
$$\displaystyle\beta_{\pi\eta}^{st}=\gamma_{KK}^{st}=0.$$
(67)
Appendix B Results
In the results we have denoted the once subtracted integral
$$\overline{J}_{PQ}(s)=\frac{s}{16\pi^{2}}\int_{\Sigma}^{\infty}\frac{dx}{x}%
\frac{1}{x-s}\frac{\lambda^{1/2}_{PQ}(x)}{x}$$
(68)
in the S channel (analogically for the other crossed channels) and similarly the twice subtracted integral
$$\overline{\overline{J}}_{PQ}(s)=\frac{s^{2}}{16\pi^{2}}\int_{\Sigma}^{\infty}%
\frac{dx}{x^{2}}\frac{1}{x-s}\frac{\lambda^{1/2}_{PQ}(x)}{x}.$$
(69)
The amplitudes using the first iteration of the reconstruction
procedure read:
$\bullet\quad\eta\eta\rightarrow\eta\eta$
$$\begin{split}\displaystyle A_{\eta\eta}&\displaystyle=\frac{1}{3F_{\pi}^{2}}%
\alpha_{\eta\eta}\left(4M_{\eta}^{2}-M_{\pi}^{2}\right)+\frac{1}{3F_{\pi}^{4}}%
\delta_{\eta\eta}(s^{2}+t^{2}+u^{2})\\
&\displaystyle\quad+\left[\frac{1}{6F_{\pi}^{4}}\left(Z_{\eta\eta}^{\eta\eta}%
\overline{J}_{\eta\eta}(s)+Z^{\pi\pi}_{\eta\eta}\overline{J}_{\pi\pi}(s)+Z^{KK%
}_{\eta\eta}(s)\overline{J}_{KK}(s)\right)\right]+[s\leftrightarrow t]+[s%
\leftrightarrow u]+O\!\left(p^{5}\right).\end{split}$$
(70)
$$\displaystyle Z_{\eta\eta}^{\eta\eta}$$
$$\displaystyle=\frac{1}{3}\alpha_{\eta\eta}^{2}\left(M_{\pi}^{2}-4M_{\eta}^{2}%
\right)^{2},$$
(71)
$$\displaystyle Z^{\pi\pi}_{\eta\eta}$$
$$\displaystyle=\alpha_{\pi\eta}^{2}M_{\pi}^{4},$$
(72)
$$\displaystyle Z^{KK}_{\eta\eta}(s)$$
$$\displaystyle=\frac{3}{4}\left(3\beta_{\eta K}s-2\left(\beta_{\eta K}+\frac{1}%
{3}\alpha_{\eta K}\right)M_{K}^{2}+2\left(\alpha_{\eta K}-\beta_{\eta K}\right%
)M_{\eta}^{2}\right)^{2}.$$
(73)
$\bullet\quad\pi^{0}\eta\rightarrow\pi^{0}\eta$
$$\begin{split}\displaystyle A_{\pi\eta}&\displaystyle=\frac{1}{3F_{\pi}^{2}}%
\Big{(}\beta_{\pi\eta}(3t-2M_{\eta}^{2}-2M_{\pi}^{2})+M_{\pi}^{2}\alpha_{\pi%
\eta}\Big{)}+\frac{1}{3F_{\pi}^{4}}\left(\delta_{\pi\eta}(s^{2}+u^{2})+%
\varepsilon_{\pi\eta}t^{2}\right)\\
&\displaystyle\quad+\frac{1}{72F_{\pi}^{4}}\Big{(}Z^{KK}_{\pi\eta}(t)\overline%
{J}_{KK}(t)+Z^{\eta\eta}_{\pi\eta}\overline{J}_{\eta\eta}(t)+Z^{\pi\pi}_{\pi%
\eta}(t)\overline{J}_{\pi\pi}(t)\Big{)}\\
&\displaystyle\quad+\left[\frac{1}{9F_{\pi}^{4}}\left(Y_{\pi\eta}^{\pi\eta}%
\overline{J}_{\pi\eta}(s)+Y^{KK}_{\pi\eta}(s)\overline{J}_{KK}(s)\right)\right%
]+[s\leftrightarrow u]+O\!\left(p^{5}\right).\end{split}$$
(74)
$$\displaystyle\begin{split}\displaystyle Z^{KK}_{\pi\eta}(t)&\displaystyle=%
\left(9\beta_{\eta K}t-2(3\beta_{\eta K}+\alpha_{\eta K})M_{K}^{2}+6(\alpha_{%
\eta K}-\beta_{\eta K})M_{\eta}^{2}\right)\\
&\displaystyle\qquad\times\left(3\beta_{\pi K}t+2(1-\beta_{\pi K})(M_{K}^{2}+M%
_{\pi}^{2})+4(\alpha_{\pi K}-1)M_{\pi}M_{K}\right),\end{split}$$
(75)
$$\displaystyle Z^{\eta\eta}_{\pi\eta}$$
$$\displaystyle=-4\alpha_{\pi\eta}\alpha_{\eta\eta}M_{\pi}^{2}\left(M_{\pi}^{2}-%
4M_{\eta}^{2}\right),$$
(76)
$$\displaystyle Z^{\pi\pi}_{\pi\eta}(t)$$
$$\displaystyle=4\alpha_{\pi\eta}M_{\pi}^{2}\left(6\beta_{\pi\pi}t+(5\alpha_{\pi%
\pi}-8\beta_{\pi\pi})M_{\pi}^{2}\right),$$
(77)
$$\displaystyle Y_{\pi\eta}^{\pi\eta}$$
$$\displaystyle=\alpha_{\pi\eta}^{2}M_{\pi}^{4},$$
(78)
$$\displaystyle Y^{KK}_{\pi\eta}(s)$$
$$\displaystyle=\frac{3}{8}\left(3\beta_{\eta\pi K}s-2(1+\beta_{\eta\pi K})M_{K}%
^{2}+(1-\alpha_{\eta\pi K}-\beta_{\eta\pi K})M_{\pi}^{2}+(1-\beta_{\eta\pi K})%
M_{\eta}^{2}\right)^{2}.$$
(79)
$\bullet\quad\pi^{+}\pi^{-}\rightarrow\pi^{0}\pi^{0}$
$$\begin{split}\displaystyle A_{\pi\pi}&\displaystyle=\frac{1}{3F_{\pi}^{2}}%
\left(\beta_{\pi\pi}\left(3s-4M_{\pi}^{2}\right)+\alpha_{\pi\pi}M_{\pi}^{2}%
\right)+\frac{1}{F_{\pi}^{4}}\left(\delta_{\pi\pi}s^{2}+\varepsilon_{\pi\pi}(t%
^{2}+u^{2})\right)\\
&\displaystyle\quad+\frac{1}{72F_{\pi}^{4}}\left(Z_{\pi\pi}^{\pi\pi}(s)%
\overline{J}_{\pi\pi}(s)+Z^{\eta\eta}_{\pi\pi}\overline{J}_{\eta\eta}(s)+Z^{KK%
}_{\pi\pi}(s)\overline{J}_{KK}(s)\right)\\
&\displaystyle\quad+\left[\frac{1}{72F_{\pi}^{4}}\left(Y^{KK}_{\pi\pi}(t,u)%
\overline{J}_{KK}(t)+Y_{\pi\pi}^{\pi\pi}(t,u)\overline{J}_{\pi\pi}(t)\right)%
\right]+[t\leftrightarrow u]+O\!\left(p^{5}\right).\end{split}$$
(80)
$$\displaystyle Z_{\pi\pi}^{\pi\pi}(s)$$
$$\displaystyle=4\left(3\beta_{\pi\pi}s+(7\alpha_{\pi\pi}-4\beta_{\pi\pi})M_{\pi%
}^{2}\right)\left(3\beta_{\pi\pi}s+(\alpha_{\pi\pi}-4\beta_{\pi\pi})M_{\pi}^{2%
}\right),$$
(81)
$$\displaystyle Z^{\eta\eta}_{\pi\pi}$$
$$\displaystyle=4\alpha_{\pi\eta}^{2}M_{\pi}^{4},$$
(82)
$$\displaystyle Z^{KK}_{\pi\pi}(s)$$
$$\displaystyle=\left(3\beta_{\pi K}s+2(1-\beta_{\pi K})(M_{K}^{2}+M_{\pi}^{2})+%
4(\alpha_{\pi K}-1)M_{\pi}M_{K}\right)^{2},$$
(83)
$$\displaystyle Y^{KK}_{\pi\pi}(t,u)$$
$$\displaystyle=3\gamma_{\pi K}^{2}(t-4M_{K}^{2})(4M_{\pi}^{2}-t-2u),$$
(84)
$$\displaystyle Y_{\pi\pi}^{\pi\pi}(t,u)$$
$$\displaystyle=4\left(3\beta_{\pi\pi}^{2}t(t-u)+6\beta_{\pi\pi}(2\beta_{\pi\pi}%
u-\alpha_{\pi\pi}t)M_{\pi}^{2}+2(\alpha_{\pi\pi}^{2}+4\beta_{\pi\pi}(\alpha_{%
\pi\pi}-2\beta_{\pi\pi}))M_{\pi}^{4}\right).$$
(85)
$\bullet\quad{\overline{K^{0}}}K^{0}\rightarrow\eta\eta$
$$\begin{split}\displaystyle A_{\eta K}&\displaystyle=\frac{1}{4F_{\pi}^{2}}%
\left(\beta_{\eta K}(3s-2M_{K}^{2}-2M_{\eta}^{2})+\alpha_{\eta K}\left(2M_{%
\eta}^{2}-\frac{2}{3}M_{K}^{2}\right)\right)+\frac{1}{4F_{\pi}^{4}}\left(%
\delta_{\eta K}s^{2}+\varepsilon_{\eta K}(t^{2}+u^{2})\right)\\
&\displaystyle\quad+\frac{1}{24F_{\pi}^{4}}\left(Z_{\eta K}^{\eta\eta}(s)%
\overline{J}_{\eta\eta}(s)+Z_{\eta K}^{\pi\pi}(s)\overline{J}_{\pi\pi}(s)+Z_{%
\eta K}^{KK}(s)\overline{J}_{KK}(s)\right)\\
&\displaystyle\quad+\left[\frac{1}{32F_{\pi}^{4}}\left(Y_{\eta K}^{\eta K}(t,u%
)\overline{J}_{\eta K}(t)+Y_{\eta K}^{K\pi}(t,u)\overline{J}_{K\pi}(t)\right)%
\right.\\
&\displaystyle\qquad\quad-\frac{3}{32F_{\pi}^{4}}\frac{1}{t}\left(X_{\eta K}^{%
\eta K}(u)\overline{J}_{\eta K}(t)+X_{\eta K}^{K\pi}(u)\overline{J}_{K\pi}(t)%
\right)\\
&\displaystyle\quad\qquad\qquad+\left.\frac{3}{16F_{\pi}^{4}}\frac{1}{t^{2}}%
\Delta_{\scriptscriptstyle K\eta}^{2}\left(W_{\eta K}^{\eta K}\overline{%
\overline{J}}_{\eta K}(t)+W_{\eta K}^{K\pi}\overline{\overline{J}}_{K\pi}(t)%
\right)\right]+\left[t\leftrightarrow u\right]+O\!\left(p^{5}\right).\end{split}$$
(86)
$$\displaystyle W_{\eta K}^{\eta K}$$
$$\displaystyle=\Delta_{\scriptscriptstyle K\eta}^{2}\beta^{2}_{\eta K},$$
(87)
$$\displaystyle W_{\eta K}^{K\pi}$$
$$\displaystyle=\Delta_{\scriptscriptstyle K\pi}^{2}\beta^{2}_{\eta\pi K},$$
(88)
$$\displaystyle X_{\eta K}^{\eta K}(u)$$
$$\displaystyle=\left(\beta_{\eta K}u+2\left(\beta_{\eta K}-\frac{2}{3}\alpha_{%
\eta K}\right)M_{K}^{2}+2(\beta_{\eta K}+2\alpha_{\eta K})M_{\eta}^{2}\right)%
\beta_{\eta K}\Delta_{\scriptscriptstyle K\eta}^{2},$$
(89)
$$\displaystyle\begin{split}\displaystyle X_{\eta K}^{K\pi}(u)&\displaystyle=%
\beta_{\eta\pi K}\Big{(}\beta_{\eta\pi K}\left(M_{K}^{2}-M_{\pi}^{2}\right)^{2%
}u+2\left(\beta_{\eta\pi K}-2\right)M_{K}^{6}-2\left(2\beta_{\eta\pi K}-3%
\right)M_{K}^{4}M_{\eta}^{2}\\
&\displaystyle\qquad+2\left(3+\beta_{\eta\pi K}-\alpha_{\eta\pi K}\right)M_{K}%
^{4}M_{\pi}^{2}-2\left(\beta_{\eta\pi K}+1-\alpha_{\eta\pi K}\right)M_{\pi}^{4%
}M_{K}^{2}-2M_{\eta}^{4}M_{K}^{2}\\
&\displaystyle\qquad\quad+2\left(\alpha_{\eta\pi K}-4\right)M_{K}^{2}M_{\eta}^%
{2}M_{\pi}^{2}-2\left(\alpha_{\eta\pi K}-1\right)M_{\pi}^{4}M_{\eta}^{2}+2(%
\beta_{\eta\pi K}+1)M_{\eta}^{4}M_{\pi}^{2}\Big{)},\end{split}$$
(90)
$$\displaystyle Z_{\eta K}^{\eta\eta}(s)$$
$$\displaystyle=\left(3\beta_{\eta K}s-2(\beta_{\eta K}+\frac{1}{3}\alpha_{\eta K%
})M_{K}^{2}-2(\beta_{\eta K}-\alpha_{\eta K})M_{\eta}^{2}\right)\alpha_{\eta%
\eta}\left(4M_{\eta}^{2}-M_{\pi}^{2}\right),$$
(91)
$$\displaystyle Z_{\eta K}^{\pi\pi}(s)$$
$$\displaystyle=\alpha_{\pi\eta}M_{\pi}^{2}\left(3\beta_{\pi K}s+2(1-\beta_{\pi K%
})(M_{K}^{2}+M_{\pi}^{2})+4(\alpha_{\pi K}-1)M_{\pi}M_{K}\right),$$
(92)
$$\displaystyle\begin{split}\displaystyle Z_{\eta K}^{KK}(s)&\displaystyle=3%
\left(3\beta_{\eta K}s-2\left(\beta_{\eta K}+\frac{1}{3}\alpha_{\eta K}\right)%
M_{K}^{2}-2(\beta_{\eta K}-\alpha_{\eta K})M_{\eta}^{2}\right)\\
&\displaystyle\qquad\times\left(\frac{3}{2}\beta_{KK}s+2(\alpha_{KK}-\beta_{KK%
})M_{K}^{2}\right),\end{split}$$
(93)
$$\displaystyle\begin{split}\displaystyle Y_{\eta K}^{\eta K}(t,u)&\displaystyle%
=\frac{1}{9}\Big{(}27\beta_{\eta K}^{2}t(t-u)-6\left(8\alpha_{\eta K}^{2}-8%
\alpha_{\eta K}\beta_{\eta K}+39\beta_{\eta K}^{2}\right)M_{K}^{2}M_{\eta}^{2}%
\\
&\displaystyle\qquad+9\left(8\alpha_{\eta K}^{2}+8\alpha_{\eta K}\beta_{\eta K%
}+5\beta_{\eta K}^{2}\right)M_{\eta}^{4}+\left(8\alpha_{\eta K}^{2}-24\alpha_{%
\eta K}\beta_{\eta K}+45\beta_{\eta K}^{2}\right)M_{K}^{4}\\
&\displaystyle\quad\qquad+36\alpha_{\eta K}\beta_{\eta K}\left(M_{K}^{2}-3M_{%
\eta}^{2}\right)t+54\beta_{\eta K}^{2}\left(M_{K}^{2}+M_{\eta}^{2}\right)u\Big%
{)},\end{split}$$
(94)
$$\displaystyle\begin{split}\displaystyle Y_{\eta K}^{K\pi}(t,u)&\displaystyle=3%
\beta_{\eta\pi K}^{2}t(t-u)+\left(5\beta_{\eta\pi K}^{2}-8\beta_{\eta\pi K}+8%
\right)M_{K}^{4}+2\left(\beta_{\eta\pi K}^{2}+\beta_{\eta\pi K}+1\right)M_{%
\eta}^{4}\\
&\displaystyle\quad-8\left(2\beta_{\eta\pi K}^{2}+1\right)M_{K}^{2}M_{\eta}^{2%
}+\left(2\beta_{\eta\pi K}\left(1-\alpha_{\eta\pi K}\right)+2\left(1-\alpha_{%
\eta\pi K}\right)^{2}-\beta_{\eta\pi K}^{2}\right)M_{\pi}^{4}\\
&\displaystyle\quad+2\left(2\left(\beta_{\eta\pi K}^{2}+\beta_{\eta\pi K}+1-%
\alpha_{\eta\pi K}\right)-\alpha_{\eta\pi K}\beta_{\eta\pi K}\right)M_{\eta}^{%
2}M_{\pi}^{2}\\
&\displaystyle\quad-2\left(5\beta_{\eta\pi K}^{2}+2\alpha_{\eta\pi K}\beta_{%
\eta\pi K}+4\left(1-\alpha_{\eta\pi K}\right)\right)M_{K}^{2}M_{\pi}^{2}\\
&\displaystyle\quad-6\beta_{\eta\pi K}\left(M_{\eta}^{2}-2M_{K}^{2}-\left(%
\alpha_{\eta\pi K}-1\right)M_{\pi}^{2}\right)t+6\beta_{\eta\pi K}^{2}\left(M_{%
K}^{2}+M_{\pi}^{2}\right)u.\end{split}$$
(95)
$\bullet\quad{\overline{K^{0}}}K^{0}\rightarrow\pi^{0}\eta$
$$\begin{split}\displaystyle A_{\eta\pi K}&\displaystyle=\frac{1}{4\sqrt{3}F_{%
\pi}^{2}}V_{\eta\pi K}(s)+\frac{1}{4\sqrt{3}F_{\pi}^{4}}\left(\delta_{\eta\pi K%
}s^{2}+\varepsilon_{\eta\pi K}(t^{2}+u^{2})\right)\\
&\displaystyle\quad+\frac{1}{24\sqrt{3}F_{\pi}^{4}}\left(Z_{\eta\pi K}^{\eta%
\pi}(s)\overline{J}_{\eta\pi}(s)+Z_{\eta\pi K}^{KK}\overline{J}_{KK}(s)\right)%
V_{\eta\pi K}(s)\\
&\displaystyle\quad+\Big{[}-\frac{1}{96\sqrt{3}F_{\pi}^{4}}\left(Y_{\eta\pi K}%
^{\eta K}(t,u)\overline{J}_{\eta K}(t)+Y_{\eta\pi K}^{K\pi}(t,u)\overline{J}_{%
K\pi}(t)\right)\\
&\displaystyle\quad\qquad-\frac{1}{32\sqrt{3}F_{\pi}^{4}}\frac{1}{t}\left(X_{%
\eta\pi K}^{\eta K}(u)\overline{J}_{\eta K}(t)+X_{\eta\pi K}^{K\pi}(u)%
\overline{J}_{K\pi}(t)\right)\\
&\displaystyle\quad\qquad\qquad+\frac{1}{16\sqrt{3}F_{\pi}^{4}}\frac{1}{t^{2}}%
U_{\eta\pi K}\left.\left(W_{\eta\pi K}^{\eta K}\overline{\overline{J}}_{\eta K%
}(t)+W_{\eta\pi K}^{K\pi}\overline{\overline{J}}_{K\pi}(t)\right)\right]+[t%
\leftrightarrow u]+O\!\left(p^{5}\right).\end{split}$$
(96)
$$\displaystyle U_{\eta\pi K}$$
$$\displaystyle=\beta_{\eta\pi K}\Delta_{\scriptscriptstyle K\eta}\Delta_{%
\scriptscriptstyle K\pi},$$
(97)
$$\displaystyle V_{\eta\pi K}(s)$$
$$\displaystyle=\beta_{\eta\pi K}\left(3s-2M_{K}^{2}-M_{\pi}^{2}-M_{\eta}^{2}%
\right)-\left(2M_{K}^{2}-M_{\pi}^{2}-M_{\eta}^{2}+\alpha_{\eta\pi K}M_{\pi}^{2%
}\right),$$
(98)
$$\displaystyle Z_{\eta\pi K}^{\eta\pi}$$
$$\displaystyle=2\alpha_{\pi\eta}M_{\pi}^{2},$$
(99)
$$\displaystyle Z_{\eta\pi K}^{KK}(s)$$
$$\displaystyle=\frac{3}{2}\beta_{KK}s+2(\alpha_{KK}-\beta_{KK})M_{K}^{2},$$
(100)
$$\displaystyle\begin{split}\displaystyle Y_{\eta\pi K}^{\eta K}(t,u)&%
\displaystyle=9\beta_{\eta\pi K}\beta_{\eta K}(t(u-t)-2(M_{\eta}^{2}+M_{K}^{2}%
)u)-3(\beta_{\eta\pi K}+2)(2\alpha_{\eta K}+\beta_{\eta K})M_{\eta}^{4}\\
&\displaystyle\quad+\left(3\beta_{\eta K}(4-5\beta_{\eta\pi K})+4\alpha_{\eta K%
}(\beta_{\eta\pi K}-2)\right)M_{K}^{4}\\
&\displaystyle\quad+2(\alpha_{\eta K}(2-2\alpha_{\eta\pi K}+\beta_{\eta\pi K})%
+3\beta_{\eta K}(\alpha_{\eta\pi K}-1+4\beta_{\eta\pi K}))M_{K}^{2}M_{\pi}^{2}%
\\
&\displaystyle\quad+(\alpha_{\eta K}(28-10\beta_{\eta\pi K})+6\beta_{\eta K}(9%
\beta_{\eta\pi K}+1))M_{K}^{2}M_{\eta}^{2}\\
&\displaystyle\quad+6((\alpha_{\eta\pi K}-1)(2\alpha_{\eta K}+\beta_{\eta K})-%
\beta_{\eta\pi K}(\alpha_{\eta K}+2\beta_{\eta K}))M_{\eta}^{2}M_{\pi}^{2}\\
&\displaystyle\quad-3t\big{(}2(3\beta_{\eta K}+\alpha_{\eta K}\beta_{\eta\pi K%
})M_{K}^{2}-3(\beta_{\eta K}+2\alpha_{\eta K}\beta_{\eta\pi K})M_{\eta}^{2}+3%
\beta_{\eta K}(\alpha_{\eta\pi K}-1)M_{\pi}^{2}\big{)},\end{split}$$
(101)
$$\displaystyle\begin{split}\displaystyle Y_{\eta\pi K}^{K\pi}(t,u)&%
\displaystyle=3\beta_{\eta\pi K}t\left((10\gamma_{\pi K}-\beta_{\pi K})t+(%
\beta_{\pi K}+2\gamma_{\pi K})u\right)+3(\beta_{\pi K}-2\gamma_{\pi K}(3+2%
\beta_{\eta\pi K}))tM_{\eta}^{2}\\
&\displaystyle\quad-6\beta_{\eta\pi K}(\beta_{\pi K}+2\gamma_{\pi K})(M_{K}^{2%
}+M_{\pi}^{2})u-6(2\gamma_{\pi K}(4\beta_{\eta\pi K}-3)+\beta_{\pi K}-\beta_{%
\eta\pi K})tM_{K}^{2}\\
&\displaystyle\quad-3(2\beta_{\eta\pi K}(6\gamma_{\pi K}-1)+(1-\alpha_{\eta\pi
K%
})(6\gamma_{\pi K}-\beta_{\pi K}))tM_{\pi}^{2}\\
&\displaystyle\quad+(2+\beta_{\eta\pi K}-2\alpha_{\eta\pi K})(6\gamma_{\pi K}-%
\beta_{\pi K}-2)M_{\pi}^{4}\\
&\displaystyle\quad+\left(\beta_{\pi K}(4-5\beta_{\eta\pi K})+2\gamma_{\pi K}(%
21\beta_{\eta\pi K}-12)+8-4\beta_{\eta\pi K}\right)M_{K}^{4}\\
&\displaystyle\quad+2(\beta_{\eta\pi K}(4\beta_{\pi K}-6\gamma_{\pi K}-1)-%
\beta_{\pi K}+6\gamma_{\pi K}-2)M_{K}^{2}M_{\eta}^{2}\\
&\displaystyle\quad-2(\beta_{\pi K}(2\beta_{\eta\pi K}+1)-6\gamma_{\pi K}(3%
\beta_{\eta\pi K}+1)+2+\beta_{\eta\pi K})M_{\pi}^{2}M_{\eta}^{2}\\
&\displaystyle\quad+4(\alpha_{\pi K}-1)(V_{\eta\pi K}(t)+3(2M_{K}^{2}-M_{\pi}^%
{2}-M_{\eta}^{2}+\alpha_{\eta\pi K}M_{\pi}^{2}))M_{K}M_{\pi}\\
&\displaystyle\quad+2(3\beta_{\eta\pi K}(3\beta_{\pi K}+4\gamma_{\pi K}-1)+(%
\alpha_{\eta\pi K}+1)(\beta_{\pi K}-6\gamma_{\pi K}+2))M_{K}^{2}M_{\pi}^{2},%
\end{split}$$
(102)
$$\displaystyle\begin{split}\displaystyle X_{\eta\pi K}^{\eta K}(u)&%
\displaystyle=\Delta_{\scriptscriptstyle\eta K}\Big{(}3\beta_{\eta K}\beta_{%
\eta\pi K}(M_{\eta}^{2}-M_{K}^{2})u+3\beta_{\eta K}M_{\eta}^{4}+(2\alpha_{\eta
K%
}\beta_{\eta\pi K}+6\beta_{\eta K}\left(1-\beta_{\eta\pi K}\right))M_{K}^{4}\\
&\displaystyle\qquad\quad+3(\beta_{\eta K}(1-\alpha_{\eta\pi K})+2\beta_{\eta%
\pi K}(\alpha_{\eta K}+\beta_{\eta K}))M_{\eta}^{2}M_{\pi}^{2}\\
&\displaystyle\qquad\quad\quad+(3\beta_{\eta K}(\alpha_{\eta\pi K}-1)+2\beta_{%
\eta\pi K}(3\beta_{\eta K}-\alpha_{\eta K}))M_{K}^{2}M_{\pi}^{2}\\
&\displaystyle\qquad\quad\quad\quad-3(3\beta_{\eta K}+2\beta_{\eta\pi K}(\beta%
_{\eta K}+\alpha_{\eta K}))M_{K}^{2}M_{\eta}^{2}\Big{)},\end{split}$$
(103)
$$\displaystyle\begin{split}\displaystyle X_{\eta\pi K}^{K\pi}(u)&\displaystyle=%
\Delta_{\scriptscriptstyle\pi K}\Big{(}\beta_{\eta\pi K}(\beta_{\pi K}+2\gamma%
_{\pi K})\left(M_{\pi}^{2}-M_{K}^{2}\right)u+4\beta_{\eta\pi K}(1-\alpha_{\pi K%
})(M_{K}^{2}-M_{\eta}^{2})M_{K}M_{\pi}\\
&\displaystyle\qquad\quad+2\beta_{\eta\pi K}(2\gamma_{\pi K}-\beta_{\pi K}-1)(%
M_{K}^{4}-M_{K}^{2}M_{\eta}^{2}-M_{\pi}^{2}M_{\eta}^{2}+M_{K}^{2}M_{\pi}^{2})%
\\
&\displaystyle\qquad\quad\quad+(\beta_{\pi K}+2\gamma_{\pi K})(2M_{K}^{4}-M_{K%
}^{2}M_{\eta}^{2}+M_{\pi}^{2}M_{\eta}^{2}+(\alpha_{\eta\pi K}-3)M_{K}^{2}M_{%
\pi}^{2})\\
&\displaystyle\qquad\quad\quad\quad+(\beta_{\pi K}+2\gamma_{\pi K})(1-\alpha_{%
\eta\pi K})M_{\pi}^{4}\Big{)},\end{split}$$
(104)
$$\displaystyle W_{\eta\pi K}^{\eta K}$$
$$\displaystyle=3\Delta_{\scriptscriptstyle K\eta}^{2}\beta_{\eta K},$$
(105)
$$\displaystyle W_{\eta\pi K}^{K\pi}$$
$$\displaystyle=\Delta_{\scriptscriptstyle K\pi}^{2}(\beta_{\pi K}+2\gamma_{\pi K%
}).$$
(106)
$\bullet\quad\pi^{0}\pi^{0}\rightarrow{\overline{K^{0}}}K^{0}$
$$\begin{split}\displaystyle A_{\pi K}&\displaystyle=\frac{1}{12F_{\pi}^{2}}%
\left(\beta_{\pi K}\left(3s-2M_{K}^{2}-2M_{\pi}^{2}\right)+2(M_{K}-M_{\pi})^{2%
}+4\alpha_{\pi K}M_{\pi}M_{K}\right)\\
&\displaystyle\quad+\frac{1}{12F_{\pi}^{4}}\left(\delta_{\pi K}s^{2}+%
\varepsilon_{\pi K}(t^{2}+u^{2})\right)\\
&\displaystyle\quad+\frac{1}{72F_{\pi}^{4}}\Big{(}Z_{\pi K}^{\pi\pi}(s)%
\overline{J}_{\pi\pi}(s)+Z_{\pi K}^{\eta\eta}(s)\overline{J}_{\eta\eta}(s)+Z_{%
\pi K}^{KK}(s)\overline{J}_{KK}(s)\Big{)}\\
&\displaystyle\quad+\bigg{[}\frac{1}{96F_{\pi}^{4}}\left(Y_{\pi K}^{\eta K}(t,%
u)\overline{J}_{\eta K}(t)+Y_{\pi K}^{\pi K}(t,u)\overline{J}_{\pi K}(t)\right%
)\\
&\displaystyle\quad\qquad+\frac{1}{32F_{\pi}^{4}}\frac{1}{t}\left(X_{\pi K}^{%
\eta K}(u)\overline{J}_{\eta K}(t)+X_{\pi K}^{\pi K}(u)\overline{J}_{\pi K}(t)%
\right)\\
&\displaystyle\quad\qquad\qquad+\frac{1}{16F_{\pi}^{4}}\frac{1}{t^{2}}\Delta_{%
\scriptscriptstyle K\pi}^{2}\left(W_{\pi K}^{\eta K}\overline{\overline{J}}_{%
\eta K}(t)+W_{\pi K}^{\pi K}\overline{\overline{J}}_{\pi K}(t)\right)\bigg{]}+%
[t\leftrightarrow u]+O\!\left(p^{5}\right).\end{split}$$
(107)
$$\displaystyle\begin{split}\displaystyle Z_{\pi K}^{\pi\pi}(s)&\displaystyle=%
\left(3\beta_{\pi K}s+2(1-\beta_{\pi K})(M_{K}^{2}+M_{\pi}^{2})+4(\alpha_{\pi K%
}-1)M_{\pi}M_{K}\right)\\
&\displaystyle\qquad\times\left(6\beta_{\pi\pi}s+(5\alpha_{\pi\pi}-8\beta_{\pi%
\pi})M_{\pi}^{2}\right),\end{split}$$
(108)
$$\displaystyle Z_{\pi K}^{\eta\eta}(s)$$
$$\displaystyle=3\alpha_{\pi\eta}M_{\pi}^{2}\left(3\beta_{\eta K}s-2\left(\beta_%
{\eta K}+\frac{1}{3}\alpha_{\eta K}\right)M_{K}^{2}+2(\alpha_{\eta K}-\beta_{%
\eta K})M_{\eta}^{2}\right),$$
(109)
$$\displaystyle\begin{split}\displaystyle Z_{\pi K}^{KK}(s)&\displaystyle=3\left%
(3\beta_{\pi K}s+2(1-\beta_{\pi K})(M_{K}^{2}+M_{\pi}^{2})+4(\alpha_{\pi K}-1)%
M_{\pi}M_{K}\right)\\
&\displaystyle\qquad\times\left(\frac{3}{2}\beta_{KK}s+2(\alpha_{KK}-\beta_{KK%
})M_{K}^{2}\right),\end{split}$$
(110)
$$\displaystyle\begin{split}\displaystyle Y_{\pi K}^{\eta K}(t,u)&\displaystyle=%
3\beta_{\eta K}^{2}t(t-u)-6\beta_{\eta\pi K}\left(M_{\eta}^{2}-2M_{K}^{2}+(1-%
\alpha_{\eta\pi K})M_{\pi}^{2}\right)t+6\beta_{\eta\pi K}^{2}(M_{\eta}^{2}+M_{%
K}^{2})u\\
&\displaystyle\quad+\left(5\beta_{\eta\pi K}^{2}+8(1-\beta_{\eta\pi K})\right)%
M_{K}^{4}+\left(2+2\beta_{\eta\pi K}-\beta_{\eta\pi K}^{2}\right)M_{\eta}^{4}-%
2(5\beta_{\eta\pi K}^{2}+4)M_{K}^{2}M_{\eta}^{2}\\
&\displaystyle\quad+2\left(\alpha_{\eta\pi K}(\alpha_{\eta\pi K}-\beta_{\eta%
\pi K}-2)+\beta_{\eta\pi K}^{2}+\beta_{\eta\pi K}+1\right)M_{\pi}^{4}\\
&\displaystyle\quad-4(\beta_{\eta\pi K}(4\beta_{\eta\pi K}+\alpha_{\eta\pi K})%
+2(1-\alpha_{\eta\pi K}))M_{K}^{2}M_{\pi}^{2}\\
&\displaystyle\quad+2(2(\beta_{\eta\pi K}^{2}+\beta_{\eta\pi K}+1)-\alpha_{%
\eta\pi K}(\beta_{\eta\pi K}+2))M_{\pi}^{2}M_{\eta}^{2},\end{split}$$
(111)
$$\displaystyle\begin{split}\displaystyle Y_{\pi K}^{\pi K}(t,u)&\displaystyle=%
\frac{1}{3}\Big{(}3t\left(\left(\beta_{\pi K}^{2}+26\gamma_{\pi K}^{2}\right)t%
-\left(\beta_{\pi K}^{2}+2\gamma_{\pi K}^{2}\right)u\right)+6\left(\beta_{\pi K%
}^{2}+2\gamma_{\pi K}^{2}\right)\left(M_{K}^{2}+M_{\pi}^{2}\right)u\\
&\displaystyle\qquad-12\left(\left(8\gamma_{\pi K}^{2}+\beta_{\pi K}\right)%
\left(M_{K}^{2}+M_{\pi}^{2}\right)+2\beta_{\pi K}\left(\alpha_{\pi K}-1\right)%
M_{K}M_{\pi}\right)t\\
&\displaystyle\qquad\quad+2\left(16\alpha_{\pi K}\left(\alpha_{\pi K}-2\right)%
+\beta_{\pi K}\left(8-13\beta_{\pi K}\right)+6\left(4+13\gamma_{\pi K}^{2}%
\right)\right)M_{K}^{2}M_{\pi}^{2}\\
&\displaystyle\qquad\quad\quad+\left(5\beta_{\pi K}^{2}+8\left(\beta_{\pi K}+1%
\right)-30\gamma_{\pi K}^{2}\right)\left(M_{K}^{4}+M_{\pi}^{4}\right)\\
&\displaystyle\qquad\quad\quad\quad+16\left(\alpha_{\pi K}-1\right)\left(\beta%
_{\pi K}+2\right)M_{K}M_{\pi}\left(M_{K}^{2}+M_{\pi}^{2}\right)\Big{)},\end{split}$$
(112)
$$\displaystyle\begin{split}\displaystyle X_{\pi K}^{\eta K}(u)&\displaystyle=%
\beta_{\eta\pi K}\Big{(}-\beta_{\eta\pi K}\Delta_{\scriptscriptstyle K\eta}^{2%
}u+2\left(2-\beta_{\eta\pi K}\right)M_{K}^{6}+2M_{\eta}^{4}\left(\left(\beta_{%
\eta\pi K}+1\right)M_{K}^{2}-M_{\pi}^{2}\right)\\
&\displaystyle\qquad+2M_{K}^{2}M_{\pi}^{2}M_{\eta}^{2}\left(4-\alpha_{\eta\pi K%
}\right)+2M_{K}^{4}\left(\left(\alpha_{\eta\pi K}+2\beta_{\eta\pi K}-3\right)M%
_{\pi}^{2}-\left(\beta_{\eta\pi K}+3\right)M_{\eta}^{2}\right)\\
&\displaystyle\quad\qquad+2M_{\pi}^{4}\left(\left(\alpha_{\eta\pi K}-\beta_{%
\eta\pi K}-1\right)M_{\eta}^{2}+\left(1-\alpha_{\eta\pi K}\right)M_{K}^{2}%
\right)\Big{)},\end{split}$$
(113)
$$\displaystyle\begin{split}\displaystyle X_{\pi K}^{\pi K}(u)&\displaystyle=-%
\frac{1}{3}\Delta_{\scriptscriptstyle K\pi}^{2}\Big{(}\left(\beta_{\pi K}^{2}+%
2\gamma_{\pi K}^{2}\right)u+8\left(\alpha_{\pi K}-1\right)\beta_{\pi K}M_{K}M_%
{\pi}\\
&\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+2\left(\beta_{%
\pi K}^{2}+2\beta_{\pi K}-6\gamma_{\pi K}^{2}\right)\left(M_{K}^{2}+M_{\pi}^{2%
}\right)\Big{)},\end{split}$$
(114)
$$\displaystyle W_{\pi K}^{\eta K}$$
$$\displaystyle=\Delta_{\scriptscriptstyle K\eta}^{2}\beta^{2}_{\eta\pi K},$$
(115)
$$\displaystyle W_{\pi K}^{\pi K}$$
$$\displaystyle=\frac{1}{3}\Delta_{\scriptscriptstyle K\pi}^{2}\left(\beta^{2}_{%
\pi K}+2\gamma_{\pi K}^{2}\right).$$
(116)
$\bullet\quad\pi^{-}\pi^{0}\rightarrow K^{-}K^{0}$
$$\begin{split}\displaystyle B_{\pi K}&\displaystyle=\frac{1}{4F_{\pi}^{2}}%
\gamma_{\pi K}(t-u)+\frac{1}{4F_{\pi}^{4}}\varphi_{\pi K}s(t-u)\\
&\displaystyle\quad+\frac{1}{48F_{\pi}^{4}}\gamma_{\pi K}(t-u)\left(2\overline%
{J}_{\pi\pi}(s)\beta_{\pi\pi}(s-4M_{\pi}^{2})+\overline{J}_{KK}(s)\beta_{KK}(s%
-4M_{K}^{2})\right)\\
&\displaystyle\quad+\bigg{[}\frac{1}{96F_{\pi}^{4}}\left(Y_{\pi Kch}^{\eta K}(%
t,u)\overline{J}_{\eta K}(t)+Y_{\pi Kch}^{\pi K}(t,u)\overline{J}_{\pi K}(t)%
\right)\\
&\displaystyle\quad\qquad+\frac{1}{32F_{\pi}^{4}}\frac{1}{t}\left(X_{\pi Kch}^%
{\eta K}(u)\overline{J}_{\eta K}(t)+X_{\pi Kch}^{\pi K}(u)\overline{J}_{\pi K}%
(t)\right)\\
&\displaystyle\quad\qquad\qquad+\frac{1}{16F_{\pi}^{4}}\frac{1}{t^{2}}\Delta_{%
\scriptscriptstyle K\pi}^{2}\left(W_{\pi Kch}^{\eta K}\overline{\overline{J}}_%
{\eta K}(t)+W_{\pi Kch}^{\pi K}\overline{\overline{J}}_{\pi K}(t)\right)\bigg{%
]}-[t\leftrightarrow u]+O\!\left(p^{5}\right).\end{split}$$
(117)
$$\displaystyle Y_{\pi Kch}^{\eta K}(t,u)$$
$$\displaystyle=Y_{\pi K}^{\eta K}(t,u),$$
(118)
$$\displaystyle\begin{split}\displaystyle Y_{\pi Kch}^{\pi K}(t,u)&\displaystyle%
=-\gamma_{\pi K}\Big{(}t\left((10\beta_{\pi K}-13\gamma_{\pi K})t+(2\beta_{\pi
K%
}+\gamma_{\pi K})u\right)\\
&\displaystyle\qquad\quad-4((4(\beta_{\pi K}-\gamma_{\pi K})+3)(M_{K}^{2}+M_{%
\pi}^{2})+6(\alpha_{\pi K}-1)M_{K}M_{\pi})t\\
&\displaystyle\qquad\quad\quad-2(2\beta_{\pi K}+\gamma_{\pi K})(M_{K}^{2}+M_{%
\pi}^{2})u+(14\beta_{\pi K}+5\gamma_{\pi K}+8)(M_{K}^{4}+M_{\pi}^{4})\\
&\displaystyle\qquad\quad\quad\quad+2(2\beta_{\pi K}-13\gamma_{\pi K}+8)M_{K}^%
{2}M_{\pi}^{2}+16(\alpha_{\pi K}-1)M_{K}M_{\pi}(M_{K}^{2}+M_{\pi}^{2})\Big{)},%
\end{split}$$
(119)
$$\displaystyle X_{\pi Kch}^{\eta K}(u)$$
$$\displaystyle=X_{\pi K}^{\eta K}(u),$$
(120)
$$\displaystyle\begin{split}\displaystyle X_{\pi Kch}^{\pi K}(u)&\displaystyle=-%
\frac{1}{3}\gamma_{\pi K}(M_{K}^{2}-M_{\pi}^{2})^{2}\Big{(}\left(2\beta_{\pi K%
}+\gamma_{\pi K}\right)u+8\left(\alpha_{\pi K}-1\right)M_{K}M_{\pi}\\
&\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad+2\left(2-2\beta_{\pi K%
}-3\gamma_{\pi K}\right)\left(M_{K}^{2}+M_{\pi}^{2}\right)\Big{)},\end{split}$$
(121)
$$\displaystyle W_{\pi Kch}^{\eta K}$$
$$\displaystyle=W_{\pi K}^{\eta K},$$
(122)
$$\displaystyle W_{\pi Kch}^{\pi K}$$
$$\displaystyle=\frac{1}{3}\gamma_{\pi K}\Delta_{\scriptscriptstyle K\pi}^{2}%
\left(2\beta_{\pi K}+\gamma_{\pi K}\right).$$
(123)
$\bullet\quad K^{-}K^{+}\rightarrow{\overline{K^{0}}}K^{0}$
$$\begin{split}\displaystyle A_{KK}&\displaystyle=\frac{1}{6F_{\pi}^{2}}\left(%
\beta_{KK}\left(4M_{K}^{2}-3u\right)+3\gamma_{KK}(s-t)+2\alpha_{KK}M_{K}^{2}%
\right)+\frac{1}{6F_{\pi}^{4}}\left(\delta_{KK}s^{2}+\varepsilon_{KK}t^{2}+%
\varphi_{KK}st\right)\\
&\displaystyle\quad+\frac{1}{288F_{\pi}^{4}}\Big{(}Z_{KK}^{KK}(s,t)\overline{J%
}_{KK}(s)+Z^{\pi\eta}_{KK}(s)\overline{J}_{\pi\eta}(s)+Z^{\eta\eta}_{KK}(s)%
\overline{J}_{\eta\eta}(s)+Z_{KK}^{\pi\pi}(s,t)\overline{J}_{\pi\pi}(s)\Big{)}%
\\
&\displaystyle\quad+\frac{1}{72F_{\pi}^{4}}\left(Y_{KK}^{KK}(s,t)\overline{J}_%
{KK}(t)+Y_{KK}^{\pi\eta}(t)\overline{J}_{\pi\eta}(t)+Y_{KK}^{\pi\pi}(s,t)%
\overline{J}_{\pi\pi}(t)\right)\\
&\displaystyle\quad+\frac{1}{36F_{\pi}^{4}}\overline{J}_{KK}(u)\left(3\beta_{%
KK}u-2(\alpha_{KK}+2\beta_{KK})M_{K}^{2}\right)^{2}+O\!\left(p^{5}\right).\end%
{split}$$
(124)
$$\displaystyle Z_{KK}^{KK}(s,t)$$
$$\displaystyle=8\Big{(}\left(3\beta_{KK}s+4(\alpha_{KK}-\beta_{KK})M_{K}^{2}%
\right)^{2}+3\beta_{KK}^{2}(s-4M_{K}^{2})(s+2t-4M_{K}^{2})\Big{)}$$
(125)
$$\displaystyle Z^{\pi\eta}_{KK}(s)$$
$$\displaystyle=-6\Big{(}3\beta_{\eta\pi K}s-2(1+\beta_{\eta\pi K})M_{K}^{2}+(1-%
\alpha_{\eta\pi K}-\beta_{\eta\pi K})M_{\pi}^{2}+(1-\beta_{\eta\pi K})M_{\eta}%
^{2}\Big{)}^{2}$$
(126)
$$\displaystyle Z^{\eta\eta}_{KK}(s)$$
$$\displaystyle=\left(9\beta_{\eta K}s-2(3\beta_{\eta K}+\alpha_{\eta K})M_{K}^{%
2}+6(\alpha_{\eta K}-\beta_{\eta K})M_{\eta}^{2}\right)^{2}$$
(127)
$$\displaystyle\begin{split}\displaystyle Z_{KK}^{\pi\pi}(s,t)&\displaystyle=3%
\Big{(}s\left((9\beta_{\pi K}^{2}-2\gamma_{\pi K}^{2})s-4\gamma_{\pi K}^{2}t%
\right)+4(1-\beta_{\pi K})^{2}(M_{K}^{4}+M_{\pi}^{4})\\
&\displaystyle\qquad+4s\left(3\beta_{\pi K}(1-\beta_{\pi K})+2\gamma_{\pi K}^{%
2}\right)(M_{K}^{2}+M_{\pi}^{2})\\
&\displaystyle\qquad\quad+8(2\gamma_{\pi K}^{2}t+((1-\beta_{\pi K})^{2}+2(1-%
\alpha_{\pi K})^{2}-4\gamma_{\pi K}^{2})M_{K}^{2})M_{\pi}^{2}\\
&\displaystyle\qquad\quad\quad+8(\alpha_{\pi K}-1)(3\beta_{\pi K}s+2(1-\beta_{%
\pi K})(M_{K}^{2}+M_{\pi}^{2}))M_{\pi}M_{K}\Big{)}\end{split}$$
(128)
$$\displaystyle Y_{KK}^{KK}(s,t)$$
$$\displaystyle=\frac{1}{16}Z_{KK}^{KK}(t,s)$$
(129)
$$\displaystyle Y_{KK}^{\pi\eta}(t)$$
$$\displaystyle=3\Big{(}3\beta_{\eta\pi K}t-2(1+\beta_{\eta\pi K})M_{K}^{2}+(1-%
\alpha_{\eta\pi K}-\beta_{\eta\pi K})M_{\pi}^{2}+(1-\beta_{\eta\pi K})M_{\eta}%
^{2}\Big{)}^{2}$$
(130)
$$\displaystyle Y_{KK}^{\pi\pi}(s,t)$$
$$\displaystyle=3\gamma_{\pi K}^{2}(2s+t-4M_{K}^{2})(t-4M_{\pi}^{2})$$
(131)
Appendix C Standard chiral perturbation theory $O\!\left(p^{4}\right)$ values of the polynomial parameters
In this appendix we give the values of our polynomial parameters which reproduce the $O\!\left(p^{4}\right)$ results of the standard chiral perturbation theory. To that end we have used the computation of [10], which contains all the considered amplitudes and its advantage is also that their results are in the unitary form.
The form of our results is in terms of physical observables and thus has to be scale-independent and the same for all the possible regularization schemes. The only thing which can change by an eventual change of the scheme is the relation between the parameters of our parametrization and
the (renormalised) constants of the Lagrangian theory (LEC). By the change of the scale, the values of the LEC change, but their combinations giving the value of our parameters remain scale-independent - what can be another test of the results obtained from the Lagrangian theory. A further difference between (Lagrangian theory) results of different authors can rise from a different choice of the way they parameterize the $O\!\left(p^{2}\right)$ constants (bare masses and decay constants) using the physical parameters, e.g. in [10] they expand $F_{K}$ and $F_{\eta}$ decay constants in terms of $F_{\pi}$, $L_{4}^{r}$ and $L_{5}^{r}$.
In [10] they have used the Gell-Mann-Okubo relation (GMO) to get their results more simplified. In the standard chiral power counting the GMO formula has correction of the $O\!\left(p^{4}\right)$ order and thus we could also use it to simplify our results in some places, where it would give only correction of the $O\!\left(p^{6}\right)$ order at least. However, to let these relations be closely connected to the results [10], we have not done it and the only place where we refer to the GMO formula (and its $O\!\left(p^{4}\right)$ order correction) is in those places where we want to emphasize the validity of the $O\!\left(p^{2}\right)$ values of the parameters from Appendix A. Nevertheless, the use of
$$\Delta_{\mathrm{GMO}}(M_{\eta}^{2}-M_{\pi}^{2})=4M_{K}^{2}-M_{\pi}^{2}-3M_{%
\eta}^{2},$$
(132)
which is in the standard power counting of the $O\!\left(p^{4}\right)$ order, can also be understood just as a (more complicated) notation of the right-hand side of this definition.
Other objects appearing in the relations are the chiral logarithms, given in accordance with [10] by
$$\mu_{i}=\frac{M_{i}^{2}}{32\pi^{2}F_{\pi}^{2}}\log\frac{M_{i}^{2}}{\mu^{2}}%
\qquad\text{with }i=\pi,K,\eta.$$
(133)
(As has been already stated, its dependence on the scale $\mu$ is compensated on the right-hand side of the following relations by the scale-dependence of LEC $L_{i}^{r}$ as listed in [9].)
By the comparison of [10] with our results from the previous appendix we get the following relations:
$\bullet\quad\eta\eta$
$$\displaystyle\delta_{\eta\eta}$$
$$\displaystyle=12(2L_{1}^{r}+2L_{2}^{r}+L_{3})-\frac{27F_{\pi}^{2}\mu_{K}}{4M_{%
K}^{2}}-\frac{27}{128\pi^{2}}\,,$$
(134)
$$\displaystyle\begin{split}\displaystyle F_{\pi}^{2}(\alpha_{\eta\eta}-1)(M_{%
\pi}^{2}-4M_{\eta}^{2})&\displaystyle=-\frac{4}{3}F_{\pi}^{2}\Delta_{\mathrm{%
GMO}}(M_{\eta}^{2}-M_{\pi}^{2})+96M_{\eta}^{4}(L_{4}^{r}-3L_{6}^{r})\\
&\displaystyle+8L_{5}^{r}\left(3M_{\pi}^{4}-10M_{\eta}^{2}M_{\pi}^{2}+13M_{%
\eta}^{4}\right)\\
&\displaystyle-192L_{7}\left(M_{\pi}^{4}-3M_{\eta}^{2}M_{\pi}^{2}+2M_{\eta}^{4%
}\right)-48L_{8}^{r}\left(2M_{\pi}^{4}-6M_{\eta}^{2}M_{\pi}^{2}+7M_{\eta}^{4}%
\right)\\
&\displaystyle+\frac{\left(7M_{\pi}^{4}-24M_{\pi}^{2}M_{\eta}^{2}-54M_{\eta}^{%
4}\right)\mu_{K}F_{\pi}^{2}}{3M_{K}^{2}}+\frac{\left(48M_{\eta}^{2}-7M_{\pi}^{%
2}\right)\mu_{\pi}F_{\pi}^{2}}{3}\\
&\displaystyle+\frac{\left(M_{\pi}^{4}-8M_{\eta}^{2}M_{\pi}^{2}+24M_{\eta}^{4}%
\right)\mu_{\eta}F_{\pi}^{2}}{M_{\eta}^{2}}+\frac{80M_{K}^{4}-176M_{\eta}^{2}M%
_{K}^{2}+103M_{\eta}^{4}}{32\pi^{2}}\,.\end{split}$$
(135)
$\bullet\quad\pi\eta$
$$\displaystyle\varepsilon_{\pi\eta}$$
$$\displaystyle=24L_{1}^{r}+4L_{3}-\frac{9F_{\pi}^{2}\mu_{K}}{4M_{K}^{2}}-\frac{%
9}{128\pi^{2}}\,,$$
(136)
$$\displaystyle\delta_{\pi\eta}$$
$$\displaystyle=12L_{2}^{r}+4L_{3}-\frac{9F_{\pi}^{2}\mu_{K}}{4M_{K}^{2}}-\frac{%
9}{128\pi^{2}}\,,$$
(137)
$$\displaystyle F_{\pi}^{2}\beta_{\pi\eta}$$
$$\displaystyle=8L_{4}^{r}\left(M_{\pi}^{2}+M_{\eta}^{2}\right)-\frac{M_{\pi}^{2%
}+M_{\eta}^{2}}{32\pi^{2}}-\frac{\left(M_{\pi}^{2}+3M_{\eta}^{2}\right)F_{\pi}%
^{2}\mu_{K}}{3M_{K}^{2}}-\frac{2F_{\pi}^{2}\mu_{\pi}}{3}\,,$$
(138)
$$\displaystyle\begin{split}\displaystyle F_{\pi}^{2}(\alpha_{\pi\eta}-1)M_{\pi}%
^{2}&\displaystyle=16L_{4}^{r}\left(M_{\pi}^{4}-4M_{\eta}^{2}M_{\pi}^{2}+M_{%
\eta}^{4}\right)+16M_{\eta}^{2}M_{\pi}^{2}(-L_{5}^{r}+6L_{6}^{r})\\
&\displaystyle+96L_{7}\left(M_{\pi}^{2}-M_{\eta}^{2}\right)M_{\pi}^{2}+48L_{8}%
^{r}M_{\pi}^{4}\\
&\displaystyle-\frac{\left(5M_{\pi}^{4}-28M_{\eta}^{2}M_{\pi}^{2}+6M_{\eta}^{4%
}\right)F_{\pi}^{2}\mu_{K}}{3M_{K}^{2}}+\frac{\left(-17M_{\pi}^{4}+9M_{\eta}^{%
2}M_{\pi}^{2}+4M_{\eta}^{4}\right)F_{\pi}^{2}\mu_{\pi}}{3\left(M_{\pi}^{2}-M_{%
\eta}^{2}\right)}\\
&\displaystyle+\frac{\left(M_{\pi}^{4}-M_{\eta}^{2}M_{\pi}^{2}+4M_{\eta}^{4}%
\right)M_{\pi}^{2}F_{\pi}^{2}\mu_{\eta}}{3M_{\eta}^{2}(M_{\pi}^{2}-M_{\eta}^{2%
})}-\frac{8M_{\pi}^{4}-11M_{\eta}^{2}M_{\pi}^{2}+6M_{\eta}^{4}}{96\pi^{2}}\,.%
\end{split}$$
(139)
$\bullet\quad\pi\pi$
$$\displaystyle\varepsilon_{\pi\pi}$$
$$\displaystyle=4L_{2}^{r}+\frac{1}{12}F_{\pi}^{2}\left(-\frac{\mu_{K}}{M_{K}^{2%
}}-\frac{8\mu_{\pi}}{M_{\pi}^{2}}\right)-\frac{7}{384\pi^{2}}\,,$$
(140)
$$\displaystyle\delta_{\pi\pi}$$
$$\displaystyle=8L_{1}^{r}+4L_{3}+\frac{1}{12}F_{\pi}^{2}\left(-\frac{\mu_{K}}{M%
_{K}^{2}}-\frac{8\mu_{\pi}}{M_{\pi}^{2}}\right)-\frac{13}{384\pi^{2}}\,,$$
(141)
$$\displaystyle F_{\pi}^{2}(\beta_{\pi\pi}-1)$$
$$\displaystyle=8\left(2L_{4}^{r}+L_{5}^{r}\right)M_{\pi}^{2}-\frac{F_{\pi}^{2}%
\mu_{K}M_{\pi}^{2}}{M_{K}^{2}}-\frac{5M_{\pi}^{2}}{32\pi^{2}}-4F_{\pi}^{2}\mu_%
{\pi},$$
(142)
$$\displaystyle F_{\pi}^{2}(\alpha_{\pi\pi}-1)M_{\pi}^{2}$$
$$\displaystyle=-16\left(2L_{4}^{r}+L_{5}^{r}-6L_{6}^{r}-3L_{8}^{r}\right)M_{\pi%
}^{4}-\frac{F_{\pi}^{2}\mu_{\eta}M_{\pi}^{4}}{3M_{\eta}^{2}}-\frac{F_{\pi}^{2}%
\mu_{K}M_{\pi}^{4}}{M_{K}^{2}}-\frac{7M_{\pi}^{4}}{96\pi^{2}}-F_{\pi}^{2}\mu_{%
\pi}M_{\pi}^{2}.$$
(143)
$\bullet\quad\eta K$
$$\displaystyle\varepsilon_{\eta K}$$
$$\displaystyle=16L_{2}^{r}+\frac{4}{3}L_{3}-\frac{F_{\pi}^{2}(2\mu_{K}+\mu_{\pi%
}+3\mu_{\eta})}{2\left(M_{K}^{2}-M_{\eta}^{2}\right)}+\frac{1}{64\pi^{2}}\,,$$
(144)
$$\displaystyle\delta_{\eta K}$$
$$\displaystyle=32L_{1}^{r}+\frac{40}{3}L_{3}+\frac{\left(9M_{\eta}^{2}-8M_{K}^{%
2}\right)F_{\pi}^{2}\mu_{K}}{2M_{K}^{2}\left(M_{K}^{2}-M_{\eta}^{2}\right)}+%
\frac{F_{\pi}^{2}\left(\mu_{\pi}-3\mu_{\eta}\right)}{4(M_{K}^{2}-M_{\eta}^{2})%
}-\frac{11}{64\pi^{2}}\,,$$
(145)
$$\displaystyle\begin{split}\displaystyle F_{\pi}^{2}(\beta_{\eta K}-1)&%
\displaystyle=\frac{32}{3}\left(M_{K}^{2}+M_{\eta}^{2}\right)L_{4}^{r}+8L_{5}^%
{r}M_{\pi}^{2}-\frac{M_{\eta}^{2}\left(7M_{K}^{2}-9M_{\eta}^{2}\right)F_{\pi}^%
{2}\mu_{K}}{3M_{K}^{2}(M_{K}^{2}-M_{\eta}^{2})}\\
&\displaystyle+\frac{\left(13M_{\eta}^{2}-11M_{K}^{2}\right)F_{\pi}^{2}\mu_{%
\pi}}{3\left(M_{K}^{2}-M_{\eta}^{2}\right)}+\frac{\left(4M_{K}^{4}-15M_{\eta}^%
{2}M_{K}^{2}+7M_{\eta}^{4}\right)F_{\pi}^{2}\mu_{\eta}}{3M_{\eta}^{2}(M_{K}^{2%
}-M_{\eta}^{2})}\\
&\displaystyle-\frac{17M_{\eta}^{2}+37M_{K}^{2}}{192\pi^{2}}\,,\end{split}$$
(146)
$$\displaystyle\begin{split}\displaystyle F_{\pi}^{2}(\alpha_{\eta K}-1)\left(3M%
_{\eta}^{2}-M_{K}^{2}\right)&\displaystyle=F_{\pi}^{2}\Delta_{\mathrm{GMO}}(M_%
{\eta}^{2}-M_{\pi}^{2})+32L_{4}^{r}\left(M_{K}^{4}-4M_{\eta}^{2}M_{K}^{2}+M_{%
\eta}^{4}\right)\\
&\displaystyle+192L_{6}^{r}M_{K}^{2}M_{\eta}^{2}+4L_{5}^{r}\left(-3M_{\pi}^{4}%
+6M_{K}^{2}M_{\pi}^{2}+M_{\eta}^{2}M_{\pi}^{2}-12M_{\eta}^{4}\right)\\
&\displaystyle+48L_{7}\left(M_{\pi}^{4}-4M_{\eta}^{2}M_{\pi}^{2}+3M_{\eta}^{4}%
\right)+24L_{8}^{r}\left(M_{\pi}^{4}-3M_{\eta}^{2}M_{\pi}^{2}+6M_{\eta}^{4}%
\right)\\
&\displaystyle-\frac{\left(14M_{K}^{6}-84M_{\eta}^{2}M_{K}^{4}+102M_{\eta}^{4}%
M_{K}^{2}-27M_{\eta}^{6}\right)F_{\pi}^{2}\mu_{K}}{3(M_{K}^{2}-M_{\eta}^{2})}%
\\
&\displaystyle-\frac{\left(M_{K}^{4}+12M_{\eta}^{2}M_{K}^{2}-14M_{\eta}^{4}%
\right)F_{\pi}^{2}\mu_{\pi}}{2(M_{K}^{2}-M_{\eta}^{2})}\\
&\displaystyle+\frac{\left(8M_{K}^{6}-41M_{\eta}^{2}M_{K}^{4}+104M_{\eta}^{4}M%
_{K}^{2}-78M_{\eta}^{6}\right)F_{\pi}^{2}\mu_{\eta}}{6M_{\eta}^{2}(M_{K}^{2}-M%
_{\eta}^{2})}\\
&\displaystyle-\frac{31M_{K}^{4}-55M_{\eta}^{2}M_{K}^{2}+39M_{\eta}^{4}}{96\pi%
^{2}}\,.\end{split}$$
(147)
$\bullet\quad\eta\pi K$
$$\displaystyle\varepsilon_{\eta\pi K}$$
$$\displaystyle=-4L_{3}+\frac{6\mu_{K}F_{\pi}^{2}}{M_{K}^{2}-M_{\pi}^{2}}-\frac{%
3\mu_{\pi}F_{\pi}^{2}}{2\left(M_{K}^{2}-M_{\pi}^{2}\right)}+\frac{3\mu_{\eta}F%
_{\pi}^{2}}{2\left(M_{K}^{2}-M_{\eta}^{2}\right)}+\frac{1}{64\pi^{2}}\,,$$
(148)
$$\displaystyle\delta_{\eta\pi K}$$
$$\displaystyle=8L_{3}+\frac{3\left(M_{\pi}^{2}-2M_{K}^{2}\right)\mu_{K}F_{\pi}^%
{2}}{2M_{K}^{2}\left(M_{K}^{2}-M_{\pi}^{2}\right)}-\frac{3\mu_{\pi}F_{\pi}^{2}%
}{4\left(M_{K}^{2}-M_{\pi}^{2}\right)}-\frac{3\mu_{\eta}F_{\pi}^{2}}{4\left(M_%
{K}^{2}-M_{\eta}^{2}\right)}-\frac{5}{64\pi^{2}}\,,$$
(149)
$$\displaystyle\begin{split}\displaystyle F_{\pi}^{2}(\beta_{\eta\pi K}-1)&%
\displaystyle=\frac{4}{3}L_{3}\Delta_{\mathrm{GMO}}(M_{\eta}^{2}-M_{\pi}^{2})+%
8M_{\pi}^{2}L_{5}^{r}\\
&\displaystyle+\frac{\left(-32M_{K}^{4}+2M_{K}^{2}(2M_{\pi}^{2}-3M_{\eta}^{2})%
+3M_{\pi}^{2}(M_{\pi}^{2}+M_{\eta}^{2})\right)F_{\pi}^{2}\mu_{K}}{6\left(M_{K}%
^{2}-M_{\pi}^{2}\right)}\\
&\displaystyle+\frac{\left(-40M_{K}^{2}+9M_{\eta}^{2}+33M_{\pi}^{2}\right)F_{%
\pi}^{2}\mu_{\pi}}{12\left(M_{K}^{2}-M_{\pi}^{2}\right)}-\frac{\left(M_{\pi}^{%
2}+5M_{\eta}^{2}\right)F_{\pi}^{2}\mu_{\eta}}{4(M_{K}^{2}-M_{\eta}^{2})}+\frac%
{-57M_{K}^{2}+4M_{\pi}^{2}+27M_{\eta}^{2}}{192\pi^{2}}\,,\end{split}$$
(150)
$$\displaystyle\begin{split}\displaystyle F_{\pi}^{2}\alpha_{\eta\pi K}M_{\pi}^{%
2}&\displaystyle=-\frac{F_{\pi}^{2}}{3}\Delta_{\mathrm{GMO}}(M_{\eta}^{2}-M_{%
\pi}^{2})+\frac{8}{3}L_{3}\left(8M_{K}^{2}+M_{\pi}^{2}+M_{\eta}^{2}\right)%
\Delta_{\mathrm{GMO}}(M_{\eta}^{2}-M_{\pi}^{2})\\
&\displaystyle+4\left(M_{\pi}^{4}-4M_{K}^{2}M_{\pi}^{2}+4M_{\eta}^{2}M_{\pi}^{%
2}-M_{\eta}^{4}\right)L_{5}^{r}+24\left(M_{\eta}^{4}-M_{\pi}^{4}\right)(2L_{7}%
+L_{8}^{r})\\
&\displaystyle\hskip-28.452756pt+\frac{\left(40M_{K}^{6}+8\left(M_{\eta}^{2}-1%
0M_{\pi}^{2}\right)M_{K}^{4}-3M_{\pi}^{2}(M_{\pi}^{2}+M_{\eta}^{2})^{2}+2M_{K}%
^{2}(17M_{\pi}^{4}+25M_{\pi}^{2}M_{\eta}^{2}-6M_{\eta}^{4})\right)F_{\pi}^{2}%
\mu_{K}}{6\left(M_{K}^{2}-M_{\pi}^{2}\right)}\\
&\displaystyle+\frac{\left(-70M_{K}^{4}+2\left(22M_{\pi}^{2}+19M_{\eta}^{2}%
\right)M_{K}^{2}-19M_{\pi}^{4}+9M_{\eta}^{4}-12M_{\pi}^{2}M_{\eta}^{2}\right)F%
_{\pi}^{2}\mu_{\pi}}{6\left(M_{K}^{2}-M_{\pi}^{2}\right)}\\
&\displaystyle+\frac{\left(14M_{K}^{4}-M_{\pi}^{4}+21M_{\eta}^{4}+6M_{\pi}^{2}%
M_{\eta}^{2}-2M_{K}^{2}\left(M_{\pi}^{2}+18M_{\eta}^{2}\right)\right)F_{\pi}^{%
2}\mu_{\eta}}{2\left(M_{K}^{2}-M_{\eta}^{2}\right)}\\
&\displaystyle+\frac{-408M_{K}^{4}+\left(73M_{\pi}^{2}+387M_{\eta}^{2}\right)M%
_{K}^{2}+8M_{\pi}^{4}-63M_{\eta}^{4}-M_{\pi}^{2}M_{\eta}^{2}}{192\pi^{2}}\,.%
\end{split}$$
(151)
$\bullet\quad\pi K$
$$\displaystyle\varphi_{\pi K}$$
$$\displaystyle=-4L_{3}+\frac{\left(M_{\pi}^{2}-4M_{K}^{2}\right)F_{\pi}^{2}\mu_%
{K}}{6M_{K}^{2}\left(M_{K}^{2}-M_{\pi}^{2}\right)}+\frac{\left(M_{\pi}^{2}-4M_%
{K}^{2}\right)F_{\pi}^{2}\mu_{\pi}}{12M_{\pi}^{2}\left(M_{K}^{2}-M_{\pi}^{2}%
\right)}-\frac{F_{\pi}^{2}\mu_{\eta}}{M_{\pi}^{2}-M_{\eta}^{2}}\,,$$
(152)
$$\displaystyle\varepsilon_{\pi K}$$
$$\displaystyle=12(4L_{2}^{r}+L_{3})-\frac{3F_{\pi}^{2}\mu_{K}}{M_{K}^{2}-M_{\pi%
}^{2}}+\frac{15F_{\pi}^{2}\mu_{\pi}}{2(M_{K}^{2}-M_{\pi}^{2})}+\frac{6F_{\pi}^%
{2}\mu_{\eta}}{M_{\pi}^{2}-M_{\eta}^{2}}+\frac{1}{64\pi^{2}}\,,$$
(153)
$$\displaystyle\begin{split}\displaystyle\delta_{\pi K}&\displaystyle=24(4L_{1}^%
{r}+L_{3})+\frac{3\left(3M_{\pi}^{2}-4M_{K}^{2}\right)F_{\pi}^{2}\mu_{K}}{2M_{%
K}^{2}\left(M_{K}^{2}-M_{\pi}^{2}\right)}+\frac{3\left(7M_{\pi}^{2}-8M_{K}^{2}%
\right)F_{\pi}^{2}\mu_{\pi}}{4M_{\pi}^{2}\left(M_{K}^{2}-M_{\pi}^{2}\right)}\\
&\displaystyle-\frac{3F_{\pi}^{2}\mu_{\eta}}{M_{\pi}^{2}-M_{\eta}^{2}}-\frac{2%
3}{64\pi^{2}}\,,\end{split}$$
(154)
$$\displaystyle\begin{split}\displaystyle F_{\pi}^{2}(\gamma_{\pi K}-1)&%
\displaystyle=8M_{\pi}^{2}L_{5}^{r}-\frac{2M_{K}^{2}F_{\pi}^{2}\mu_{K}}{M_{K}^%
{2}-M_{\pi}^{2}}-\frac{\left(M_{K}^{2}-5M_{\pi}^{2}\right)F_{\pi}^{2}\mu_{\pi}%
}{2(M_{K}^{2}-M_{\pi}^{2})}\\
&\displaystyle+\frac{\left(2M_{K}^{2}+M_{\pi}^{2}-3M_{\eta}^{2}\right)F_{\pi}^%
{2}\mu_{\eta}}{M_{\pi}^{2}-M_{\eta}^{2}}-\frac{5M_{K}^{2}+M_{\pi}^{2}}{192\pi^%
{2}}\,,\end{split}$$
(155)
$$\displaystyle\begin{split}\displaystyle F_{\pi}^{2}(\beta_{\pi K}-1)&%
\displaystyle=32\left(M_{K}^{2}+M_{\pi}^{2}\right)L_{4}^{r}+8L_{5}^{r}M_{\pi}^%
{2}-\frac{15M_{\pi}^{2}+19M_{K}^{2}}{64\pi^{2}F_{\pi}^{2}}\\
&\displaystyle+\frac{\left(3M_{\pi}^{4}+3M_{K}^{2}M_{\pi}^{2}-8M_{K}^{4}\right%
)F_{\pi}^{2}\mu_{K}}{M_{K}^{2}(M_{K}^{2}-M_{\pi}^{2})}-\frac{\left(3M_{\pi}^{4%
}-3M_{K}^{2}M_{\pi}^{2}-4M_{K}^{4}\right)F_{\pi}^{2}\mu_{\pi}}{2M_{\pi}^{2}(M_%
{K}^{2}-M_{\pi}^{2})}\\
&\displaystyle-\frac{\left(2M_{\pi}^{4}+9M_{\eta}^{4}+\left(4M_{K}^{2}+M_{\pi}%
^{2}\right)M_{\eta}^{2}\right)F_{\pi}^{2}\mu_{\eta}}{2M_{\eta}^{2}\left(M_{\pi%
}^{2}-M_{\eta}^{2}\right)}\,,\end{split}$$
(156)
$$\displaystyle\begin{split}\displaystyle F_{\pi}^{2}(\alpha_{\pi K}-1)M_{\pi}M_%
{K}&\displaystyle=16L_{4}^{r}\left(M_{K}^{4}-4M_{\pi}^{2}M_{K}^{2}+M_{\pi}^{4}%
\right)+4L_{5}^{r}M_{\pi}^{2}\left(M_{\pi}^{2}-5M_{K}^{2}\right)+48M_{K}^{2}M_%
{\pi}^{2}(2L_{6}^{r}+L_{8}^{r})\\
&\displaystyle-\frac{21M_{K}^{4}-25M_{\pi}^{2}M_{K}^{2}+21M_{\pi}^{4}}{192\pi^%
{2}}+\frac{\left(2M_{K}^{6}-4M_{K}^{4}M_{\pi}^{2}+6M_{\pi}^{4}M_{K}^{2}-3M_{%
\pi}^{6}\right)F_{\pi}^{2}\mu_{K}}{2M_{K}^{2}(M_{K}^{2}-M_{\pi}^{2})}\\
&\displaystyle-\frac{\left(8M_{K}^{6}-13M_{K}^{4}M_{\pi}^{2}+12M_{\pi}^{4}M_{K%
}^{2}-6M_{\pi}^{6}\right)F_{\pi}^{2}\mu_{\pi}}{4M_{\pi}^{2}(M_{K}^{2}-M_{\pi}^%
{2})}\\
&\displaystyle\hskip-22.762205pt\hskip-42.679134pt-\frac{\left(17M_{\pi}^{6}-2%
4M_{K}^{2}M_{\pi}^{4}+3\left(27M_{\eta}^{2}+7M_{\pi}^{2}-36M_{K}^{2}\right)M_{%
\eta}^{4}+\left(24M_{K}^{4}+12M_{\pi}^{2}M_{K}^{2}+M_{\pi}^{4}\right)M_{\eta}^%
{2}\right)F_{\pi}^{2}\mu_{\eta}}{24M_{\eta}^{2}\left(M_{\pi}^{2}-M_{\eta}^{2}%
\right)}\,.\end{split}$$
(157)
$\bullet\quad KK$
$$\displaystyle\varphi_{KK}$$
$$\displaystyle=24L_{2}^{r}-\frac{17F_{\pi}^{2}\mu_{K}}{4M_{K}^{2}}-\frac{F_{\pi%
}^{2}\mu_{\pi}}{4M_{\pi}^{2}}-\frac{7}{64\pi^{2}}\,,$$
(158)
$$\displaystyle\varepsilon_{KK}$$
$$\displaystyle=12(2L_{2}^{r}+L_{3})-\frac{F_{\pi}^{2}\mu_{K}}{4M_{K}^{2}}-\frac%
{F_{\pi}^{2}}{8}\left(\frac{2}{M_{\pi}^{2}}-\frac{27}{M_{K}^{2}-M_{\pi}^{2}}%
\right)\mu_{\pi}+\frac{9F_{\pi}^{2}\mu_{\eta}}{2\left(M_{\pi}^{2}-M_{\eta}^{2}%
\right)}-\frac{1}{64\pi^{2}}\,,$$
(159)
$$\displaystyle\begin{split}\displaystyle\delta_{KK}&\displaystyle=12(4L_{1}^{r}%
+L_{3})-\frac{11F_{\pi}^{2}\mu_{K}}{4M_{K}^{2}}-\frac{F_{\pi}^{2}}{16}\left(%
\frac{10}{M_{\pi}^{2}}+\frac{27}{M_{K}^{2}-M_{\pi}^{2}}\right)\mu_{\pi}\\
&\displaystyle-\frac{9}{8}F_{\pi}^{2}\left(\frac{3}{M_{\eta}^{2}}+\frac{2}{M_{%
\pi}^{2}-M_{\eta}^{2}}\right)\mu_{\eta}-\frac{29}{128\pi^{2}}\,,\end{split}$$
(160)
$$\displaystyle\begin{split}\displaystyle F_{\pi}^{2}\gamma_{KK}&\displaystyle=1%
6M_{K}^{2}L_{4}^{r}-\frac{7M_{K}^{2}}{64\pi^{2}}-\frac{3F_{\pi}^{2}\mu_{K}}{2}%
-\frac{3M_{K}^{2}\left(2M_{K}^{2}+M_{\pi}^{2}\right)F_{\pi}^{2}\mu_{\pi}}{8M_{%
\pi}^{2}(M_{K}^{2}-M_{\pi}^{2})}\\
&\displaystyle+\frac{\left(M_{\pi}^{4}-3M_{\eta}^{4}+2M_{\pi}^{2}M_{\eta}^{2}+%
3\left(M_{\eta}^{2}-3M_{\pi}^{2}\right)M_{K}^{2}\right)F_{\pi}^{2}\mu_{\eta}}{%
4M_{\eta}^{2}\left(M_{\pi}^{2}-M_{\eta}^{2}\right)}\,,\end{split}$$
(161)
$$\displaystyle\begin{split}\displaystyle F_{\pi}^{2}(\beta_{KK}-1)&%
\displaystyle=16M_{K}^{2}L_{4}^{r}+8L_{5}^{r}M_{\pi}^{2}-\frac{9M_{K}^{2}}{64%
\pi^{2}}-\frac{3F_{\pi}^{2}\mu_{K}}{2}-\frac{\left(6M_{K}^{4}+11M_{\pi}^{2}M_{%
K}^{2}-20M_{\pi}^{4}\right)F_{\pi}^{2}\mu_{\pi}}{8M_{\pi}^{2}(M_{K}^{2}-M_{\pi%
}^{2})}\\
&\displaystyle+\frac{\left(M_{\pi}^{4}-9M_{\eta}^{4}+8M_{\pi}^{2}M_{\eta}^{2}+%
M_{K}^{2}\left(11M_{\eta}^{2}-9M_{\pi}^{2}\right)\right)F_{\pi}^{2}\mu_{\eta}}%
{4M_{\eta}^{2}\left(M_{\pi}^{2}-M_{\eta}^{2}\right)}\,,\end{split}$$
(162)
$$\displaystyle\begin{split}\displaystyle F_{\pi}^{2}(\alpha_{KK}-1)M_{K}^{2}&%
\displaystyle=16M_{K}^{4}(-2L_{4}^{r}+6L_{6}^{r}+3L_{8}^{r})+8L_{5}^{r}M_{K}^{%
2}\left(M_{\pi}^{2}-3M_{K}^{2}\right)\\
&\displaystyle-\frac{7M_{K}^{4}}{192\pi^{2}}-\frac{3M_{K}^{2}F_{\pi}^{2}\mu_{K%
}}{2}-\frac{M_{K}^{2}\left(6M_{K}^{4}+17M_{\pi}^{2}M_{K}^{2}-20M_{\pi}^{4}%
\right)F_{\pi}^{2}\mu_{\pi}}{8M_{\pi}^{2}(M_{K}^{2}-M_{\pi}^{2})}\\
&\displaystyle\hskip-42.679134pt-\frac{\left(M_{\pi}^{6}+2M_{\eta}^{2}M_{\pi}^%
{4}-3M_{\eta}^{4}M_{\pi}^{2}+3M_{K}^{4}(9M_{\pi}^{2}-7M_{\eta}^{2})-6M_{K}^{2}%
\left(2M_{\pi}^{4}+5M_{\eta}^{2}M_{\pi}^{2}-7M_{\eta}^{4}\right)\right)F_{\pi}%
^{2}\mu_{\eta}}{12M_{\eta}^{2}\left(M_{\eta}^{2}-M_{\pi}^{2}\right)}\,.\end{split}$$
(163)
In [10] they believe $K^{+}K^{-}\rightarrow K^{+}K^{-}$ to be independent on $K^{+}K^{-}\rightarrow K^{0}{\overline{K^{0}}}$, but we know that isospin structure (Fierz-like identities) and the crossing symmetry dictate the relation between these two processes given by (54) together with (53). Therefore, we have used the values of our parameters obtained from their $K^{+}K^{-}\rightarrow K^{0}{\overline{K^{0}}}$ amplitude to explicitly check their $K^{+}K^{-}\rightarrow K^{+}K^{-}$ result $T_{ch}$.
Appendix D Short comment on the assumptions of the analyticity of the amplitude and the dispersion
relations
The most important assumptions of the theorem – the existence of
dispersion relations and the analyticity of the amplitude and its
absorptive parts are results of a complicated theory of analytic
properties of scattering amplitudes, which is even older than QCD
itself. A good and still valid summary of it is the article by Sommer
[17].
We will not address this theory in more detail, just summarising
results interesting for us (details can be found in [17]
and [18]).
From the principles of axiomatic field theory (even without use of the unitarity of
S-matrix), Lehmann has proven that the amplitude with $s$ fixed at some physical value is
holomorphic in some finite region of the $u$-plane (if $s$ is above
the physical threshold, this region is the so-called small Lehmann
ellipse). Absorptive parts of amplitudes ($\operatorname{Im}{A(s,u)}$ for
$s\geq\Sigma$ and similarly for $t$) are holomorphic in the large
Lehmann ellipses in the $u$-plane (depending on $s$) and there this
absorptive part has also a convergent partial wave decomposition.
Further, the $N$-times subtracted $u$-fixed dispersion relations can
be proven on the intersection of those large Lehmann ellipses for
$s\geq\Sigma$ and $t\geq\tau$ (if not empty). Since its semiminor
axis tends to zero for $s\rightarrow\infty$, this intersection is
just an interval on the negative real $u$-axis.
Taking into account the unitarity, Martin and others have succeeded in
enlarging the validity of the dispersion relations into the circle
$|u|<R$ with some fixed radius $R$.
There exist further methods to enlarge the region of validity as
well as the analyticity domain. Using the ones given in
[17] we end up with the results of Table IV there. We are
interested in their specific application for our processes. If we
assume the isospin conservation and that all the mesons $\pi,K,\eta$ are thereby stable (and forget about resonances) we get the
validity of fixed $u$ dispersion relations within the regions
depicted in Fig. 1 (the intersections of Lehmann ellipses are
marked by bold lines there). These regions are glued ellipses and have
been obtained analytically in [18] using procedures from
[17]. Let us remind that these regions are only the minimal
domains, where the dispersive relations are valid, proven directly
from the axiomatic theory. Using further methods we could also
extend these regions - e.g. we have not taken into account the
further specific crossing (and Bose) symmetries of some of the
amplitudes. Finally let us once more emphasise that the described
method is ineffective if we allow e.g. the $\eta\rightarrow 3\pi$
decay and such cases have to be proven for unphysical masses of $\eta$ and then analytically continued in them.
Another assumption of our theorem was the existence of a point $s$, where the amplitude is analytic with respect to
all the values of variable $u$, for which the theorem should be valid - but in all the cases one can show that there exists a value $s$ for
which the small Lehmann ellipse is larger than the regions from the Fig. 1 and so we can conclude that both the assumptions
of the theorem are valid within these regions.
References
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J. Bijnens, arXiv:hep-ph/0604043.
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S. Weinberg, Physica A 96 (1979) 327.
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M. Knecht and J. Stern, arXiv:hep-ph/9411253.
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B. Ananthanarayan, P. Buettiker and B. Moussallam, Eur. Phys. J. C 22 (2001) 133 [arXiv:hep-ph/0106230].
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M. Froissart,
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K. Kampf, M. Knecht, J. Novotný, M. Zdráhal, arXiv:0810.1906 [hep-ph].
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[18]
M. Zdráhal, Ph.D. thesis - in preparation, Charles University. |
Description of nuclear octupole and quadrupole deformation
close to the axial symmetry and phase transitions in the octupole mode
P.G. Bizzeti
A.M. Bizzeti-Sona
Dipartimento di Fisica, Universitá di Firenze
I.N.F.N., Sezione di Firenze
Via G. Sansone 1, 50019 Sesto Fiorentino (Firenze), Italy
(December 4, 2020)
Abstract
The dynamics of nuclear collective motion is investigated in the case
of reflection-asymmetric shapes.
The model is based on a
new parameterization of the octupole and quadrupole degrees of
freedom, valid for nuclei close to the axial symmetry.
Amplitudes of oscillation in other degrees of freedom different from
the axial ones are assumed to be small, but not frozen to zero.
The case of nuclei which already possess a permanent quadrupole
deformation is discussed in some more detail and a simple solution is
obtained at the critical point of the phase transition
between harmonic octupole oscillation and a permanent asymmetric shape.
The results are compared with experimental data of the Thorium
isotopic chain. The isotope ${}^{226}$Th is found to be close to
the critical point.
Octupole deformation. Phase transition.
pacs: 21.60.ev
I Introduction
Phase transitions in nuclear shapes have been recently observed
at the boundary between regions characterized by different intrinsic
shapes of quadrupole deformation.
It has been shown by Iachello Iachello (2000a, b, 2001)
that new dynamic symmetries, called E(5), X(5) and Y(5) hold,
respectively, at the critical point
between spherical shape and $\gamma$–unstable deformation,
between spherical and deformed
axially symmetric shape and between deformed axial and triaxial shape.
Here, we are mainly interested in the second case.
First examples of X(5)
symmetry in transitional nuclei have been found
in ${}^{152}$Sm Casten and Zamfir (2001) and ${}^{150}$Nd
Krucken et al. (2002).
Other candidates for the X(5)
symmetry in different nuclear regions have been reported
later Bizzeti and Bizzeti-Sona (2002); Hutter et al. (2003); Clark et al. (2003); Fransen et al. (2004); Bizzeti (2003); Bijker et al. (2003, 2004); Tonev et al. (2004).
From the theoretical point of view, slightly different potentials have
been explored by Caprio Caprio (2004) and by Bonatsos
et al. Bonatsos et al. (2004), while the evolution of the $s=2$ band
when the lower border of the square well potential is displaced from
zero has been investigated by Pietralla and Gorbachenko Pietralla and Gorbachenko (2004).
A similar phase transition (i.e.,
from shape oscillation to
permanent deformation, conserving the axial symmetry of the system)
could take place also in the octupole degree of freedom. We have
found a possible example of such a phase transition
in the Thorium isotope chain, with the critical
point close to the mass 226.
In this case, the octupole mode is combined with a stable
quadrupole deformation.
Preliminary results have been reported in
two recent Conferences Bizzeti and Bizzeti-Sona (2004); Bizzeti (2003).
In order to provide a theoretical frame where to discuss the different
aspects of the octupole motion, we introduce a new
parameterization of the quadrupole and octupole degrees of freedom,
valid in conditions close to the axial symmetry. In this limit, a model
similar to the classical one by Bohr Bohr (1952) has been developed.
This is the subject of the first part of this paper.
In the second part, the model is used to discuss the evolution of the
octupole mode along the isotopic chain of Thorium, and the results are
compared with the experimental data.
II Theoretical frame for combined
quadrupole and
octupole excitations
II.1 Previous investigations of the Octupole
plus Quadrupole deformation
Reflection–asymmetric nuclear shapes have been discussed in a number
of papers, either in terms of surface Quadrupole + Octupole
deformation (Bohr geometrical approach) or with an extended
Interacting Boson Model (algebraic approach). The latter,
proposed in 1985 by Engel and
Iachello Engel and Iachello (1985), has been recently used by Alonso et
al. Alonso et al. (1995), Raduta et al. Raduta and Ionescu (2003); Raduta et al. (1996a),
Zamfir and Kusnezov Zamfir and Kusnezov (2001, 2003).
An alternative approach assuming $\alpha$–cluster configurations
has been discussed by Shneidman et al. Shneidman et al. (2002).
In the frame of the geometrical approach, a
number of theoretical investigations of the octupole vibrations
around a stable quadrupole deformation have been reported in the
last 50 years Rohozinski (1988); Butler and Nazarewicz (1996); Lipas and Davidson (1961); Denisov and Dzyublik (1995); Jolos and von Brentano (1999); Minkov et al. (2000, 2004).
Most of them, however, are limited to the case of axial symmetry.
This approach has been criticized, e.g., by Donner and
Greiner Donner and Greiner (1966), who have stressed the fact that all terms of a
given tensor order must be taken into account for a consistent
treatment.
To do this, Donner and Greiner renounce to the use of an “intrinsic
frame” referred to the principal axes of the overall tensor of
inertia and choose to define the octupole amplitudes in the
“intrinsic frame” of the quadrupole mode alone.
In this approach, definite predictions have been obtained at the limit
where the octupole deformations are “small” in comparison with the
quadrupole ones Eisenberg and Greiner (1987).
II.2 The new parameterization
Here we adopt a different approach, which can be useful also in the
case of comparable octupole and quadrupole deformation, close to the
axial–symmetry limit. Namely, we
choose as “intrinsic” reference frame the principal axes of
the overall tensor of inertia, as it results from the combined
quadrupole and octupole deformation.
The definitions of quadrupole and octupole amplitudes
$a^{(\lambda)}_{\mu}$,
with $\lambda=2,3$ and
$a^{(\lambda)}_{-\mu}=(-)^{\mu}\ {a^{(\lambda)}_{\mu}}^{*}$,
are recalled in the Appendix A. All these amplitudes are
defined in the (non inertial) intrinsic frame. To this purpose, in the
case of the quadrupole mode alone, it is
enough to assume
$a^{(2)}_{2}\!=a\!^{(2)}_{-2}$ real and $a^{(2)}_{\pm 1}\!=\!0$, with
the standard parameterization in terms of $\beta_{2}$ and $\gamma_{2}$:
$$\displaystyle a^{(2)}_{0}$$
$$\displaystyle=$$
$$\displaystyle\beta_{2}\cos\gamma_{2}$$
(1a)
$$\displaystyle a^{(2)}_{1}$$
$$\displaystyle=$$
$$\displaystyle 0$$
(1b)
$$\displaystyle a^{(2)}_{2}$$
$$\displaystyle=$$
$$\displaystyle\sqrt{1/2}\ \beta_{2}\ \sin\gamma_{2}\ .$$
(1c)
For the octupole mode
alone, a parameterization suitable to this purpose has been proposed
in 1999 by Wexler and Dussel Wexler and Dussel (1999). We adopt here a very
similar one:
$$\displaystyle a^{(3)}_{0}$$
$$\displaystyle=$$
$$\displaystyle\beta_{3}\ \cos\gamma_{3}$$
(2a)
$$\displaystyle a^{(3)}_{1}$$
$$\displaystyle=$$
$$\displaystyle-(5/2)\ \left(X+iY\right)\ \sin\gamma_{3}$$
(2b)
$$\displaystyle a^{(3)}_{2}$$
$$\displaystyle=$$
$$\displaystyle\sqrt{1/2}\ \beta_{3}\ \sin\gamma_{3}$$
(2c)
$$\displaystyle a^{(3)}_{3}$$
$$\displaystyle=$$
$$\displaystyle X\left[\cos\gamma_{3}+(\sqrt{15}/2)\ \sin\gamma_{3}\right]$$
(2d)
$$\displaystyle+$$
$$\displaystyle i\ Y\left[\cos\gamma_{3}-(\sqrt{15}/2)\ \sin\gamma_{3}\right]\ .$$
With this choice, the tensor of inertia turns out to be diagonal (see
Appendix A).
In both cases, one has to consider, in addition to the intrinsic
variables ($\beta_{2},\ \gamma_{2}$ for the quadrupole, or
$\beta_{3},\ \gamma_{3}\ ,X\ ,Y$ for the octupole), the three Euler angles
defining the orientation of the intrinsic frame in the laboratory
frame, in order to reach a number of parameters equal to the number of
degrees of freedom (5 for the quadrupole, 7 for the octupole).
Unfortunately, the situation is not so simple when quadrupole and
octupole modes are considered together, as the intrinsic frames of the
two modes do not necessarily coincide.
We shall limit our discussion to situations close to the axial
symmetry limit – in which, obviously, the two frames coincide – and
define a parametrization which automatically sets to zero the three
products of inertia $\mathcal{J}_{\kappa,\kappa^{\prime}}$
($\kappa\neq\kappa^{\prime}$) up to the first order in the
amplitudes of non–axial modes.
To this purpose we put
$$a^{(\lambda)}_{\mu}=\bar{a}^{(\lambda)}_{\mu}+\tilde{a}^{(\lambda)}_{\mu}$$
(3)
where $\bar{a}^{(\lambda)}_{\mu}$ are defined according to the
eq.s (1,2) and $\tilde{a}^{(\lambda)}_{\mu}$ are
correction terms, which are assumed to be small compared to the axial
amplitudes $a^{(\lambda)}_{0}$, but of the same order of magnitude as the
other non-axial terms.
It will be enough to consider these corrections only
for those amplitudes which, according to eq.s (1,2),
are either zero or small of the second order: the imaginary part
of $a^{(2)}_{2},\ a^{(3)}_{2}$ and the
real and imaginary parts of $a^{(2)}_{1},\ a^{(3)}_{1}$.
The six “new” first–order amplitudes added to those of
eq.s (1,2) are, however, not independent from one
another, if we choose as the reference systeme the one in which
the three
products of inertia turn out to be zero.
The expressions of the inertia tensor as a function of the
deformation parameters, obtained with the Bohr
assumptions Bohr (1952) of not–too–big deformations and
irrotational flow, are given in the Appendix A. In order to
simplify the notations, from now on we consider the inertia parameters
$B_{2},\ B_{3}$ included in our definitions of the amplitudes $a^{(\lambda)}_{\mu}$, which therefore correspond to
$\sqrt{B_{\lambda}}\ a^{(\lambda)}_{\mu}$ in the original Bohr notations.
From the eq.s (56), and retaining only terms
of the first order
in the small amplitudes $a^{(\lambda)}_{\mu}$ with $\mu\neq 0$, we
obtain the conditions
$$\displaystyle\mathcal{J}_{12}$$
$$\displaystyle=$$
$$\displaystyle-2\sqrt{6}\left(\beta_{2}{\rm Im}\ \tilde{a}^{(2)}_{2}+\sqrt{5}%
\beta_{3}{\rm Im}\ \tilde{a}^{(3)}_{2}\right)=0$$
(4a)
$$\displaystyle\mathcal{J}_{13}$$
$$\displaystyle+$$
$$\displaystyle i\mathcal{J}_{23}\ =\ \sqrt{6}\ \left(\beta_{2}\tilde{a}^{(2)}_{%
1}+\sqrt{2}\beta_{3}\tilde{a}^{(3)}_{1}\right)=0$$
(4b)
which are satisfied (at the leading order) if we put
$$\begin{array}[c]{lp{0.1mm}lp{0mm}l}\tilde{a}^{(2)}_{1}=\!\frac{-\sqrt{2}\beta_%
{3}}{\sqrt{\beta_{2}^{2}+2\beta_{3}^{2}}}\left(\eta+i\zeta\right)&\hbox{}&%
\tilde{a}^{(3)}_{1}=\!\frac{\beta_{2}}{\sqrt{\beta_{2}^{2}+2\beta_{3}^{2}}}%
\left(\eta+i\zeta\right)\\
{\rm Im}\;\tilde{a}^{(2)}_{2}=\!\frac{-\sqrt{5}\beta_{3}}{\sqrt{\beta_{2}^{2}+%
5\beta_{3}^{2}}}\xi&\hbox{}&{\rm Im}\;\tilde{a}^{(3)}_{2}=\!\frac{\beta_{2}}{%
\sqrt{\beta_{2}^{2}+5\beta_{3}^{2}}}\xi\end{array}$$
(5)
with the new parameters $\eta,\ \zeta$ and $\xi$ small of the first
order.
It is clear that only the ratios of the relevant amplitudes are
constrained by the eq.s (4). The definition of the new variables
given in each line of eq.s (5) contain therefore an arbitrary
factor. Our choice (and in particular for the square–root factors at
the denominators) has some distinguished advantage, which will result
clear from the classical expression of the kinetic energy, discussed
in the next paragraph.
In the intrinsic reference frame, and at the same
order of approximation, the
values of the three principal moments of inertia can be derived from
eq.s (54,55):
$$\displaystyle\mathcal{J}_{1}$$
$$\displaystyle=$$
$$\displaystyle 3(\beta_{2}^{2}+2\beta_{3}^{2})+2\sqrt{3}(\beta_{2}^{2}\gamma_{2%
}+\sqrt{5}\beta_{3}^{2}\gamma_{3})$$
(6a)
$$\displaystyle\mathcal{J}_{2}$$
$$\displaystyle=$$
$$\displaystyle 3(\beta_{2}^{2}+2\beta_{3}^{2})-2\sqrt{3}(\beta_{2}^{2}\gamma_{2%
}+\sqrt{5}\beta_{3}^{2}\gamma_{3})$$
(6b)
$$\displaystyle\mathcal{J}_{3}$$
$$\displaystyle=$$
$$\displaystyle 4(\beta_{2}^{2}\gamma_{2}^{2}+\beta_{3}^{2}\gamma_{3}^{2})+18(X^%
{2}+Y^{2})$$
(6c)
$$\displaystyle+2(\eta^{2}+\zeta^{2})+8\xi^{2}\ .$$
With the amplitudes given by eq.s (1,2) the principal
axes of the quadrupole would coincide with those of the octupole.
It is not necessarily so with our more general assumptions.
When $a^{(\lambda)}_{1}\neq 0$, the
axis 3 of the tensor of inertia for the quadrupole mode alone does not
coincide with that of the octupole. If $a^{(\lambda)}_{1}=0$, but
Im$\ a^{(\lambda)}_{2}\neq 0$, the misalignment concerns the other two
principal axes perpendicular to the common axis 3.
II.3 The classical expression of the kinetic energy
Now it is possible to express the classical kinetic energy (as given
by Bohr hydrodynamical model) in terms of the new variables and
of the intrinsic components $q_{\kappa}$
of the angular velocity. The classical expression has the form
$$T=\frac{1}{2}\sum\dot{Q}_{\mu}\mathcal{G}_{\mu\nu}\dot{Q}_{\nu}$$
(7)
where $\dot{Q}\equiv\{\dot{\xi}_{1},\dot{\xi}_{2},...,\dot{\xi}_{9},q_{1},q_{2},q_{3}\}$, $\dot{\xi}_{\mu}\ (\mu=1,...,9)$ are the
time derivative of the nine parameters we have just defined, and
$q_{1},\ q_{2},\ q_{3}$ are the intrinsic components of the angular velocity
of the intrinsic system with respect to an inertial frame.
The elements of the matrix $\mathcal{G}$
(leading terms and
relevant first-order terms) are shown in the Table 1.
The determinant $G={\rm Det}\ \mathcal{G}$ takes the form
$$G\propto\beta_{2}^{2}\beta_{3}^{2}\big{(}\beta_{2}^{2}+2\beta_{3}^{2}\big{)}^{%
2}\big{(}\beta_{2}^{2}+5\beta_{3}^{2}\big{)}^{-1}\big{(}\beta_{2}^{2}\gamma_{2%
}+\sqrt{5}\ \beta_{3}^{2}\gamma_{3}\big{)}^{2}$$
and, at the limit $\beta_{3}\ll\beta_{2}$, results to be proportional
to $\beta_{2}^{8}$, and therefore
consistent with that of the Bohr model for a pure quadrupole motion.
This is a consequence of our choice of the normalization factors in
the eq.s (5). This choice has other advantages: all the non diagonal
terms involving the time derivatives of $\beta_{2}$ or $\beta_{3}$ and
either the derivative of one
of the other intrinsic amplitudes or $q_{3}$ turn out to be zero in the
present approximation. Other non diagonal elements are small (of the
first order) in the “small” amplitudes $\gamma_{2},\ \gamma_{3},\ X,\ Y,\ \xi,\ \eta,\ \zeta$. In situations close to the axial symmetry, they
have negligible effect on the results (see Appendix B),
with the only exception of
elements of the
last line and column. The latter, in fact, are still small of the
first order, but must be compared with the diagonal element ${\mathcal{J}}_{3}$,
which is small of the second order in the “small” non–axial
amplitudes.
These terms play an important role in the treatment of the intrinsic
component of the angular momentum along the approximate
axial–symmetry axis, which will be discussed in the next paragraph.
II.4 Intrinsic components of the angular momentum
According to the classical mechanics, the components $L_{1}$, $L_{2}$ and
$L_{3}$ of the angular momentum in the intrinsic frame
are obtained as the derivatives of the total kinetic energy
with respect to the corresponding intrinsic component of the
angular velocity:
$$L_{k}=\frac{\partial T}{\partial q_{k}}\ .$$
(8)
The part of the kinetic energy depending on the component $q_{k}$
has the form
$$T_{k}(q_{k})=\frac{1}{2}\mathcal{J}_{k}q_{k}^{2}+F_{k}q_{k}$$
(9)
where $F_{k}$ is a function of the dynamical variables $\xi_{i}$
and of their
derivatives with respect to the time, and is
small of the first order
according to our definition. As for the moments of inertia,
$\mathcal{J}_{3}$ is
small of the second order, while $\mathcal{J}_{1}$,
$\mathcal{J}_{2}$ are not small.
According to eq.(8), we have
$$\displaystyle L_{k}$$
$$\displaystyle=$$
$$\displaystyle\mathcal{J}_{k}q_{k}+F_{k}$$
$$\displaystyle q_{k}$$
$$\displaystyle=$$
$$\displaystyle\frac{L_{k}-F_{k}}{\mathcal{J}_{k}}$$
(10)
$$\displaystyle T_{k}(L_{k})$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}\mathcal{J}_{k}\ \left[\frac{1}{\mathcal{J}_{k}}\big{(%
}L_{k}-F_{k}\big{)}\right]^{2}+F_{k}\left[\frac{1}{\mathcal{J}_{k}}\big{(}L_{k%
}-F_{k}\big{)}\right]$$
(11)
$$\displaystyle=$$
$$\displaystyle\frac{L_{k}^{2}}{2\mathcal{J}_{k}}-\frac{F_{k}^{2}}{2\mathcal{J}_%
{k}}\ .$$
For $k=1,2$ the second term is small of the second order and can
be neglected.
It is not so for $k=3$, as $\mathcal{J}_{3}$ is also small and of
the same order as $F_{k}^{2}$.
In more detail, we have
$$\displaystyle L_{3}$$
$$\displaystyle=$$
$$\displaystyle\mathcal{J}_{3}q_{3}+\Big{[}\ \frac{\sqrt{8}\ \beta_{2}\beta_{3}}%
{\sqrt{\beta_{2}^{2}+5\beta_{3}^{2}}}\big{(}\gamma\dot{\xi}-\xi\dot{\gamma}%
\big{)}$$
(12)
$$\displaystyle+$$
$$\displaystyle 6\big{(}Y\dot{X}-X\dot{Y}\big{)}+2\big{(}\zeta\dot{\eta}-\eta%
\dot{\zeta}\big{)}\ \Big{]}$$
where we have put
$$\gamma=\sqrt{5}\ \gamma_{2}-\gamma_{3}\ .$$
(13)
At this point, it will be convenient to express the variables
$\gamma_{2}$ and $\gamma_{3}$ as linear combinations of two new variables,
one of which is, obviously, $\gamma=\sqrt{5}\gamma_{2}-\gamma_{3}$.
The other one, that we call $\gamma_{0}$, can be chosen proportional to
the linear combination which enters in the expression of the
determinant $G$,
$$\gamma_{0}=c\left(\beta_{2}^{2}\gamma_{2}+\sqrt{5}\ \beta_{3}^{2}\gamma_{3}%
\right).$$
(14)
With this choice, we obtain
$$\displaystyle\gamma_{2}$$
$$\displaystyle=$$
$$\displaystyle\frac{\gamma_{0}/c+\sqrt{5}\ \beta_{3}^{2}\gamma}{\beta_{2}^{2}+5%
\beta_{3}^{2}}$$
$$\displaystyle\gamma_{3}$$
$$\displaystyle=$$
$$\displaystyle\frac{\sqrt{5}\ \gamma_{0}/c-\beta_{2}^{2}\gamma}{\beta_{2}^{2}+5%
\beta_{3}^{2}}$$
(15)
and, at the leading order,
$$\beta_{2}^{2}\dot{\gamma}_{2}^{2}+\beta_{3}^{2}\dot{\gamma}_{3}^{2}=\frac{%
\beta_{2}^{2}\beta_{3}^{2}}{\beta_{2}^{2}+5\beta_{3}^{2}}\dot{\gamma}^{2}+%
\frac{1}{\beta_{2}^{2}+5\beta_{3}^{2}}\ (\dot{\gamma}_{0}/c\big{)}^{2}\ .$$
(16)
In deriving the eq. 16, the factor $1/c$ has been considered
constant. We may note,however, that the same result holds at
the leading order also if $1/c$ is a function of $\beta_{2}$ and/or
$\beta_{3}$. In fact, terms involving the time derivative of $1/c$ also
contain the “small” quantity $\gamma_{0}$, and their effect is
negligible in the present approximation (see Appendix B).
E.g., one could choose for $1/c$ a quadratic expression in
$\beta_{2},\ \beta_{3}$, in order to obtain for $\gamma_{0}$ an
adimensional quantity, like $\gamma_{2},\ \gamma_{3}$ and $\gamma$.
The same argument applies for possible redefinitions of other
“small” variables, like $\gamma$.
The expression (12) can be somewhat simplified with the
substitutions
$$\displaystyle X$$
$$\displaystyle=$$
$$\displaystyle w\sin\vartheta$$
(17a)
$$\displaystyle Y$$
$$\displaystyle=$$
$$\displaystyle w\cos\vartheta$$
(17b)
$$\displaystyle\eta$$
$$\displaystyle=$$
$$\displaystyle v\sin\varphi$$
(17c)
$$\displaystyle\zeta$$
$$\displaystyle=$$
$$\displaystyle v\cos\varphi$$
(17d)
$$\displaystyle\xi$$
$$\displaystyle=$$
$$\displaystyle u\sin\chi$$
(17e)
$$\displaystyle\gamma$$
$$\displaystyle=$$
$$\displaystyle\sqrt{2}\ \left(\sqrt{\beta_{2}^{2}+5\beta_{3}^{2}}\ /\ \beta_{2}%
\beta_{3}\right)u\cos\chi$$
(17f)
$$\displaystyle\frac{\gamma_{0}}{c}$$
$$\displaystyle=$$
$$\displaystyle f(\beta_{2},\beta_{3})\sqrt{\beta_{2}^{2}+5\beta_{3}^{2}}\ u_{0}$$
(17g)
which gives for the determinant of the matrix $\mathcal{G}$
$$G={\rm Det}\;\mathcal{G}=2304\big{(}\beta_{2}^{2}+2\beta_{3}^{2}\big{)}^{2}\ u%
_{0}^{2}\ v^{2}u^{2}w^{2}\ f^{2}(\beta_{2},\beta_{3}).$$
(18)
The choice of the function $f(\beta_{2},\beta_{3})$ is irrelevant for
what concerns the angular momentum. Non diagonal terms (small of the
first order) would depend on this choice, but their effect is
negligible (see Appendix B). As a criterion to define the
form of the function $f$, we observe that for permanent quadrupole
deformation $\beta_{2}=\bar{\beta}_{2}$ and at the limit $\beta_{3}^{2}\ll\beta_{2}^{2}$ our value of $G$ must agree with the result given, at this
limit, by Eisenberg and Greiner Eisenberg and Greiner (1987). This happens if the
function $f$ we have left undetermined tends to a constant when
$\beta_{2}\rightarrow\bar{\beta}_{2}$.
We adopt here the simplest possible choice,
i.e. $f(\beta_{2},\beta_{3})=1$, to obtain the matrix
$\mathcal{G}$
given in the Table 2, and (at the leading order)
$$\displaystyle G$$
$$\displaystyle=$$
$$\displaystyle{\rm Det}\;\mathcal{G}=2304\big{(}\beta_{2}^{2}+2\beta_{3}^{2}%
\big{)}^{2}\ u_{0}^{2}\ v^{2}u^{2}w^{2}$$
(19)
$$\displaystyle\mathcal{J}_{3}$$
$$\displaystyle=$$
$$\displaystyle 4u_{0}^{2}+2v^{2}+8u^{2}+18w^{2}$$
(20)
$$\displaystyle 2T$$
$$\displaystyle=$$
$$\displaystyle\dot{\beta}_{2}^{2}+\dot{\beta}_{3}^{2}+\dot{u}_{0}^{2}$$
(21)
$$\displaystyle+$$
$$\displaystyle 2\big{(}\dot{v}^{2}+v^{2}\dot{\varphi}^{2}\big{)}+2\big{(}\dot{u%
}^{2}+u^{2}\dot{\chi}^{2}\big{)}+2\big{(}\dot{w}^{2}+w^{2}\dot{\vartheta}^{2}%
\big{)}$$
$$\displaystyle+$$
$$\displaystyle 2q_{3}\left[2v^{2}\dot{\varphi}+4u^{2}\dot{\chi}+6w^{2}\dot{%
\vartheta}\right]$$
$$\displaystyle+$$
$$\displaystyle\mathcal{J}_{1}q_{1}^{2}+\mathcal{J}_{2}q_{2}^{2}+\mathcal{J}_{3}%
q_{3}^{2}\ .$$
II.5 Classification of elementary
excitations with respect to $K^{\pi}$
We can now deduce the intrinsic components of the angular momentum:
$$\displaystyle L_{1}$$
$$\displaystyle=$$
$$\displaystyle\mathcal{J}_{1}q_{1}$$
(22a)
$$\displaystyle L_{2}$$
$$\displaystyle=$$
$$\displaystyle\mathcal{J}_{2}q_{2}$$
(22b)
$$\displaystyle L_{3}$$
$$\displaystyle=$$
$$\displaystyle\mathcal{J}_{3}q_{3}+\left[2v^{2}\dot{\varphi}+4u^{2}\dot{\chi}+6%
w^{2}\dot{\vartheta}\right]$$
(22c)
$$\displaystyle=$$
$$\displaystyle 2v^{2}\big{(}q_{3}+\dot{\varphi}\big{)}+4u^{2}\big{(}2q_{3}+\dot%
{\chi}\big{)}+6w^{2}\big{(}3q_{3}+\dot{\vartheta}\big{)}$$
$$\displaystyle+4u_{0}^{2}q_{3}\ .$$
In the same way, we can obtain the classical moments conjugate to
$\chi$, $\vartheta$ and $\varphi$
(we observe that none of these variables appears in the expressions of
$G$ or $\mathcal{J}_{3}$):
$$\displaystyle p_{\varphi}$$
$$\displaystyle=$$
$$\displaystyle 2v^{2}(\dot{\varphi}+q_{3})$$
(22d)
$$\displaystyle p_{\chi}$$
$$\displaystyle=$$
$$\displaystyle 2u^{2}\left(\dot{\chi}+2q_{3}\right)$$
(22e)
$$\displaystyle p_{\vartheta}$$
$$\displaystyle=$$
$$\displaystyle 2w^{2}(\dot{\vartheta}+3q_{3})\ .$$
(22f)
Now we can solve the system of equations
(22) with respect to
the variables $q_{1},\ q_{2},\ q_{3},\ \dot{\varphi},\ \dot{\chi},\ \dot{\vartheta}$. We obtain
$$\displaystyle q_{1}$$
$$\displaystyle=$$
$$\displaystyle L_{1}/\mathcal{J}_{1}$$
(23a)
$$\displaystyle q_{2}$$
$$\displaystyle=$$
$$\displaystyle L_{2}/\mathcal{J}_{2}$$
(23b)
$$\displaystyle q_{3}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{u_{0}}\big{(}L_{3}-p_{\varphi}-2p_{\chi}-3p_{\vartheta}%
\big{)}$$
(23c)
$$\displaystyle\dot{\varphi}$$
$$\displaystyle=$$
$$\displaystyle\frac{p_{\varphi}}{2v_{0}^{2}}-\frac{1}{u_{0}^{2}}\big{(}L_{3}-p_%
{\varphi}-2p_{\chi}-3p_{\vartheta}\big{)}$$
(23d)
$$\displaystyle\dot{\chi}$$
$$\displaystyle=$$
$$\displaystyle\frac{p_{\chi}}{2u^{2}}-\frac{2}{u_{0}^{2}}\big{(}L_{3}-p_{%
\varphi}-2p_{\chi}-3p_{\vartheta}\big{)}$$
(23e)
$$\displaystyle\dot{\vartheta}$$
$$\displaystyle=$$
$$\displaystyle\frac{p_{\vartheta}}{2\chi_{0}^{2}}-\frac{3}{u_{0}^{2}}\big{(}L_{%
3}-p_{\varphi}-2p_{\chi}-3p_{\vartheta}\big{)}$$
(23f)
The equations (23) have a very simple meaning in the case
where the potential energy does not depend on the variables $\varphi$,
$\chi$ or $\vartheta$. In such a case (a sort of model
$\varphi$-$\chi$-$\vartheta$–instable, in the sense of the
$\gamma$–instable model by Wilets and Jean Wilets and Jean (1956))
the conjugate moments of these three angular variables are
constants of the
motion, with integer eigenvalues $n_{\varphi}$, $n_{\chi}$ and $n_{\vartheta}$
(in units of $\hbar$).
Moreover, if we assume that $u_{0}\rightarrow 0$,
the third component $q_{3}$ of the angular velocity tends to
$\infty$ unless $L_{3}-p_{\varphi}-2p_{\chi}-3p_{\vartheta}=0$
(eq. (23c)).
In this case, the operator $L_{3}$ is diagonal, with eigenvalues
$K=n_{\varphi}+2n_{\chi}+3n_{\vartheta}$,
and the three degrees of freedom corresponding to
$\varphi$, $\chi$ and $\vartheta$ can be associated to non-axial
excitation modes with $K=$1, 2 and 3, respectively.
To investigate the character of the degree of freedom described by the
parameter $u_{0}$, and for a deeper understanding of the nature of
the other degrees of freedom, it is necessary to express the complete
Hamiltonian in the frame of a definite model which, although not
unique, is at least completely self-consistent at the limit close to
the axial symmetry.
In fact, it is now possible to use the Pauli prescriptions
Pauli (1933) to construct the
quantum operator $\hat{T}$ corresponding to the classical kinetic
energy $T$ of eq. (21). In doing this, we make use of the
partial inversion of the matrix $\mathcal{G}$ given by
the solution (23) of the linear system (22),
and note that none of the variables $\varphi$, $\chi$, $\vartheta$
or $q_{3}$
enter in the expression of $G$:
$$\displaystyle\frac{2}{\hbar^{2}}\ \hat{T}$$
$$\displaystyle=$$
$$\displaystyle-\left\{\frac{1}{\beta_{2}^{2}+2\beta_{3}^{2}}\frac{\partial}{%
\partial\beta_{2}}\left[\big{(}\beta_{2}^{2}+2\beta_{3}^{2}\big{)}\frac{%
\partial}{\partial\beta_{2}}\right]\right.$$
(24)
$$\displaystyle+$$
$$\displaystyle\frac{1}{\beta_{2}^{2}+2\beta_{3}^{2}}\frac{\partial}{\partial%
\beta_{3}}\left[\big{(}\beta_{2}^{2}+2\beta_{3}^{2}\big{)}\frac{\partial}{%
\partial\beta_{3}}\right]$$
$$\displaystyle+$$
$$\displaystyle\frac{1}{u_{0}}\frac{\partial}{\partial u_{0}}\left[u_{0}\frac{%
\partial}{\partial u_{0}}\right]+\frac{1}{2v}\frac{\partial}{\partial v}\left[%
v\frac{\partial}{\partial v}\right]$$
$$\displaystyle+$$
$$\displaystyle\frac{1}{2u}\frac{\partial}{\partial u}\left[u\frac{\partial}{%
\partial u}\right]+\frac{1}{2w}\frac{\partial}{\partial w}\left[w\frac{%
\partial}{\partial w}\right]$$
$$\displaystyle+$$
$$\displaystyle\left.\ \frac{1}{2v^{2}}\frac{\partial^{2}}{\partial\varphi^{2}}+%
\frac{1}{2u^{2}}\frac{\partial 2}{\partial\chi 2}+\frac{1}{2w^{2}}\frac{%
\partial 2}{\partial\vartheta 2}\right\}$$
$$\displaystyle+$$
$$\displaystyle\frac{1}{4u_{0}^{2}}\ \left[\hat{L}_{3}-\frac{\partial}{\partial%
\varphi}-2\frac{\partial}{\partial\chi}-3\frac{\partial}{\partial\vartheta}%
\right]^{2}$$
$$\displaystyle+$$
$$\displaystyle\frac{1}{2\mathcal{J}_{1}}\ \hat{L}_{1}^{2}+\frac{1}{2\mathcal{J}%
_{2}}\ \hat{L}_{2}^{2}\ .$$
This is, admittedly, only a semi-classical discussion. However, the
formal quantum treatment in the frame of the Pauli procedure,
shown in the Appendix C, gives exactly the same result.
Until now, no assumption has been made on the form of the
potential-energy operator, which will determine the particular model.
A few general remarks on this subject are contained in the Appendix
D. We now assume that the potential energy can be separated
in the sum of a term depending only on $u_{0}$ and another
containing the other dynamical variables. In this case the
differential equation in the variable $u_{0}$ is approximately
decoupled from the rest. One obtains the Schrödinger equation
$$\displaystyle\left\{\frac{1}{u_{0}}\frac{\partial}{\partial u_{0}}\left[u_{0}%
\frac{\partial}{\partial u_{0}}\right]\right.$$
$$\displaystyle+$$
$$\displaystyle\frac{2}{\hbar^{2}}\left[E_{u_{0}}-U(u_{0})\right]$$
$$\displaystyle-\ \frac{1}{u_{0}^{2}}$$
$$\displaystyle\left.\left[\frac{\Omega_{u_{0}}}{2}\right]^{2}\right\}\phi(u_{0}%
)=0$$
(25)
where we have put
$$\Omega_{u_{0}}=K-n_{\varphi}-2n_{\chi}-3n_{\vartheta}\ .$$
(26)
If we assume, for simplicity, a harmonic form for the potential
$U(u_{0})=\frac{1}{2}Cu_{0}^{2}$, the eq. (25) is the
radial equation of a bidimensional harmonic oscillator. For the
existence of a solution, it is required that
$$\displaystyle\Omega_{u_{0}}$$
$$\displaystyle=$$
$$\displaystyle 2n_{u_{0}}$$
$$\displaystyle E_{u_{0}}$$
$$\displaystyle=$$
$$\displaystyle(N_{u_{0}}+1)\hbar\omega_{u_{0}}$$
(27)
with $n_{u_{0}}$ positive or negative integer and
the integer $N_{u_{0}}\geq|n_{u_{0}}|$.
Excitations in the degree of freedom corresponding to the variable
$u_{0}$ carry, therefore, two units of angular momentum in the
direction of the 3rd axis of the intrinsic reference frame.
We could extend our model to include all the intrinsic variables
different from $\beta_{2},\ \beta_{3}$. We assume a potential energy
corresponding to the sum of independent harmonic potentials in the
variables $v,\ u,\ w$ and $u_{0}$, plus a term depending
on $\beta_{2}$ and $\beta_{3}$ (at the moment, we do not need to define the
form of this term). We also assume that the equations in
$\beta_{2},\ \beta_{3}$ can be approximately decoupled from those of the
other variables and that, in the latter, $\beta_{2}$ and $\beta_{3}$
can be replaced by their average values. It is easy to verify that the
differential equations in the pair of variables $v,\ \varphi$
(or $u,\ \chi$ or $w,\ \vartheta$) correspond again to a
bidimensional harmonic oscillator and that, as long as we
neglect the rotation–vibration coupling, the eigenvalue
$K$ of the intrinsic component $L_{3}$ of the angular momentum is given
by
$$K=n_{\varphi}+2n_{\chi}+3n_{\vartheta}+2n_{u_{0}}\ .$$
(28)
The energy eigenvalues are, for the equation in the variables
$v,\ \varphi$,
$$E_{v}=(N_{v}+1)\ \hbar\omega_{v}\hskip 22.762205pt{\rm with}\ N_{v}\geq|n_{%
\varphi}|$$
(29)
and have a similar form for the other two oscillators.
It remains to consider the character of the different dynamical
variables with respect to the parity operator.
We know that the parity of the amplitude $a^{(\lambda)}_{\mu}$ is
$(-1)^{\lambda}$. Therefore, $\beta_{2}$ and $\gamma_{2}$ are even, while
$\beta_{3}$, $X$ and $Y$ are odd. As for $\gamma_{3}$, we observe that
$\beta_{3}\sin\gamma_{3}$ is odd, and therefore $\gamma_{3}$ must be
even.
As a consequence, are also even the linear combinations $\gamma$ and
$\gamma_{0}/c$ defined in the eq. (13,14).
The new variables $\eta,\ \zeta$ and $\xi$ are odd, as (e.g.)
$\beta_{2}\xi$ is an octupole amplitude and therefore is odd, while
$\beta_{3}\xi$ must be even. Finally, on the basis of eq.s (17)
we realize that $v,\ u$ and $w$ must be odd
(while $\varphi,\chi$ and $\vartheta$ are even). We have therefore
identified elementary excitations corresponding to
$K^{\pi}=1^{-},2^{-},3^{-}$ and $2^{+}$. Excitations with $K=0$ (of positive or
negative parity) conserve the axial symmetry, and are related to the
variables $\beta_{2}$ and $\beta_{3}$. A particular example will be
discussed in the following Sections.
III A specific model:
Axial octupole vibrations in nuclei with
permanent quadrupole deformation
Specific assumptions on the form of the
potential–energy terms for all the variables describing the
quadrupole and octupole degrees of freedom are necessary in order to
obtain definite predictions, also if these are limited to the axial
modes.
We discuss here, as an example, the case of axial octupole excitations
in nuclei which already possess a stable quadrupole deformation. In
this case, one obtains relatively simple results, suitable for
comparison with experimental data. This comparison will be performed
in the next Section.
Following the usual treatment of vibration + rotation, we
put $\beta_{2}=\bar{\beta}_{2}+\beta_{2}^{\prime}$, with $\bar{\beta}_{2}=$
constant and $|\beta_{2}^{\prime}|\ll|\bar{\beta}_{2}|$.
The new variable $\beta_{2}^{\prime}$ is therefore assumed to be small
(of the first order) as all other variables, with the exception of
$\beta_{3}$.
With this choice, and assuming that the variables introduced in the
eq. (17) are suitable to describe the other degrees of freedom,
the matrix $\mathcal{G}$ takes a form similar to that of Table II (with
$\beta_{2}$ substituted by $\bar{\beta}_{2}$ and $\dot{\beta}_{2}$ by
$\dot{\beta}_{2}^{\prime}$) and – at the lowest significant order –
results to be diagonal with respect to the variables $\dot{\beta}_{3}$,
$\dot{\beta}_{2}^{\prime}$, $q_{1}$ and $q_{2}$.
Moreover, in our model, the amplitude of oscillation for all
degrees of freedom different from $\beta_{3}$ are constrained to very
small values: this fact implies strong restoring forces and,
therefore, oscillation frequencies much larger than for $\beta_{3}$.
In the limit of small amplitude of the octupole oscillations, this
case has been discussed, e.g., by Eisenberg and
Greiner Eisenberg and Greiner (1987).
In their approach, the “intrinsic” reference frame is chosen to
coincide with the principal axis of the quadrupole deformation tensor,
but the differences between their approach and ours tend to disappear
for $|\beta_{3}|\ll\bar{\beta}_{2}$. The model we try to develop should
be able to describe (axial) octupole vibrations of finite amplitude,
but its limit for $|\beta_{3}|\ll\bar{\beta}_{2}$ must obviously agree
with the results of Eisenberg and Greiner.
The quantum-mechanical equation of motion for $\beta_{3}$ can be
obtained with the Pauli prescription Pauli (1933), with the additional
assumption that the equations involving $\beta_{3}$
and $\beta_{2}^{\prime}$ or the angular-momentum components $L_{1},\ L_{2}$ are
effectively decoupled from those containing the other dynamical
variables and/or the $L_{3}$ operator. The latter equations could
possibly be complicated, and substantially coupled with one another
and with the angular momentum component $L_{3}$ along the (approximate)
symmetry axis. A short discussion of this subject, with some
simplifying assumptions, has been given in the previous
section II.4.
At the moment, we assume that terms involving $\beta_{2}^{\prime},\ L_{3}$
and other dynamical variables different from $\beta_{3}$ contribute to
the total energy with their own eigenvalue,
independent from the
eigenfunction in the $\beta_{3}$ degree of freedom, and we only consider
their lowest–energy state. We also assume that this state has $K=0$
(and neglect, as usual, the possible rotation-vibration coupling). In
this case the complete wavefunction has the form
$$\displaystyle\Psi$$
$$\displaystyle\propto$$
$$\displaystyle\psi(\beta_{3})\Phi_{0}(\beta_{2}^{\prime},\gamma_{0},w,\rho,\xi_%
{0},\vartheta,\chi,\varphi)\mathcal{D}^{J}_{M,0}$$
(30)
$$\displaystyle\propto$$
$$\displaystyle\psi(\beta_{3})\Phi_{0}\ Y_{JM}(\hat{\Omega})$$
where $\mathcal{D}^{J}_{M,M^{\prime}}$ are the Wigner matrices and
$\psi(-\beta_{3})=(-)^{J}\psi(\beta_{3})$. The differential equation
for the wavefunction $\psi(\beta_{3})$, obtained with the Pauli
prescription, has the form
$$\displaystyle\left[-\frac{\hbar^{2}}{2}\right.$$
$$\displaystyle G^{-1/2}$$
$$\displaystyle\frac{\rm d}{{\rm d}\beta_{3}}G^{1/2}\frac{\rm d}{{\rm d}\beta_{3%
}}+V(\beta_{3})$$
(31)
$$\displaystyle+$$
$$\displaystyle\left.\!\frac{\hbar^{2}}{2}\frac{J(J+1)}{\mathcal{J}_{1}}\right]%
\psi(\beta_{3})=E\psi(\beta_{3})$$
where $V(\beta_{3})$ is the potential-energy term.
The expression of the determinant $G$ is not uniquely defined, as it
also depends on the part of the matrix of inertia $\mathcal{G}$
involving all
other dynamical variables. This is a general problem for all models
where part of the degrees of freedom are ignored (and, in collective
models, a number of dynamical variables describing the details of
nucleon degrees of freedom, are certainly ignored).
With the present choice of dynamical variables (Table 2),
one gets
$G=$Det$\;\mathcal{G}=2304\ u_{0}^{2}\ v^{2}u^{2}w^{2}(\bar{\beta}_{2}^{2}+2\beta_{3}^%
{2})^{2}$,
and the eq. (31) becomes
$$\displaystyle\left[-\frac{\hbar^{2}}{2}\!\right.$$
$$\displaystyle\!\left(\frac{{\rm d}^{2}}{{\rm d}\beta_{3}^{2}}+\frac{4\beta_{3}%
}{\bar{\beta}_{2}^{2}+2\beta_{3}^{2}}\frac{\rm d}{{\rm d}\beta_{3}}\right)+V(%
\beta_{3})$$
(32)
$$\displaystyle+$$
$$\displaystyle\left.\frac{\hbar^{2}}{2}\frac{J(J+1)}{\mathcal{J}_{1}}\right]%
\psi(\beta_{3})=E\psi(\beta_{3})\ .$$
(33)
This is the equation we have used in ref.s Bizzeti and Bizzeti-Sona (2004); Bizzeti (2003).
A different choice of the dynamical variables would have brought to a
different result. E.g., with the variables used in the
Table 1,
one obtains
$G\propto\bar{\beta}_{2}^{2}\beta_{3}^{2}(\bar{\beta}_{2}^{2}+2\beta_{3}^{2})^{%
2}(\bar{\beta}_{2}^{2}+5\beta_{3}^{2})^{-1}(\bar{\beta}_{2}^{2}\gamma_{2}+%
\sqrt{5}\beta_{3}^{2}\gamma_{3})^{2}$.
However, in this case the limit $|\beta_{3}|\ll|\bar{\beta}_{2}|$
would not correspond to the Eisenberg–Greiner result, due to the
presence of the factor $\beta_{3}^{2}$ in the expression of the
determinant $G$. In fact, when the spherical symmetry is broken by the
permanent quadrupole deformation, and at the limit of small octupole
deformation, the octupole amplitudes $a^{(3)}_{\mu}$ are decoupled from
one another Eisenberg and Greiner (1987) and it would have been more reasonable
to chose a
dynamical variable $u_{3}=\beta_{3}\gamma_{3}$ in the place of $\gamma_{3}$.
With this substitution, the factor $\beta_{3}^{2}$ disappears.
It is convenient to express the differential equation (33) in
terms of the adimensional quantities
$x=\sqrt{2}\beta_{3}/\bar{\beta}_{2}$ and $\epsilon=\frac{1}{\hbar^{2}}\bar{\beta}_{2}^{2}E$,
$v=\frac{1}{\hbar^{2}}\bar{\beta}_{2}^{2}V$. One obtains
$$\frac{{\rm d}^{2}\psi(x)}{{\rm d}x^{2}}+\frac{2x}{1+x^{2}}\frac{\rm d\psi(x)}{%
{\rm d}x}+\left[\epsilon-\frac{J(J+1)}{6(1+x^{2})}-v(x)\right]\psi(x)\!=\!0$$
(34)
where $\psi(-x)=(-)^{J}\psi(x)$.
This equation reduces to that of Eisenberg and Greiner Eisenberg and Greiner (1987)
when $x\ll 1$.
As for the potential $v(x)$, we have explored two possible
forms: a quadratic term111After completion of this work, we
have been informed that a quadratic potential plus a centrifugal
term with variable moment of inertia has also been considered in the
model by Minkov et al. Minkov et al. (2004).
$v(x)=\frac{1}{2}cx^{2}$ or a
square–well potential, as it has been adopted Iachello (2000b)
at the critical point in the X(5) model ( $v(x)=0$ for
$|x|<b$ and $=+\infty$ outside, so that $\psi(\pm b)=0$ ).
In both cases, there is a free parameter to be determined from the
comparison with empirical data.
We now discuss in particular the second case.
We may note that for $V=0$ the eq. (34) is formally
equivalent to that of spheroidal oblate wavefunctions (see
eq. 21.6.3 of ref. Abramowitz and Stegun (1970)) with the parameters $m,\ \lambda$ and
$C$ redefined as $m=0$, $C^{2}=\epsilon$ and
$\lambda=J(J+1)/6-\epsilon$. Here, however, the solution is confined
in the interval $-b<x<b$ and the equation has been solved
numerically.
For a given $b$ and for every value of $J$, one obtains a complete set
of orthogonal eigenfunctions, with an integrating factor $(1+x^{2})$.
These eigenfunctions can be characterized by the quantum
number $s=\nu+1$, where $\nu$ is the number of zeroes in the open
interval $0<x<b$. A few examples of wavefunctions corresponding to
the square–well potential with $b=1.73$ and for different values of
$s$ and $J$ are depicted in the Fig. 1.
The dependence of the eigenvalues on the parameter $b$ is illustrated
by the Fig. 2, where the ratios $E(J)/E(2)$ are shown for the
g.s. band ($s=1$).
Other possible choices of the set of independent
dynamical variables would have brought to a different equation, but
the difference would have concerned the coefficient of the
first–derivative term the eq. (34),
with very small effect on the results, as long
as $b$ is in a range of “reasonable” values.
To exemplify the effect of this term, results obtained with the
coefficient of the first derivative put to zero are also shown – as
dotted lines – in the Figure 2. Differences between the two
sets of
results turn out to be very limited for small values of $b$ (at least,
up to $b\approx 2$).
There is at least one case (${}^{226}$Th) in which our results
with the square–well
potential are in good agreement with the level scheme, for low-lying
states of positive and negative parity, while for ${}^{228}$Th
a better agreement is obtained with the quadratic potential.
The possible interpretation of this result as evidence for a phase
transition in the octupole degree of freedom is discussed in the
next Section.
IV Evidence of phase transition
in the octupole degree of freedom
IV.1 The Radium and Thorium isotopic chain
A phase transition in the nuclear shape manifests itself as a
relatively sharp change of a proper order parameter –
e.g., the ratio $R=E(4^{+})/E(2^{+})$ – as a function of a
driving
parameter which can be, in our case, the number of neutrons in the
isotopes of a given element or the number of protons along an isotone
chain. Due to the finite number of degrees of freedom, the transition
region has a finite width around the critical point, and extends over
several nuclides in the chain. In the case of transitions between
spherical shape and axial quadrupole deformation, the X(5) symmetry,
valid at the critical point, predicts
$E(4^{+})/E(2^{+})=2.91$ and we can use this criterion to locate the
critical point of the phase transition.
The situation is more complex when the quadrupole and the octupole
degrees of freedom must be considered at the same time. Fig. 3
shows, as a function of the neutron number of Ra and Th isotopes, a
few parameters which can be used as indicators of the quadrupole and
octupole collectivity. As for the quadrupole mode, the decrease of
$E(2^{+})$ with increasing $N$ (Fig. 3$a$) shows a corresponding
increase of collectivity.
Moreover, in the Fig. 3$b$
we observe the transition between the vibrational (or not
collective) behavior of the lighter isotopes of the chain
and a clear rotational
behavior ($R\approx 10/3$) above $A=226$, with a critical point
which can be located around $A=224$.
The ratio $E(1^{-})/E(2^{+})$, depicted in Fig. 3$d$, shows that the
relative importance of the octupole collectivity increases with
decreasing $N$ and reaches its maximum in the region below $N=138$,
where the
critical point of the phase transition in the quadrupole mode could be
located on the basis of Fig. 3$b$.
Heavier isotopes show evidence of octupole vibrations
(of different $K$) around a quadrupole-deformed core Guntner et al. (2000); Raduta et al. (1996b).
Lighter isotopes ($N<132$)
appear not to be deformed in their lower-$J$ states.
However, at larger angular momentum a rotational-like band develops,
and this band has the alternate-parity pattern typical of a
stable octupole deformation Smith et al. (1995).
The model introduced in the first part of this paper, and developed in
the section III for the particular case of a permanent quadrupole
deformation, assumes that non–axial amplitudes are constrained to
very small values by the large restoring forces. This implies that
excitation of one of the non–axial degrees of freedom leads to high
excitation energy, compared to that of
the first $1^{-}$ level of the $K=0$ band. Experimental data of
Fig. 3$c$
show that this is actually the case for the light Thorium isotopes, at
least up to $A=228$. In fact, the first $1^{-}$ level is not far from
the first
$2^{+}$ and much lower than levels belonging to negative–parity bands
with $K\neq 0$ (as the lowest $2^{-}$ or the second $1^{-}$).
IV.2 Comparison with experimental
data for ${}^{226,228}$Th
Our model assumes a permanent quadrupole deformation. Therefore, it
can be useful only for relatively heavy Th isotopes
(Fig. 3$a,b$). The quadrupole-deformed
region extends above the mass 224 (which could correspond to the
critical point of the phase
transition, having $E(4^{+})/E(2^{+})\approx 2.91$). Heavier Th isotopes
(with $A\geq 230$) show negative-parity bands built on the different
states of octupole vibration, from $K=0$ to $K=3$, with band heads much
higher than the first $2^{+}$ (Fig. 3$c$).
Only for lower $A$, the $1^{-}$ band head
of the $K^{\pi}=0^{-}$ band decreases well below the band heads of all
other octupole bands, and higher levels of the $K^{\pi}=0^{-}$ band
merge with those
of positive-parity of the ground-state band (Fig. 4),
approaching (but not
reaching) the pattern expected for a rigid, reflection asymmetric
rotor.
The region of possible validity of our model is therefore restricted
to ${}^{226}$Th and ${}^{228}$Th.
The ratios $E(J^{\pi})/E(2^{+})$ for the low-lying states of
${}^{226}$Th and ${}^{228}$Th are depicted in the Fig. 4, and
compared with the predictions of different models. The rigid-rotor
model cannot account for the position of the lowest negative-parity
levels, and overestimates the excitation energy for all the high-spin
states. Instead, a rather good agreement is obtained with the present
model, if one assumes, in the case of ${}^{226}$Th, a square-well
potential (as the one hypothesized by Iachello in his X(5) model)
and, in the case of ${}^{228}$Th, a harmonic restoring force.
In both cases, the free parameter of the model has been adjusted to
reproduce the position of the $1^{-}$ level. As shown in the Fig.
4 (dotted line) a much better agreement with the high-spin
levels of ${}^{226}$Th is obtained with a slightly different value of
the parameter ($b=1.87$ instead of 1.73),
at the expense of a very limited discrepancy
for the $1^{-}$ level.
Therefore, for what concerns the level energies of the ground-state
band (including in it also the odd-$J$, negative-parity states),
${}^{226}$Th seems to present the expected behavior of a nucleus with
permanent quadrupole deformation and close to the critical point of the
phase transition in the octupole mode.
IV.3
Other possible tests of the critical-point behavior
A considerable amount of experimental information has been reported,
in the last few years, on possible
candidates Casten and Zamfir (2001); Krucken et al. (2002); Bizzeti and Bizzeti-Sona (2002); Hutter et al. (2003); Clark et al. (2003); Fransen et al. (2004); Bijker et al. (2003, 2004); Tonev et al. (2004)
for the dynamical
symmetry X(5)(phase transition point in the quadrupole mode).
It is now clear that the agreement between experimental and calculated
energies for the ground-state band does not automatically imply that a
similar agreement exists also for other observables, like the excitation
energy of the second $0^{+}$ level (the band head of the $s=2$ band)
and the in-band and inter-band transition probabilities.
In several transitional nuclei, the excitation energies in the
ground-state band are in excellent agreement with the X(5)
predictions but the calculated ratios of the $B$(E2) transition
probabilities fail to reproduce the experimental ones Clark et al. (2003),
unless an ad-hoc second-order term is included in the E2
transition operator Arias (2001); Caprio (2004).
It is therefore important to test the predictions of our models
also for what concerns such observables.
The low–lying level scheme of ${}^{226}$Th is shown in the
Fig. 5, together with the one resulting from the present
model, with the value of $b$ adjusted in order to reproduce the
empirical value of the ratio $E(1^{-})/E(2^{+})$.
At the moment, only the first two levels of the $s=2$ band
($0^{+}_{2},\ 2^{+}_{2}$)
are known and their excitation energies are somewhat higher than the
values predicted at the critical point.
We can observe, however, that also in the best X(5) nuclei
Casten and Zamfir (2001); Krucken et al. (2002); Tonev et al. (2004) the
position of the levels of the $s=2$ band deviates somewhat from the
model predictions (although in the opposite direction).
In our opinion, a similar
qualitative agreement is obtained also in the present case. The
negative–parity levels of the $s=2$ band are predicted to lie at
higher energies, and could be difficult to observe.
Absolute values of the transition strengths are not available for
${}^{226}$Th, but some relevant information is provided by the branching
ratios in the level decays. In the tables 3,4,
experimental ratios of the reduced transition strengths for E1 or E2
transitions coming from the same level are compared with the model
prediction at the critical point.
The electric dipole moment would vanish for the collective motion of
a fluid with uniform charge density, as the center of charge would
coincide with the center of mass. Therefore,the observed E1 transition
amplitudes are entirely due to the non uniformity of the
nuclear charge distribution Tsvekov et al. (2002).
To calculate the value of
$B$(E1)$=(i||\mathcal{M}(E1)||f)^{2}/(2J_{i}+1)$,
the E1 transition operator has been assumed to have the
form Bohr and Mottelson (1957, 1958); Strutinski (1957); Lipas (1963)
$$\mathcal{M}_{\mu}(E1)=C_{1}\beta_{2}\beta_{3}Y_{1,\mu}\ ,$$
(35)
with the constant factor $C_{1}$ depending on the nuclear charge
polarizability.
The E2 transition operator for the in-band transition and at the limit
close to the axial symmetry has been taken in the simple form
$$\mathcal{M}_{\mu}(E2)=C_{2}(\beta_{0})Y_{2,\mu}\ ,$$
(36)
neglecting the (weak) dependence on $\beta_{3}^{2}$.
Therefore, the theoretical ratio of the reduced strengths for
transitions of different multipolarity (E1 and E2) is determined apart
from a constant factor, which must be fixed by comparison with the
experimental data. The average of the ratios
$B$(E1)/$B$(E2) for the transitions coming from the $10^{+}$ and $11^{-}$
levels has been used for normalization of the theoretical values given
in the Table 4.
V Conclusions
A theoretical scheme for the description of quadrupole plus octupole
excitations close to the axial symmetry limit has been developed in
the Section II and specialized in the Section III to the
simpler case of a permanent (and axially symmetric) quadrupole
deformation. In principle, the model should be able to describe the
wide field of reflection–asymmetric nuclear shapes, close to (but
not coincident with!) the axial symmetry limit. Calculations of
nuclear shapes in the frame of the HBF-Cranking model Nazarewicz and Olanders (1985) for
nuclei of the Radium – Thorium region find a large variety of
results, including proper potentials
for quadrupole–octupole vibrations around a spherical
shape or for octupole vibrations around a deformed, reflection symmetric
shape, and also situations with a rather flat minimum of the potential
along a line at constant $\beta_{2}$ with $|\beta_{3}|<\beta_{3}^{max}$.
The latter case
is just what is expected for our “critical point” of the
phase transition in the octupole mode.
In the Section IV, we have investigated the
evolution of the nuclear shape along the isotopic chain of Thorium,
and shown that evidence of phase transition exists, not only in the
quadrupole mode but also in the octupole mode around a stable
quadrupole deformation. The model developed in the Section III
results to be able to account for the experimental data of ${}^{226}$Th
(at the critical point of the phase transition in the
octupole mode), and also of its
neighbor ${}^{228}$Th (characterized by axial octupole
vibrations). More and improved experimental data on E2 and E1
transition strengths would be necessary for a more stringent test of
the model predictions.
Further developments of the calculations are in program, to provide
detailed predictions for other significant cases: first of all, those
of ${}^{224}$Ra and ${}^{224}$Th, whose positive–parity levels show an
energy sequence very close to Iachello X(5) predictions
for the critical point of phase transition between
spherical shape and axially symmetric quadrupole deformation.
Appendix A Summary of the general formalism
The general formalism to describe collective states of
rotation/vibration in nuclei is discussed e.g. in
ref. Bohr (1952).
The nuclear surface is described in polar coordinates as
$$r(\theta,\phi)=R_{0}\left[1+\sum_{\lambda}\sum_{\mu=-\lambda,\lambda}\alpha^{(%
\lambda)}_{\mu}Y^{*}_{\lambda,\mu}(\theta,\phi)\right],$$
(37)
with the condition $\alpha^{(\lambda)}_{-\mu}=(-1)^{\mu}{\alpha^{(\lambda)}_{\mu}}^{*}$.
In the sum, the values of $\lambda$ are now limited to 2,3.
A term with $\lambda=1$ should be included in order to
maintain fixed
the position of the center of mass Eisenberg and Greiner (1987):
$$\displaystyle\alpha^{(1)}_{\mu}$$
$$\displaystyle=$$
$$\displaystyle\frac{3}{\sqrt{16\pi}}\ \sum_{\lambda,\lambda^{\prime}}\sqrt{(2%
\lambda+1)(2\lambda^{\prime}+1)}$$
(38)
$$\displaystyle\cdot\left(\begin{array}[]{ccc}\lambda&\lambda^{\prime}&1\\
0&0&0\end{array}\right)\ \left[\alpha^{(\lambda)}\otimes\alpha^{(\lambda^{%
\prime})}\right]^{(1)}_{\mu}\ .$$
This term, however, is inessential
for the following discussion, and will be omitted.
For not-too-large deformation, in fact, the amplitudes
$\alpha^{(1)}_{\mu}$ are much smaller than the others, and their effect
results to be negligible for most purposes (with the noticeable
exception of the E1 transition amplitudes). At the same level of
accuracy,
one can also neglect the slight variation of $R_{0}$ necessary to keep
the volume exactly constant.
In the Bohr model that we are considering here, the classical
expression of the collective kinetic energy is
$$T=\frac{1}{2}B_{2}\sum_{\mu=-2,2}\left|\dot{\alpha}^{(2)}_{\mu}\right|^{2}+%
\frac{1}{2}B_{3}\sum_{\mu=-3,3}\left|\dot{\alpha}^{(3)}_{\mu}\right|^{2}\ .$$
(39)
In order to simplify the notation in the following, we include the
inertia coefficient $B_{\lambda}$ in the definition of the collective
variables $a^{(\lambda)}_{\mu}$.
In the literature, this symbol is usually reserved
to the variables defined in the intrinsic reference frame and, from
now on, we will always use this reference frame for the collective
variables $a^{(\lambda)}_{\mu}$:
$$\sqrt{B_{\lambda}}\ \alpha^{(\lambda)}_{\mu}=\sum_{\nu}a^{(\lambda)}_{\nu}{D^{%
(\lambda)}_{\mu\nu}}^{*}(\theta_{i})$$
(40)
where $D^{(\lambda)}$ are the Wigner matrices and $\theta_{i}$ the Euler angles.
The axis of the intrinsic reference frame are defined along the
principal axis of the inertia tensor.
The expression of the kinetic energy
in terms of the time derivatives
of the intrinsic deformation variables $a^{(\lambda)}_{\mu}$
and of the angular velocity $\vec{q}$ of the intrinsic frame with
respect to an inertial frame is discussed, e.g., in the ref.s
Bohr (1952); Eisenberg and Greiner (1987).
If only the quadrupole mode is considered, the total kinetic energy
can be expressed as the sum of a vibrational term (in the intrinsic
frame) and a rotational term. If the octupole mode is also considered,
a rotation-vibration coupling term must be added Eisenberg and Greiner (1987); Wexler and Dussel (1999):
$$T=T_{\rm vib}+T_{\rm rot}+T_{\rm coup}$$
(41)
where
$$\displaystyle T_{\rm vib}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}\sum_{\lambda,\mu}\left|\dot{a}^{(\lambda)}_{\mu}%
\right|^{2}$$
(42)
$$\displaystyle T_{\rm rot}$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{2}\sum_{\lambda,\nu,\nu^{\prime}}a^{(\lambda)}_{\nu}{a^{%
(\lambda)}_{\nu^{\prime}}}^{*}\sum_{k,k^{\prime}}q_{k}q_{k^{\prime}}\left(M^{%
\{\lambda\}}_{k}M^{\{\lambda\}}_{k^{\prime}}\right)_{\nu\nu^{\prime}}$$
(43)
$$\displaystyle\equiv$$
$$\displaystyle\frac{1}{2}\sum_{k,k^{\prime}}q_{k}q_{k^{\prime}}\mathcal{J}_{kk^%
{\prime}}\ ,$$
$$\displaystyle T_{\rm coup}$$
$$\displaystyle=$$
$$\displaystyle\frac{i}{2}\sum_{\lambda,\nu,\nu^{\prime},k}q_{k}\left[\dot{a}^{(%
\lambda)}_{\nu}{a^{(\lambda)}_{\nu^{\prime}}}^{*}-a^{(\lambda)}_{\nu}{{\dot{a}%
}^{(\lambda)*}_{\nu^{\prime}}}\right]\left(M^{\{\lambda\}}_{k}\right)_{\nu,\nu%
^{\prime}}$$
$$\displaystyle=$$
$$\displaystyle i$$
$$\displaystyle\!\sum_{\lambda}\!\sqrt{3(2\lambda\!+\!1)}\left[q^{(1)}\!\otimes%
\!\left[a^{(\lambda)}\!\otimes\dot{a}^{(\lambda)}\right]^{(1)}\right]^{(0)}_{0%
}\!.$$
(44)
Here, $q_{k}$ ($k=1,2,3$) are the Cartesian components of the angular
velocity $\vec{q}$ along the axes of the intrinsic frame, while
the $M^{\{\lambda\}}_{k}$ are $(2\lambda+1)$–dimensional matrices
giving the quantum–mechanical representation of the Cartesian
components of an angular momentum $M=\lambda$ in the intrinsic frame,
subject to commutation rules of the form
$$M^{\{\lambda\}}_{1}M^{\{\lambda\}}_{2}-M^{\{\lambda\}}_{2}M^{\{\lambda\}}_{1}=%
-iM^{\{\lambda\}}_{3}\ .$$
(45)
We assume, as usually,
$$\displaystyle\left(\right.$$
$$\displaystyle M^{\{\lambda\}}_{3}\left.\right)_{\nu\nu^{\prime}}=\nu\delta_{%
\nu,\nu^{\prime}}$$
$$\displaystyle\left(\right.$$
$$\displaystyle M^{\{\lambda\}}_{1}\left.\right)_{\nu\nu^{\prime}}=\frac{1}{2}\ %
\left(\sqrt{(\lambda-\nu)\ (\lambda+\nu+1)}\ \delta_{\nu^{\prime},\nu+1}\right.$$
$$\displaystyle+\left.\sqrt{(\lambda+\nu)\ (\lambda-\nu+1)}\ \delta_{\nu^{\prime%
},\nu-1}\right)$$
$$\displaystyle\left(\right.$$
$$\displaystyle M^{\{\lambda\}}_{2}\left.\right)_{\nu\nu^{\prime}}=\frac{1}{2}%
\left(-i\ \sqrt{(\lambda-\nu)\ (\lambda+\nu+1)}\ \delta_{\nu^{\prime},\nu+1}\right.$$
$$\displaystyle+\left.i\ \sqrt{(\lambda+\nu)\ (\lambda-\nu+1)}\ \delta_{\nu^{%
\prime},\nu-1}\right).$$
Taking into account the properties of the $M^{\{\lambda\}}_{k}$,
it is possible to obtain the explicit expression for the diagonal and
non diagonal elements of the tensor of inertia Bohr (1952); Wexler and Dussel (1999)
$$\mathcal{J}_{kk^{\prime}}=\frac{1}{2}\sum_{\lambda,\nu,\nu^{\prime}}a^{(%
\lambda)}_{\nu}{a^{(\lambda)}_{\nu^{\prime}}}^{*}\left(M^{\{\lambda\}}_{k}M^{%
\{\lambda\}}_{k^{\prime}}+M^{\{\lambda\}}_{k^{\prime}}M^{\{\lambda\}}_{k}%
\right)_{\nu\nu^{\prime}}\ .$$
(46)
The spin operators $M^{\{\lambda\}}_{k}$ transform
as the Cartesian components of a vector $\vec{M}^{\{\lambda\}}$ under
rotation in the ordinary space.
Taking into account
the commutation rule (45), we can define the irreducible
tensor components of $\vec{M}^{\{\lambda\}}$ as
$$\displaystyle M^{(1)\{\lambda\}}_{0}$$
$$\displaystyle=$$
$$\displaystyle M^{\{\lambda\}}_{3}$$
$$\displaystyle M^{(1)\{\lambda\}}_{\pm 1}$$
$$\displaystyle=$$
$$\displaystyle\pm\frac{1}{\sqrt{2}}\left(M^{\{\lambda\}}_{1}\mp i\ M^{\{\lambda%
\}}_{2}\right)$$
(47)
and express the products of two Cartesian components
as the sum of products of two tensor components.
To this purpose, we
define, for each value of $\lambda$,
the irreducible tensor product
$$\displaystyle T^{(J)}_{m}$$
$$\displaystyle=$$
$$\displaystyle\left[M^{(1)}\otimes M^{(1)}\right]^{(J)}_{m}$$
(48)
$$\displaystyle\equiv$$
$$\displaystyle\sum_{\nu,\nu^{\prime}}(1\nu 1\nu^{\prime}|Jm)M^{(1)}_{\nu}\ M^{(%
1)}_{\nu^{\prime}}$$
(where the common suffix $\{\lambda\}$ has been dropped,
for sake of simplicity).
We now introduce the reduced matrix elements
of the tensor operator $T^{(J)}$,
$$(\lambda||T^{(J)}||\lambda)=(-1)^{J}\sqrt{2J+1}\left\{\!\begin{array}[]{ccc}1&%
1&J\\
\lambda&\lambda&\lambda\end{array}\!\right\}\lambda(\lambda+1)(2\lambda+1)$$
(49)
to obtain
$$\displaystyle\left(M^{(1)}_{\mu}\ M^{(1)}_{\mu^{\prime}}\right)_{\nu,\nu^{%
\prime}}$$
$$\displaystyle=$$
$$\displaystyle\sum_{Jm}(1\mu 1\mu^{\prime}|Jm)\ \left(T^{(J)}_{m}\right)_{\nu,%
\nu^{\prime}}$$
$$\displaystyle=$$
$$\displaystyle\sum_{Jm}(1\mu 1\mu^{\prime}|Jm)(\lambda||T^{(J)}||\lambda)$$
$$\displaystyle\times\frac{(-1)^{\lambda-\nu+m}}{\sqrt{2J+1}}\ (\lambda\nu%
\lambda-\nu^{\prime}|Jm)\ .$$
We now observe that
$${a^{(\lambda)}_{\nu^{\prime}}}^{*}=(-1)^{\nu^{\prime}}a^{(\lambda)}_{-\nu^{%
\prime}}$$
(51)
and that
$$\left[a^{(\lambda)}\otimes a^{(\lambda)}\right]^{(J)}_{m}=\sum_{\nu,\nu^{%
\prime}}a^{(\lambda)}_{\nu}a^{(\lambda)}_{-\nu^{\prime}}\ (\lambda\nu\lambda-%
\nu^{\prime}|Jm)$$
(52)
to obtain
$$\displaystyle\sum_{\nu,\nu^{\prime}}$$
$$\displaystyle a^{(\lambda)}_{\nu}{a^{(\lambda)}_{\nu^{\prime}}}^{*}\left(M^{(1%
)\{\lambda\}}_{\mu}M^{(1)\{\lambda\}}_{\mu^{\prime}}\right)_{\nu\nu^{\prime}}$$
(53)
$$\displaystyle=\sum_{J,m}\frac{(-1)^{\lambda}}{\sqrt{2J+1}}(1\mu 1\mu^{\prime}|%
Jm)\ (\lambda||T^{(J)}||\lambda)$$
$$\displaystyle\ \times\ \left[a^{(\lambda)}\otimes a^{(\lambda)}\right]^{(J)}_{%
m}\ .$$
The rank of the tensor product $T^{(J)}$ of the two identical
vectors $M^{(1)\{\lambda\}}$ must be even, and therefore the possible
values of $J$ are limited to 0 and 2.
Now we can substitute, in the eq. (46),
the cartesian components of the
angular momentum with its tensor components
defined in the eq.s (47).
The possible values of $m$ (and $J$) contributing to
the sum are limited to $m=0$ (and therefore $J=$0 or 2) in the case of
$(M_{3})^{2}$, to $m=0$ or 2 ($J=$0 or 2) for
$(M_{1})^{2}$, $(M_{2})^{2}$ and
$(M_{1}M_{2})$, to $m=1$ ($J=2$)
for $(M_{3}M_{1})$ and $(M_{3}M_{2})$.
One obtains
$$\displaystyle\mathcal{J}_{1}$$
$$\displaystyle=$$
$$\displaystyle\sum_{\lambda}\left\{C_{0}(\lambda)\left[a^{(\lambda)}\otimes a^{%
(\lambda)}\right]^{(0)}_{0}\right.$$
(54a)
$$\displaystyle+$$
$$\displaystyle\frac{1}{\sqrt{6}}C_{2}(\lambda)\left[a^{(\lambda)}\otimes a^{(%
\lambda)}\right]^{(2)}_{0}$$
$$\displaystyle-$$
$$\displaystyle\left.C_{2}(\lambda){\rm Re}\left[a^{(\lambda)}\otimes a^{(%
\lambda)}\right]^{(2)}_{2}\right\}\ ,$$
$$\displaystyle\mathcal{J}_{2}$$
$$\displaystyle=$$
$$\displaystyle\sum_{\lambda}\left\{C_{0}(\lambda)\left[a^{(\lambda)}\otimes a^{%
(\lambda)}\right]^{(0)}_{0}\right.$$
(54b)
$$\displaystyle+$$
$$\displaystyle\frac{1}{\sqrt{6}}C_{2}(\lambda)\left[a^{(\lambda)}\otimes a^{(%
\lambda)}\right]^{(2)}_{0}$$
$$\displaystyle+$$
$$\displaystyle\left.C_{2}(\lambda){\rm Re}\left[a^{(\lambda)}\otimes a^{(%
\lambda)}\right]^{(2)}_{2}\right\}\ ,$$
$$\displaystyle\mathcal{J}_{3}$$
$$\displaystyle=$$
$$\displaystyle\sum_{\lambda}\left\{C_{0}(\lambda)\left[a^{(\lambda)}\otimes a^{%
(\lambda)}\right]^{(0)}_{0}\right.$$
(54c)
$$\displaystyle-$$
$$\displaystyle\sqrt{\frac{2}{3}}\left.C_{2}(\lambda)\left[a^{(\lambda)}\otimes a%
^{(\lambda)}\right]^{(2)}_{0}\right\}\ ,$$
$$\displaystyle\mathcal{J}_{12}$$
$$\displaystyle=$$
$$\displaystyle\sum_{\lambda}C_{2}(\lambda)\ {\rm Im}\left[a^{(\lambda)}\otimes a%
^{(\lambda)}\right]^{(2)}_{2}\ ,$$
(54d)
$$\displaystyle\mathcal{J}_{13}$$
$$\displaystyle=$$
$$\displaystyle\sum_{\lambda}C_{2}(\lambda)\ {\rm Re}\left[a^{(\lambda)}\otimes a%
^{(\lambda)}\right]^{(2)}_{1}\ ,$$
(54e)
$$\displaystyle\mathcal{J}_{23}$$
$$\displaystyle=$$
$$\displaystyle\sum_{\lambda}C_{2}(\lambda)\ {\rm Im}\left[a^{(\lambda)}\otimes a%
^{(\lambda)}\right]^{(2)}_{1}.$$
(54f)
where
$$\displaystyle C_{0}(\lambda)$$
$$\displaystyle=$$
$$\displaystyle(\!-\!1)^{\lambda}\ \frac{\lambda(\lambda+1)}{3}\sqrt{2\lambda+1}$$
$$\displaystyle C_{2}(\lambda)$$
$$\displaystyle=$$
$$\displaystyle(\!-\!1)^{\lambda+1}\ \sqrt{\frac{\lambda(\lambda\!+\!1)(2\lambda%
\!+\!\!3)(4\lambda^{2}-1)}{30}}$$
and therefore
$$\begin{array}[c]{lp{8mm}l}C_{0}(2)=2\ \sqrt{5}\ ,&\hbox{}&C_{0}(3)=-4\ \sqrt{7%
}\ ,\\
C_{2}(2)=-\sqrt{21}\ ,&\hbox{}&C_{2}(3)=3\ \sqrt{14}\ .\end{array}$$
(55)
If we chose as the intrinsic reference frame the principal axis of the
tensor of inertia, we must put $\mathcal{J}_{12}=\mathcal{J}_{13}=\mathcal{J}_{23}=0$. Using Eq.s (1,2,3)
we obtain, up to the first order
$$\displaystyle\mathcal{J}_{12}$$
$$\displaystyle=$$
$$\displaystyle-2\ \sqrt{21}\ {\rm Im}\ \left[\bar{a}^{(2)}\otimes\tilde{a}^{(2)%
}\right]^{(2)}_{2}$$
(56a)
$$\displaystyle+$$
$$\displaystyle 6\ \sqrt{14}\ {\rm Im}\ \left[\bar{a}^{(3)}\otimes\tilde{a}^{(3)%
}\right]^{(2)}_{2}=0,$$
$$\displaystyle\mathcal{J}_{13}$$
$$\displaystyle+$$
$$\displaystyle i\mathcal{J}_{23}=-2\ \sqrt{21}\ \left[\bar{a}^{(2)}\otimes%
\tilde{a}^{(2)}\right]^{(2)}_{1}$$
(56b)
$$\displaystyle+$$
$$\displaystyle 6\ \sqrt{14}\ \left[\bar{a}^{(3)}\otimes\tilde{a}^{(3)}\right]^{%
(2)}_{1}=0\ .$$
Here we make use of the fact that the zero–order terms are
automatically set to zero if the $\bar{a}^{(\lambda)}_{\mu}$ are defined
according to eq.s (1,2). By inserting these definitions
in the above equations, and retaining only the first–order terms,
one obtains
$$\displaystyle\left[\bar{a}^{(2)}\otimes\tilde{a}^{(2)}\right]^{(2)}_{1}$$
$$\displaystyle\approx$$
$$\displaystyle(2,0,2,1|2,1)\ \beta_{2}\ \tilde{a}^{(2)}_{1}$$
$$\displaystyle=$$
$$\displaystyle-\frac{1}{\sqrt{14}}\ \beta_{2}\ \tilde{a}^{(2)}_{1}\ ,$$
$$\displaystyle{\rm Im}\left[\bar{a}^{(2)}\otimes\tilde{a}^{(2)}\right]^{(2)}_{2}$$
$$\displaystyle\approx$$
$$\displaystyle(2,0,2,2|2,2)\ \beta_{2}\ {\rm Im}\ \tilde{a}^{(2)}_{2}$$
$$\displaystyle=$$
$$\displaystyle\sqrt{\frac{2}{7}}\ \beta_{2}\ {\rm Im}\ \tilde{a}^{(2)}_{2}\ ,$$
$$\displaystyle\left[\bar{a}^{(3)}\otimes\tilde{a}^{(3)}\right]^{(2)}_{1}$$
$$\displaystyle\approx$$
$$\displaystyle(3,0,3,1|2,1)\ \beta_{3}\ \tilde{a}^{(3)}_{1}$$
$$\displaystyle=$$
$$\displaystyle\sqrt{\frac{1}{42}}\ \beta_{3}\ \tilde{a}^{(3)}_{1},$$
$$\displaystyle{\rm Im}\left[\bar{a}^{(3)}\otimes\tilde{a}^{(3)}\right]^{(2)}_{2}$$
$$\displaystyle\approx$$
$$\displaystyle(3,0,3,2|2,2)\ \beta_{3}\ {\rm Im}\ \tilde{a}^{(3)}_{2}$$
$$\displaystyle=$$
$$\displaystyle-\sqrt{\frac{5}{21}}\ \beta_{3}\ {\rm Im}\ \tilde{a}^{(3)}_{2}.$$
We obtain, therefore, from eq.s (56)
$$\displaystyle\beta_{2}\tilde{a}^{(2)}_{1}$$
$$\displaystyle=$$
$$\displaystyle-\sqrt{2}\ \beta_{3}\tilde{a}^{(3)}_{1},$$
$$\displaystyle\beta_{2}{\rm Im}\ \tilde{a}^{(2)}_{2}$$
$$\displaystyle=$$
$$\displaystyle-\sqrt{5}\ \beta_{3}{\rm Im}\ \tilde{a}^{(3)}_{2}.$$
(57)
Appendix B Effect of non diagonal (first-order) terms
The matrix $\mathcal{G}$, as it results from the Table 1,
contains zero-order terms only in its principal diagonal (whose
last element, however, is small of the second order). Non diagonal
terms
have been expanded in series up to the first order in the “small”
dynamical variables (all of them, apart from $\beta_{2}$ and $\beta_{3}$).
We shall call $\xi_{k}$ a generic “small” variable, different from
$\beta_{\lambda}$ $(\lambda=2,3)$.
Terms of the last line and column (those related to the third intrinsic
component of the angular velocity) are discussed in the
Section II.4.
Here, we consider a simpler problem: the inversion of a
matrix $\mathcal{G}$ which has finite values for all terms
in the principal
diagonal, and only “small” values for all others.
In the zero-order approximation, the inverse matrix
$\mathcal{A=G}^{-1}$
is diagonal, with diagonal elements $A_{\mu\mu}=1/G_{\mu\mu}$.
The first–order approximation gives the non–diagonal elements
$$A_{\mu\nu}=A_{\nu\mu}=-\frac{G_{\mu\nu}}{G_{\mu\mu}G_{\nu\nu}}.$$
(58)
We are interested in particular on the effect of non diagonal terms on
the coefficients of the derivatives with respect to $\beta_{2}$ or
$\beta_{3}$.
Terms involving the derivatives with respect to
$\beta_{\lambda}$ and $\xi_{k}$ have the form
$$G^{-\frac{1}{2}}\frac{\partial}{\partial\xi_{k}}\left[G^{\frac{1}{2}}A_{k%
\lambda}\frac{\partial}{\partial\beta_{\lambda}}\right]+G^{-\frac{1}{2}}\frac{%
\partial}{\partial\beta_{\lambda}}\left[G^{\frac{1}{2}}A_{\lambda k}\frac{%
\partial}{\partial\xi_{k}}\right].$$
(59)
The non diagonal matrix elements of the matrix $\mathcal{A}$ are of the
first order in the “small” variables. They have therefore the form
$f^{(i)}_{\lambda k}(\beta_{2},\beta_{3})\xi_{i}$, where
$f^{(i)}_{\lambda k}(\beta_{2},\beta_{3})=\partial\mathcal{A}_{\lambda,k}/\ %
\partial\xi_{i}$,
or – possibly – are the sum of
several terms like that.
Substituting this expression in the
Eq. (59) one obtains
$$\displaystyle G$$
$${}^{-\frac{1}{2}}\frac{\partial}{\partial\xi_{k}}\left[G^{\frac{1}{2}}A_{k%
\lambda}\frac{\partial}{\partial\beta_{\lambda}}\right]+G^{-\frac{1}{2}}\frac{%
\partial}{\partial\beta_{\lambda}}\left[G^{\frac{1}{2}}A_{\lambda k}\frac{%
\partial}{\partial\xi_{k}}\right]$$
(60)
$$\displaystyle=$$
$$\displaystyle f^{(i)}_{\lambda k}(\beta_{2},\beta_{3})\left[2\xi_{i}\frac{%
\partial}{\partial\xi_{k}}+\frac{\xi_{i}}{2G}\ \left(\frac{\partial G}{%
\partial\xi_{k}}\right)+\delta_{ik}\right]\frac{\partial}{\partial\beta_{%
\lambda}}$$
$$\displaystyle+$$
$$\displaystyle\xi_{i}\left[f^{(i)}_{\lambda k}(\beta_{2},\beta_{3})\frac{1}{2G}%
\ \left(\frac{\partial G}{\partial\beta_{\lambda}}\right)+\left(\frac{\partial
f%
^{(i)}_{\lambda k}}{\partial\beta_{\lambda}}\right)\right]\frac{\partial}{%
\partial\xi_{k}}\ .$$
The last line of Eq. (60) only contains the partial-derivative
operator with respect to the small variable $\xi_{k}$.
Compared with the diagonal term involving the corresponding
second-derivative operator, it contains a
small factor $\xi_{i}$ more, and can be neglected.
In the first line
of the expression, all terms are potentially of the same order of
magnitude of
the leading ones, and in principle could not be neglected.
The first of the terms in the square brackets
contains the partial derivative with respect
to $\xi_{k}$. In the spirit of the adiabatic approach, we try to
estimate the expectation value of this term for the ground–state
wavefunction in all the variables, with the exception of $\beta_{2}$ and
$\beta_{3}$.
If we assume that the potential is harmonic for the ensemble of these
variables, the ground–state wavefunction can be expected to be close
to a multivariate Gaussian function. If, moreover, $\xi_{k}$
and $\xi_{i}$ are
uncorrelated, the expectation value of $\xi_{i}\partial/\partial\xi_{k}$
is zero, as $\left<\xi_{i}\right>=0$.
For the case $i=k$, instead, the expectation value is
$$\displaystyle\left<2\xi\frac{\partial}{\partial\xi}\right>$$
$$\displaystyle\approx$$
$$\displaystyle 2\ \frac{\int\exp(-\alpha\xi^{2}/2)\xi\frac{\partial}{\partial%
\xi}\exp(-\alpha\xi^{2}/2){\rm d}\xi}{\int\exp(-\alpha\xi^{2}){\rm d}\xi}$$
(61)
$$\displaystyle=$$
$$\displaystyle\frac{\int\exp(-\alpha\xi^{2}/2)\left(-2\alpha\xi^{2}\right)\exp(%
-\alpha\xi^{2}/2){\rm d}\xi}{\int\exp(-\alpha\xi^{2}){\rm d}\xi}$$
$$\displaystyle=$$
$$\displaystyle-1$$
(62)
and (in this approximation) cancels the third term, $\delta_{ik}$.
In conclusion, the coefficient of $\partial/\partial\beta_{\lambda}$
coming from the non–diagonal terms of the matrix ${\mathcal{G}}$
can be approximated with a sum of expressions like
$$\ f^{(i)}_{\lambda k}(\beta_{2},\beta_{3})\ \left<\frac{\xi_{i}}{G}\ \frac{%
\partial G}{\partial\xi_{k}}\right>$$
(63)
which approximately vanishes for $i\neq k$, and also
vanishes for $i=k$ unless both
$\mathcal{A}_{\lambda k}$ and $G$ depend explicitly on $\xi_{k}$.
Moreover, also in this latter case,
it is possible to eliminate non-diagonal elements of the matrix
$\mathcal{G}$ of the form $\mathcal{G}_{\lambda k}=\ g_{\lambda k}(\beta_{2},\beta_{3})\ \xi_{k}$, with a slight
change in the definition of $\beta_{\lambda}$, without any other
effect at the present order of approximation.
Namely, it is sufficient to substitute the dynamical variable
$\beta_{\lambda}$ with the new variable
$$\beta^{\rm o}_{\lambda}=\beta_{\lambda}+\frac{1}{2}g_{\lambda k}(\beta_{2},%
\beta_{3})\xi_{k}^{2}$$
(64)
to obtain
$$\dot{\beta}_{\lambda}=\dot{\beta}^{\rm o}_{\lambda}-g_{\lambda k}(\beta_{2},%
\beta_{3})\xi_{k}\dot{\xi}_{k}+0(\xi_{k}^{2})$$
(65)
and therefore, up to the second order in $\xi_{k}$,
$$\displaystyle\dot{\beta}_{\lambda}^{2}$$
$$\displaystyle+2g_{\lambda k}(\beta_{2},\beta_{3})\xi_{k}\dot{\beta}_{\lambda}%
\dot{\xi}_{k}$$
(66)
$$\displaystyle=$$
$$\displaystyle\left(\dot{\beta}^{\rm o}_{\lambda}-g_{\lambda k}(\beta_{2},\beta%
_{3})\xi_{k}\dot{\xi}_{k}\right)^{2}+2g_{\lambda k}(\beta_{2},\beta_{3})\xi_{k%
}\dot{\beta}_{\lambda}\dot{\xi}$$
$$\displaystyle\approx$$
$$\displaystyle\left(\dot{\beta}^{\rm o}_{\lambda}\right)^{2}.$$
This argument can be easily extended to the case in which the
dependence of $\mathcal{A}_{\lambda k}$ on $\xi_{k}$ comes from the
dependence on $\xi_{k}$ of one of the terms at the denominator in the
eq. (58).
Appendix C Quantization according to the Pauli rule
In this Appendix, we discuss some aspects of the quantization of the
kinetic energy expression given, e.g., in the
eq. (21),
by means of the Pauli procedure: namely, if
the classical expression of the kinetic energy in terms of the time
derivatives of the dynamical variables $\xi_{\mu}$ is
$$T=\frac{1}{2}\sum_{\mu,\nu}\mathcal{G}_{\mu,\nu}\dot{\xi}_{\mu}\dot{\xi}_{\nu}$$
(67)
the corresponding quantum operator has the form
$$\hat{T}=-\frac{\hbar^{2}}{2}G^{-1/2}\sum_{\mu,\nu}\frac{\partial}{\partial\xi_%
{\mu}}\ G^{1/2}\mathcal{A}_{\mu,\nu}\frac{\partial}{\partial\xi_{\nu}}$$
(68)
where $G={\rm Det}\ \mathcal{G}$ and $\mathcal{A}=\mathcal{G}^{-1}$.
The choice of the
“best set” of
dynamical variables is, in part, related to the expression of the
potential-energy term, and it is not obvious that it will eventually
coincide with the one discussed in the section II.5. However, it
can be useful to explore the properties of the kinetic-energy
operator in the particular model in which the matrix of coefficients
${\mathcal{G}}$ is exactly that of the Table 2, with the
non-diagonal terms confined in one single line (and column) of the
lower $6\times 6$ sub-matrix.
From now on, we limit our
discussion to the lowest $6\times 6$ sub-matrix $\mathcal{G}_{1}$ of the
matrix $\mathcal{G}$ (and of the matrices derived from $\mathcal{G}$,
Tables 6 ,7) as
the six corresponding variables – $\varphi,\ \chi,\ \vartheta,\ \theta_{1},\ \theta_{2},\ \theta_{3}$ – are effectively decoupled from
the others.
The formal procedure is discussed, e.g., in the Ch.s 5 and
6 of the ref. Eisenberg and Greiner (1987).
The first pass is the substitution of the intrinsic components
$q_{1},\ q_{2},\ q_{3}$ of the angular velocity with the time derivatives of
the three Euler angles $\theta_{1},\ \theta_{2},\ \theta_{3}$:
$$q_{k}=\sum_{i}V_{ki}\dot{\theta}_{i}$$
(69)
with
$$V=\left|\begin{array}[]{ccc}-\cos\theta_{3}\sin\theta_{2}&\sin\theta_{3}&0\\
\phantom{-}\sin\theta_{3}\sin\theta_{2}&\cos\theta_{3}&0\\
\cos\theta_{2}&0&1\\
\end{array}\right|\ .$$
(70)
As a consequence of this substitution, the $6\times 6$ matrix
$\mathcal{G}_{1}$ transforms according to the relation
$$\mathcal{G}_{1}^{\prime}=\widetilde{W}\ \ \mathcal{G}_{1}\ W,\hskip 56.905512%
ptW=\left|\begin{array}[]{cc}\{1\}&\{0\}\\
\{0\}&V\end{array}\right|$$
(71)
(where $\widetilde{W}$ is the transpose of the matrix $W$ and $\{1\}$,
$\{0\}$ are the $3\times 3$ unit matrix and null matrix) and
takes the form shown in the Table 5.
The next step is the inversion of the matrix $\mathcal{G}_{1}^{\prime}$.
In doing this, the explicit form (20) of $\mathcal{J}_{3}$
has been introduced
and second-order terms in the small amplitudes have been
neglected.
At this point, it is necessary to introduce the intrinsic components
of the angular momentum operator ($L_{1},\ L_{2},\ L_{3}$) in the place of
the derivatives with respect to the Euler angles. The expression of
$L_{k}$ in terms of $\partial/\partial\theta_{k}$ and vice-versa are
given, e.g. in the Ch. 5 of ref Eisenberg and Greiner (1987). One gets
$$-i\frac{\partial}{\partial\theta_{k}}=\sum V_{ik}\hat{L}_{i}$$
(72)
with the matrix $V$ given by the eq. (70).
It is also convenient to define the quantum operators
$$\displaystyle\hat{p}_{\varphi}$$
$$\displaystyle=$$
$$\displaystyle-i\partial\ /\ \partial\varphi$$
$$\displaystyle\hat{p}_{\chi}$$
$$\displaystyle=$$
$$\displaystyle-i\partial\ /\ \partial\chi$$
$$\displaystyle\hat{p}_{\vartheta}$$
$$\displaystyle=$$
$$\displaystyle-i\partial\ /\ \partial\vartheta$$
(73)
to obtain
$$\displaystyle-i$$
$$\displaystyle\left\{\frac{\partial}{\partial\varphi},\ \frac{\partial}{%
\partial\chi},\ \frac{\partial}{\partial\vartheta},\ \frac{\partial}{\partial%
\theta_{1}},\ \frac{\partial}{\partial\theta_{2}},\ \frac{\partial}{\partial%
\theta_{3}}\right\}$$
(74)
$$\displaystyle=\ \widetilde{W}\ \left\{\hat{p}_{\varphi},\ \hat{p}_{\chi},\ %
\hat{p}_{\vartheta},\ \hat{L}_{1},\ \hat{L}_{2},\ \hat{L}_{3}\right\}.$$
The introduction of a set of conjugate momenta $\hat{p}_{k}$ such that
$$-i\frac{\partial}{\partial\xi_{k}}=\sum_{i}W_{ik}\hat{p}_{i}$$
(75)
in the eq. (68) gives
$$\displaystyle\hat{T}$$
$$\displaystyle=$$
$$\displaystyle\frac{\hbar^{2}}{2}\left[\sum_{\mu,\nu}(W\mathcal{A}\widetilde{W}%
)_{\mu,\nu}\ \hat{p}_{\mu}\hat{p}_{\nu}\right.$$
(76)
$$\displaystyle\left.+i\ G^{-1/2}\sum_{\mu,\nu}\frac{\partial\ G^{1/2}(\mathcal{%
A}\widetilde{W})_{\mu,\nu}}{\partial\xi_{\mu}}\ \hat{p}_{\nu}\right].$$
In our case, $\mathcal{A}=(\mathcal{G}_{1}^{\prime} )^{-1}$,
$\mathcal{G}_{1}^{\prime}=\widetilde{W}\mathcal{G}_{1}W$, so that
$$W\mathcal{A}\widetilde{W}=W\left(W^{-1}\mathcal{G}_{1}^{-1}\widetilde{W}^{-1}%
\right)\widetilde{W}=\mathcal{G}_{1}^{-1}$$
(77)
and the first term of the eq. (76) takes the form given in the
Table 7.
The second term of the eq. (76) vanishes.
In fact, only the derivatives with respect to $\theta_{2}$ or
$\theta_{3}$ could give a contribution to the sum, as the other
variables do not appear in the elements of the sub-matrix. Moreover,
the determinant $G_{1}$ is simply proportional to $\sin^{2}\theta_{2}$
(with the proportionality factor depending on dynamical variables
which are outside the present sub-space), and
the term $G^{1/2}$ in the equation (76) can be replaced by
$\sin\theta_{2}$.
The matrix $i(\mathcal{G}_{1}^{\prime})^{-1}\widetilde{W}\sin\theta_{2}$
is given in the
Table 6. The sum of the derivatives of each element of
the 5th row, with respect to $\theta_{2}$, plus the corresponding one of
the 6th row, with respect to $\theta_{3}$, would give the coefficient of
the corresponding momentum operator, but
it is easy to verify they cancel each other.
Appendix D The possible forms of the potential term
The form of the approximate potential–energy expression depends on
the details of the underlying microscopic structure that the model
should
try to simulate. There are, however, some general rules to which the
expression of the potential energy must conform: it must be invariant
under space rotation, time reversal, and parity.
Irreducible tensors of different rank have been constructed with each of
the basic tensors $a^{(2)}$ and $a^{(3)}$, and scalar products of
tensors of equal rank have been considered.
In order to be invariant under time reversal, each term must
contain an even number of octupole amplitudes $a^{(3)}_{\mu}$.
Moreover, due to symmetry, only tensors
$T^{(K)}_{\lambda\lambda}=[a^{(\lambda)}\otimes a^{(\lambda)}]^{(K)}$
of even rank $K$ can be obtained
with the coupling of two identical tensors $a^{(\lambda)}$.
There are therefore two independent invariants of order 2,
$\big{(}a^{(2)}\cdot a^{(2)}\big{)}$ and $\big{(}a^{(3)}\cdot a^{(3)}\big{)}$,
two of order 3, $\big{(}a^{(2)}\cdot T^{(2)}_{22}\big{)}$
and $\big{(}a^{(2)}\cdot T^{(2)}_{33}\big{)}$,
and seven of order 4, namely
$\big{(}T^{(K)}_{22}\cdot T^{(K)}_{22}\big{)}$ ($K=2,4$),
$\big{(}T^{(K)}_{33}\cdot T^{(K)}_{33}\big{)}$ ($K=2,4,6$),
and $\big{(}T^{(K)}_{22}\cdot T^{(K)}_{33}\big{)}$ ($K=2,4$).
Additional fourth-order invariants of the form
$\big{(}[a^{(2)}\otimes a^{(3)}]^{(K)}\cdot[a^{(2)}\otimes a^{(3)}]^{(K)}\big{)}$ ($K=1-5$), are not independent from the above
ones, and can be expressed as linear combination of them with the
standard rules of angular momentum recoupling.
Invariant expressions up to the fourth order in the amplitudes
$a^{(\lambda)}_{\mu}$ ($\lambda=2,3$) are shown in the
Table 8, in terms of the dynamical variables
$\beta_{2}$, $\gamma_{2}$, $\beta_{3}$, $\gamma_{3}$, $w^{2}=(X^{2}+Y^{2})$,
$\xi$, and $v^{2}=(\eta^{2}+\zeta^{2})$. Expressions corresponding to
different choices of the dynamical variables can be easily obtained.
Here, only the terms up to the second order in the series expansion of
the “small” (non-axial) amplitudes are given. In this approximation,
the fourth-order invariants built with the quadrupole amplitudes
($C_{1},\ C_{2}$)
result to be proportional to each other and to the square of the
corresponding second-order invariant ($A_{1}^{2}$).
Moreover, as it is clear from the Table 8,
the three fourth-order invariants built with the octupole
amplitudes ($C_{3},\ C_{4},\ C_{5}$)
and the square of the corresponding second-order invariant ($A_{2}^{2}$)
are not linearly independent of one another, and provide only two
independent relations.
One can finally observe that the variables $X,\ Y,\ \eta$ and $\zeta$
always appear only in the combinations $(X^{2}+Y^{2})$,
$(\eta^{2}+\zeta^{2})$.
As long as the expression of the potential energy does only depend
on the invariants up to the fourth order,
the angles $\vartheta$ and $\varphi$ defined in the Eq. (17)
are not subject to a restoring force.
The situation is more complicated for the three variables related to
the $\mu=\pm 2$ components of $a^{\lambda}_{\mu}$. Also in this case,
however, it is possible to construct combinations of invariants that
contain only linear combinations of the squares of the variables
$\xi$, $\gamma=\sqrt{5}\gamma_{2}-\gamma_{3}$, and
$\gamma_{0}=\beta_{2}^{2}\gamma_{2}+\sqrt{5}\beta_{3}^{2}\gamma_{3}$.
Moreover, in this case, the first two of them do appear in the
combination
$\beta_{2}\beta_{3}\gamma^{2}+2\big{(}\beta_{2}^{2}+5\beta_{3}^{2}\big{)}\xi^{2}$.
It is therefore possible to imaging a situation in which the potential
energy is independent also of the angle $\chi$ (although this is is not
a direct consequence of the model, as it is the case for the angles
$\vartheta$ and $\varphi$).
Acknowledgements.
We have the pleasure to thank
Prof. F. Iachello and Prof. B.R. Mottelson for helpful discussions.
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Reconfiguration of Non-crossing Spanning Trees
Oswin Aichholzer
University of Technology Graz, Austria. [email protected]
Brad Ballinger
Cal Poly Humboldt, USA. [email protected]
Therese Biedl
University of Waterloo, Canada. {biedl,alubiw,jayson.lynch}@uwaterloo.ca
Mirela Damian
Villanova University, USA. [email protected]
Erik D. Demaine
MIT Computer Science and Artificial Intelligence Laboratory, USA. [email protected]
Matias Korman
Siemens Electronic Design Automation, USA. [email protected]
Anna Lubiw
University of Waterloo, Canada. {biedl,alubiw,jayson.lynch}@uwaterloo.ca
Jayson Lynch
University of Waterloo, Canada. {biedl,alubiw,jayson.lynch}@uwaterloo.ca
Josef Tkadlec
Harvard Unversity, USA. [email protected]
Yushi Uno
Osaka Metropolitan University, Japan. [email protected]
Abstract
For a set $P$ of $n$ points in the plane in general position,
a non-crossing spanning tree is a spanning tree of the points
where every edge is a straight-line segment between a pair of points and
no two edges intersect except at a common endpoint.
We study the problem of reconfiguring one non-crossing spanning tree of $P$ to another
using a sequence of flips where each flip removes one edge
and adds one new edge so that the result is again a non-crossing spanning tree of $P$.
There is a known upper bound of $2n-4$ flips [Avis and Fukuda, 1996] and a lower bound of $1.5n-5$ flips.
We give a reconfiguration algorithm that uses at most $2n-3$ flips but reduces that to $1.5n-2$ flips when one tree is a path and either: the points are in convex position; or the path is monotone in some direction.
For points in convex position, we prove an upper bound of
$2d-\Omega(\log d)$ where $d$ is half the size of the symmetric difference between the trees.
We also examine whether the happy edges (those common to the initial and final trees) need to flip,
and we find exact minimum flip distances for small point sets using exhaustive search.
1 Introduction
Let $P$ be a set of $n$ points in the plane in general position.
A non-crossing spanning tree is a spanning tree of $P$
whose edges are straight line segments between pairs of points such that no two edges intersect except at a common endpoint.
A reconfiguration step or flip removes one edge of a non-crossing spanning tree and adds one new edge so that the result is again a non-crossing spanning tree of $P$.
We study the problem of
reconfiguring one non-crossing spanning of $P$ to another
via a sequence of flips.
Researchers often consider three problems about reconfiguration, which are most easily expressed in terms of the reconfiguration graph that has a vertex for each configuration (in our case, each non-crossing spanning tree) and an edge for each reconfiguration step. The problems are:
(1) connectivity of the reconfiguration graph—is reconfiguration always possible?
(2) diameter of the reconfiguration graph—how many flips are needed for reconfiguration in the worst case? and (3) distance in the reconfiguration graph—what is the complexity of finding the minimum number of flips to reconfigure between two given configurations?
For reconfiguration of non-crossing spanning trees,
Avis and Fukuda [10, Section 3.7] proved
that reconfiguration is always possible,
and that at most $2n-4$ flips are needed.
Hernando et al. [20] proved a lower bound of $1.5n-5$ flips.
Even for the special case of points in convex position, there are no better upper or lower bounds known.
Our two main results
make some progress in reducing the diameter upper bounds.
(1) For points in general position, we give a reconfiguration algorithm
that uses at most $2n-3$ flips but reduces that to $1.5n-2$ flips
in two cases: (1) when the points are in convex position and one tree is a path; (2) for general point sets when one tree is a monotone path.
The algorithm first flips one tree
to a downward tree
(with each vertex connected to a unique higher vertex)
and the other
tree to an upward tree (defined symmetrically)
using $n-2$ flips—this is where we save when one tree is a path. After that, we give an algorithm to flip from a downward tree $T_{D}$ to an upward tree $T_{U}$ using at most $n-1$
“perfect” flips each of which removes an edge of $T_{D}$ and adds an edge of $T_{U}$. The algorithm is simple to describe, but proving that intermediate trees remain non-crossing is non-trivial.
We also show that $1.5n-5$ flips
may be required, even in the two special cases.
See Section 2.
(2) For points in convex position, we improve the upper bound on the number of required flips to
$2d-\Omega(\log d)$ where $d$ is half the size of the symmetric difference between the trees.
So $d$ flips are needed in any flip sequence, and $2d$ is an upper bound.
The idea is to find an edge $e$ of one tree that is crossed by at most (roughly) $d/2$ edges of the other tree,
flip all but one of the crossing edges temporarily to the convex hull (this will end up costing 2 flips per edge), and then flip the last crossing edge to $e$. Repeating this saves us one flip, compared to the $2d$ bound, for each of the (roughly) $\log d$ repetitions.
See Section 3.
Notably, neither of our algorithms uses the common—but perhaps limited—technique of
observing that the diameter is at most twice the radius, and bounding the radius of the reconfiguration
graph by identifying a “canonical” configuration that every other configuration can be flipped to. Rather, our
algorithms find reconfiguration sequences tailored to the specific input trees.
In hopes of
making further progress on the diameter and distance problems,
we address the question of which edges need to be involved in a minimum flip sequence from an initial non-crossing spanning tree $T_{I}$ to a final non-crossing spanning tree $T_{F}$.
We say that the edges of $T_{I}\cap T_{F}$ are happy edges, and we formulate the Happy Edge Conjecture that for points in convex position, there is a minimum flip sequence that never flips happy edges.
We prove the conjecture for happy convex hull edges.
See Section 4.
More generally, we say that a reconfiguration problem has the “happy element property” if elements that are common to the initial and final configurations can remain fixed in a minimum flip sequence.
Reconfiguration problems that satisfy the happy element property seem easier.
For example, the happy element property holds for reconfiguring
spanning trees in a graph (and indeed for matroids more generally), and the distance problem is easy.
On the other hand,
the happy element property
fails for reconfiguring triangulations of a point set in the plane, and for the problem of token swapping on a tree [12],
and in both cases,
this is the key to constructing gadgets
to prove that the distance problem is NP-hard [26, 31, 4].
As an aside, we note that
for reconfiguring triangulations of a set of points in convex position—where the distance problem is the famous open question of rotation distance in binary trees—the happy element property holds [35], which may be why no one has managed to prove that the distance problem is NP-hard.
Finally,
we implemented a combinatorial search program to compute the diameter (maximum reconfiguration distance between two trees) and radius of the reconfiguration graph for points in convex position.
For $6\leq n\leq 12$ the diameter is $\lfloor 1.5n-4\rfloor$ and the radius is $n-2$. In addition we provide the same information for the special case when the initial and final trees are non-crossing spanning paths, though intermediate configurations may be more general trees. We also verify
the Happy Edge Conjecture for $n\leq 10$ points in convex position.
See Section 5.
1.1 Background and Related Results
Reconfiguration is about changing one structure to another, either through continuous motion or through discrete changes.
In mathematics, the topic has a vast and deep history, for example in knot theory,
and the study of bounds on the simplex method for linear programming.
Recently, reconfiguration via discrete steps has become a focused research area, see the surveys by van den Heuvel [36] and Nishimura [29].
Examples include sorting a list by swapping pairs of adjacent elements, solving a Rubik’s cube, or changing one colouring of a graph to another.
With discrete reconfiguration steps, the reconfiguration graph is well-defined. Besides questions of diameter and distance in the reconfiguration graph, there is also research on enumeration via a Hamiltonian cycle in the reconfiguration graph, see the recent survey [27], and on mixing properties to find random configurations, see [33].
Our work is about reconfiguring one graph to another.
Various reconfiguration steps have been considered, for example exchanging one vertex for another (for reconfiguration of paths [14], independent sets [23, 11], etc.), or exchanging multiple edges
(for reconfiguration of matchings [13]).
However, we concentrate on elementary steps (often called edge flips) that exchange one edge for one other edge.
A main example of this is reconfiguring one spanning tree of a graph to another, which can always be accomplished by “perfect” flips that add an edge of the final tree and delete an edge of the initial tree—more generally, such a perfect exchange sequence is possible when reconfiguring bases of a matroid.
Our focus is on geometric graphs whose vertices are points in the plane
and whose edges are non-crossing line segments between the points.
In this setting, one well-studied problem is reconfiguring between triangulations of a point set in the plane, see the survey by Bose and Hurtado [15]. Here, a flip replaces
an edge in a convex quadrilateral by the other diagonal of the quadrilateral.
For the special case of $n$ points in convex position this is equivalent to rotation of an edge in a given (abstract) rooted binary tree, which is of interest in the study of splay trees, and the study of phylogenetic trees in computational biology.
While an upper bound for the reconfiguration distance of $2n-10$ is known to be tight for $n>12$ [32, 35],
the complexity of computing the shortest distance between
two triangulations of a convex point set
(equivalently between two given binary trees) is still unknown. See [6, 26, 31] for related hardness results for the flip-distance of triangulations of point sets and simple polygons.
Another well-studied problem for geometric graphs is reconfiguration of non-crossing perfect matchings. Here a flip typically exchanges matching and non-matching edges in non-crossing cycles, and there are results for a single cycle of unbounded length [21], and for multiple cycles [3, 34].
For points in convex position, the flip operation on a single cycle of length 4 suffices to connect the reconfiguration graph [19], but this is open for general point sets.
For a more general flip operation that connects any two disjoint matchings whose union is non-crossing, the reconfiguration graph is connected for points in convex position [1] (and, notably, the diameter is less than twice the radius), but connectivity is open for general point sets [3, 22].
The specific geometric graphs we study are non-crossing (or “plane”) spanning trees of a set of points in the plane.
For points in convex position, these have been explored for enumeration [30], and for a duality with quadrangulations and consequent lattice properties, e.g., see [9] and related literature.
For non-crossing spanning trees of a general point set in the plane,
there
are
several basic reconfiguration operations that can be used to transform these trees into each other. The one we use in this work is the simple edge exchange of an edge $e$ by an edge $e^{\prime}$ as described above. If
we require that
$e$ and $e^{\prime}$ do not cross, then this operation is called a compatible edge exchange. Even more restricted is an edge rotation, where
$e=uv$ and $e^{\prime}=uw$
share a common vertex $u$.
If the triangle $u,v,w$ is not intersected by an edge of the two involved trees, then this is called an empty triangle edge rotation. The most restricted operation is the edge slide (see also Section 4.3) where the edge $vw$ has to exist in both trees. The name comes from viewing this transformation as sliding one end of the edge $e$ along $vw$ to $e^{\prime}$ (and rotating the other end around $u$) and at no time crossing through any other edge of the trees.
For an overview and detailed comparison of the described operations see Nichols et al. [28].
It has been shown that the reconfiguration graph of non-crossing spanning trees is even connected
for the “as-local-as-possible” edge slide operation [2]. See also [7] where a tight bound of $\Theta(n^{2})$ steps for the diameter is show. This implies that also for the other reconfiguration operations the flip graph is connected.
For edge exchange, compatible edge exchange, and edge rotation a linear upper bound for the diameter is known [10], while for empty triangle edge rotations an upper bound of $O(n\log n)$ has been shown recently [28].
For all operations (except edge slides) the best known lower bound is $1.5n-5$ [20].
There are several variants for the reconfiguration of spanning trees. For example, the operations can be performed in parallel, that is, as long as the exchanges (slides etc) do not interfere with each other, they can be done in one step; see [28] for an overview of results.
In a similar direction of a more global operation we can say that two non-crossing spanning trees are compatible, if their union is still crossing free. A single reconfiguration step then transforms one such tree into the other. A lower bound of $\Omega(\log n/\log\log n)$ [16] and an upper bound of $O(\log n)$ [2] for the diameter of this reconfiguration graph has been shown.
Another variation is that the edges are labeled (independent of the vertex labels), and if an edge is exchanged the replacement edge takes that label. In this way two geometrically identical trees can have a rather large transformation distance.
For labeled triangulations
there is a good characterization of when reconfiguration is possible, and a polynomial bound on the number of steps required [25].
The reconfiguration of non-crossing spanning paths (where each intermediate configuration must be a path) has also been considered. For points in convex position, the diameter of the reconfiguration graph is $2n-6$ for $n\geq 5$ [8, 17]. Surprisingly, up to now it remains an open problem if the reconfiguration graph of non-crossing spanning paths is connected for general point sets [5].
1.2 Definitions and Terminology
Let $P$ be a set of $n$ points in
general position, meaning that no three points are collinear.
The points of $P$ are in convex position if the boundary of the convex hull of $P$ contains all the points of $P$.
An edge is a line segment joining two points of $P$, and a spanning tree $T$ of $P$ is a set of $n-1$ edges that form a tree. Two edges cross if they intersect but do not share a common endpoint.
A non-crossing spanning tree is a spanning tree such that no two of its edges cross.
When we write “two non-crossing spanning trees,” we mean that each tree is non-crossing but we allow edges of one tree to cross edges of the other tree.
We sometimes consider the special case where a non-crossing spanning tree of $P$ is a path. A path is monotone if there is a direction in the plane such that
the order of points along the path matches the order of points in that direction.
For a spanning tree of a graph,
a flip removes one edge and adds one new edge to obtain a new spanning tree, i.e., spanning trees $T$ and $T^{\prime}$ are related by a flip if $T^{\prime}=T\setminus\{e\}\cup\{e^{\prime}\}$, where $e\in T$ and $e^{\prime}\notin T$.
The same definition applies
to non-crossing spanning trees: If $T$ is a non-crossing spanning tree of $P$, and $T^{\prime}=T\setminus\{e\}\cup\{e^{\prime}\}$ is a non-crossing spanning tree of $P$, then we say that $T$ and $T^{\prime}$ are related by a flip. We allow $e$ and $e^{\prime}$ to cross.
Let $T_{I}$ and $T_{F}$ be initial and final non-crossing spanning trees of $P$. A flip sequence from $T_{I}$ to $T_{F}$ is a sequence of flips that starts with $T_{I}$ and ends with $T_{F}$ and such that each intermediate tree is a non-crossing spanning tree.
We say that $T_{I}$ can be reconfigured to $T_{F}$
using $k$ flips (or “in $k$ steps”) if there is a reconfiguration sequence of length at most $k$.
The flip distance from $T_{I}$ to $T_{F}$ is the minimum length of a flip sequence.
The edges of $T_{I}\cap T_{F}$ are called happy edges.
Thus, $T_{I}\cup T_{F}$ consists of the happy edges together with the symmetric difference $(T_{I}\setminus T_{F})\cup(T_{F}\setminus T_{I})$. We have $|T_{I}\setminus T_{F}|=|T_{F}\setminus T_{I}|$. A flip sequence of length $|T_{I}\setminus T_{F}|$ is called a perfect flip sequence. In a perfect flip sequence, every flip removes an edge of $T_{I}$ and adds an edge of $T_{F}$—these are called perfect flips.
2 A two-phase reconfiguration approach
In this section we give a new algorithm to reconfigure between two non-crossing spanning trees on $n$ points using at most $2n-3$ flips. This is basically the same as the upper bound of $2n-4$ originally achieved by Avis and Fukuda [10], but the advantage of our new algorithm is that it gives a bound of $1.5n-2$ flips when one tree is a path and either: (1) the path is monotone; or (2) the points are in convex position.
Furthermore, for these two cases, we show a lower bound of $1.5n-5$ flips, so the bounds are tight up to the additive constant.
Before proceeding, we mention one way in which our upper bound result differs from some other reconfiguration bounds.
Many of those bounds (i.e., upper bounds on the diameter $d$ of the reconfiguration graph) are actually bounds on the radius $r$ of the reconfiguration graph.
The idea is to identify a
“canonical” configuration and prove that its distance to any other configuration is at most $r$, thus proving that the diameter $d$ is at most $2r$. For example, Avis and Fukuda’s $2n$ bound is achieved via a canonical star centered at a convex hull point. As another example,
the bound of $O(n^{2})$ flips between triangulations of a point set can be proved using the Delaunay triangulation as a canonical configuration, and
the bound of $2n$ flips for the special case of points in convex position uses a canonical star triangulation [35].
For some reconfiguration graphs $d$ is equal to $2r$ (e.g., for the Rubik’s cube, because of the underlying permutation group). However, in general, $d$ can be less than $2r$, in which case,
using a canonical configuration will not give the best diameter bound.
Indeed, our result does not use a canonical configuration, and we do not bound the radius of the reconfiguration graph.
Our algorithm has two phases. In the first phase, we reconfigure the input trees in a total of at most $n-2$ flips so that one is “upward” and one is “downward” (this is where we save if one tree is a path). In the second phase we show that an upward tree can be reconfigured to a downward tree in $n-1$ flips. We begin by defining these terms.
Let $P$ be a set of $n$ points in general position. Order the points $v_{1},\ldots,v_{n}$ by increasing $y$-coordinate
(if necessary, we slightly perturb the point set to ensure that no two $y$-coordinates are identical).
Let $T$ be a non-crossing spanning tree of $P$.
Imagining the edges as directed upward,
we call a vertex $v_{i}$ a sink if there are no edges in $T$ connecting $v_{i}$ to a higher vertex $v_{j},j>i$, and we call $v_{i}$ a source if there are no edges connecting $v_{i}$ to a lower vertex $v_{k},k<i$.
We call $T$ a downward tree if it has only one sink (which must then be $v_{n}$) and we call $T$ an upward tree if it has only one source (which must then be $v_{1}$).
Observe that in a downward tree every vertex except $v_{n}$ has exactly one edge connected to a higher vertex, and in an upward tree every vertex except $v_{1}$ has exactly one edge connected to a lower vertex.
2.1 Phase 1: Reconfiguring to upward/downward trees
We first bound the number of flips needed to reconfigure a single tree $T$ to be upward or downward.
If a tree has $t$ sinks, then we need at least $t-1$ flips to reconfigure it to a downward tree—we show that $t-1$ flips suffice. Note that $t$ is at most $n-1$ since $v_{1}$ cannot be a sink (this bound is realized by a star at $v_{1}$).
Theorem 1.
Let $T$ be a non-crossing spanning tree with $s$ sources and $t$ sinks.
$T$ can be reconfigured to a downward tree with $t-1\leq n-2$ flips.
$T$ can be reconfigured to an upward tree with $s-1\leq n-2$ flips.
Furthermore, these reconfiguration sequences do not flip any edge of the form
$v_{i}v_{i+1}$ where $1\leq i<n$.
Proof.
We give the proof for a downward tree, since the other case is symmetric.
The proof is by induction on $t$. In the base case, $t=1$ and the tree is downward and so no flips are needed.
Otherwise, let $v_{i},1<i<n$ be a sink. The plan is to decrease $t$ by adding an edge going upward from $v_{i}$ and removing some edge $v_{k}v_{l},k<l$ from the resulting cycle while ensuring that $v_{k}$ does not become a sink.
If there is an edge $v_{i}v_{j}$ that does not cross any edge of $T$, we say that $v_{i}$ sees $v_{j}$.
We argue that $v_{i}$ sees some vertex $v_{j}$ with $j>i$.
If $v_{i}$ sees $v_{n}$, then choose $j=n$. Otherwise the upward ray directed from $v_{i}$ to $v_{n}$ hits some edge $e$ before it reaches $v_{n}$. Continuously rotate the ray around $v_{i}$ towards the higher endpoint of $e$ until the ray reaches the endpoint or is blocked by some other vertex.
In either case this gives us a vertex $v_{j}$ visible from $v_{i}$ and higher than $v_{i}$.
For example, in Figure 1, the sink $v_{5}$ sees $v_{7}$.
Adding the edge $v_{i}v_{j}$ to $T$ creates a cycle. Let $v_{k}$ be the lowest vertex in the cycle. Then $v_{k}$ has two upward edges, and in particular, $k<i$. Remove the edge $v_{k}v_{l}$ that goes higher up. Then $v_{k}$ does not become a sink, and furthermore, if the edge $v_{k}v_{k+1}$ is in $T$, then we do not remove it.
∎
Since no vertex is both a sink and a source, any tree has $s+t\leq n$, which yields the following result that will be useful later on when we reconfigure between a path and a tree.
Corollary 2.
Let $T$ be a non-crossing spanning tree. Then either $T$ can be reconfigured to a downward tree in $0.5n-1$ flips or $T$ can be reconfigured to an upward tree in $0.5n-1$ flips. Furthermore, these reconfiguration sequences do not flip any edge of the form
$v_{i}v_{i+1}$ where $1\leq i<n$.
We next bound the number of flips needed to reconfigure two given trees into
opposite trees,
meaning that one tree is upward and one is downward.
By Theorem 1, we can easily do this in at most $2n-4$ flips (using $n-2$ flips to reconfigure each tree independently). We now show that $n-2$ flips suffice to reconfigure the two trees into opposite trees.
Theorem 3.
Given two non-crossing spanning trees on the same point set,
we can flip them into opposite trees
in at most $n-2$ flips.
Proof.
Let the trees be $T_{1}$ and $T_{2}$, and
let $s_{i}$ and $t_{i}$ be the number of sources and sinks of $T_{i}$, for $i=1,2$.
Since $s_{i}+t_{i}\leq n$,
we have
$s_{1}+t_{1}+s_{2}+t_{2}\leq 2n$.
This implies that
$s_{1}+t_{2}\leq n$ or $t_{1}+s_{2}\leq n$.
In the former case use Theorem 1 to flip $T_{1}$ upward and $T_{2}$ downward in $s_{1}-1+t_{2}-1\leq n-2$ flips; otherwise flip $T_{1}$ downward and $T_{2}$ upward in $t_{1}-1+s_{2}-1\leq n-2$ flips.
∎
2.2 Phase 2: Reconfiguring an upward tree into a downward tree
In this section we show how to reconfigure
from an initial downward tree $T_{I}$ to a final upward tree $T_{F}$ on a general point set using only perfect flips.
Thus the total number of flips will be $|T_{I}\setminus T_{F}|$.
The sequence of flips is simple to describe, and it will be obvious that each flip yields a spanning tree. What needs proving is that each intermediate tree is non-crossing.
To simplify the description of the algorithm, imagine $T_{I}$ colored red and $T_{F}$ colored blue. Refer to Figure 2.
Recall that
$v_{1},\ldots,v_{n}$ is the ordering of the points by increasing $y$-coordinate.
Define $b_{i}$ to be the (unique) blue edge in $T_{F}$ going down from $v_{i}$, $i=2,\dots,n$.
An unhappy edge is an edge of $T_{I}\setminus T_{F}$, i.e., it is red but not blue.
Reconfiguration algorithm.
Let $T_{1}=T_{I}$.
For $i=2,\ldots,n$ we create a tree $T_{i}$ that contains $b_{2},\ldots,b_{i}$ and all the happy edges.
If $b_{i}$ is happy, then $b_{i}$ is already in the current tree and we simply set $T_{i}:=T_{i-1}$.
Otherwise, consider the cycle formed by adding $b_{i}$ to $T_{i-1}$ and, in this cycle,
let $r_{i}$ be the unhappy red edge with the lowest bottom endpoint.
Note that $r_{i}$ exists, otherwise all edges in the cycle would be blue.
Set $T_{i}:=T_{i-1}\cup\{b_{i}\}\setminus\{r_{i}\}$.
This reconfiguration algorithm, applied to the trees $T_{I}$ and $T_{F}$ from Figure 2, is depicted in Figure 3.
Theorem 4.
Given a red downward tree $T_{I}$ and a blue upward tree $T_{F}$ on a general point set, the reconfiguration algorithm described above flips $T_{I}$ to $T_{F}$ using $|T_{I}\setminus T_{F}|$ perfect flips.
Proof.
It is clear that each $T_{i}$ is a spanning tree, and that each flip is perfect, so the number of flips is $|T_{I}\setminus T_{F}|$. In particular, a happy edge is never removed, so $T_{i}$ contains all happy edges.
We must show that each $T_{i}$ is non-crossing. By induction, it suffices to show that if step $i\geq 2$ adds edge $b_{i}$ and removes edge $r_{i}$, then $b_{i}$ does not cross any edge of $T_{i-1}$ except possibly $r_{i}$. We examine the structure of $T_{i-1}$.
Let $B_{i-1}$ be the subtree with edges $b_{2},\ldots,b_{i-1}$.
Note that $B_{i-1}$ is connected.
By construction, $T_{i-1}$ contains $B_{i-1}$ and all the other edges of $T_{i-1}$ are red. Let $R_{i-1}$ consist of vertices $v_{i},\ldots,v_{n}$ and the edges of $T_{i-1}$ induced on those vertices. The edges of $R_{i-1}$ are red, and $R_{i-1}$ consists of some connected components (possibly isolated vertices).
In $T_{i-1}$ each component of $R_{i-1}$ has exactly one red connector-edge joining it to $B_{i-1}$.
Thus $T_{i-1}$ consists of $B_{i-1}$, $R_{i-1}$, and the connector-edges for the components of $R_{i-1}$.
Now consider the flip performed to create $T_{i}$ by adding edge $b_{i}$ and removing edge $r_{i}$. Since $b_{i}$ is a blue edge, it cannot cross any edge of $B_{i-1}$.
Since $b_{i}$’s topmost vertex is $v_{i}$, $b_{i}$ cannot cross any edge of $R_{i-1}$.
Furthermore, $b_{i}$ cannot cross a happy edge. Thus the only remaining possibility is for $b_{i}$ to cross an unhappy connector-edge.
We will prove:
Claim 5.
If $R_{i-1}$ is disconnected, then all connector-edges are happy.
Assuming the claim, we only need to show that $b_{i}$ is non-crossing when $R_{i-1}$ is connected. Then there is only one connector-edge $r$ joining $R_{i-1}$ to $B_{i-1}$. If $r$ is happy, then $b_{i}$ cannot cross it. So assume that $r$ is unhappy. Refer to Figure 4(a).
Now, the cycle $\gamma$ in $T_{i-1}\cup\{b_{i}\}$ contains $r$, since $b_{i}$ and $r$ are the only edges between $R_{i-1}$ and $B_{i-1}$. Of the red edges in $\gamma$, $r$ is the one with the lowest bottom endpoint. This implies that $r$ is chosen as $r_{i}$, and removed.
Therefore $b_{i}$ does not cross any edges of $T_{i}$, so $T_{i}$ is non-crossing.
It remains to prove the claim. Let $C$ be a connected component of $R_{i-1}$, and let $r$ be its connector-edge. Refer to Figure 4(b). We must prove that $r$ is happy.
In the initial red tree $T_{I}$, the vertices $v_{i},\ldots,v_{n}$ induce a connected subtree, so $C$ and $R_{i-1}\setminus C$ were once connected. Suppose they first became disconnected by the removal of red edge $r_{j}$ in step $j$ of the algorithm, for some $j<i$.
Consider the blue edge $b_{j}$ that was added in step $j$ of the algorithm, and the cycle $\gamma_{j}$ in $T_{j-1}\cup\{b_{j}\}$. Now $\gamma_{j}$ must contain another edge, call it $e$, with one endpoint in $C$ and one endpoint, $v_{k}$, not in $C$.
Note that $e$ is red since it has an endpoint in $C$, and note that $v_{k}$ is not in $R_{i-1}\setminus C$ otherwise $C$ and $R_{i-1}\setminus C$ would not be disconnected after the removal of $r_{j}$.
Therefore $v_{k}$ must lie in $B_{i-1}$, i.e., $k\leq i-1$. If $e$ is unhappy, then in step $j$ the algorithm would prefer to remove $e$ instead of $r_{j}$ since $e$ is a red edge in $\gamma_{j}$ with a lower bottom endpoint.
So $e$ is happy, which means that the algorithm never removes it, and it is contained in $T_{i-1}$. Therefore $e$ must be equal to $r$, the unique connector-edge in $T_{i-1}$ between $C$ and $B_{i-1}$. Therefore $r$ is happy.
∎
2.3 Two-phase reconfiguration algorithm
We can now combine the results of Sections 2.1 and 2.2 to develop a new two-phase reconfiguration algorithm between two non-crossing spanning trees $T_{I}$ and $T_{F}$:
1.
In the first phase we reconfigure $T_{I}$ into $T^{\prime}_{I}$ and $T_{F}$ into $T^{\prime}_{F}$ such that $T^{\prime}_{I}$ and $T^{\prime}_{F}$ are opposite trees (one upward and one downward), using
Theorem 3.
2.
In the second phase we reconfigure $T^{\prime}_{I}$ into $T^{\prime}_{F}$ using only perfect flips,
as given by Theorem 4.
(Note, however, that the happy edges in $T^{\prime}_{I}$ and $T^{\prime}_{F}$ may differ from the ones in $T_{I}$ and $T_{F}$, since the first phase does not preserve happy edges).
Finally, we concatenate the reconfiguration sequences from $T_{I}$ to $T^{\prime}_{I}$, from $T^{\prime}_{I}$ to $T^{\prime}_{F}$, and the reverse of the sequence from $T_{F}$ to $T^{\prime}_{F}$.
Theorem 6.
If $T_{I}$ and $T_{F}$ are non-crossing spanning trees on a general point set, then the algorithm presented above reconfigures $T_{I}$ to $T_{F}$ in at most
$2n-3$ flips.
Proof.
By Theorem 3, the first phase of the algorithm takes at most $n-2$ flips.
By Theorem 4, the second phase uses
at most $n-1$ flips.
It follows that the total number of flips is at most $2n-3$.
∎
Theorem 7.
For a general point set, if $T_{I}$ is a non-crossing spanning tree and $T_{F}$ is a non-crossing path that is monotone in some direction, then $T_{I}$ can be reconfigured to $T_{F}$ in
at most $1.5n-2-h$
flips, where $h=|T_{I}\cap T_{F}|$ is the number of happy edges.
Furthermore, there is a lower bound of $1.5n-5$ flips, even for points in convex position and if one tree is a monotone path.
Proof.
Rotate the plane so that $T_{F}$ is $y$-monotone. Note that $T_{F}$ is then
both an upward and a downward tree. We thus have the flexibility to turn $T_{I}$ into either an upward or a downward tree in the first phase of the algorithm. By Corollary 2,
$T_{I}$ can be turned into an upward or downward tree $T^{\prime}_{I}$ in at most $0.5n-1$ flips. Furthermore, since $T_{F}$ is a $y$-monotone path, any edge in
$T_{I}\cap T_{F}$ has the form $v_{i}v_{i+1}$ for some $1\leq i<n$, and thus, by
Corollary 2,
these edges do not flip, which implies that they are still in $T^{\prime}_{I}\cap T_{F}$, so $|T^{\prime}_{I}\cap T_{F}|\geq h$.
The second phase of the algorithm uses
$|T_{I}^{\prime}\setminus T_{F}|\leq n-1-h$
perfect flips to reconfigure $T^{\prime}_{I}$ into $T_{F}$. Hence the total number of flips is at most $1.5n-2-h$.
For the lower bound, see Lemma 8 below.
∎
Lemma 8.
On any set of $n\geq 4$ points in convex position, for $n$ even, there exists a non-crossing spanning tree $T_{I}$ and a
non-crossing path $T_{F}$ such that
reconfiguring $T_{I}$ to $T_{F}$ requires at least $1.5n-5$ flips, and this bound is tight.
Proof.
Our construction is depicted in Figure 5.
Note that the tree $T_{I}$ is the same as in the lower bound of
$1.5n-5$ proved by Hernando et al. [20], but their tree $T_{F}$ was not a path.
The construction is as follows (refer to Figure 5(a) and (c)). The points $v_{1}\ldots v_{n}$ (ordered by increasing $y$-coordinate) are placed in convex position on alternate sides of $v_{1}v_{n}$.
$T_{I}$ contains edges $v_{1}v_{2i+1}$ and $v_{n}v_{2i}$ for $i=1,\ldots,n/2-1$, and $v_{1}v_{n}$.
Note that $v_{1}$ and $v_{n}$ have degree $n/2$ each. $T_{F}$ is a path that connects vertices in order (so it includes edges $v_{i}v_{i+1}$, for $i=1,\ldots,n-1$).
Note that every non-hull edge of $T_{F}$ (in blue) crosses at least $n/2-1$ edges of $T_{I}$ (in red).
Indeed, edges of the form
$v_{2i+1}v_{2i}$ cross exactly $n/2$ edges:
$n/2-i-1$ edges incident to $v_{1}$, plus $i+1$ edges incident to $v_{n}$, minus $1$ because $v_{1}v_{n}$ is included in both counts.
Edges of the form $v_{2i}v_{2i+1}$ cross one less edge (specifically $v_{1}v_{2i+1}$).
Thus, any valid reconfiguration from $T_{I}$ to $T_{F}$ must flip $n/2-1$ edges of $T_{I}$ out of the way before the first of the $n-3$ non-hull edges of $T_{F}$ is added. After that, we need at least one flip for each of the remaining $n-4$ non-hull edges of $T_{F}$. Thus the total number of flips is at least $n/2-1+(n-4)=1.5n-5$.
We note that our two-phase reconfiguration algorithm uses $1.5n-3$ flips for this instance, but there is a flip sequence of length $1.5n-5$: first flip $v_{2}v_{n}$ to $v_{1}v_{2}$ to create a happy hull edge, then connect $v_{n}$ to all $v_{i}$ for odd $i\geq 7$ by performing the flips $v_{1}v_{i}$ to $v_{n}v_{i}$ in order of decreasing $i$. The number of flips thus far is $n/2-2$ (note that $v_{1}v_{3}$ and $v_{1}v_{5}$ stay in place). The resulting tree (shown in Figure 5b) has two happy hull edges. We show that this tree can be reconfigured into $T_{F}$ using perfect flips only (so the number of flips is $n-3$). Flip $v_{1}v_{n}$ to non-hull edge $v_{4}v_{5}$ and view the resulting tree as the union of an upward tree rooted at $v_{1}$ and a downward subtree rooted at $v_{n}$ (sharing $v_{4}v_{5}$). These two subtrees are separated by $v_{4}v_{5}$ and therefore can be independently reconfigured into their corresponding subtrees in $T_{F}$ using perfect flips, as given by Theorem 4. Thus the total number of flips is $(n/2-2)+(n-3)=1.5n-5$, proving this bound tight.
∎
Theorem 9.
For points in convex position, if $T_{I}$ is a non-crossing spanning tree and $T_{F}$ is a path, then $T_{I}$ can be reconfigured to $T_{F}$ in at most $1.5n-2-h$
flips, where $h=|T_{I}\cap T_{F}|$ is the number of happy edges.
Furthermore, there is a lower bound of $1.5n-5$ flips.
Proof.
When points are in convex position, two edges cross (a geometric property) if and only if their endpoints alternate in the cyclic ordering of points around the convex hull (a combinatorial property).
This insight allows us to show that the path $T_{F}$ is “equivalent to” a monotone path, which means that we can use the previous Theorem 7.
In particular, let the ordering of points in $T_{F}$ be $v_{1},\ldots,v_{n}$. We claim that the above algorithms can be applied using this ordering in place of the ordering of points by $y$-coordinate. Thus, a sink in $T_{I}$ is a point $v_{i}$ with no edge to a later vertex in the ordering, and etc.
One could justify this by examining the steps of the algorithms (we relied on geometry only to show that we can add a non-crossing edge “upward” from a sink, which becomes easy for points in convex position). As an alternative, we
make the argument formal by showing how to perturb the points so that $T_{F}$ becomes a monotone path while preserving the ordering of points around the convex hull—which justifies that the flips for the perturbed points are correct for the original points.
First adjust the points so that they lie on a circle with $v_{1}$ lowest at $y$-coordinate 1 and $v_{n}$ highest at $y$-coordinate $n$.
The convex hull separates into two chains
from $v_{1}$ to $v_{n}$.
Observe that $T_{F}$ visits the points of each chain in order from bottom to top (if $a$ appears before $b$ on one chain but $T_{F}$ visits $b$ before $a$, then the subpaths from $v_{1}$ to $b$ and from $a$ to $v_{n}$ would cross). Thus, we can place $v_{i}$ at $y$-coordinate $i$ while preserving the ordering of points around the circle.
We can now apply Theorem 7 to $T_{I}$ and $T_{F}$ on the perturbed points. This gives a sequence of at most $1.5n-h-2$ flips to reconfigure $T_{I}$ into $T_{F}$ and the flip sequence is still correct on the original points, thus proving the upper bound claimed by the theorem.
For the lower bound, note that the points in Figure 5 are in convex position and $T_{F}$ is a path. Thus Lemma 8 (which employs the example from Figure 5) settles the lower bound claim.
∎
3 Improving the Upper Bound for a Convex Point Set
In this section we
show that for $n$ points in convex position, reconfiguration between two non-crossing spanning trees can always be done with fewer than $2n$
flips.
Theorem 10.
There is an algorithm to reconfigure between
an initial non-crossing spanning tree $T_{I}$ and a final non-crossing spanning tree $T_{F}$
on $n$ points in convex position using
at most
$2d-\Omega(\log d)$ flips, where $d=|T_{I}\setminus T_{F}|$.
Before proving the theorem we note
that previous reconfiguration algorithms do not respect this bound.
Avis and Fukuda [10, Section 3.7] proved an upper bound of $2n$ minus a constant by reconfiguring each spanning tree $T_{i}$ to a star $S$ using $|S\setminus T_{i}|$ flips. When $T_{i}$ is a path, $|S\cap T_{i}|\leq 2$, so $|S\setminus T_{i}|\geq n-3$ and their method takes at least $2n-6$ flips. Similarly, the method of flipping both trees to a canonical path around the convex hull takes at least $2n-6$ flips when $T_{1}$ and $T_{2}$ are paths with only two edges on the convex hull.
Although paths behave badly for these canonicalization methods, they are actually easy cases
as we showed in
Section 2.
As in that section, we do not use a canonical tree to prove Theorem 10—instead, the flips are tailored to the specific initial and final trees.
Throughout this section, we assume points in convex position.
Consider
the
symmetric difference
$D=(T_{I}\setminus T_{F})\cup(T_{F}\setminus T_{I})$,
so $|D|=2d$.
It is easy to reconfigure $T_{I}$ to $T_{F}$ using $2d$ flips—we use $d$ flips to move the edges of $T_{I}\setminus T_{F}$ to the convex hull, giving an intermediate tree $T$, and, from the other end, use $d$ flips to move the edges of $T_{F}\setminus T_{I}$ to the same tree $T$. The plan is to save $\Omega(\log d)$ of these flips by using that many perfect flips
(recall that
a perfect flip exchanges an edge of $T_{I}\setminus T_{F}$ directly with an edge of $T_{F}\setminus T_{I}$).
In more detail, the idea is to find an edge $e\in D$ that is crossed by at most (roughly) $d/2$ edges of the other tree. We flip all but one of the crossing edges out of the way to the convex hull, and—if $e$ is chosen carefully—we show that we can perform one flip from $e$ to the last crossing edge, thus providing one perfect flip after at most $d/2$ flips.
Repeating this approach
$\log d$ times
gives our claimed bound.
We first show how to find an edge $e$ with not too many crossings. To do this, we define “minimal” edges.
An edge joining points $u$ and $v$
determines two subsets of points, those clockwise from $u$ to $v$ and those clockwise from $v$ to $u$ (both sets include $u$ and $v$). We call these the sides of the edge.
An edge is contained in a side if both endpoints are in the side.
We call a side minimal if it contains no edge of the symmetric difference $D$, and call an edge $e\in D$ minimal if at least one of its sides is minimal.
Note that if $D$ is non-empty then it contains at least one minimal edge (possibly a convex hull edge).
We need the following property of minimal edges.
Claim 11.
Let $e=uv$ be a minimal edge of $D$. Let $Q$ be a minimal side of $e$, and let $\bar{Q}$ be the other side. Suppose $e\in T_{F}$. Then
$T_{I}\cap Q$ consists of exactly two connected components, one containing $u$ and one containing $v$.
Proof.
The set $T_{F}\cap Q$ is a non-crossing tree
consisting of edge $e$ and two subtrees $T_{u}$ containing $u$ and $T_{v}$ containing $v$.
Since $e$ is minimal, there are no edges of $D$ in $Q$ except for $e$ itself.
This means that
$T_{I}\cap Q$
consists of $T_{u}$ and $T_{v}$,
and since $e\notin T_{I}$, these two components of $T_{I}\cap Q$ are disconnected in $Q$.
See Figure 6(a).
∎
Our algorithm will operate on a minimal edge $e$. To guarantee the savings in flips, we need a minimal edge with not too many crossings.
Lemma 12.
If $D$ is non-empty, then there is a minimal edge with at most
$\lfloor(d+3)/2\rfloor$ crossings.
Proof.
Clearly the lemma holds if some minimal edge is not crossed at all (e.g., a convex hull edge in $D$), so we assume that all minimal edges have a crossing.
Let $a$ be a minimal edge of $D$.
Suppose $a\in T_{F}$.
Let $A$ be a minimal side of $a$, and let $\bar{A}$ be the other side.
Our plan is to find a second minimal edge $b\in T_{F}$ such that $b$ is inside $\bar{A}$ and $b$ has
a minimal side $B$ that is contained in $\bar{A}$. We will then argue that $a$ or $b$ satisfies the lemma.
See Figure 6(b).
If $\bar{A}$ is minimal, then set $b:=a$ and $B=\bar{A}$.
Otherwise,
let $b$ be an edge of $T_{F}\setminus T_{I}$ in $\bar{A}$ whose $B$ side (the side in $\bar{A}$)
contains no other edge of $T_{F}\setminus T_{I}$.
Note that $b$ exists, and that all the edges of $T_{F}$ in $B$ (except $b$) lie in $T_{I}$.
If $b$ is not minimal, then $B$ contains
a minimal edge $c$ (which must then be in $T_{I}\setminus T_{F}$), and $c$ is not crossed by any edge of $T_{F}$,
because
such an edge would either have to cross $b$, which is impossible since $b\in T_{F}$, or lie in $B$, which is impossible since
all the edges of $T_{F}$ in $B\setminus\{b\}$
are in $T_{F}\cap T_{I}$.
But we assumed that all minimal edges are crossed, so $c$ cannot exist, and so $b$ must be minimal.
Let $k_{a}$ be the number of edges
of $T_{I}$ crossing $a$, let $k_{b}$ be the number of edges of $T_{I}$ crossing $b$, and let $k_{ab}$ be the number of edges of $T_{I}$ crossing both $a$ and $b$.
Observe that $k_{a}+k_{b}\leq d+k_{ab}$.
We claim that $k_{ab}\leq 3$. Then $k_{a}+k_{b}\leq d+3$ so $\min\{k_{a},k_{b}\}\leq(d+3)/2$, which
will complete
the proof since the number of crossings is an integer.
By Claim 11, $T_{I}\cap A$ has two connected components and $T_{I}\cap B$ has two connected components.
Now 4 connected components in a tree can have at most three edges joining them, which
implies that $k_{ab}\leq 3$.
(Note that this argument is correct even for $a=b$, though we get a sharper bound since $k_{a}=k_{b}=k_{ab}\leq 3$.)
∎
Algorithm.
Choose a minimal edge $e$ with $k$ crossings, where $0\leq k\leq(d+3)/2$ (as guaranteed by Lemma 12).
Suppose $e\in T_{F}$, so the crossing edges belong to $T_{I}$. The case where $e\in T_{I}$ is symmetric. In either case the plan is to perform some flips on $T_{I}$ and some on $T_{F}$ to reduce the difference $d$
by $k$ (or by 1, if $k=0$) and apply the algorithm recursively to the resulting instance. Note that the algorithm constructs a flip sequence by adding flips at both ends of the sequence.
If $k=0$ then add $e$ to $T_{I}$. This creates a cycle, and the cycle must have an edge $f$ in $T_{I}\setminus T_{F}$. Remove $f$. This produces a new tree $T_{I}$. We have performed one perfect flip and reduced $d$ by 1.
Now recurse.
Next suppose $k\geq 1$.
Let $e=uv$.
Let $Q$ be the minimal side of $e$ and let $\bar{Q}$ be the other side (both sets include $u$ and $v$).
Let $f_{1},\ldots,f_{k}$ be the edges that cross $e$. We will flip all but the last crossing edge to the convex hull.
For $i=1,\ldots k-1$ we flip $f_{i}$ as follows.
1.
Perform a flip in $T_{I}$ by removing $f_{i}$ and adding a convex hull edge $g$ that lies in $\bar{Q}$. (The existence of $g$ is proved below.)
2.
If $g\in T_{F}$ then this was a perfect flip and we have performed one perfect flip and reduced $d$ by 1.
3.
Otherwise (if $g\notin T_{F}$), perform a flip in $T_{F}$ by adding $g$ and removing an edge $h\in T_{F}\setminus T_{I}$,
that lies in $\bar{Q}$ and is not equal to $e$.
(The existence of $h$ is proved below.)
At this point, only $f_{k}$ crosses $e$.
Perform one flip in $T_{I}$ to remove $f_{k}$ and add $e$.
(Correctness proved below.)
Now, apply the algorithm recursively to the resulting $T_{I},T_{F}$.
This completes the description of the algorithm.
Correctness.
We must prove that $g$ and $h$ exist and that the final flip is valid.
First note that $e$ remains a minimal edge
after each flip performed inside the loop
because we never add or remove edges inside $Q$.
We need one more invariant of the loop.
Claim 13.
Throughout the loop $u$ and $v$ are disconnected in $T_{I}\cap{\bar{Q}}$.
Proof.
Suppose there is a path $\pi$ from $u$ to $v$ in $T_{I}\cap{\bar{Q}}$.
Now consider the edge $f_{k}$ which crosses $e$, say from $x\in Q$ to $y\in{\bar{Q}}$. Since $f_{k}$ cannot cross $\pi$, we have $y\in\pi$.
By Claim 11, $T_{I}\cap Q$ consists of two components, one containing $u$ and one containing $v$. Suppose, without loss of generality, that $x$ lies in the component containing $u$. Then there is a path from $x$ to $u$ in $T_{I}\cap Q$ and a path from $u$ to $y$ in $T_{I}\cap{\bar{Q}}$, and these paths together with $f_{k}$ make a cycle in $T_{I}$, a contradiction.
∎
First we prove that $g$ exists in Step 1. Removing $f_{i}$ from $T_{I}$ disconnects $T_{I}$ into two pieces. There are two convex hull edges that connect the two pieces. By Claim 11, $T_{I}\cap Q$ consists of two connected components, one containing $u$ and one containing $v$. Thus at most one of the convex hull edges lies in $Q$, so at least one lies in $\bar{Q}$.
Next we prove that $h$ exists in Step 3.
Adding $g$ to $T_{F}$ creates a cycle $\gamma$ in $T_{F}$ and this cycle must lie in $\bar{Q}$ (because $e\in T_{F}$) and must contain at least one edge of $T_{F}\setminus T_{I}$ (because $T_{I}$ does not contain a cycle). If $e$ were the only edge of $T_{F}\setminus T_{I}$ in $\gamma$, then $u$ and $v$ would be joined by a path in $T_{I}\cap{\bar{Q}}$, contradicting Claim 13. Thus $h$ exists.
Finally, we prove that the last flip in $T_{I}$ (to remove $f_{k}$ and add $e$) is valid. Removing $f_{k}$ leaves $u$ and $v$ disconnected in $Q$ by Claim 11 and disconnected in $\bar{Q}$ by Claim 13. Adding $e$ reconnects them, and yields a non-crossing spanning tree.
Analysis.
We now prove that the algorithm uses at most the claimed number of flips.
Observation 14.
In each recursive call:
if $k=0$, then the algorithm performs one perfect flip and reduces $d$ by 1;
and if $k>0$, then the algorithm performs at most $2k-1$ flips (one for $f_{k}$ and at most 2 for each other $f_{i}$) and reduces $d$ by $k$ (in each loop iteration, $g$ joins the happy set $T_{I}\cap T_{F}$ and in the final step $e$ joins the happy set).
Lemma 15.
The number of flips performed by the algorithm is at most $2d-\lfloor\log(d{+}3)\rfloor+1$.
Proof.
We prove this by induction on $d$. In the base case $d=0$ we perform $0=2d-\lfloor\log(d{+}3)\rfloor+1$ flips.
Now assume $d\geq 1$,
and consider what happens in the first recursive call of the algorithm, see Observation 14.
If the algorithm chooses an edge with $k=0$ crossings, then the algorithm performs one perfect flip.
The resulting instance has a difference set of size $d^{\prime}=d-1$ and induction applies, so in
the total number of flips we perform is at most
$$\displaystyle 1+2d^{\prime}-\lfloor\log(d^{\prime}{+}3)\rfloor+1=2d-\lfloor\log(d{+}2)\rfloor\leq 2d-\lfloor\log(d{+}3)\rfloor+1,$$
which proves the result in this case.
Now suppose that the algorithm chooses an edge with $k\geq 1$ crossings, where $k\leq\lfloor(d{+}3)/2\rfloor$.
The algorithm performs at most $2k-1$ flips and the resulting instance has a difference set
of size $d^{\prime}=d-k$ and therefore
$d^{\prime}{+}3\geq d{+}3-\lfloor(d{+}3)/2\rfloor=\lceil(d{+}3)/2\rceil\geq(d{+}3)/2$.
By induction, the total number
of flips that we perform is hence at most
$$\displaystyle(2k-1)+(2d^{\prime}-\lfloor\log(d^{\prime}{+}3)\rfloor+1)$$
$$\displaystyle\leq(2k-1)+(2(d-k)-\lfloor\log((d{+}3)/2)\rfloor+1)$$
$$\displaystyle\leq~{}2d-\lfloor\log((d{+}3)/2)\rfloor$$
$$\displaystyle\leq 2d-\lfloor\log(d{+}3)\rfloor+1$$
as desired.
∎
This completes the proof of Theorem 10.
4 The Happy Edge Conjecture
In this section we make some conjectures and prove some preliminary results in attempts to characterize which edges need to be flipped in minimum flip sequences for non-crossing spanning trees.
Recall that
an edge $e$ is happy if
$e$ lies in $T_{I}\cap T_{F}$.
We make the following
conjecture for points in convex position. In fact, we do not have a counterexample even for general point sets, though our guess is that the conjecture fails in the general case.
Conjecture 16.
[Happy Edge Conjecture for Convex Point Sets]
For any point set $P$ in convex position and any two non-crossing spanning trees $T_{I}$ and $T_{F}$ of $P$, there is a minimum flip sequence from $T_{I}$ to $T_{F}$ such that
no happy edge is flipped during the sequence.
In this section we first prove this conjecture for the case of happy edges on the convex hull. Then in Section 4.1 we make some stronger conjectures about which extra edges (outside $T_{I}$ and $T_{F}$) might be needed in minimum flip sequences. In Section 4.2 we show that even if no extra edges are needed, it may be tricky to find a minimum flip sequence—or, at least, a greedy approach fails.
Finally, in Section 4.3 we prove that the Happy Edge Conjecture fails if we restrict the flips to “slides” where one endpoint of the flipped edge is fixed and the other endpoint moves along an adjacent tree edge.
If the Happy Edge Conjecture is false then a minimum flip sequence might need to remove an edge and later add it back.
We are able to prove something about such “remove-add” subsequences, even for general point sets:
Proposition 17.
Consider any point set $P$ and any two non-crossing spanning trees $T_{I}$ and $T_{F}$ on $P$ and any minimum flip sequence from $T_{I}$ to $T_{F}$. If some edge $e$ is removed and later added back, then
some flip during that subsequence must
add an edge $f$ that crosses $e$.
Before proving this Proposition, we note the implication that the Happy Edge Conjecture is true for convex hull edges:
Corollary 18.
Conjecture 16 is true for happy edges on the convex hull.
Furthermore, every minimum flip sequence keeps the happy convex hull edges throughout the sequence.
Proof.
Let $e$ be a happy convex hull edge.
Suppose for a contradiction that there is a minimum flip sequence in which $e$ is removed. Note that $e$ must be added back, since it is in $T_{F}$. By Proposition 17, the flip sequence must use an edge $f$ that crosses $e$. But that is impossible because $e$ is a convex hull edge so nothing crosses it.
∎
Proof of Proposition 17.
Consider a flip sequence from $T_{I}$ to $T_{F}$ and suppose that an edge $e$ is removed and later added back, and that no edge crossing $e$ is added during that subsequence. We will make a shorter flip sequence. The argument is similar to the “normalization” technique used by Sleator et al. [35] to prove the happy edge result for flips in triangulations of a convex point set.
Let $T_{0},\ldots,T_{k}$ be the trees in the subsequence, where $T_{0}$ and $T_{k}$ contain $e$, but none of the intervening trees do. Suppose that none of the trees $T_{i}$ contains an edge that crosses $e$.
We will construct a shorter flip sequence from $T_{0}$ to $T_{k}$.
For each $i$, $0\leq i\leq k$ consider adding $e$ to $T_{i}$. For $i\neq 0,k$, this creates a cycle $\gamma_{i}$. Let $f_{i}$ be the first edge of $\gamma_{i}$ that is removed during the flip sequence from $T_{i}$ to $T_{k}$. Note that $f_{i}$ exists since $T_{k}$ contains $e$, so it cannot contain all of $\gamma_{i}$.
Define $N_{i}=T_{i}\cup\{e\}\setminus\{f_{i}\}$ for $1\leq i\leq k-1$, and define $N_{0}:=T_{0}$.
Observe that $N_{i}$ is a spanning tree, and is non-crossing because no edge of $T_{i}$ crosses $e$ by hypothesis.
Furthermore,
$N_{k-1}=T_{k}$
because the flip from $T_{k-1}$ to $T_{k}$ is exactly the same as the flip from $T_{k-1}$ to $N_{k-1}$.
We claim that $N_{0},\ldots,N_{k-1}$ is a flip sequence. This will complete the proof, since it is a shorter flip sequence from $T_{0}$ to $T_{k}$.
Consider $N_{i}$ and $N_{i+1}$. Suppose that the flip from $T_{i}$ to $T_{i+1}$ adds $g$ and removes $h$.
$$\begin{tikzcd}$$
Recall that $\gamma_{i}$ is the cycle containing $e$ in $T_{i}\cup e$.
If $h$ belongs to $\gamma_{i}$ then $f_{i}=h$, and then to get from $N_{i}$ to $N_{i+1}$ we add $g$ and remove $f_{i+1}$.
Next, suppose that $h$ does not belong to $\gamma_{i}$. Then the cycle $\gamma_{i}$ still exists in $T_{i+1}$. Now, $\gamma_{i+1}$ is the unique cycle in $T_{i+1}\cup e$. Thus $\gamma_{i+1}=\gamma_{i}$. Furthermore, $f_{i+1}$ is by definition the first edge removed from $\gamma_{i+1}$ in the flip sequence from $T_{i+1}$ to $T_{k}$. Thus $f_{i+1}=f_{i}$. Therefore, to get from $N_{i}$ to $N_{i+1}$ we add $g$ and remove $h$.
This shows that a single flip changes $N_{i}$ to $N_{i+1}$, which completes the proof.
∎
Note that the proof of Proposition 17 produces a strictly shorter flip sequence. But to prove the Happy Edge Conjecture (Conjecture 16) it would suffice to produce a flip sequence of the same length.
One possible approach is to consider how
remove-add pairs and add-remove pairs
interleave in a flip sequence.
Proposition 17 shows that
a remove-add pair for edge $e$ must contain an add-remove pair for $f$ inside it.
We may need to understand
how the order of flips can be rearranged in a flip sequence.
Such flip order rearrangements
are at the heart of results on triangulation flips, both for convex point sets [35, 32] and for general point sets [24].
4.1 Extra edges used in flip sequences
Any flip sequence from $T_{I}$ to $T_{F}$ must involve flips that remove edges of $T_{I}\setminus T_{F}$ and flips that add edges of $T_{F}\setminus T_{I}$.
Recall that in a perfect flip sequence, these are the only moves and they pair up perfectly, so the number of flips is $|T_{I}\setminus T_{F}|$.
Theorem 4 gives one situation where a perfect flip sequence is possible, but typically (e.g., in the example of Figure 5)
we must add edges not in $T_{F}$, and later remove them.
More formally, an edge outside $T_{I}\cup T_{F}$ that is used in a flip sequence is called a parking edge,
with the idea that we “park” edges there temporarily.
We make two further successively stronger conjectures. They may not hold, but disproving them would give more insight.
Conjecture 19.
For any point set $P$ in convex position and any two non-crossing spanning trees $T_{I}$ and $T_{F}$ of $P$ there is a minimum flip sequence from $T_{I}$ to $T_{F}$ that never
uses a parking edge that crosses an edge of $T_{F}$.
Conjecture 20.
For a point set $P$ in convex position and any two non-crossing spanning trees $T_{I}$ and $T_{F}$ on $P$ there is a minimum flip sequence from $T_{I}$ to $T_{F}$ that only uses
parking edges from the convex hull.
Our experiments verify Conjecture 20 for $n\leq 10$ points, (see Observation 22).
We note that Conjecture 20 cannot hold for general point sets (there just aren’t enough convex hull edges). However, we do not know if Conjecture 19 fails for general point sets.
Claim 21.
Conjecture 20 $\Longrightarrow$ Conjecture 19 $\Longrightarrow$ Conjecture 16.
Proof.
The first implication is clear.
For the second implication we use Proposition 17. Consider the minimum flip sequence promised by Conjecture 19. If there is a happy edge $e\in T_{I}\cap T_{F}$ that is removed during this flip sequence, then by Proposition 17, the flip sequence must add an edge $f$ that crosses $e$. But then $f$ is a parking edge that crosses an edge of $T_{F}$, a contradiction.
∎
4.2 Finding a perfect flip sequence—greedy fails
It is an open question whether there is a polynomial time algorithm to find [the length of] a minimum flip sequence between two given non-crossing spanning trees $T_{I}$ and $T_{F}$.
A more limited goal is testing whether there is a flip sequence of length $|T_{I}\setminus T_{F}|$—i.e., whether there is a perfect flip sequence. This is also open.
In Figure 7 we give an example to show that
a greedy approach to finding a perfect flip sequence may fail.
In this example there is a perfect flip sequence but a poor choice of perfect flips leads to a dead-end configuration where no further perfect flips are possible. Note that choosing perfect flips involves pairing edges of $T_{I}\setminus T_{F}$ with edges of $T_{F}\setminus T_{I}$ as well as ordering the pairs.
4.3 The Happy Edge Conjecture fails for edge slides
Researchers have examined various restricted types of flips for non-crossing spanning trees, see [28].
An edge slide is the most
restricted
flip operation possible: it keeps one endpoint of the flipped edge fixed and moves the
other one along an adjacent tree edge without intersecting any of the other edges or vertices of the tree. In other words, the edge that is removed, the edge that is inserted,
and the edge along which the slide takes place form an empty triangle. Aichholzer et al. [7] proved that for any set $P$ of $n$ points in the plane it is possible to transform between any two non-crossing spanning trees of $P$
using $O(n^{2})$ edge slides. The authors also give an example to show that $\Omega(n^{2})$ slides might be required even if the two spanning trees differ in only two edges. This example already implies that for point sets in general position the Happy Edge Conjecture fails for edge slides. We will show that this is also the case for points in convex position.
Figure 8 shows an example of two plane spanning trees $T_{I}$ and $T_{F}$ on $8$ points in convex position which can be transformed into each other with 8 slides, shown at the bottom of the figure. To obtain this short sequence we temporarily use an edge which is common to both trees to connect the two vertices $a$ and $b$. Thus this sequence contains a non-happy slide operation, that is, an edge that is common to both, $T_{I}$ and $T_{F}$ is moved. When flipping from tree $T_{I}$ to tree $T_{F}$ by using only happy slide operations there are some useful observations. First, there can not be an edge directly connecting $a$ and $b$, as this would cause a cycle. This implies that any edge which connects a vertex $v_{i}$, $1\leq i\leq 4$, with $b$ needs at least two slides to connect $a$ to some (possible different) vertex $v_{j}$. Moreover, the first of these edges that gets connected to $a$ needs at least three slides, as at the beginning this is the shortest path connecting $b$ to $a$. Thus in total we need at least $3+2+2+2=9$ happy slide operations. Figure 8(top) shows such a sequence. It is not hard to see that this example can be generalized to larger $n$ and implies that the Happy Edge Conjecture fails for points in convex position.
5 Exhaustive search over small point sets in convex position
For small point sets in convex position we investigated the minimum flip distance between non-crossing spanning trees by exhaustive computer search. Table 1 summarizes the obtained results. For $n=3,\ldots,12$ we give the number of non-crossing spanning trees (which is sequence A001764 in the On-Line Encyclopedia of Integer Sequences, https://oeis.org) and the number of reconfiguration
steps
between them. Moreover, we computed the maximum reconfiguration distance between two trees
(the diameter of the reconfiguration graph)
as well as the radius of the reconfiguration graph.
We provide the same information for the special case when the trees are non-crossing spanning paths. Note that in this case the intermediate graphs can still be non-crossing spanning trees. For the case where all intermediate graphs are also non-crossing spanning paths the diameter of the reconfiguration graph for points in convex position is known to be $2n-6$ for $n\geq 5$ [8, 17].
Results.
Our computations show that
for small sets in convex position the radius of each reconfiguration graph is strictly larger than half
the diameter.
More precisely, for $6\leq n\leq 12$ the diameter is $\lfloor 1.5n-4\rfloor$ and the radius is $n-2$ which would give an upper bound for the diameter of only $2n-4$.
But this might be an artefact of small numbers: compare for example the result of Sleator, Tarjan, and Thurston which give the upper bound of $2n-10$ for the rotation distance of binary trees which is tight only for $n\geq 13$ [32, 35]. That the radius seems not to be suitable for obtaining a tight bound for the diameter also supports our way of bounding the diameter of the reconfiguration graph by not using a central canonical tree.
In addition to the results shown in the table, we checked, for $n\leq 10$, which edges are exchanged, in order to test the Happy Edge Conjecture (Conjecture 16) and whether only parking edges on the convex hull are used (Conjecture 20).
Observation 22.
For $n\leq 10$ points in convex position (1) the Happy Edge Conjecture is true, and (2) there are always minimum flip sequences that only use parking edges on the convex hull.
Methods.
These computations are rather time consuming, as in principle for any pair of non-crossing spanning trees (paths) the flip distance has to be computed. For an unweighted and undirected graph $G$ with $n^{\prime}$ nodes (non-crossing spanning trees in our case) and $m^{\prime}$ edges (edge exchanges in our case) the standard algorithm to compute the diameter of $G$ is to apply breath first search (BFS) for each node of $G$. The time requirement for this simple solution is $O(n^{\prime}m^{\prime})$. There exist several algorithms which achieve a better running time for graphs of real world applications, see e.g., [18], but in the worst case they still need $O(n^{\prime}m^{\prime})$ time. The basic idea behind these approaches is to compute the eccentricity $e(v)$ of a node $v\in G$ (which is the radius as seen from this node $v$), and compare this with the largest distance $d$ between two nodes found so far. If $e(v)=d/2$ we know that the diameter of the graph is $d$ and the algorithm terminates. The difference between the various algorithms is how the nodes for computing the eccentricity and the lower bound for the diameter are chosen and the performance of the approaches are usually tested by applying them to a set of examples.
However, it turned out that by the structure of our reconfiguration graphs these approaches do not perform better than the simple
textbook
solution.
Because the radius of the reconfiguration graph is strictly larger than half the diameter in our test cases, no algorithmic shortcut is possible.
To still be able to compute the diameter of the rather large graphs (for $n=12$ the reconfiguration graph has $8~{}414~{}650$ nodes and $193~{}666~{}176$ edges) we make use of the inherent symmetries of our graphs. For every tree $T$ we can cyclically shift the labels of the vertices (by $1$ to $n-1$ steps) and/or mirror the tree to obtain another non-crossing spanning tree $T^{\prime}$ of the convex point set. All trees that can be obtained this way can be grouped together. While every tree is needed in the reconfiguration graph to correctly compute shortest reconfiguration distances, by symmetry a call of BFS for any tree from the same group will result in the same eccentricity. It is thus sufficient to call BFS only for one tree of each group. For $n$ points this reduces the number of calls by almost a factor of $2n$, as the size of the group can be up to $2n$ (some trees are self-symmetric in different ways, thus some groups have a cardinality less than $2n$).
For our experiments on which edges are exchanged (for Observation 22),
the computations get even more involved. The reason is that these properties of edges are defined by the initial and final tree. So it can happen that a short sub-path is valid only for some, but not all, pairs of trees where we would like to use it. Moreover, for similar reasons we can not make full use of the above described symmetry.
This is the reason why we have been able to test our conjectures only for sets with up to $n=10$ points.
6 Conclusions and Open Questions
We conclude with some open questions:
1.
We gave two algorithms to find flip sequences for non-crossing spanning trees and we bounded the length of the flip sequence. The algorithms run in polynomial time, but it would be good to optimize the run-times.
2.
A main open question is to close the gap between $1.5n$ and $2n$ for the leading term
of the diameter of the reconfiguration graph of non-crossing spanning trees.
3.
A less-explored problem is to find the radius of the reconfiguration graph (in the worst case, as a function of $n$, the number of points).
Is there a lower bound of $n-c$ on the radius of the reconfiguration graph for some small constant $c$?
4.
Prove or disprove the Happy Edge Conjecture.
5.
Is the distance problem (to find the minimum flip distance between two non-crossing spanning trees) NP-complete for general point sets? For convex point sets?
A first step towards an NP-hardness reduction would be to find instances where the Happy Edge Conjecture fails.
6.
An easier question is to test whether there is a perfect flip sequence between two non-crossing spanning trees. Can that be done in polynomial time, at least for points in convex position?
7.
Suppose the Happy Edge Conjecture turns out to be false. Is the following problem NP-hard?
Given two trees, is there a minimum flip sequence between them that does not flip happy edges?
8.
Suppose we have a minimum flip sequence that does not flip happy edges and does not use parking edges (i.e., the flips only involve edges of the difference set $D=(T_{I}\setminus T_{F})\cup(T_{F}\setminus T_{I})$). Is it a perfect flip sequence?
9.
All the questions above can be asked for the other versions of flips between non-crossing spanning trees (as discussed in Section 1.1 and surveyed in [28]).
10.
For the convex case, what if we only care about the cyclic order of points around the convex hull, i.e., we may freely relabel the points so long as we preserve the cyclic order of the labels. This “cyclic flip distance” may be less than the standard flip distance.
For example, two stars rooted at different vertices have cyclic flip distance $0$ but standard flip distance $n-2$.
Acknowledgements
This research was begun in the Fall 2019 Waterloo problem solving session run by Anna Lubiw and Therese Biedl. For preliminary progress on the Happy Edge Conjecture, we thank the participants, Pavle Bulatovic, Bhargavi Dameracharla, Owen Merkel, Anurag Murty Naredla, and Graeme Stroud.
Work on the topic continued in the 2020 and 2021 Barbados workshops run by Erik Demaine and we thank the other participants for many valuable discussions.
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Embedding large graphs into a random graph
Asaf Ferber
Department of Mathematics, Yale University, and Department of Mathematics, MIT. Emails:
[email protected], and [email protected].
Kyle Luh
Department of Mathematics, Yale University. Email: [email protected].
Oanh Nguyen
Department of Mathematics, Yale University. Email: [email protected].
(December 5, 2020)
Abstract
In this paper we consider the problem of embedding almost spanning, bounded degree graphs in a random graph. In particular, let $\Delta\geq 5$ and let $H$ be a graph on $(1-o(1))n$ vertices and with maximum degree $\Delta$. We show that a random graph $G_{n,p}$ with high probability contains a copy of $H$, provided that $p\gg(n^{-1}\log^{1/\Delta}n)^{2/(\Delta+1)}$. Our assumption on $p$ is optimal (even up to the $polylogn$ factor, which is the conjectured one for the spanning case) and the result verifies the so-called Kahn-Kalai Conjecture for this class of graphs.
1 Introduction
Ever since its introduction by Erdős and Rényi in 1960 [8], the random graph model has been one of the main objects of
study in probabilistic combinatorics. Given a positive integer $n$
and a real number $p\in[0,1]$, the binomial random graph $G_{n,p}$ is a random variable taking values in the set
of all labeled graphs on the vertex set $[n]$. We can describe the
probability distribution of $G_{n,p}$ by saying that each pair of
elements of $[n]$ forms an edge in $G_{n,p}$ independently, with
probability $p$. Usually, the questions considered in this model
have the following generic form: given some graph property ${\cal P}$ and a function $p=p(n)\in[0,1]$, determine if $G_{n,p}$
satisfies ${\cal P}$ with high probability (whp), i.e., if the
probability that $G_{n,p}$ satisfies ${\cal P}$ tends to $1$ as $n$
tends to infinity.
The core meta-problem in the area is the study of
the evolution of $G_{n,p}$, i.e., analyzing how $G_{n,p}$
behaves with respect to certain graph properties as $p$ traverses
the interval $[0,1]$. This task is inherently connected with the
problem of finding thresholds guaranteeing almost sure
appearance (or almost sure non-appearance) of certain subgraphs in
$G_{n,p}$.
Recall that for a monotone property $\mathcal{P}_{n}$ (for example, “$G_{n,p}$ contains a fixed graph $H$”), a function $q(n)$ is a threshold function for $\mathcal{P}_{n}$ if and only if
$$\Pr\left[G_{n,p}\text{ satisfies }\mathcal{P}_{n}\right]\rightarrow\left\{%
\begin{array}[]{cc}1&\text{if }p(n)/q(n)=\omega(1)\\
0&\text{ if }p(n)/q(n)=o(1).\end{array}\right.$$
A classical result for random graphs is that a threshold for $G_{n,p}$ to
contain a given (fixed) graph $H$ as a subgraph, is
$n^{-1/m(H)}$, where $m(H)=\max\{|E(Hâ²)|/|V(Hâ²)|\mid Hâ²\subseteq H\}$.
The case where $H$ is “balanced” (that is, $m(H)=|E(H)|/|V(H)|$) had been proven in [8], and for a general $H$, one can find a proof in [5], pages 257-274.
Note that equivalently one can take as a threshold function the least $p(n)$ such that for each $H^{\prime}\subseteq H$, the expected number of copies of $H^{\prime}$ in $G_{n,p}$ is at least one. This particular function has been named as the expected threshold by Kahn and Kalai in [11].
Following [11], note that it also makes sense if, instead of a fixed $H$,
we consider a sequence $\{H_{n}\}_{n}$, of graphs with $|V(H_{n})|=n$. Formally,
for an arbitrary $H$, Kahn and Kalai defined $p_{E}(H)$ to be the least $p$ such that, for every spanning $H^{\prime}\subseteq H$, $(|V(H^{\prime})|!/|Aut(H^{\prime})|)p^{|E(H^{\prime})}|\geq 1$. Then, even for a fixed graph $H$, the expected threshold is the same as $p_{E}(H_{n})$ if we consider $H_{n}$ as $H$ plus $n-|V(H)|$ many isolated vertices.
For the class of large graphs (that is, whenever the number of vertices of $H$ is allowed to grow with $n$), one cannot expect $p_{E}(H)$ to capture the true threshold behavior. For example, suppose that $H_{n}$ are Hamilton cycles (that is, a cycle on $n$ vertices). In this case, on one hand, the expected number of Hamilton cycles in $G_{n,p}$ is clearly $\mu_{n}=\frac{(n-1)!}{2}p^{n}$, and therefore the expectation threshold is of order $1/n$. On the other hand, it is well known [13], that a threshold function for $H_{n}$ is of order $\log n/n$ (in fact, more precise results are known. The interested reader is referred to [5] and its relevant references). A similar phenomena holds for perfect matchings (that is, a collection of $\lfloor n/2\rfloor$ pairwise disjoint edges), as was proven in [8].
These examples lead Kahn and Kalai to the beautiful conjecture that in fact, one cannot lose more than a $\Theta(\log|V(H)|)$ factor in $p_{E}(H_{n})$. Specifically, they conjectured the following (Conjecture 2. in [11]):
Conjecture 1.1.
For any $\{H_{n}\}_{n}$, a threshold function for the property “$G_{n,p}$ contains $H_{n}$ ” is $O(p_{E}(H_{n})\log|V(H_{n})|)$.
Given a graph $H$ on $h$ vertices, we say that a graph $G$ on $n$ vertices contains an $H$-factor if it contains $\lfloor n/h\rfloor$ vertex disjoint copies of $H$. For example, if $H$ is an edge, then an $H$-factor is simply a matching of size $\lfloor n/2\rfloor$.
Perhaps the most impressive result supporting Conjecture 1.1 is due to Johansson, Kahn and Vu [10]. In their seminal paper, Johansson, Kahn and Vu determined the threshold behavior of $H$-factors. In particular, they showed that if $H=K_{\Delta+1}$ then
$$\displaystyle p(\Delta)=\left(n^{-1}\ln^{1/\Delta}n\right)^{\frac{2}{\Delta+1}}$$
(1)
is a threshold function for the existence of an $H$-factor. Their results naturally extend to hypergraphs and as a corollary they proved the long standing conjecture, which is known as Shamir’s Problem, of determining the threshold behavior of perfect matchings in random hypergraphs (this result garnered them the prestigious Fulkerson Prize!).
As their proof relies heavily on the facts that the connected components are small and that an $H$-factor has a lot of symmetries, it is natural to consider the following, more general, problem.
Problem 1.2.
Suppose that $H$ is an arbitrary graph on $n$ vertices with maximum degree $\Delta$. What is the threshold behavior of the property “contains $H$”?
Let us denote by $\mathcal{H}(n,\Delta)$ the family of all such graphs. A common belief is that the “worst case threshold” for $H\in\mathcal{H}(n,\Delta)$ is attained when $H$ is a $K_{\Delta+1}$ factor, as this is the the most “locally dense” graph in this family. It is thus natural to conjecture the following:
Conjecture 1.3.
Let $\Delta\in\mathbb{N}$, $H\in\mathcal{H}(n,\Delta)$, and $p=\omega\left(n^{-1}\ln^{1/\Delta}n\right)^{\frac{2}{\Delta+1}}$. Then, whp $G_{n,p}$ contains a copy of $H$.
Remark 1.4.
It is worth mentioning the following:
1.
If true, Conjecture 1.3 is optimal for the family $\mathcal{H}(n,\Delta)$ as it is tight for $K_{\Delta+1}$-factors.
2.
It may be the case that for some specific members $H\in\mathcal{H}(n,\Delta)$, one can do much better. For example, if $H\in\mathcal{H}(n,2)$ is a Hamilton cycle, as mentioned above, the threshold function for $H$ is roughly $\log n/n$, where from (1) we obtain a bound of roughly $n^{-2/3}$. Another example is the following remarkable recent result due to Montgomery [12]. He showed that for every $\Delta=O(1)$ and $H\in\mathcal{H}(n,\Delta)$ which is a tree, a threshold function for $H$ is of order of magnitude at most $\log^{5}n/n$.
In fact, we believe that something stronger is true. Let $\mathcal{F}$ be a family of graphs. A graph $G$ is called $\mathcal{F}$-universal if it contains every member of $\mathcal{F}$. We believe that the following is true.
Conjecture 1.5.
Let $\Delta\in\mathbb{N}$ and $p=\omega\left(n^{-1}\ln^{1/\Delta}n\right)^{\frac{2}{\Delta+1}}$. Then, whp $G_{n,p}$ is $\mathcal{H}(n,\Delta)$-universal.
As the problem of embedding large graphs in $G_{n,p}$ is fundamental, it has been studied quite extensively, particularly with respect to special graphs such as forests, planar graphs, and Hamilton cycles. Perhaps the first known result for general graphs from $\mathcal{H}(n,\Delta)$ is due to Alon and Füredi [3] from 1992. They introduced an embedding technique based on matchings, and showed that for a given
graph $H$ on $n$ vertices and with maximum degree $\Delta(H)=\Delta$, a
typical $G_{n,p}$ contains a copy of $H$, provided that
$p=\omega\left(n^{-1/\Delta}(\log n)^{1/\Delta}\right)$. Note that this bound on $p$ is natural, as in this range it is easy to see that whp every subset of size $\Delta$ has common neighbors in $G_{n,p}$. Therefore, this bound
has become a benchmark in the field and much subsequent work on
embedding spanning or almost-spanning structures of maximum degree
$\Delta$ in random graphs achieves this threshold.
Among all the examples we mention the following. Regarding the “universality problem”, Alon, Capalbo, Kohayakawa, Rödl, Ruciński and Szemerédi [2] showed that for any $\varepsilon>0$ and any natural number $\Delta$ there exists a constant $c>0$ such that the random graph $G_{np}$ is whp $\mathcal{H}((1-\varepsilon)n,\Delta)$-universal for $p\geq c(\log n/n)^{1/\Delta}$. This result was improved to $\mathcal{H}(n,\Delta)$ by Dellamonica, Kohayakawa, Rödl and Ruciński [7] and the $p$ has been slightly improved by Nenadov, Peter, and the first author [9] in case all graphs are “not locally dense”. Recently, Conlon, Nenadov, Skorić and the first author [6] managed to improve the edge probability to $p=n^{-1/(\Delta-1)}\log^{a}n$, for the family $\mathcal{H}((1-o(1))n,\Delta)$. Regarding the “embedding one graph” problem, it is worth mentioning the very nice result of Riordan [14], who managed to prove Conjecture 1.3 up to a factor of $n^{\Theta(1/\Delta^{2})}$ using the second moment method (note that his bound converges to the truth when $\Delta$ goes to infinity).
In this paper we solve Conjecture 1.3 for almost spanning graphs. That is, we prove the following:
Theorem 1.6.
Let $\varepsilon,\delta>0$ be any constants and let $\Delta\geq 5$ be an integer. Then, there exists an $N_{0}$ such that for all $n\geq N_{0}$ the following holds.
For every $H\in\mathcal{H}((1-\varepsilon)n,\Delta)$, with probability at least $1-\delta$ the random graph $G_{n,p}$ contains a copy of $H$, provided that $p=\omega\left(n^{-1}\log^{1/\Delta}n\right)^{2/(\Delta+1)}$
Note that for $\Delta=2$ it is relatively simple to obtain this result, and $\Delta=3$ is obtained from [6]. Our proof breaks down for $\Delta=4$, and it looks like a new idea is required for settling this case.
Another thing to be mentioned is that it seems like the $\log^{2/(\Delta(\Delta+1))}$ factor is redundant for the almost spanning case (for example, it is not needed if one tries to embed $(1-\varepsilon)\frac{n}{\Delta+1}$ copies of $K_{\Delta+1}$) but we have not managed to get rid of it.
Outline of the proof. Our proof strategy is relatively simple. Given a graph $H\in\mathcal{H}((1-o(1))n,\Delta)$, we first partition it into a “sparse” part $H^{\prime}$ and a “dense” part (see Section 2.2). The “sparse” part is going to be embedded using a result of Riordan [14] which is stated in Section 2.3. The “dense” part is basically the heart of our proof. While partitioning $H$, we make sure that the “dense” part consists of constant many “batches” of small graphs to be embedded iteratively. In each round, similar to the approach taken in [6], we make use of a hypergraph matching theorem of Aharoni and Haxell [1] (see Theorem 2.6) and Janson’s inequality to show that one can extend the current embedding.
1.1 Notation
Given two induced subgraph $S,S^{\prime}$ of a graph $G$, we let $dist_{G}(S,S^{\prime})$ denote the distance between them. That is, the length of the shortest path in $G$ connecting some vertex of $S$ to a vertex of $S^{\prime}$ (for example, if $V(S)\cap V(S^{\prime})\neq\emptyset$ then $dist_{G}(S,S^{\prime})=0$).
For a graph $G$, $V(G)$ and $E(G)$ denote its vertex set and edge set respectively. The number of vertices is denoted by $v(G)=|V(G)|$ and the number of edges $e(G)=|E(G)|$. For two vertices $x,y\in G$, if $xy\in E(G)$ then we sometimes abbreviate to $x\sim_{G}y$ (or $x\sim y$ if there is no risk of confusion).
For two sequences $f_{n}$ and $g_{n}$ of positive numbers, we say that $f_{n}=O(g_{n})$ or $g_{n}=\Omega(f_{n})$ if there exists a constant $C$ such that $f_{n}\leq Cg_{n}$ for all $n$. We say that $f_{n}=o(g_{n})$ or $g_{n}=\omega(f_{n})$ if $\lim_{n\to\infty}\frac{f_{n}}{g_{n}}=0$.
2 Auxiliary results
In this section we present some tools and auxiliary results to be used in the proof of our main result.
2.1 Probability estimates
We will use lower tail estimates for random
variables which count the number of copies of certain graphs in a
random graph. The following version of
Janson’s inequality, tailored for graphs, is the main estimate we will be using. This particular
statement follows immediately from Theorems $8.1.1$ and $8.1.2$ in
[4].
Theorem 2.1 (Janson’s inequality).
Let $p\in(0,1)$ and consider a family $\{H_{i}\}_{i\in\mathcal{I}}$ of subgraphs of the complete graph on the vertex set
$[n]$. For each $i\in\mathcal{I}$, let $X_{i}$
denote the indicator random variable for the event that $H_{i}\subseteq G_{n,p}$ and, for each ordered pair $(i,j)\in\mathcal{I}\times\mathcal{I}$ with $i\neq j$, write $H_{i}\sim H_{j}$ if $E(H_{i})\cap E(H_{j})\neq\emptyset$. Then, for
$$\displaystyle X$$
$$\displaystyle=\sum_{i\in\mathcal{I}}X_{i},$$
$$\displaystyle\mu$$
$$\displaystyle=\mathbb{E}[X]=\sum_{i\in\mathcal{I}}p^{e(H_{i})},\text{ and }$$
$$\displaystyle\delta$$
$$\displaystyle=\sum_{\begin{subarray}{c}(i,j)\in\mathcal{I}\times\mathcal{I}\\
H_{i}\sim H_{j}\end{subarray}}\mathbb{E}[X_{i}X_{j}]=\sum_{\begin{subarray}{c}%
(i,j)\in\mathcal{I}\times\mathcal{I}\\
H_{i}\sim H_{j}\end{subarray}}p^{e(H_{i})+e(H_{j})-e(H_{i}\cap H_{j})}$$
and any $0<\gamma<1$,
$$\Pr[X<(1-\gamma)\mu]\leq e^{-\frac{\gamma^{2}\mu^{2}}{2(\mu+\delta)}}.$$
2.2 Partitioning $H$
In order to embed a graph $H\in\mathcal{H}((1-o(1))n,\Delta)$ in $G_{n,p}$, we first wish to partition it “nicely” in a way which will be convenient for us to describe the embedding. Before stating it formally, we need the following definition.
Definition 2.2.
Let $H\in\mathcal{H}(m,\Delta)$, let $\mathcal{S}_{1},\ldots,\mathcal{S}_{k}$ be a collection of families of induced subgraphs of $H$ and let $H^{\prime}:=H\setminus\left(\cup_{i}\cup_{S\in\mathcal{S}_{i}}S\right)$. We say that the partition $(H^{\prime},\mathcal{S}_{1},\ldots,\mathcal{S}_{k})$ is $\alpha$-good for some $\alpha>0$ if and only if the following hold.
1.
For every $i=1,\dots,k$, all the graphs in $\mathcal{S}_{i}$ are isomorphic in $H$. More precisely, for any two graphs $S$ and $S^{\prime}$ in $\mathcal{S}_{i}$, there exists a way to label the vertices $V(S)=\{v_{1},\dots,v_{g}\}$ and $V(S^{\prime})=\{v_{1}^{\prime},\dots,v_{g}^{\prime}\}$ such that $v_{j}\sim v_{l}$ if and only if $v_{j}^{\prime}\sim v_{l}^{\prime}$ and $\deg_{H}(v_{j})=\deg_{H}(v_{j}^{\prime})$ for any $j$.
2.
Each $\mathcal{S}_{i}$ contains at most $\alpha n$ graphs.
3.
For every $1\leq i\leq k$ and every graph $S\in\mathcal{S}_{i}$, $S$ is minimal with respect to the property: $\frac{e(S)}{v(S)-2}>\frac{\Delta+1}{2}$. That is, $v(S)\geq 3$ and
$$\frac{e(S)}{v(S)-2}>\frac{\Delta+1}{2}\geq\frac{e(S^{\prime})}{v(S^{\prime})-2}$$
for all proper subgraphs $S^{\prime}\subset S$ with $v(S^{\prime})\geq 3$;
4.
Every two distinct graphs in $\cup_{i}\mathcal{S}_{i}$ are vertex disjoint;
5.
For every $i$ and every $S,S^{\prime}\in\mathcal{S}_{i}$ we have $dist_{H}(S,S^{\prime})\geq 3$;
6.
The graph $H^{\prime}$ has “small” density, namely, $\frac{e(S)}{v(S)-2}\leq\frac{\Delta+1}{2}$ for every subgraph $S\subseteq H^{\prime}$ with $v(S)\geq 3$.
Lemma 2.3.
Let $\alpha\in(0,1)$, $\Delta\in\mathbb{N}$ and let $H\in\mathcal{H}(m,\Delta)$ with $m$ being sufficiently large. Then, there exists an $\alpha$-good partition $(H^{\prime},\mathcal{S}_{1},\ldots,\mathcal{S}_{k})$ of $H$ with $k\leq k(\alpha,\Delta)$ where $k(\alpha,\Delta)$ is some constant depending only on $\alpha$ and $\Delta$.
Proof.
First, we find the graph $H^{\prime}$ greedily as follows.
Let $\mathcal{S}:=\emptyset$ and $H_{0}:=H$. As long as there exists an induced subgraph $S\subseteq H_{i}$ with at least three vertices and $e(S)/(v(S)-2)>(\Delta+1)/2$, pick a minimal (with respect to inclusion) such graph $S$ and update $\mathcal{S}:=\mathcal{S}\cup\{S\}$ and $H_{i+1}:=H_{i}\setminus V(S)$. Note that this procedure must terminate at some step $\ell$, and let $H^{\prime}:=H_{\ell}$. Observe that $e(S)/(v(S)-2)\leq(\Delta+1)/2$ for every $S\subseteq H^{\prime}$ (this verifies $6.$ in the definition of $\alpha$ goodness).
Second, note that since $\Delta(H)\leq\Delta$, it follows that for every subgraph $S\subseteq H$ we have $e(S)\leq v(S)\Delta/2$. In particular, for every $S\in\mathcal{S}$, together with the inequality $e(S)/(v(S)-2)>(\Delta+1)/2$ we conclude that $v(S)\leq 2\Delta+1$.
Third, define an auxiliary graph $\mathcal{A}$ with vertex set $\mathcal{S}$, where for $S$ and $S^{\prime}$ in $\mathcal{S}$, $SS^{\prime}$ is an edge in $\mathcal{A}$ if and only if $dist_{H}(S,S^{\prime})\leq 2$. Note that since $\Delta(H)\leq\Delta$ and since $v(S)\leq 2\Delta+1$ for every $S\in\mathcal{S}$, it follows that the maximum degree of $\mathcal{A}$ is $\Delta(\mathcal{A})\leq(2\Delta+1)\Delta^{2}\leq 3\Delta^{3}$. Therefore, one can (greedily) find a proper coloring of $\mathcal{S}$ using $c=3\Delta^{3}+1$ colors. Let $\mathcal{T}_{i}$ denote the color class $i$, and observe that $\mathcal{A}[\mathcal{T}_{i}]$ is an independent set in $\mathcal{A}$. In particular, we have that for every $i$ and every $S,S^{\prime}\in\mathcal{T}_{i}$, $dist_{H}(S,S^{\prime})\geq 3$.
Next, since $v(S)\leq 2\Delta+1$ for all $S\in\mathcal{S}$, it follows that there are at most (say) $s:=\Delta^{2\Delta+1}2^{{2\Delta+1\choose 2}}$ possible distinct such graphs (up to graph isomorphism in $H$), and therefore, one can further divide each of the classes $\mathcal{T}_{i}$ into at most $s$ subsets, each of which consists of isomorphic graphs.
Finally, if any of these subsets contain more than $\alpha n$ graphs, we further partition each of them into at most $\alpha^{-1}$ subsets. All in all, by relabeling, for $k\leq sc\alpha^{-1}$, one obtains an $\alpha$-good partitioning $\mathcal{S}_{1},\ldots,\mathcal{S}_{k}$ and $H^{\prime}$ as desired.
∎
2.3 Embedding the “sparse” part
Let $H\in\mathcal{H}(m,\Delta)$ and let $(H^{\prime},\mathcal{S}_{1},\ldots,\mathcal{S}_{k})$ be an $\alpha$-good partitioning of $H$ as obtained by Lemma 2.3. Note that, by definition of $\alpha$-goodness, for every $S\subseteq H$ with at least $3$ vertices, one has
$$e(S)/(v(S)-2)\leq(\Delta+1)/2.$$
In order to embed $H^{\prime}$, we make use of the following theorem due to Riordan (an immediate corollary of [14, Theorem 5]).
Theorem 2.4.
Let $\varepsilon,\delta>0$ and let $\Delta\geq 2$ be a positive integer. Then, there exists an $N_{0}$ such that for all $n\geq N_{0}$ the following holds.
For every $H\in\mathcal{H}(n,\Delta)$, let
$$\gamma(H)=\max_{S\subset H,v(S)\geq 3}\frac{e(S)}{v(S)-2}.$$
Then with probability at least $1-\delta$ the random graph $G_{n,M}$ contains a copy of $H$, provided that $M=p^{\prime}{n\choose 2}$ with $p^{\prime}\geq n^{-1/\gamma}\log\log n$.
Note that the property “containing $H$” is a monotone increasing property (that is, whenever a graph $G$ satisfies it, every graph $G^{\prime}\supseteq G$ satisfies it as well), and since whp $e(G_{n,p})\geq(1-o(1))\binom{n}{2}p$, it follows that the same conclusion of Theorem 2.4 holds for $G_{n,p}$, provided that (say) $p\geq 2n^{-1/\gamma}\log\log n$.
2.4 Embedding the small dense graphs
The following technical lemma is the primary ingredient in the proof of our main result. Roughly speaking, together with a hypergraph matching theorem (see Theorem 2.6), given an $\alpha$-good partitioning of $H$, we wish to find a tool which enables us to embed the graphs from the sets $\mathcal{S}_{i}$. Before making the exact statement, we need the following preparation.
Let $\varepsilon>0$ and $\Delta\geq 5$. Let $H\in\mathcal{H}((1-\varepsilon)n,\Delta)$, and let $(H^{\prime},\mathcal{S}_{1},\ldots,\mathcal{S}_{k})$ be an $\alpha$-good partition of $H$ where $\alpha=\varepsilon/(1000\Delta^{2})$ and $k\leq k(\alpha,\Delta)$(the existence of such a partition is ensured by Lemma 2.3). Our strategy in the proof of Theorem 1.6 is to expose $G_{n,p}$ in $k+1$ rounds $0\leq h\leq k$ as follows. In round $h=0$ we embed $H^{\prime}$ into a $G_{0}:=G_{n,q}$, where $q$ is the unique number $q\in(0,1)$ satisfying $(1-q)^{k+1}=1-p$. For each $0\leq h\leq k-1$, in round $h+1$ we extend our current embedding $f$ of $H_{h}:=H^{\prime}\cup\left(\cup_{i\leq h}\mathcal{S}_{i}\right)$ into an embedding of $H_{h+1}:=H_{h}\cup(\cup\mathcal{S}_{h+1})$ using the edge-set of a graph $G_{h+1}=G_{n,q}$ which is independent of all the other $G_{j}$s. Here we abuse notation and let $f$ denote the partial embedding at each stage. Our goal in this section is to prove the following key lemma which enables us to extend $f$.
Lemma 2.5.
Under the above setting, for every $h\leq k-1$, assuming that $f$ is an embedding of $H_{h}$ into $\cup_{i\leq h}G_{i}$, whp the embedding $f$ can be extended into an embedding of $H_{h+1}$ using the edges of $G_{h+1}$.
Clearly, given an embedding of $H^{\prime}$ into $G_{0}$, by iterating Lemma 2.5 $k$ times, one can (whp) extend $f$ into an embedding of $H$ into $\cup_{h=0}^{k}G_{h}$ which has the exact same distribution as a $G_{n,p}$.
In order to prove Lemma 2.5, we make use of the following Hall-type argument for finding large matchings in hypergraphs due to Aharoni and Haxel [1]. This idea is based on the one in [6], although here we have to be much more careful in order to obtain the correct $p$. Bellow, we describe how to use this theorem in our setting.
Theorem 2.6 (Hall’s criterion for hypergraphs, Corollary 1.2 [1]).
Let $\{L_{1}\dots L_{t}\}$ be a family of $s$-uniform hypergraphs on the same vertex set.
If, for every $\mathcal{I}\subseteq[t]$, the hypergraph $\bigcup_{i\in\mathcal{I}}L_{i}$ contains a matching of size greater than $s(|\mathcal{I}|-1)$, then there exists a function $g:[t]\rightarrow\bigcup_{i=1}^{t}E(L_{i})$ such that $g(i)\in E(L_{i})$ and $g(i)\cap g(j)=\emptyset$ for $i\neq j$.
In order to apply Theorem 2.6 to our setting we proceed as follows.
First, let us enumerate $\mathcal{S}_{h+1}=\{S_{1},\ldots,S_{t}\}$ and recall that since our partitioning is $\alpha$-good, all the graphs $S_{i}$ in $\mathcal{S}_{h+1}$ are isomorphic (henceforth, we refer to all the graphs $S_{i}$ as $S$). Moreover, letting $s=v(S)$, from the definition of $\alpha$-goodness and the proof of Lemma 2.3, we have
$$t\leq\alpha n,$$
(2)
$$s\leq 2\Delta+1,$$
(3)
and for any proper subgraph $S^{\prime}$ of $S$, one has
$$\frac{e(S^{\prime})}{v(S^{\prime})-2}\leq\frac{\Delta+1}{2}<\frac{e(S)}{s-2}.$$
(4)
Second, suppose that $V(S)=[s]$, and for every $i\leq t$, let us enumerate the vertex set of $S_{i}$ as $v_{i1},\dots,v_{is}$ in such a way that the map $\varphi(j)=v_{ij}$ is a graph isomorphism between $S$ and $S_{i}$, for all $i$. Let $D_{h}=[n]\setminus f(V(H_{h}))$. Since $v(H_{h})\leq(1-\varepsilon)n$, it follows that $|D_{h}|\geq\varepsilon n$. Let us choose
$$\alpha=\frac{\varepsilon}{2(2\Delta+1)^{2}}.$$
(5)
Clearly, we have that
$$|D_{h}|-s^{2}t\geq\varepsilon n-s^{2}\alpha n\geq\varepsilon n/2,$$
(6)
and this will be useful in Lemma 2.7 below.
Third, let $W_{ij}$ be the (image of the) set of neighbors of $v_{ij}$ in $H_{h}$. Note that all the $W_{ij}$s are fixed because $H_{h}$ has already been embedded and there are no edges between graphs in $\mathcal{S}_{h+1}$. Moreover, by $5.$ of Definition 2.2, the family $\{W_{ij}\}_{ij\in[t]\times[s]}$ satisfies
(i)
$W_{ij}\subseteq[n]\setminus D_{h}$, and $|W_{ij}|=|W_{i^{\prime}j}|\leq d_{j}:=\Delta-\text{indeg}(v_{ij})$ where $\text{indeg}(v_{ij})$ is the degree of $v_{ij}$ in $S_{i}$, and
(ii)
$W_{ij}\cap W_{i^{\prime}j^{\prime}}=\emptyset$ for all $i\neq i^{\prime}$.
Next, for each $1\leq i\leq t$, let $L_{i}$ be the $s$-uniform hypergraph on the vertex set $D_{h}$ where $(v_{1},\dots,v_{s})\in D_{h}^{s}$ is an hyperedge if and only if, in $G_{h+1}=G_{n,q}$ (the new random graph with edge-probabillity $q$ to be generated), $v_{j}$ is connected to every vertex in $W_{ij}$ and $v_{ij}v_{ij^{\prime}}\in E(S_{i})$ implies $v_{j}v_{j^{\prime}}\in E(G_{h+1})$. That is, every hyperedge $e$ of $L_{i}$ corresponds to an embedding of $S_{i}$ which extends $f$. A moment’s thought now reveals that extending $f$ into an embedding of $H_{h+1}$ is equivalent to showing that there exists a function $g:[t]\rightarrow\bigcup_{i=1}^{t}E(L_{i})$ such that $g(i)\in E(L_{i})$ and $g(i)\cap g(j)=\emptyset$ for every $i\neq j$. To this end, in order to check that the assumptions of Theorem 2.6 are satisfied (and therefore, such a $g$ exists), we need the following technical lemma.
Lemma 2.7.
For every $r\leq t$ and $1\leq h\leq k$, with probability at least $1-\exp\left(-\omega(r\log n)\right)$, the following holds. For every $D^{\prime}\subseteq D_{h}$
of size $|D^{\prime}|\geq|D_{h}|-s^{2}r$, and every $\mathcal{I}\subseteq[t]$ of size $|\mathcal{I}|=r$, $G_{h+1}$ contains a copy $S^{\prime}=(v_{1},\ldots,v_{s})$ of $S$ for which, $v_{j}\in D^{\prime}$ for all $1\leq j\leq s$ and for some $i\in\mathcal{I}$, $W_{ij}$ is contained in the neighborhood of $v_{j}$ for all $1\leq j\leq s$.
In the rest of this section we prove Lemma 2.7.
Proof of Lemma 2.7.
Let us fix $r\leq t$, $D^{\prime}$ and $\mathcal{I}$ as in the assumptions of the lemma. Note that $s^{2}r\leq s^{2}t\leq\epsilon n/2$ and $|D_{h}|\leq n$. Therefore, there are at most $\binom{n}{n-s^{2}r}\binom{n}{r}\leq\exp\left(s^{2}r\log n+r\log n\right)=\exp%
\left(\Theta(r\log n)\right)$ ways to choose $D^{\prime}$ and $\mathcal{I}$. Since the the tail probability we aim to show is $\exp\left(-\omega(r\log n)\right)$, this allows us to take a union bound.
Next, for a copy $S^{\prime}=(v_{1},\ldots,v_{s})$ of $S$, we define the graph $S^{\prime}\oplus i$ with vertex set and edge set
$$\displaystyle V(S\oplus i)=V(S)\cup\bigcup_{j=1}^{s}W_{ij},\quad\text{and}%
\quad E(S\oplus i)=E(S)\cup\{v_{j}v\mid j\in[s]\text{ and }v\in W_{ij}\}.$$
Furthermore, we decompose $D^{\prime}$ into $s$ disjoint sets $V_{1},\ldots,V_{s}\subseteq D^{\prime}$ each of which has size roughly $|D^{\prime}|/s$, and define the family of canonical graphs $\mathcal{S}$ as
$$\mathcal{S}:=\{S^{\prime}=(v_{1},\ldots,v_{s}):S^{\prime}\mbox{ is a copy of }%
S\mbox{ and }v_{j}\in V_{j}\;\text{for all}\;j\in[s]\}$$
and set
$$\mathcal{S}^{+}:=\{S^{\prime}\oplus i:S^{\prime}\in\mathcal{S}\;\text{and}\;i%
\in[r]\}.$$
Observe that if $G$ contains any graph from $\mathcal{S}^{+}$, then $G$ contains the desired copy. In order to bound the probability for $G$ not to contain such a graph, we apply Janson’s inequality (Theorem 2.1). Thus, we need to estimate the parameters $\mu$ and $\delta$ appearing in Theorem 2.1.
For every $S^{\prime}=(v_{1},\ldots,v_{s})\in\mathcal{S}$ we have
$$\Delta s=\sum_{j=1}^{s}(d_{j}+\text{indeg}(u_{j}))=\sum_{j=1}^{s}d_{j}+2e(S).$$
(7)
Note that each graph $S^{+}\in\mathcal{S}^{+}$ appears in $G_{h}$ with probability $q^{e(S^{+})}$, where $e(S^{+})=e(S)+\sum_{j=1}^{s}|W_{ij}|\leq e(S)+\sum_{j=1}^{n}d_{j}$. Clearly,
$$\mu=|\mathcal{S}^{+}|q^{e(S^{+})}=r\left(\frac{|D^{\prime}|}{s}\right)^{s}q^{e%
(S^{+})}.$$
It is worth observing that the last equation is not exact as the sizes of the sets $V_{j}$ may differ by $1$. However, this will not affect our calculations and we abstain from correcting this to avoid clutter in the notation.
Our goal is to show that $\mu=\omega(r\log n)$ and $\delta=o(\mu^{2}r^{-1}\log^{-1}n)$. Indeed, assuming so, by Janson’s inequality we obtain
$$\mathbb{P}(G\text{ does not contain any }S^{+}\in\mathcal{S}^{+})\leq\exp\left%
(-\frac{\mu^{2}}{4(\mu+\delta)}\right)=\exp(-\omega(r\log n))$$
as desired.
First, we show that $\mu=\omega(r\log n)$. By our choice of $\alpha$ in (5), it follows that $|D^{\prime}|\geq\varepsilon n/2$. Moreover, as $p=o(1)$, $k$ is a constant, and $(1-q)^{k+1}=1-p$, it follows that $q=\Theta(p)$. Therefore, we have that
$$\mu=\Omega(rn^{s}q^{e(S)+\sum_{j=1}^{s}d_{j}})=\omega\left(rn^{s-\frac{2e(S)+2%
\sum d_{j}}{\Delta+1}}(\log n)^{2(e(S)+\sum d_{j})/(\Delta(\Delta+1))}\right).$$
In order to estimate $\mu$, we show that
$$\Delta(\Delta+1)\leq 2e(S)+2\sum d_{j}\leq(\Delta+1)s,$$
(8)
from which we conclude that
$$\mu=\omega\left(r\log n)\right),$$
as desired.
By equation (7), (8) is equivalent to proving that
$$\Delta(\Delta+1-s)\leq\sum_{j=1}^{s}d_{j}\leq s.$$
(9)
Recall that by property (4) we have
$$\frac{e(S)}{s-2}>\frac{\Delta+1}{2}.$$
Therefore, combining this with (7), we obtain that
$$2e(S)>(\Delta+1)s-2(\Delta+1)=2e(S)+\sum d_{j}+s-2(\Delta+1),$$
which yields
$$s+\sum d_{j}\leq 2\Delta+1.$$
(10)
If $s\geq\Delta+1$, then clearly $0\leq\sum d_{j}\leq\Delta<s$, proving (9).
If $s=\Delta$, then by (10), $\sum d_{j}\leq\Delta+1$. From (7), $\sum d_{j}$ has the same parity as $\Delta s=\Delta^{2}$ and therefore it cannot be $\Delta+1$. Moreover, since $e(S)\leq{s\choose 2}$, from (7), we also find that $\sum d_{j}\geq\Delta$. Thus, $\sum d_{j}=\Delta$, proving (9).
Finally, we show that $s\leq\Delta-1$ cannot happen. Assume for contradiction that $s\leq\Delta-1$. If $s=3$, due to the assumptions $\frac{e(S)}{s-2}>\frac{\Delta+1}{2}$ and $\Delta\geq 5$, $e(S)$ has to be at least $4$ which is impossible. Hence, $4\leq s\leq\Delta-1$. Plugging $e(S)\leq{s\choose 2}$ into (4) gives $\frac{s(s-1)}{s-2}>\Delta+1$, or equivalently, $s^{2}-(\Delta+2)s+2(\Delta+1)>0$. Since $\Delta\geq 5$, the equation $x^{2}-(\Delta+2)x+2(\Delta+1)=0$ has two real roots
$$x_{1}=\frac{\Delta+2-\sqrt{\Delta^{2}-4\Delta-4}}{2}$$
and
$$x_{2}=\frac{\Delta+2+\sqrt{\Delta^{2}-4\Delta-4}}{2}.$$
By a simple calculation one can show that $x_{2}\geq\Delta-1\geq s$ when $\Delta\geq 5$ and $x_{1}<4\leq s$. Thus, $s$ is bounded between $x_{1}$ and $x_{2}$. This gives a contradiction and completes the proof of (9) and hence (8).
To prove that $\delta=o(\mu^{2}r^{-1}\log^{-1}n)$, we have
$$\delta=\sum_{i,j\in[r]}\sum_{\begin{subarray}{c}S^{\prime},S^{\prime\prime}\in%
\mathcal{S}\\
S^{\prime}\oplus i\sim S^{\prime\prime}\oplus j\end{subarray}}q^{e(S^{\prime}%
\oplus i)+e(S^{\prime\prime}\oplus j)-e((S^{\prime}\oplus i)\cap(S^{\prime%
\prime}\oplus j))}.$$
We consider the two cases $i=j$ and $i\neq j$.
First, let us now consider the case $S^{\prime}\oplus i\sim S^{\prime\prime}\oplus i$, for some $i\in[r]$ and distinct graphs $S^{\prime},S^{\prime\prime}\in\mathcal{S}$. Let $J:=S^{\prime}\cap S^{\prime\prime}$ and observe that $v(J)\geq 1$ and $v(J)\leq s-1$. Let $\mathcal{J}_{1}$ be the family consisting of all possible graphs of the form $S^{\prime}\cap S^{\prime\prime}$. That is,
$$\mathcal{J}_{1}:=\{J=S^{\prime}\cap S^{\prime\prime}:S^{\prime},S^{\prime%
\prime}\in\mathcal{S}\;\text{and}\;v(J)\in\{1,\ldots,s-1\}\}.$$
We have
$$e(S^{\prime}\oplus i\cap S^{\prime\prime}\oplus i)=e(J)+\sum_{j\in J}|W_{i,j}|%
\leq e(J)+\sum_{j\in J}d_{j}.$$
With these observations in hand, we can bound the contribution of such pairs to $\delta$ as follows:
$$\displaystyle\delta_{1}=\sum_{i\in[r]}\sum_{\begin{subarray}{c}S^{\prime},S^{%
\prime\prime}\in\mathcal{S}\\
v(S^{\prime}\cap S^{\prime\prime})\geq 1\end{subarray}}q^{e(S^{\prime}\oplus i%
)+e(S^{\prime\prime}\oplus i)-e(S^{\prime}\oplus i\cap S^{\prime\prime}\oplus i%
)}\leq\sum_{i\in[r]}\sum_{J\in\mathcal{J}_{1}}\sum_{\begin{subarray}{c}S^{%
\prime},S^{\prime\prime}\in\mathcal{S}\\
S^{\prime}\cap S^{\prime\prime}=J\end{subarray}}q^{2e({S^{+}})-(e(J)+\sum_{j%
\in J}d_{j})}$$
and so
$$\displaystyle\delta_{1}\leq r\sum_{J\in\mathcal{J}_{1}}\left(\frac{|D^{\prime}%
|}{s}\right)^{2(s-v(J))}q^{2e({S^{+}})-(e(J)+\sum_{j\in J}d_{j})}=\frac{\mu^{2%
}}{r}\sum_{J\in\mathcal{J}_{1}}\left(\frac{|D^{\prime}|}{s}\right)^{-2v(J)}q^{%
-(e(J)+\sum_{j\in J}d_{j})}.$$
(11)
Let us now show that the power $e(J)+\sum_{j\in J}d_{j}$ cannot be large. Since $I:=S\setminus J$ is a proper subgraph of $S$, we have from assumption (4),
$$\frac{e(I)}{v(I)-2}\leq\frac{\Delta+1}{2}<\frac{e(S)}{s-2}.$$
Rearranging, one gets
$$\displaystyle(\Delta+1)v(I)-2e(I)\geq 2(\Delta+1)>(\Delta+1)s-2e(S),$$
which gives
$$\Delta v(J)<2e(J,I)+2e(J)-v(J)$$
where $e(J,I)$ is the number of edges connecting $I$ and $J$.
Combining that with
$$\Delta v(J)=2e(J)+\sum_{j\in J}d_{j}+e(J,I),$$
yields
$$\displaystyle\sum_{j\in J}d_{j}\leq e(J,I)-v(J)-1$$
and so
$$\displaystyle e(J)+\sum_{j\in J}d_{j}\leq\frac{2e(J)+\sum_{j\in J}d_{j}+e(J,I)%
-v(J)-1}{2}=\frac{(\Delta-1)v(J)-1}{2}<\frac{(\Delta-1)v(J)}{2}.$$
Plugging this bound into (11), we obtain
$$\displaystyle\delta_{1}$$
$$\displaystyle\leq$$
$$\displaystyle\frac{\mu^{2}}{r}\sum_{J\in\mathcal{J}_{1}}\left(\frac{|D^{\prime%
}|}{s}\right)^{-2v(J)}q^{-\frac{\Delta-1}{2}v(J)}.$$
Note that there are at most $\binom{s}{v_{J}}\left(|D^{\prime}|/s\right)^{v_{J}}$ graphs $J\in\mathcal{J}_{1}$ on $v_{J}$ vertices. Thus, we have
$$\displaystyle\delta_{1}\leq\frac{\mu^{2}}{r}\sum_{v_{J}=1}^{s-1}\binom{s}{v_{J%
}}\left(\frac{|D^{\prime}|}{s}\right)^{-v_{J}}q^{-\frac{(\Delta-1)v(J)}{2}}=O(%
\mu^{2}r^{-1}n^{-\frac{2}{\Delta+1}})=o(\mu^{2}r^{-1}\log^{-1}n).$$
Next, if $S^{\prime}\oplus i\sim S^{\prime\prime}\oplus j$ for $i\neq j$ and $S^{\prime},S^{\prime\prime}\in\mathcal{S}$, then we have $(S^{\prime}\oplus i)\cap(S^{\prime\prime}\oplus j)=S^{\prime}\cap S^{\prime\prime}$. Let $J:=S^{\prime}\cap S^{\prime\prime}$ and
observe that $e(J)\geq 1$, as otherwise $S^{\prime}\oplus i$ and $S^{\prime\prime}\oplus j$ would not have any edges in common. Let $\mathcal{J}_{2}$ be the
family consisting of all possible graphs of the form $S^{\prime}\cap S^{\prime\prime}$,
$$\mathcal{J}_{2}:=\{J=S^{\prime}\cap S^{\prime\prime}:S^{\prime},S^{\prime%
\prime}\in\mathcal{S}\;\text{and}\;e(J)\geq 1\}.$$
The contribution of such pairs to $\delta$ is
$$\delta_{2}=\sum_{i\neq j}\sum_{\begin{subarray}{c}S^{\prime},S^{\prime\prime}%
\in\mathcal{S}\\
e(S^{\prime}\cap S^{\prime\prime})\geq 1\end{subarray}}q^{e(S^{\prime}\oplus i%
)+e(S^{\prime\prime}\oplus j)-e(S^{\prime}\cap S^{\prime\prime})}=\sum_{i\neq j%
}\sum_{J\in\mathcal{J}_{2}}\sum_{\begin{subarray}{c}S^{\prime},S^{\prime\prime%
}\in\mathcal{S}\\
S^{\prime}\cap S^{\prime\prime}=J\end{subarray}}q^{2e({S^{+}})-e(J)}$$
and can by bounded by
$$\delta_{2}\leq r^{2}\sum_{J\in\mathcal{J}_{2},2\leq v(J)\leq s}\left(\frac{|D^%
{\prime}|}{s}\right)^{2(s-v(J))}q^{2e({S^{+}})-e(J)}=\mu^{2}\sum_{J\in\mathcal%
{J}_{2},2\leq v(J)\leq s}\left(\frac{|D^{\prime}|}{s}\right)^{-2v(J)}q^{-e(J)}$$
which can be further split into two terms when $S^{\prime}\neq S^{\prime\prime}$ and when $S^{\prime}=S^{\prime\prime}$,
$$\delta_{2}\leq\mu^{2}\sum_{J\in\mathcal{J}_{2},2\leq v(J)<s}\left(\frac{|D^{%
\prime}|}{s}\right)^{-2v(J)}q^{-e(J)}+\mu^{2}\sum_{S\in\mathcal{S}}n^{-2s}q^{-%
e(S)}=:\delta_{2,1}+\delta_{2,2}.$$
For $3\leq v(J)<s$ and $J\subset S$ with $S\in\mathcal{S}$, by property (4), we have
$$e(J)\leq\frac{\Delta+1}{2}(v(J)-2).$$
When $v(J)=2$, $e(J)=1$ by the definition of $\mathcal{J}_{2}$.
Thus,
$$\displaystyle\delta_{2,1}$$
$$\displaystyle=O(\mu^{2})n^{-2}q^{-1}+O(\mu^{2})\sum_{v_{J}=3}^{s-1}n^{-v_{J}}q%
^{-\frac{\Delta+1}{2}(v(J)-2)}=o\left(\mu^{2}n^{-\frac{2\Delta}{\Delta+1}}%
\right)=o\left(\mu^{2}r^{-1}\log^{-1}n\right).$$
For $\delta_{2,2}$, we have
$$\delta_{2,2}=O\left(\mu^{2}n^{-s}q^{-e(S)}\right)=o\left(\mu^{2}n^{-s+\frac{2e%
(S)}{\Delta+1}}(\log n)^{-\frac{2e(S)}{\Delta(\Delta+1)}}\right).$$
Recall that we showed in the proof of the lower bound for $\mu$ that $s\geq\Delta$.
If $s\geq\Delta+1$ then
$$n^{-s+\frac{2e(S)}{\Delta+1}}\leq n^{-s+\frac{\Delta s}{\Delta+1}}=n^{-\frac{s%
}{\Delta+1}}\leq n^{-1}\leq r^{-1}.$$
In these estimates, unless $s=\Delta+1$ and $e(S)=s\Delta/2$, we have
$$n^{-s+\frac{2e(S)}{\Delta+1}}\leq n^{-1-\frac{1}{\Delta+1}}\leq r^{-1}\log^{-1%
}n.$$
Hence,
$$\delta_{2,2}=o\left(\mu^{2}r^{-1}\log^{-1}n\right).$$
In the case that $s=\Delta+1$ and $e(S)=s\Delta/2$, that is $S$ is a clique on $\Delta+1$ vertices, we still have
$$\delta_{2,2}=o\left(\mu^{2}n^{-s+\frac{2e(S)}{\Delta+1}}(\log n)^{-\frac{2e(S)%
}{\Delta(\Delta+1)}}\right)=o\left(\mu^{2}r^{-1}\log^{-1}n\right).$$
If $s=\Delta$ then
$$e(S)\leq{s\choose 2}=(\Delta-1)\Delta/2.$$
Thus,
$$n^{-s+\frac{2e(S)}{\Delta+1}}\leq n^{-\frac{2\Delta}{\Delta+1}}\leq n^{-1-%
\frac{\Delta-1}{\Delta+1}}\leq r^{-1}n^{-\frac{\Delta-1}{\Delta+1}}.$$
In all cases, we obtain
$$\delta_{2}=o(\mu^{2}r^{-1}\log^{-1}n).$$
To sum up, we have
$$\delta\leq\delta_{1}+\delta_{2}=O(\mu^{2}r^{-1}\log^{-1}n),$$
completing the proof of the lemma.
∎
Now we are ready to prove Lemma 2.5.
Proof.
All that is left to do is to check that the condition in Theorem 2.6 is satisfied. That is, for every $\mathcal{I}\subseteq[t]$, the hypergraph $\bigcup_{i\in\mathcal{I}}H_{i}$ contains a matching of size greater than $s(|\mathcal{I}|-1)$. Let $\mathcal{I}\subseteq[t]$ and let $r:=|\mathcal{I}|$. Assume that the conclusion of Lemma 2.7 holds and choose $D^{\prime}=D_{h}$. Clearly, one can find a hyperedge $e_{1}\in L_{i_{1}}$ for some $i_{1}\in\mathcal{I}$. Let $D^{\prime}:=D\setminus e$ and observe that $|D^{\prime}|\geq|D|-s$. Then, by Lemma 2.7, there exists another edge $e_{2}\in L_{i_{2}}$ for some $i_{2}\in\mathcal{I}$. Repeating this argument $s(r-1)$ times, using the fact that the set $D^{\prime}$ always has size at least $|D|-s^{2}(r-1)\geq|D|-s^{2}r$, gives us the desired result.
∎
3 Proof of Theorem 1.6
In this section we prove Theorem 1.6.
Proof.
Actually, we have already described more or less the whole proof in Section 2.4 but for completeness, we will repeat the main steps of the argument (for more details, the reader can read the first part of Section 2.4 again).
Let $H\in\mathcal{H}((1-\varepsilon)n,\Delta)$ and let $(H^{\prime},\mathcal{S}_{1},\ldots,S_{k})$ be an $\alpha$-good partition of $H$ as obtained by Lemma 2.3 with respect to $\alpha=\frac{\varepsilon}{2(2\Delta+1)^{2}}$ (recall that $k:=k(\Delta,\alpha)$ is a constant). Let us fix $p=\beta(n)(n^{-1}\log^{1/\Delta})^{2/(\Delta+1)}$, where $\beta(n)\rightarrow\infty$ arbitrarily slowly whenever $n$ goes to infinity. Note that as “containing $H$” is a monotone increasing property, it is enough to prove the theorem for this $p$.
Let $q$ be the unique number $0<q<1$ for which
$$(1-q)^{k+1}=1-p,$$
and observe that as $p=o(1)$ and $k$ is a constant, it follows that $q=(1+o(1))p/k=\Omega(p)$.
We embed $H$ in $k+1$ rounds, where in each round $0\leq h\leq k$ we expose the edges of a graph $G_{h}:=G_{n,q}$, independently of the previous exposures, and extend a current partial embedding $f$ of $H^{\prime}\cup\left(\bigcup_{1\leq i<h}\mathcal{S}_{i}\right)$, using the edges of $G_{h}$. Clearly, $\cup G_{h}$ can be coupled as $G_{n,p}$. Formally, we can describe the scheme as follows:
Round $0$. Embed $H_{0}:=H^{\prime}$ into $G_{0}$. In order to do so, observe that by property $6.$ of the definition of $\alpha$-goodness, using the notation in Theorem 2.4, we have that
$$\gamma(H^{\prime})\leq(\Delta+1)/2.$$
Therefore, using Theorem 2.4 we find (whp) a copy of $H^{\prime}$ in $G_{0}$.
Round $h+1$. Extend the embedding of $H_{h}$ into an embedding of $H_{h+1}:=H_{h}\cup\mathcal{S}_{h+1}$. This is obtained whp directly from Lemma 2.5, and one can iterate this $k+1$ times. This completes the proof.
∎
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pacs: 13.20.He, 12.15.Mm, 12.60.Jv
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-123
LHCb-PAPER-2013-030
July 18, 2013
Search for the lepton-flavour violating decays $B^{0}_{s}\rightarrow e^{\pm}\mu^{\mp}$ and $B^{0}\rightarrow e^{\pm}\mu^{\mp}$
The LHCb collaboration
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C. Pavel-Nicorescu${}^{28}$,
A. Pazos Alvarez${}^{36}$,
A. Pellegrino${}^{40}$,
G. Penso${}^{24,l}$,
M. Pepe Altarelli${}^{37}$,
S. Perazzini${}^{14,c}$,
E. Perez Trigo${}^{36}$,
A. Pérez-Calero Yzquierdo${}^{35}$,
P. Perret${}^{5}$,
M. Perrin-Terrin${}^{6}$,
L. Pescatore${}^{44}$,
G. Pessina${}^{20}$,
K. Petridis${}^{52}$,
A. Petrolini${}^{19,i}$,
A. Phan${}^{58}$,
E. Picatoste Olloqui${}^{35}$,
B. Pietrzyk${}^{4}$,
T. Pilař${}^{47}$,
D. Pinci${}^{24}$,
S. Playfer${}^{49}$,
M. Plo Casasus${}^{36}$,
F. Polci${}^{8}$,
G. Polok${}^{25}$,
A. Poluektov${}^{47,33}$,
E. Polycarpo${}^{2}$,
A. Popov${}^{34}$,
D. Popov${}^{10}$,
B. Popovici${}^{28}$,
C. Potterat${}^{35}$,
A. Powell${}^{54}$,
J. Prisciandaro${}^{38}$,
A. Pritchard${}^{51}$,
C. Prouve${}^{7}$,
V. Pugatch${}^{43}$,
A. Puig Navarro${}^{38}$,
G. Punzi${}^{22,r}$,
W. Qian${}^{4}$,
J.H. Rademacker${}^{45}$,
B. Rakotomiaramanana${}^{38}$,
M.S. Rangel${}^{2}$,
I. Raniuk${}^{42}$,
N. Rauschmayr${}^{37}$,
G. Raven${}^{41}$,
S. Redford${}^{54}$,
M.M. Reid${}^{47}$,
A.C. dos Reis${}^{1}$,
S. Ricciardi${}^{48}$,
A. Richards${}^{52}$,
K. Rinnert${}^{51}$,
V. Rives Molina${}^{35}$,
D.A. Roa Romero${}^{5}$,
P. Robbe${}^{7}$,
D.A. Roberts${}^{57}$,
E. Rodrigues${}^{53}$,
P. Rodriguez Perez${}^{36}$,
S. Roiser${}^{37}$,
V. Romanovsky${}^{34}$,
A. Romero Vidal${}^{36}$,
J. Rouvinet${}^{38}$,
T. Ruf${}^{37}$,
F. Ruffini${}^{22}$,
H. Ruiz${}^{35}$,
P. Ruiz Valls${}^{35}$,
G. Sabatino${}^{24,k}$,
J.J. Saborido Silva${}^{36}$,
N. Sagidova${}^{29}$,
P. Sail${}^{50}$,
B. Saitta${}^{15,d}$,
V. Salustino Guimaraes${}^{2}$,
C. Salzmann${}^{39}$,
B. Sanmartin Sedes${}^{36}$,
M. Sannino${}^{19,i}$,
R. Santacesaria${}^{24}$,
C. Santamarina Rios${}^{36}$,
E. Santovetti${}^{23,k}$,
M. Sapunov${}^{6}$,
A. Sarti${}^{18,l}$,
C. Satriano${}^{24,m}$,
A. Satta${}^{23}$,
M. Savrie${}^{16,e}$,
D. Savrina${}^{30,31}$,
P. Schaack${}^{52}$,
M. Schiller${}^{41}$,
H. Schindler${}^{37}$,
M. Schlupp${}^{9}$,
M. Schmelling${}^{10}$,
B. Schmidt${}^{37}$,
O. Schneider${}^{38}$,
A. Schopper${}^{37}$,
M.-H. Schune${}^{7}$,
R. Schwemmer${}^{37}$,
B. Sciascia${}^{18}$,
A. Sciubba${}^{24}$,
M. Seco${}^{36}$,
A. Semennikov${}^{30}$,
K. Senderowska${}^{26}$,
I. Sepp${}^{52}$,
N. Serra${}^{39}$,
J. Serrano${}^{6}$,
P. Seyfert${}^{11}$,
M. Shapkin${}^{34}$,
I. Shapoval${}^{16,42}$,
P. Shatalov${}^{30}$,
Y. Shcheglov${}^{29}$,
T. Shears${}^{51,37}$,
L. Shekhtman${}^{33}$,
O. Shevchenko${}^{42}$,
V. Shevchenko${}^{30}$,
A. Shires${}^{52}$,
R. Silva Coutinho${}^{47}$,
M. Sirendi${}^{46}$,
T. Skwarnicki${}^{58}$,
N.A. Smith${}^{51}$,
E. Smith${}^{54,48}$,
J. Smith${}^{46}$,
M. Smith${}^{53}$,
M.D. Sokoloff${}^{56}$,
F.J.P. Soler${}^{50}$,
F. Soomro${}^{18}$,
D. Souza${}^{45}$,
B. Souza De Paula${}^{2}$,
B. Spaan${}^{9}$,
A. Sparkes${}^{49}$,
P. Spradlin${}^{50}$,
F. Stagni${}^{37}$,
S. Stahl${}^{11}$,
O. Steinkamp${}^{39}$,
S. Stevenson${}^{54}$,
S. Stoica${}^{28}$,
S. Stone${}^{58}$,
B. Storaci${}^{39}$,
M. Straticiuc${}^{28}$,
U. Straumann${}^{39}$,
V.K. Subbiah${}^{37}$,
L. Sun${}^{56}$,
S. Swientek${}^{9}$,
V. Syropoulos${}^{41}$,
M. Szczekowski${}^{27}$,
P. Szczypka${}^{38,37}$,
T. Szumlak${}^{26}$,
S. T’Jampens${}^{4}$,
M. Teklishyn${}^{7}$,
E. Teodorescu${}^{28}$,
F. Teubert${}^{37}$,
C. Thomas${}^{54}$,
E. Thomas${}^{37}$,
J. van Tilburg${}^{11}$,
V. Tisserand${}^{4}$,
M. Tobin${}^{38}$,
S. Tolk${}^{41}$,
D. Tonelli${}^{37}$,
S. Topp-Joergensen${}^{54}$,
N. Torr${}^{54}$,
E. Tournefier${}^{4,52}$,
S. Tourneur${}^{38}$,
M.T. Tran${}^{38}$,
M. Tresch${}^{39}$,
A. Tsaregorodtsev${}^{6}$,
P. Tsopelas${}^{40}$,
N. Tuning${}^{40}$,
M. Ubeda Garcia${}^{37}$,
A. Ukleja${}^{27}$,
D. Urner${}^{53}$,
A. Ustyuzhanin${}^{52,p}$,
U. Uwer${}^{11}$,
V. Vagnoni${}^{14}$,
G. Valenti${}^{14}$,
A. Vallier${}^{7}$,
M. Van Dijk${}^{45}$,
R. Vazquez Gomez${}^{18}$,
P. Vazquez Regueiro${}^{36}$,
C. Vázquez Sierra${}^{36}$,
S. Vecchi${}^{16}$,
J.J. Velthuis${}^{45}$,
M. Veltri${}^{17,g}$,
G. Veneziano${}^{38}$,
M. Vesterinen${}^{37}$,
B. Viaud${}^{7}$,
D. Vieira${}^{2}$,
X. Vilasis-Cardona${}^{35,n}$,
A. Vollhardt${}^{39}$,
D. Volyanskyy${}^{10}$,
D. Voong${}^{45}$,
A. Vorobyev${}^{29}$,
V. Vorobyev${}^{33}$,
C. Voß${}^{60}$,
H. Voss${}^{10}$,
R. Waldi${}^{60}$,
C. Wallace${}^{47}$,
R. Wallace${}^{12}$,
S. Wandernoth${}^{11}$,
J. Wang${}^{58}$,
D.R. Ward${}^{46}$,
N.K. Watson${}^{44}$,
A.D. Webber${}^{53}$,
D. Websdale${}^{52}$,
M. Whitehead${}^{47}$,
J. Wicht${}^{37}$,
J. Wiechczynski${}^{25}$,
D. Wiedner${}^{11}$,
L. Wiggers${}^{40}$,
G. Wilkinson${}^{54}$,
M.P. Williams${}^{47,48}$,
M. Williams${}^{55}$,
F.F. Wilson${}^{48}$,
J. Wimberley${}^{57}$,
J. Wishahi${}^{9}$,
M. Witek${}^{25}$,
S.A. Wotton${}^{46}$,
S. Wright${}^{46}$,
S. Wu${}^{3}$,
K. Wyllie${}^{37}$,
Y. Xie${}^{49,37}$,
Z. Xing${}^{58}$,
Z. Yang${}^{3}$,
R. Young${}^{49}$,
X. Yuan${}^{3}$,
O. Yushchenko${}^{34}$,
M. Zangoli${}^{14}$,
M. Zavertyaev${}^{10,a}$,
F. Zhang${}^{3}$,
L. Zhang${}^{58}$,
W.C. Zhang${}^{12}$,
Y. Zhang${}^{3}$,
A. Zhelezov${}^{11}$,
A. Zhokhov${}^{30}$,
L. Zhong${}^{3}$,
A. Zvyagin${}^{37}$.
${}^{1}$Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
${}^{2}$Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
${}^{3}$Center for High Energy Physics, Tsinghua University, Beijing, China
${}^{4}$LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
${}^{5}$Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-Ferrand, France
${}^{6}$CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
${}^{7}$LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
${}^{8}$LPNHE, Université Pierre et Marie Curie, Université Paris Diderot, CNRS/IN2P3, Paris, France
${}^{9}$Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
${}^{10}$Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
${}^{11}$Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany
${}^{12}$School of Physics, University College Dublin, Dublin, Ireland
${}^{13}$Sezione INFN di Bari, Bari, Italy
${}^{14}$Sezione INFN di Bologna, Bologna, Italy
${}^{15}$Sezione INFN di Cagliari, Cagliari, Italy
${}^{16}$Sezione INFN di Ferrara, Ferrara, Italy
${}^{17}$Sezione INFN di Firenze, Firenze, Italy
${}^{18}$Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
${}^{19}$Sezione INFN di Genova, Genova, Italy
${}^{20}$Sezione INFN di Milano Bicocca, Milano, Italy
${}^{21}$Sezione INFN di Padova, Padova, Italy
${}^{22}$Sezione INFN di Pisa, Pisa, Italy
${}^{23}$Sezione INFN di Roma Tor Vergata, Roma, Italy
${}^{24}$Sezione INFN di Roma La Sapienza, Roma, Italy
${}^{25}$Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of Sciences, Kraków, Poland
${}^{26}$AGH - University of Science and Technology, Faculty of Physics and Applied Computer Science, Kraków, Poland
${}^{27}$National Center for Nuclear Research (NCBJ), Warsaw, Poland
${}^{28}$Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest-Magurele, Romania
${}^{29}$Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
${}^{30}$Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
${}^{31}$Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow, Russia
${}^{32}$Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN), Moscow, Russia
${}^{33}$Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State University, Novosibirsk, Russia
${}^{34}$Institute for High Energy Physics (IHEP), Protvino, Russia
${}^{35}$Universitat de Barcelona, Barcelona, Spain
${}^{36}$Universidad de Santiago de Compostela, Santiago de Compostela, Spain
${}^{37}$European Organization for Nuclear Research (CERN), Geneva, Switzerland
${}^{38}$Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
${}^{39}$Physik-Institut, Universität Zürich, Zürich, Switzerland
${}^{40}$Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
${}^{41}$Nikhef National Institute for Subatomic Physics and VU University Amsterdam, Amsterdam, The Netherlands
${}^{42}$NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
${}^{43}$Institute for Nuclear Research of the National Academy of Sciences (KINR), Kyiv, Ukraine
${}^{44}$University of Birmingham, Birmingham, United Kingdom
${}^{45}$H.H. Wills Physics Laboratory, University of Bristol, Bristol, United Kingdom
${}^{46}$Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
${}^{47}$Department of Physics, University of Warwick, Coventry, United Kingdom
${}^{48}$STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
${}^{49}$School of Physics and Astronomy, University of Edinburgh, Edinburgh, United Kingdom
${}^{50}$School of Physics and Astronomy, University of Glasgow, Glasgow, United Kingdom
${}^{51}$Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
${}^{52}$Imperial College London, London, United Kingdom
${}^{53}$School of Physics and Astronomy, University of Manchester, Manchester, United Kingdom
${}^{54}$Department of Physics, University of Oxford, Oxford, United Kingdom
${}^{55}$Massachusetts Institute of Technology, Cambridge, MA, United States
${}^{56}$University of Cincinnati, Cincinnati, OH, United States
${}^{57}$University of Maryland, College Park, MD, United States
${}^{58}$Syracuse University, Syracuse, NY, United States
${}^{59}$Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de Janeiro, Brazil, associated to ${}^{2}$
${}^{60}$Institut für Physik, Universität Rostock, Rostock, Germany, associated to ${}^{11}$
${}^{a}$P.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS), Moscow, Russia
${}^{b}$Università di Bari, Bari, Italy
${}^{c}$Università di Bologna, Bologna, Italy
${}^{d}$Università di Cagliari, Cagliari, Italy
${}^{e}$Università di Ferrara, Ferrara, Italy
${}^{f}$Università di Firenze, Firenze, Italy
${}^{g}$Università di Urbino, Urbino, Italy
${}^{h}$Università di Modena e Reggio Emilia, Modena, Italy
${}^{i}$Università di Genova, Genova, Italy
${}^{j}$Università di Milano Bicocca, Milano, Italy
${}^{k}$Università di Roma Tor Vergata, Roma, Italy
${}^{l}$Università di Roma La Sapienza, Roma, Italy
${}^{m}$Università della Basilicata, Potenza, Italy
${}^{n}$LIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
${}^{o}$Hanoi University of Science, Hanoi, Viet Nam
${}^{p}$Institute of Physics and Technology, Moscow, Russia
${}^{q}$Università di Padova, Padova, Italy
${}^{r}$Università di Pisa, Pisa, Italy
${}^{s}$Scuola Normale Superiore, Pisa, Italy
A search for the lepton-flavour violating decays $B^{0}_{s}\to e^{\pm}\mu^{\mp}$ and $B^{0}\to e^{\pm}\mu^{\mp}$ is
performed with a data sample, corresponding to an integrated luminosity
of 1.0 $\mbox{\,fb}^{-1}$ of $pp$ collisions at $\sqrt{s}=7$ TeV, collected by the LHCb experiment.
The observed number of $B^{0}_{s}\to e^{\pm}\mu^{\mp}$ and $B^{0}\to e^{\pm}\mu^{\mp}$ candidates is consistent with background expectations.
Upper limits on the branching fractions of both decays are determined to be ${\cal B}(B^{0}_{s}\to e^{\pm}\mu^{\mp})<1.1\,(1.4)\times 10^{-8}$ and
${\cal B}(B^{0}\to e^{\pm}\mu^{\mp})<2.8\,(3.7)\times 10^{-9}$ at 90% (95%) confidence level (C.L.).
These limits are a factor of twenty lower than those set by previous experiments. Lower bounds on the Pati-Salam leptoquark masses
are also calculated, $M_{\rm LQ}(B^{0}_{s}\to e^{\pm}\mu^{\mp})>107$ ${\mathrm{\,Te\kern-1.0ptV\!/}c^{2}}$ and $M_{\rm LQ}(B^{0}\to e^{\pm}\mu^{\mp})>126$ ${\mathrm{\,Te\kern-1.0ptV\!/}c^{2}}$ at 95% C.L.,
and are a factor of two higher than the previous bounds.
Submitted to Physical Review Letters
Rare decays that are forbidden in the Standard Model (SM) probe potential contributions from new processes and
particles at a scale beyond the reach of direct searches. The decays $B^{0}_{s}\to e^{\pm}\mu^{\mp}$ and $B^{0}\to e^{\pm}\mu^{\mp}$ and
their charged conjugate processes111Inclusion of charge conjugate processes are implied throughout this Letter.
are forbidden within the SM, in which lepton flavour is conserved.
These decays are allowed in some scenarios beyond the SM that include models
with heavy singlet Dirac neutrinos ilakovic ,
supersymmetric models susy and the Pati-Salam model patisalam .
The latter predicts a new interaction to mediate transitions between
leptons and quarks via exchange of spin$-$1 gauge bosons, called Pati-Salam leptoquarks (LQ), that carry both colour and lepton
quantum numbers.
Current limits from ATLAS atlas1 ; atlas2 ; atlas3 and CMS cms1 ; cms2 ; cms3 on the masses of first, second or
third generation leptoquarks are in the range $[0.4,0.9]$ ${\mathrm{\,Te\kern-1.0ptV\!/}c^{2}}$, depending on the value of the couplings and the decay channel.
These leptoquarks arise from a coupling between a quark and lepton of the same generation.
The decays $B^{0}_{s}\to e^{\pm}\mu^{\mp}$ and $B^{0}\to e^{\pm}\mu^{\mp}$ can be mediated by other leptoquarks which couple leptons and quarks that
are not necessarily
from the same generation valencia ; blanke , such as when the $\tau$ lepton couples to a first or second quark generation.
The previous best upper limits on the branching fraction of these decays come from the CDF collaboration bemu_cdf ,
${\cal B}(B^{0}_{s}\to e^{\pm}\mu^{\mp})$ $<2.0\;(2.6)\times 10^{-7}$ and ${\cal B}(B^{0}\to e^{\pm}\mu^{\mp})$ $<6.4\;(7.9)\times 10^{-8}$ at 90% (95%) confidence level (C.L.).
These limits correspond to bounds on the masses of the corresponding Pati-Salam
leptoquarks of $M_{\rm LQ}(B^{0}_{s}\to e^{\pm}\mu^{\mp})>47.8\,(44.9)$ ${\mathrm{\,Te\kern-1.0ptV\!/}c^{2}}$ and $M_{\rm LQ}(B^{0}\to e^{\pm}\mu^{\mp})>59.3\,(56.3)$ ${\mathrm{\,Te\kern-1.0ptV\!/}c^{2}}$ at 90 (95)% C.L. bemu_cdf .
This Letter presents a search for the $B^{0}_{s}\to e^{\pm}\mu^{\mp}$ and $B^{0}\to e^{\pm}\mu^{\mp}$ lepton-flavour violating (LFV) decays performed
with a data sample, corresponding to an integrated luminosity
of 1.0 $\mbox{\,fb}^{-1}$ of $pp$ collisions at $\sqrt{s}=7$ TeV, collected by the LHCb experiment in 2011 at the Large Hadron Collider.
To avoid any potential bias, events in the signal mass region $[5.1,5.5]\,{\mathrm{\,Ge\kern-1.0ptV\!/}c^{2}}$
were not examined until all analysis choices were finalized.
The LHCb detector is a single-arm forward spectrom-
eter covering the pseudorapidity range $2<\eta<5$, and is
described in detail in Ref. LHCbdetector .
Events were simulated for this analysis using the software described
in Refs. Sjostrand:2006za ; Lange:2001uf ; Allison:2006ve ; Agostinelli:2002hh ; Golonka:2005pn ; LHCb-PROC-2011-005 ; LHCb-PROC-2011-006 .
The trigger trigger consists of a hardware stage (L0), based on information from the calorimeter
and muon systems, followed by a
software stage (HLT) that applies a full event reconstruction, and is split into two stages called HLT1 and HLT2.
Candidate $B^{0}_{(s)}\to e^{\pm}\mu^{\mp}$ decays considered in this analysis must satisfy a hardware decision that
requires the presence of a muon candidate with transverse momentum
$p_{\rm T}>1.5$${\mathrm{\,Ge\kern-1.0ptV\!/}c}$.
All tracks considered in the HLT1 are required to have $p_{\rm T}>0.5$${\mathrm{\,Ge\kern-1.0ptV\!/}c}$.
The muon track of the $B^{0}_{(s)}\to e^{\pm}\mu^{\mp}$ candidates is required to have $p_{\rm T}>1.0$${\mathrm{\,Ge\kern-1.0ptV\!/}c}$ and impact parameter, ${\rm IP}$ $>0.1$ mm.
The HLT2 consists of
exclusive, cut-based triggers for $B^{0}_{(s)}$ two-body decays, and inclusive multivariate trigger ; BBDT $b$-hadron triggers.
The $B^{0}\to K^{+}\pi^{-}$ decay is used as the normalization channel and $B^{0}_{(s)}\to h^{+}{h}^{\prime-}$ ($h^{(\prime)}=K,\pi$) decays are used as a control channel,
since both have the same event topology as the signal.
The $B^{0}\to K^{+}\pi^{-}$ yield is computed from the yield of $B^{0}_{(s)}\to h^{+}{h}^{\prime-}$ decays, and the
fraction of $B^{0}\to K^{+}\pi^{-}$ in the $B^{0}_{(s)}\to h^{+}{h}^{\prime-}$ sample, as described in Ref. roadmap .
In order to minimize the bias introduced by the trigger requirements,
only $B^{0}_{(s)}\to h^{+}{h}^{\prime-}$ candidates that are triggered independently of the presence of either of the two signal hadrons at L0 and HLT1
are considered.
The $B^{0}_{(s)}\to e^{\pm}\mu^{\mp}$ candidates that pass the trigger selection criteria are further required to have well identified electron and
muon muonid candidates.
The measured momenta of the electrons are corrected to account for loss of momentum by bremsstrahlung in the detector using the photon
energy deposition in the electromagnetic calorimeter Kstee .
The signal candidates are required to be displaced with respect to any $pp$ collision vertex (PV), and
form a secondary vertex (SV) with $\chi^{2}$ per degree of freedom smaller than 9 and separated from the PV in the downstream direction by a flight distance significance greater than 15.
Only $B^{0}_{(s)}$ candidates with an impact parameter $\chi^{2}$ ($\chi^{2}_{\rm IP}$) less than 25 are considered.
The $\chi^{2}_{\rm IP}$ of a $B^{0}_{(s)}$ candidate
is defined as the difference between the $\chi^{2}$
of the PV reconstructed with and without the tracks forming the candidate.
When more than one PV is reconstructed, that giving the smallest
$\chi^{2}_{\rm IP}$ for the $B^{0}_{(s)}$ candidate is chosen.
Only $B^{0}_{(s)}$ candidates
with invariant mass in the range $[4.9,5.9]{\mathrm{\,Ge\kern-1.0ptV\!/}c^{2}}$ are kept for further analysis.
The selection criteria for the $B^{0}_{(s)}\to h^{+}{h}^{\prime-}$ and $B^{0}\to K^{+}\pi^{-}$ candidates are identical to those of the signal, apart from those used for particle identification.
A two-stage multivariate selection based on boosted decision trees (BDT) Breiman ; AdaBoost
is applied to the $B^{0}_{(s)}\to e^{\pm}\mu^{\mp}$ candidates following the same strategy as Ref. bsmumu .
The two multivariate discriminants are trained using simulated samples,
$B^{0}_{s}\to e^{\pm}\mu^{\mp}$ for signal and $b\bar{b}\to l^{\pm}l^{\prime\mp}X$ for background (where $l^{(\prime)}$ can either be a $\mu$ or an $e$ and
$X$ is any other set of particles),
which is dominated by simultaneous semileptonic decays of both $b$ and $\overline{b}$ hadrons within the same event.
The six variables entering the first multivariate discriminant, in decreasing order of their background rejection power, are:
the angle between the momentum of the $B^{0}_{(s)}$ candidate and the direction
defined by the vector joining the secondary and primary vertices;
the $B$ candidate IP and $\chi^{2}_{\rm IP}$; the minimum IP of the two daughter tracks with respect to any PV;
the distance of closest approach between the two daughter tracks and the $\chi^{2}$ of the SV.
The requirement on the first multivariate discriminant
removes 75 % of the background while retaining 93 % of signal, as determined from simulation.
The same selection is applied to the $B^{0}\to K^{+}\pi^{-}$ normalization channel
and the efficiencies of this requirement for the signal and the normalization channel are
equal within 0.2 %.
The surviving background mainly comprises random combinations of electrons and muons from semileptonic
$b\bar{b}\to e^{\pm}\mu^{\mp}X$ decays.
In total 5766 electron-muon pairs pass the trigger and offline selection requirements.
The selected candidates are classified in a binned two-dimensional space formed by the electron-muon invariant mass
and the output of a second BDT, for which nine variables are employed.
They are, in decreasing order of their background rejection power, the $B^{0}_{(s)}$ candidate IP and $\chi^{2}_{\rm IP}$,
the number of good two-track vertices a daughter track can make with other tracks
in the event, the $B^{0}_{(s)}$ candidate decay time, its $p_{\rm T}$
and degree of isolation cdf_iso ,
the distance of closest approach between the two tracks,
the minimum $p_{\rm T}$ of the tracks,
and the cosine of the angle between the muon momentum in the muon-electron
rest frame and the vector perpendicular to the $B$ candidate
momentum and to the beam axis.
The BDT output is independent of the invariant mass for signal
inside the search window. The output is transformed such that the signal is approximately uniformly
distributed between zero and one, while the background peaks at zero.
The probability for a signal event to have a given BDT value is
obtained from data using the $B^{0}_{(s)}\to h^{+}{h}^{\prime-}$ sample bsmumu_plb .
Simulated samples of $B^{0}_{(s)}\to e^{\pm}\mu^{\mp}$ and $B^{0}_{(s)}\to h^{+}{h}^{\prime-}$ decays have been used to check that the distributions of the variables entering in the BDT
that do not depend on the bremsstrahlung radiation are in good agreement. Corrections to the BDT shape due to the presence
of the radiation emitted by the electron of the $B^{0}_{(s)}\to e^{\pm}\mu^{\mp}$ decays have been evaluated using simulation.
The number of $B^{0}_{(s)}\to h^{+}{h}^{\prime-}$ signal events in each BDT bin is determined by
fitting the $h^{+}h^{\prime-}$ invariant mass distribution LHCbnote:carbone .
The systematic uncertainty on the signal BDT probability distribution function is taken to be the maximum spread
in the fractions of yields going into each bin, obtained by fitting the same $B^{0}_{(s)}\to h^{+}{h}^{\prime-}$ dataset with different signal and background fit models.
Corrections are applied to the BDT shape in order to take into account the effect of the
different trigger requirements used for the signal and the $B^{0}_{(s)}\to h^{+}{h}^{\prime-}$ control sample.
The invariant mass line shape of the signal events is described
by a Crystal Ball function (CB) crystalball with two tails, left and right,
defined by two parameters each.The values of the parameters
depend on the momentum resolution,
the momentum scale and the amount of bremsstrahlung radiation recovered.
The signal shape parameters are obtained from simulation, but need to be reweighted to account
for their dependency on the event multiplicity,
which affects the amount of bremsstrahlung radiation recovered and differs between data and simulation.
We use the number of hits in the scintillating pad detector ($N_{\rm SPD}$) as a measure of the event multiplicity. The distribution of
$N_{\rm SPD}$ for $B^{0}_{(s)}\to e^{\pm}\mu^{\mp}$ signal
candidates is obtained from a $B^{+}\to J/\psi(\mu^{+}\mu^{-})K^{+}$ data sample, which is selected with the same trigger conditions as the signal,
ensuring a similar distribution of $N_{\rm SPD}$.
The signal mass shape parameters are determined by reweighting the $B^{0}_{(s)}\to e^{\pm}\mu^{\mp}$ simulated events with the
$N_{\rm SPD}$ distribution measured in the $B^{+}\to J/\psi(\mu^{+}\mu^{-})K^{+}$ sample.
This reweighting technique is used also for a $J/\psi\to e^{+}e^{-}$ simulated sample and the reweighted parameters are then compared
with those obtained with a $J/\psi\to e^{+}e^{-}$ sample in data.
The difference between the mean values of the $J/\psi\to e^{+}e^{-}$ mass in data and simulation (+0.16%) is applied as a
systematic shift to the peak values of the $B^{0}\to e^{\pm}\mu^{\mp}$ and $B^{0}_{s}\to e^{\pm}\mu^{\mp}$ invariant mass in simulation. A systematic uncertainty is added to the $B^{0}_{(s)}\to e^{\pm}\mu^{\mp}$ mass parameters when the differences in the values of the other mass parameters for the $J/\psi\to e^{+}e^{-}$ sample in data and SPD-reweighted simulation
are larger than their statistical uncertainties.
The signal region, defined by the invariant mass window $[5.1,5.5]$${\mathrm{\,Ge\kern-1.0ptV\!/}c^{2}}$, retains
$(85.0\pm 0.1_{\rm stat}\pm 5.0_{\rm syst})\%$ and $(82.0\pm 0.1_{\rm stat}\pm 5.0_{\rm syst})\%$
of the $B^{0}_{s}\to e^{\pm}\mu^{\mp}$ and $B^{0}\to e^{\pm}\mu^{\mp}$ signal decays, respectively. The systematic uncertainties on these fractions are evaluated
with pseudo-experiments that fluctuate each parameter of the mass lineshape according to its uncertainty.
The width of the corresponding fraction distribution is taken as the systematic uncertainty.
The $B^{0}_{s}\to e^{\pm}\mu^{\mp}$ and $B^{0}\to e^{\pm}\mu^{\mp}$ yields are translated
into branching fractions according to
$$\displaystyle{\cal B}(B^{0}_{(s)}\to e^{\pm}\mu^{\mp})$$
$$\displaystyle=$$
$$\displaystyle\frac{{\cal B}_{\rm norm}\,{\rm\epsilon_{\rm norm}}\,f_{d}}{N_{%
\rm norm}\,{\rm\epsilon_{sig}}\,f_{d(s)}}\times N_{B^{0}_{(s)}\to e^{\pm}\mu^{%
\mp}}$$
(1)
$$\displaystyle=$$
$$\displaystyle\alpha_{B^{0}_{(s)}}\times N_{B^{0}_{(s)}\to e^{\pm}\mu^{\mp}},$$
where $N_{\rm norm}=10\,120\pm 920$ is the number of signal events in the normalization
channel, and the uncertainty is the sum in quadrature of the statistical and systematic uncertainties.
The branching fraction of the normalization channel is ${\cal B}_{\rm norm}$ $=(1.94\pm 0.96)\times 10^{-5}$ PDG2012
and $N_{B^{0}_{(s)}\to e^{\pm}\mu^{\mp}}$ is the number of observed signal events.
The factors $f_{d}$ and $f_{s}$ indicate the probabilities
that a $b$ quark fragments into a $B^{0}$ or $B^{0}_{s}$ meson, respectively.
We use $f_{s}/f_{d}=0.256\pm 0.020$
measured in $pp$ collision data at $\sqrt{s}=7$ TeV LHCb-PAPER-2012-037 .
The measured dependence of $f_{s}/f_{d}$ on the $B$ meson $p_{\rm T}$ LHCb-PAPER-2012-037
is found to be negligible for this analysis.
The efficiency ${\rm\epsilon_{sig(norm)}}$ for the signal (normalization) channel is
the product of the reconstruction efficiency of the final state particles
including the geometric detector acceptance,
the selection efficiency and the trigger efficiency.
The ratios of acceptance, reconstruction and selection efficiencies are computed with
simulation. A systematic uncertainty is assigned to these ratios, to take into account the difference between the tracking efficiencies measured in data and predicted in simulation.
Reweighting techniques are used to correct distributions in the simulation that do not match
those from data, in particular for those variables that depend on $N_{\rm SPD}$.
The trigger efficiency of L0 and HLT1 on signal decays is evaluated using data, while the HLT2 efficiency is evaluated
in simulation after validation with control samples.
The electron and muon identification efficiencies are evaluated from data using the $B^{+}\to J/\psi(\mu^{+}\mu^{-})K^{+}$ and $B^{+}\to J/\psi(e^{+}e^{-})K^{+}$ control samples.
The two normalization factors $\alpha_{B^{0}_{s}}$ and $\alpha_{B^{0}}$ are determined to be
$\alpha_{B^{0}_{s}}=(1.1\pm 0.2)\times 10^{-9}$ and
$\alpha_{B^{0}}=(2.8\pm 0.5)\times 10^{-10}$.
The BDT range is divided into eight bins with boundaries at $0.0,0.25,0.4,0.5,0.6,0.7,0.8,0.9$ and 1.0.
The number of expected combinatorial background events in each BDT bin and the invariant mass signal region
is determined from data by fitting to an exponential function events in
the mass sidebands, defined by $[4.9,5.0]{\mathrm{\,Ge\kern-1.0ptV\!/}c^{2}}$ and $[5.5,5.9]{\mathrm{\,Ge\kern-1.0ptV\!/}c^{2}}$.
The BDT output for signal and combinatorial background events is shown in Fig. 1.
In the exponential function both the slope and the normalization are allowed to vary.
The systematic uncertainty on the estimated number of combinatorial
background events in the signal regions is determined
by fluctuating the number of events observed in the sidebands according to a Poisson distribution,
and by varying the exponential slope according to its uncertainty.
As a cross-check, two other models,
the sum of two exponential functions and a single exponential fitted to the right sideband only,
have been used and provide consistent background estimates inside the signal region.
The low-mass sideband and the signal region are potentially polluted by exclusive backgrounds.
The background from $B^{+}_{c}\to J/\psi(\mu^{+}\mu^{-})e^{+}\nu_{e}$ and
$B^{+}_{c}\to J/\psi(e^{+}e^{-})\mu^{+}\nu_{\mu}$
decays is evaluated assuming the branching fraction
value from Ref. Abe:1998wi .
The decays $B^{0}\to\pi^{-}l^{+}\nu_{l}$, $B^{0}_{(s)}\to h^{+}{h}^{\prime-}$,
$B^{0}_{s}\to K^{-}l^{+}\nu_{l}$, $\mathchar 28931\relax_{b}^{0}\to pl^{-}\overline{\nu}_{l}$ and $B^{+}\to\pi^{+}l^{+}l^{-}$ (where $l^{\pm}=e^{\pm}$ or $\mu^{\pm}$)
are potential backgrounds if the hadrons are misidentified as electrons or muons.
The $B^{0}\to\pi^{-}l^{+}\nu_{l}$ and $B^{0}_{(s)}\to h^{+}{h}^{\prime-}$ branching fractions are taken from Ref. PDG2012 .
The $B^{+}\to\pi^{+}l^{+}l^{-}$ branching fraction is taken from Ref. Bpimumu .
The theoretical estimates of the $\mathchar 28931\relax_{b}^{0}\to pl^{-}\overline{\nu}_{l}$ and $B^{0}_{s}\to K^{-}l^{+}\nu_{l}$
branching fractions are taken from Refs. Wang and BsKmunu , respectively.
We use the $\mathchar 28931\relax_{b}^{0}$ fragmentation fraction $f_{\mathchar 28931\relax_{b}^{0}}$ measured by LHCb Aaij:2011jp
and account for its $p_{\rm T}$ dependence.
The mass and BDT distributions of these background modes are evaluated from simulated samples, using the probabilities of misidentifying kaon, pion and proton as muon or electron as functions of momenta and transverse
momenta, which are determined from $D^{*+}\to D^{0}(\to K^{-}\pi^{+})\pi^{+}$ and $\mathchar 28931\relax\to p\pi^{-}$
data samples. The mass lineshape of the $B^{0}_{(s)}\to h^{+}h^{{}^{\prime}-}\to e^{+}\mu^{-}$ peaking background is obtained from a simulated
sample of doubly-misidentified $B^{0}_{(s)}\to h^{+}{h}^{\prime-}$ events.
Apart from $B^{0}_{(s)}\to h^{+}{h}^{\prime-}$, all background modes are
normalized relative to the $B^{+}\to J/\psi(\mu^{+}\mu^{-})K^{+}$ decay. We assume $f_{u}=f_{d}$ where $f_{u}$ is the $B^{+}$ fragmentation fraction.
The $\mathchar 28931\relax_{b}^{0}\to pl^{-}\overline{\nu}_{l}$ and the $B^{+}_{c}\to J/\psi(\mu^{+}\mu^{-})e^{+}\nu_{e}$ and
$B^{+}_{c}\to J/\psi(e^{+}e^{-})\mu^{+}\nu_{\mu}$ modes are the dominant exclusive modes in the range
BDT$>0.5$, where the combinatorial background is reduced by a factor $\sim 500$ according to simulation.
These decay modes have an invariant mass distribution
that is compatible with an exponential in the region [4.9-5.9] ${\mathrm{\,Ge\kern-1.0ptV\!/}c^{2}}$, and hence are taken into account by the exponential
fit to the mass sidebands.
In the entire BDT and mass range ($[4.9,5.9]{\mathrm{\,Ge\kern-1.0ptV\!/}c^{2}}$), $4.4\pm 0.7$
doubly misidentified $B^{0}_{(s)}\to h^{+}{h}^{\prime-}$ decays are expected, with $(87.9\pm 0.1)\%$ lying in the signal mass interval of $[5.1,5.5]{\mathrm{\,Ge\kern-1.0ptV\!/}c^{2}}.$
The distribution in the BDT output is almost uniform, therefore $1.1\pm 0.3$ events of this background are expected in the last two BDT bins, to be compared
to $4.8^{+1.0}_{-0.7}$ background events obtained from the exponential fit
in the same mass and BDT ranges.
However, since this background peaks in the signal region, its BDT and mass distributions are accounted for in the limit computation.
For each BDT bin we count the number of candidates observed in the signal region,
and compare to the expected number of signal and background events.
The distribution of the observed candidates is shown in Fig. 2.
The systematic uncertainties in the background and signal predictions in each bin
are computed by varying the normalization factor, and the mass and BDT shapes
within their Gaussian uncertainties.
The results for the $B^{0}_{s}\to e^{\pm}\mu^{\mp}$ and $B^{0}\to e^{\pm}\mu^{\mp}$ decays
are summarized in Table 1. In the high BDT range, the observed number of candidates is in agreement with the
number of expected exclusive backgrounds in the signal region.
The compatibility of the observed distribution of events
with that expected for a given branching fraction
hypothesis is computed with the $\textrm{CL}_{\textrm{s}}$ method Read_02 .
The method provides $\textrm{CL}_{\textrm{s+b}}$, a measure of the
compatibility of the observed distribution with the signal plus background
hypothesis, $\textrm{CL}_{\textrm{b}}$, a measure of the compatibility with the background-only
hypothesis, and $\textrm{CL}_{\textrm{s}}=\textrm{CL}_{\textrm{s+b}}/\textrm{CL}_{\textrm{b}}$.
The expected and observed $\textrm{CL}_{\textrm{s}}$ values are shown in Fig. 3 for the $B^{0}_{s}\to e^{\pm}\mu^{\mp}$ and $B^{0}\to e^{\pm}\mu^{\mp}$ channels,
as a function of the assumed branching fraction.
The expected and measured limits for $B^{0}_{s}\to e^{\pm}\mu^{\mp}$ and $B^{0}\to e^{\pm}\mu^{\mp}$ at 90 %
and 95 % C.L. are shown in Table 2.
Note that since the same events are used to set limits for both $B^{0}_{s}$ and $B^{0}$ decays, the results are strongly correlated.
The inclusion of systematic uncertainties increases the expected $B^{0}\to e^{\pm}\mu^{\mp}$ and $B^{0}_{s}\to e^{\pm}\mu^{\mp}$ upper limits by $\sim 20\%$.
The observed limits are $\sim 1\,\sigma$ below the expectation due to the lower than expected numbers of observed events in the
fourth and last BDT bins.
The comparison of the distributions of observed events and expected
background events results in a p-value $(1-\textrm{CL}_{\textrm{b}})$ of 18 % (17 %)
for the $B^{0}_{s}\to e^{\pm}\mu^{\mp}$ ($B^{0}\to e^{\pm}\mu^{\mp}$) decay, where the $\textrm{CL}_{\textrm{b}}$ values are those
corresponding to $\textrm{CL}_{\textrm{s+b}}=0.5$.
In the framework of the Pati-Salam model, the relation linking the $B^{0}_{(s)}\to e^{\pm}\mu^{\mp}$ branching fractions and the leptoquark mass
($M_{\rm LQ}$) valencia is
$${\cal B}(B^{0}_{(s)}\to e^{\pm}\mu^{\mp})=\pi{\alpha_{S}^{2}(M_{\rm LQ})\over M%
_{\rm LQ}^{4}}F^{2}_{B^{0}_{(s)}}m^{3}_{B^{0}_{(s)}}R^{2}{\tau_{B^{0}_{(s)}}%
\over\hbar},$$
(2)
where
$$R={m_{B^{0}_{(s)}}\over m_{b}}{\left(\alpha_{S}(M_{\rm LQ})\over\alpha_{S}(m_{%
t})\right)}^{-{4\over 7}}{\left(\alpha_{S}(m_{t})\over\alpha_{S}(m_{b})\right)%
}^{-{12\over 23}}.$$
The $B^{0}$ and $B^{0}_{s}$ masses, $m_{B^{0}}$ and $m_{B^{0}_{s}}$, and the average lifetimes, $\tau_{B^{0}}$ and $\tau_{B^{0}_{s}}$, are taken from
Ref. PDG2012 .
The factors $F_{B^{0}}=0.190\pm 0.004$ GeV and $F_{B^{0}_{s}}=0.227\pm 0.004$ GeV
are the decay constants of the $B^{0}$ and $B^{0}_{s}$ mesons Davies , and $m_{b}$ and $m_{t}$
are the bottom and top quark masses PDG2012 , respectively, computed in the
$\overline{\rm MS}$ scheme Bardeen:1978yd .
The value of $\alpha_{s}$ at an arbitrary scale $M_{\rm LQ}$ is determined
using the software package rundec RunDec .
Using the limits on the branching fractions shown in Table 2, we find the following lower bounds for the leptoquark masses
if the leptoquark links the $\tau$ lepton to the first and second quark generation,
$M_{\rm LQ}(B^{0}_{s}\to e^{\pm}\mu^{\mp})>101\,(107){\mathrm{\,Te\kern-1.0ptV%
\!/}c^{2}}$ and $M_{\rm LQ}(B^{0}\to e^{\pm}\mu^{\mp})>135\,(126){\mathrm{\,Te\kern-1.0ptV\!/}c%
^{2}}$ at 90 (95) % C.L., respectively.
When the parameters entering in Eq. 2 are fluctuated within $\pm 1\,\sigma$, the limits on the leptoquark masses change by $\sim\pm 1$ TeV.
In summary, a search for the lepton-flavour violating decays $B^{0}_{s}\to e^{\pm}\mu^{\mp}$ and $B^{0}\to e^{\pm}\mu^{\mp}$ has been performed on a data sample, corresponding to an
integrated luminosity of 1.0$\mbox{\,fb}^{-1}$, collected in $pp$ collisions at $\sqrt{s}=7$ TeV.
The data are consistent with the background-only hypothesis.
Upper limits are set on the branching fractions, ${\cal B}(B^{0}_{s}\to e^{\pm}\mu^{\mp})$ $<1.1\,(1.4)\times 10^{-8}$ and
${\cal B}(B^{0}\to e^{\pm}\mu^{\mp})$ $<2.8\,(3.7)\times 10^{-9}$ at 90 (95) % C.L., that are the most restrictive to date.
These limits translate into lower bounds on the leptoquark masses in the Pati-Salam model valencia of $M_{\rm LQ}(B^{0}_{s}\to e^{\pm}\mu^{\mp})>101\,(107){\mathrm{\,Te\kern-1.0ptV%
\!/}c^{2}}$ and
$M_{\rm LQ}(B^{0}\to e^{\pm}\mu^{\mp})>135\,(126){\mathrm{\,Te\kern-1.0ptV\!/}c%
^{2}}$ at 90 (95) % C.L., respectively. These are a factor of two higher than the previous bounds.
I Acknowledgements
We thank Diego Guadagnoli for the theory inputs.
We express our gratitude to our colleagues in the CERN
accelerator departments for the excellent performance of the LHC. We
thank the technical and administrative staff at the LHCb
institutes. We acknowledge support from CERN and from the national
agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC (China);
CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands);
SCSR (Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC
“Kurchatov Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain);
SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United
Kingdom); NSF (USA). We also acknowledge the support received from the
ERC under FP7. The Tier1 computing centres are supported by IN2P3
(France), KIT and BMBF (Germany), INFN (Italy), NWO and SURF (The
Netherlands), PIC (Spain), GridPP (United Kingdom). We are thankful
for the computing resources put at our disposal by Yandex LLC
(Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
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Motion of Contact Line of a Crystal Over the Edge of Solid Mask in Epitaxial Lateral Overgrowth
M. Khenner
Department of Mathematics
State University of New York at Buffalo
Buffalo, NY 14260
email: [email protected]
Abstract
Mathematical model that allows for direct tracking of the homoepitaxial crystal growth out of
the window etched in the solid, pre-deposited layer on the substrate is described. The growth is governed
by the normal (to the crystal-vapor interface) flux from the vapor phase and by the interface diffusion.
The model accounts for possibly inhomogeneous energy of the mask surface
and for strong anisotropies of crystal-vapor interfacial energy and kinetic mobility.
Results demonstrate that the motion of the crystal-mask contact line slows down abruptly as radius of curvature
of the mask edge approaches zero.
Numerical procedure is suggested to overcome difficulties associated with ill-posedness of the evolution
problem for the interface with strong energy anisotropy.
Keywords: Thin films, epitaxy, MOCVD, surface diffusion,
interface dynamics, contact lines, rough surfaces, wetting, regularization of ill-posed evolution
problems.
PACS$\;$ codes: 68.10.Cr, 68.35.Ja, 68.45.Gd, 68.55.-a, 81.10.Aj.
1 Introduction
The Epitaxial Lateral Overgrowth (ELO) and Selective Area Epitaxy (SAG) are commonly used
to grow micro-scale semiconductor crystals and thin films.
The micrometer-scale selective growth is well-achievable by
chemical vapor deposition or liquid phase epitaxy [1]-[4]; the ELO and SAG can be used also to
grow nanostructures by the molecular beam epitaxy [5]-[7].
Recently, the mathematical models that allow to numerically study the ELO and SAG
from vapor [8] - [10] were introduced. These continuum, geometric crystal growth
models are formulated as
free-boundary problems in
two dimensions
(that is, normal to the substrate/mask and normal to the axis of long stripe openings
etched in the mask) and, therefore,
they are capable of explicitly accounting for the mask topography and the
interface
anisotropy. The crystal growth is by the normal
flux of particles from the vapor phase to the interface; the surface diffusion redistributes the material along the
interface during the growth. Evaporation-recondensation from both the mask surface and crystal-vapor interface, and
the diffusion of adatoms along the mask surface to the tri-junction crystal-mask-vapor
are accounted for by the models.
The details of the formulation, boundary conditions and numerical method can be found
in [8, 9]. In this paper we only address issues related to the mask surface modeling
and the crystal interaction with the edge of the mask (mask corner point). Note that
the crystal growth out of the trench occurs relatively early in the ELO process; however,
our previous modeling suggests [8, 9] that the perturbation introduced to the crystal-vapor
interface by the edge of the mask exists for a long time after the actual interaction of the advancing
edge of a crystal (contact line) and the mask edge took place. Thus it seems important to understand
the details of the contact line motion over the edge.
In [8, 9] the treatment of the contact line motion relies on the assumptions of
(a) the thermodynamic equilibrium
at the tri-junction (indeed, SAG and ELO are equilibrium processes), (b) the mask surface as the
rectangular, macroscopic step, and (c) the mask as the energetically homogeneous surface:
the mask-vapor surface energy $\gamma_{mv}=const.$ The assumptions (b) and (c)
limit the ability of any model to describe the overgrowth of the crystal line from the vertical onto
the horizontal part of the mask. Indeed, (a) implies that the equilibrium contact angle $\phi$
is given by the equation
$$\gamma_{mv}-\gamma_{cm}-\gamma_{cv}\cos{\phi}+\frac{\partial\gamma_{cv}}{%
\partial\phi}\sin{\phi}=0,$$
(1.1)
where $\gamma_{cm}$ and $\gamma_{cv}$ are the energies of the crystal-mask and crystal-vapor interfaces, respectively
(see Appendix A).
Then, (c) together with the natural assumptions of constant (at the tri-junction) $\gamma_{cm}$ and
$\gamma_{cv}$ imply that $\phi$ is a constant value
at the mask. However, due to (b) the mask edge is a singular point in the sense that the contact
angle is not well defined there. In [9], this situation was dealt with by instant readjustment
of the interface $-$ $x$-axis angle (angle of the tangent, $\phi_{x}$)
to a new value once the contact point
reaches the mask edge to preserve the equilibrium constant angle $\phi$. Such instant readjustment causes the short
unphysical retraction of the contact point into the vapor phase and influences the shape of the interface
111Angle of the tangent
is necessary for the determination of
the location of the contact point on the mask by coordinate’s extrapolation from marker particles
adjacent to the contact point, through
$\tan{\phi_{x}}=\frac{dy/ds}{dx/ds},$
(1.2)
where $s$ is the arc length along the curve (crystal-vapor interface).
.
In this paper we make an attempt to model
the overgrowth event self-consistently by approximating the mask edge by a circular arc
and restricting the contact point to movement along this arc. We numerically study the limit of zero arc radius.
Since in applications the mask is often a crystalline solid itself, we allow for the anisotropy
of $\gamma_{mv}$ and thus, by virtue of (1.1), for the variable $\phi$ along the mask.
(Quartz or tungsten masks are common [2].
Sometimes a mask is not deposited at all, but the long stripes are etched
in the crystalline GaAs or sapphire substrate itself [1, 11, 12]. The long crystalline mesas
left on a surface of a substrate in this latter case are more advantageous for the epitaxy of some semiconductor
materials than the
geometrically equivalent mesas of the foreign (to the substrate) mask material.)
Apart from the interest to this problem
that stems from the ELO technology, there is a broad class of phenomena in which the material-defect
interaction
is a key component. For instance, when liquid wets the microscopically rough surface
the contact line gets pinned on surface asperities and thus becomes unstable. The latter problem, both for a single
asperity and for the random ensemble of asperities has been
studied intensively from a physico-mechanical standpoint;
see, for instance, recent works [13, 14] and
the review article by de Gennes [15]
.
The somewhat different
situation where the surface defect is the macroscopic, organized structure
was considered, for example, by Oliver et al. in [16]. In that work, the defect is
the edge of a solid disk that supports
the liquid drop on its top.
The pinning effect of the edge has been
examined theoretically and Gibbs inequality condition for the equilibrium of a drop bound
by the edge has been confirmed experimentally. Also, the various conditions for drop stability
at the edge have been examined. Authors used a circular sapphire disk with the 90${}^{\circ}$ edge and the
aluminum disks with the edges subtending a range of angles.
In [17], the finite element stability analysis of an inclined pendant drop
was performed, as well as the experiment. The latter showed that both the contact line and the contact
angle can adjust around the capillary until the Young-Laplace equation
that balances the capillary pressure
with pressure forces due to gravity [15] is satisfied.
In [18], the asymptotic solution
to the model of the contact line pinning on a single, macroscopic spherical defect was derived.
However, our bibliography search for the study in which the direct numerical tracking of the contact line
motion over the defect is performed was unsuccessful.
Obviously, the ELO problem
for the homoepitaxial crystal is simpler
than any similar problem involving a liquid, since
there is at least no need to solve the Young-Laplace equation.
Heteroepitaxial and thermal stresses, if present in a
solid-on-solid system, may significantly complicate the analysis, but they
usually are incapable of modifying the equilibrium contact angle determined by the interfacial energies alone
[19] (however, as is well known, they may alter the dynamics of the free surface of a solid film).
The latter in the case of the SAG was demonstrated in [20],
where the morphological evolution of a heteroepitaxial thin
film growing on a patterned substrate was studied by the phase-field method.
2 Formulation
The Figure 1 shows the problem geometry.
As was already noted, the length of the open, unmasked stripes is assumed much larger
than their widths since large aspect ratios are common
in the applications; thus, the 2D crystal growth model can be used.
2.1 Equations of crystal growth
Based on the classical Mullins’ approach [21], the normal velocity of the
interface is formulated as a function of the local curvature and its derivatives.
In this work, we use
$$V_{q}=D\frac{\partial^{2}}{\partial s^{2}}\left[\left(\hat{\gamma}_{cv}(\alpha%
)+\frac{\partial^{2}\hat{\gamma}_{cv}(\alpha)}{\partial\alpha^{2}}\right)K%
\right]+J\hat{M}(\alpha),$$
(2.1)
which is the simplified expression from [9]. We neglect the evaporation-recondensation term
as being unimportant for this study and assume the isotropic interfacial diffusivity.
The equation (2.1) is in the non-dimensional form.
$D$ has the meaning of the constant effective interfacial diffusivity (see Table LABEL:Table2 in the Appendix B),
$K$ is the curvature and $J$ the composite parameter that involves, in particular,
the chemical potential of the vapor phase and the mean value of the interface mobility, $M_{0}$ (the
kinetic coefficient).
$\hat{\gamma}_{cv}(\alpha)$ and $\hat{M}(\alpha)$ are the anisotropy factors of the
interface energy and mobility, respectively.
These factors are specified in Section 2.4.
The
marker particles on the interface are numerically advanced using the parametric evolution equations
for the Cartesian coordinates of particles, viz.
$$\frac{\partial x}{\partial t}=V_{q}\ \frac{\partial y}{\partial s},$$
(2.2)
$$\frac{\partial y}{\partial t}=-V_{q}\ \frac{\partial x}{\partial s}.$$
These equations are fourth-order parabolic when $V_{q}$ is given by (2.1).
2.2 Mask Energy and Shape
To derive the expression for $\gamma_{mv}$ which accounts for the additional edge energy,
we use the method of
Xin & Wong [22, 23], which they employed in [24] for studies of the grain boundary
(GB) grooving in polycrystalline
thin films. In contrast to the case of dynamically evolving, faceted
interfaces such as GB grooves, the application to the static crystalline mask surface is straightforward
since the method requires an a priori knowledge of facets orientations and lengths.
Details of the method can be found in [22, 23].
For the rectangular crystal in the thermodynamic equilibrium that represents the mask on a substrate
(half of that crystal is shown in Fig. 1,2), the result is the following nondimensional
expression (units for the surface
energy are chosen $\mu_{m}h_{m}/\Omega$, where $\Omega$ is the atomic volume and $\mu_{m}$ is the equilibrium
chemical potential
of the mask surface):
$$\gamma_{mv}=r+\left[\bar{\ell}-r+2(r-\bar{\ell})F\left(\theta-\frac{\pi}{2}%
\right)\right]\cos{\theta}+2(1-r)\left[\frac{1}{2}-F\left(\theta-\pi\right)%
\right]\sin{\theta},\quad 0\leq\theta\leq\pi.$$
(2.3)
In (2.3), $r=R/H$ is the ratio of the nondimensional circular arc radius to the nondimensional
step height, $\bar{\ell}=d/H$, where $d$ is the nondimensional half-width of the mask, and
$F(u)$ is Heaviside function (all lengths are nondimensionalized by the half-width of the stripe, $L$).
For our modeling of the ELO, we are only interested in values of
$\theta$ in $[0,\pi/2]$. The function $\gamma_{mv}(\theta)$ is plotted in Fig. 3 for $\bar{\ell}=1$
and decreasing values
of $r$. Notice that $\gamma_{mv}$ is at maximum at $\theta=\pi/4$,
since the half-mask is taken square with the rounded edge.
The sharper is the edge, the larger is the maximum value. For the 90${}^{\circ}$
edge $r$ is exactly zero, $\theta$ takes on values 0 and $\pi/2$ only, and (2.3)
gives $\gamma_{mv}(0)=\gamma_{mv}(\pi/2)=1$.
To parametrize the mask surface, we take angle $\tilde{\theta}\in\left[-\tilde{\theta}_{1},\tilde{\theta}_{2}+\pi/2\right]$
as the independent
variable (see Fig. 2). From geometry,
$$\tilde{\theta}_{1}=\mbox{arctan}\frac{H-R}{R}=\mbox{arctan}\left(\frac{1}{r}-1%
\right),$$
(2.4)
$$\tilde{\theta}_{2}=\mbox{arctan}\frac{d-R}{R}=\mbox{arctan}\left(\frac{d}{R}-1%
\right).$$
Then,
$$\theta=\left\{\begin{array}[]{l}0,\mbox{ if}\;\;-\tilde{\theta}_{1}\leq\tilde{%
\theta}<0,\\
\tilde{\theta},\mbox{ if}\;\;0\leq\tilde{\theta}\leq\pi/2,\\
\pi/2,\mbox{ if}\;\;\pi/2<\tilde{\theta}\leq\pi/2+\tilde{\theta}_{2}.\end{%
array}\right.$$
(2.5)
Also,
$$x_{m}=\left\{\begin{array}[]{l}0,\mbox{ if}\;\;-\tilde{\theta}_{1}\leq\tilde{%
\theta}<0,\\
R\left(-1+\cos{\tilde{\theta}}\right),\mbox{ if}\;\;0\leq\tilde{\theta}\leq\pi%
/2,\\
-R\left(1+\tan{\tilde{\theta}}\right),\mbox{ if}\;\;\pi/2<\tilde{\theta}\leq%
\pi/2+\tilde{\theta}_{2},\end{array}\right.$$
(2.6)
$$y_{m}=\left\{\begin{array}[]{l}H-R\left(1+\tan{\tilde{\theta}}\right),\mbox{ %
if}\;\;-\tilde{\theta}_{1}\leq\tilde{\theta}<0,\\
H-R\left(1-\sin{\tilde{\theta}}\right),\mbox{ if}\;\;0\leq\tilde{\theta}\leq%
\pi/2,\\
H,\mbox{ if}\;\;\pi/2<\tilde{\theta}\leq\pi/2+\tilde{\theta}_{2}\end{array}\right.$$
(2.7)
are the nondimensional parametric functions that describe the mask surface. From (2.5)-(2.7),
the rounded portion of the mask is described by
$$\left\{\begin{array}[]{l}x_{m}^{(arc)}=R\left(-1+\cos{\theta}\right),\\
y_{m}^{(arc)}=H-R\left(1-\sin{\theta}\right),\;0\leq\theta=\tilde{\theta}\leq%
\pi/2.\end{array}\right.$$
(2.8)
Taking $\mu_{m}h_{m}/\Omega$ for the unit of $\gamma_{cm}$, and $\gamma_{0}$ (the mean value, see (2.16) below) for
the unit
of $\gamma_{cv}$, the equation (1.1) takes the form
$$E(\gamma_{mv}-\gamma_{cm})-\hat{\gamma}_{cv}(\alpha^{*})\cos{\phi}+\frac{%
\partial\hat{\gamma}_{cv}}{\partial\phi}(\alpha^{*})\sin{\phi}=0,$$
(2.9)
where $E=\mu_{m}h_{m}/(\Omega\gamma_{0})$, $\gamma_{mv}$ is given by (2.3),
$\gamma_{cm}$ now is the nondimensional energy of the crystal-mask interface (assumed constant and isotropic),
$\hat{\gamma}_{cv}$ is the anisotropy
factor of the interfacial energy, and $\alpha^{*}$ is value of the angle of the normal to the interface at the
tri-junction. Notice that
$$\alpha^{*}=\phi+\theta.$$
(2.10)
(2.9)
is solved once in the beginning of the computation and yields values of the imposed contact angle at every
$\tilde{\theta}_{i}\in\left[-\tilde{\theta}_{1},\tilde{\theta}_{2}+\pi/2\right]%
,i=1...N_{\tilde{\theta}}$,
where $N_{\tilde{\theta}}$ is the number of equidistant nodes in the interval.
$N_{\tilde{\theta}}$ is taken in the range 500…3000, the larger the smaller is the radius of the circular arc
(since for the accurate marching of the contact point along the arc it is important to have sufficient
number of nodes there).
2.3 Updating the contact point
In this section, the procedure which updates the position of the contact point on the mask is described.
The update is needed
after all marker particles on the interface but the contact point are moved one time step forward
using the evolution equations (2.2).
Let $x_{*}$ and $y_{*}$ are the “old” coordinates of the contact point. Using $x_{*},y_{*}$, the coordinates of the
two nearest nodes on the mask: $x_{m}(i),y_{m}(i),x_{m}(i+1),y_{m}(i+1)$, and $\theta_{i,i+1},\phi_{i,i+1}$,
we first compute values of $\theta$ and $\phi$ at $(x_{*},y_{*})$ by linear interpolation. Using the latter values,
the angle of the tangent to the interface at $(x_{*},y_{*})$ is found from
$$\phi_{x}=\phi+\theta-\pi/2.$$
(2.11)
Let $(x_{1},y_{1})$ and $(x_{2},y_{2})$ are the coordinates of the first two marker particles on the interface
closest to the contact point.
In case the contact point is on the vertical part of the mask, the updated location is $(0,\hat{y}_{*})$, where
$$\hat{y}_{*}=\frac{1}{3}\left(4y_{1}-y_{2}+\left(x_{2}-4x_{1}\right)\tan{\phi_{%
x}}\right).$$
(2.12)
This follows from the second-order, one-sided finite difference approximation of the contact angle condition (1.2).
In case the contact point is on the horizontal part of the mask, the updated location is $(\hat{x}_{*},H)$, where
$$\hat{x}_{*}=\frac{1}{3}\left(4x_{1}-x_{2}+\frac{3H-4y_{1}+y_{2}}{\tan{\phi_{x}%
}}\right).$$
(2.13)
In case the contact point is on the circular arc, we
solve the quadratic equation (ref. (2.8))
$$\frac{1}{3}\left(4y_{1}-y_{2}+\left(3x_{*}-4x_{1}+x_{2}\right)\tan{\phi_{x}}%
\right)=H-R\left(1-\sqrt{1-(1+x_{*}/R)^{2}}\right)$$
(2.14)
and find two $x$-coordinates $\hat{x}_{*}^{(1)},\hat{x}_{*}^{(2)}$ and corresponding $y$-coordinates
$\hat{y}_{*}^{(1)},\hat{y}_{*}^{(2)}$. Then, if both $\hat{x}_{*}^{(1)},\hat{x}_{*}^{(2)}\geq-R$ (that is, if
both possible updated locations are on the arc), we take the pair of coordinates closest to $(x_{*},y_{*})$
for the updated location.
If only one of $(\hat{x}_{*}^{(1},\hat{y}_{*}^{(1)}),\;(\hat{x}_{*}^{(2)},\hat{y}_{*}^{(2)})$ is on the arc and the other
is not, we choose
the pair that is on the arc (closer to the “old” location).
2.4 Anisotropy
The anisotropy factor of the kinetic mobility is chosen as in [25], where the following form is
suggested and the simulations of faceted growth are performed within the framework of phase field bulk
solidification model:
$$\hat{M}(\alpha)=1-\sigma+2\sigma\tanh{\frac{\chi}{|\tan{2\left(\alpha+\beta_{M%
}\right)}|}};\quad M=M_{0}\hat{M}(\alpha).$$
(2.15)
In (2.15), $\sigma$ and $\chi$ are constant parameters that control the width and flatness of
facets, respectively; $\beta_{M}$ is the phase angle. Facets are formed at regions on the interface
where $\alpha=\pi/4-\beta_{M}+n\pi/2,\;n=0,1,...$. It was shown in [25] that the results of computations
with the four-fold anisotropy (2.15) are in good agreement with the so-called kinetic Wulff shapes
(see references in [25]; specifically for the selective area epitaxy the kinetic Wulff shapes
were constructed and compared to the experimental shapes in [26]). For the simulations discussed
in the next section we chose $\sigma=0.95,\;\chi=2,\;\beta_{M}=0$.
The anisotropy factor of the crystal-vapor interfacial energy has standard, four-fold symmetric form
$$\hat{\gamma}_{cv}(\alpha)=1+\epsilon_{\gamma}\cos{\left[4\left(\alpha+\beta_{%
\gamma}\right)\right]};\quad\gamma_{cv}=\gamma_{0}\hat{\gamma}_{cv}(\alpha),$$
(2.16)
where the constant $\epsilon_{\gamma}>0$ determines the degree of anisotropy, and $\beta_{\gamma}$ is the phase angle.
In this work we allow for strong anisotropy of $\gamma_{cv}$, e.g. $\epsilon_{\gamma}>1/15$.
In this case, the interface stiffness $G=\hat{\gamma}_{cv}+\partial^{2}\hat{\gamma}_{cv}/\partial\alpha^{2}$ is
negative in
certain intervals of values of $\alpha$, and that manifests in the appearance of
corners on the equilibrium Wulff shape of a crystal [27];
in the dynamical case, the evolution equations become backward-parabolic and unstable
where $G<0$, see for example [28] - [30].
2.4.1 Regularization
To track interface evolution with the strongly anisotropic interfacial energy, the problem must be regularized
in order to penalize spatial oscillations as well as the tendency to form corners. The regularization
we use in this work is based on the addition of curvature dependence to $\gamma_{cv}$. Regularization by curvature
was first proposed
by Herring [27] for the equilibrium shape; see [31] for the modern mathematical analysis
of this
problem. Apart from being
useful as a modeling tool, the regularization by curvature has physical ground, namely the interaction
of steps on a crystal surface in the vicinity of a corner [29].
Unlike [29, 30] and other works where the $\delta\kappa^{2},\delta=const.$ ($\kappa$ is the
dimensional curvature) term is added to
the surface energy on the problem formulation stage,
here we regard the regularization only as a numerical technique intended to keep the computation alive, and
regularize by adding same
term to $\gamma_{cv}$ given by (2.16) only at the distinct moments and only at the locations on the
interface where the
instability (tendency to form a corner) occurs. This is done as follows.
First, we continuously check the
sign of $G$ at every marker location on the interface. Second, once at the certain location $G$ becomes negative
(the corner has been formed), we interrupt the computation, restore all variables including the local interface
shape
and curvature to their previous states at positive $G$, add $\Delta K^{2},\ \Delta=\delta/(\gamma_{0}L^{2})$ term
to the nondimensional interfacial energy
$\hat{\gamma}_{cv}(\alpha)$
at this location and
solve the
linear equation $G(\Delta)=0$. The result,
$$\Delta=\frac{15\epsilon_{\gamma}\cos{\left[4\left(\alpha+\beta_{\gamma}\right)%
\right]}-1}{K^{2}+2K\partial^{2}K/\partial\alpha^{2}+2(\partial K/\partial%
\alpha)^{2}},$$
(2.17)
is deflected by 0.1$\Delta$ to ensure positive $G$.
The computation is then restarted with the regularized
$\gamma_{cv}$; if needed, the procedure is repeated. This method allows to reduce the influence of the regularization
on interface shape to a minimum by (i) applying the regularization selectively and (ii) choosing value for the
regularization constant $\Delta$ no larger than needed to allow the computation to proceed.
The maximum value $\Delta\sim 10^{-5}$ comes out from the computation, which translates into $\delta\sim 10^{-11}$
erg; thus the additional corner energy necessary to regularize the problem is indeed very small.
222In [29], parameter $\delta$ is introduced as the second partial derivative of the crystalline
step energy
with respect to the first coordinate derivative of step density; this is difficult, if not impossible to extract
from experimental data. We are not aware of a single experiment there this or related quantity is measured.
Since value is unknown, we think it makes sense to find the suitable numerical
approximation to the nondimensional counterpart as explained.
It must be noted here that the numerical method itself introduces some minor smoothing, since the interface is remeshed
every time step with the help of cubic splines [8].
The regularization by curvature is not used at the contact point and its vicinity. Instead,
to ensure stable tracking of the sensitive contact point motion over the mask
edge in the simulations with strongly anisotropic surface energy, we had to resort to the method which is
described in the next subsection.
2.4.2 Contact point regularization
We take
$$\epsilon_{\gamma}=\epsilon_{\gamma}^{(0)}+\left(\Upsilon-\epsilon_{\gamma}^{(0%
)}\right)H_{\lambda}(s-s_{c}),$$
(2.18)
where $\epsilon_{\gamma}^{(0)}$ is constant value $\epsilon_{\gamma}$ takes on the interval of the arc
length $0\leq s<s_{c}-\lambda$,
$\Upsilon>\epsilon_{\gamma}^{(0)}$ is constant value $\epsilon_{\gamma}$ takes on the
interval $s_{c}+\lambda<s\leq S$, and
$H_{\lambda}(s-s_{c})$ is the smoothed Heaviside function (see, for instance, [32]),
viz.
$$H_{\lambda}(s-s_{c})=\left\{\begin{array}[]{l}0,\mbox{ if}\;s-s_{c}<-\lambda,%
\\
\frac{1}{2}\left(1+\frac{s-s_{c}}{\lambda}+\frac{1}{\pi}\sin{\frac{\pi(s-s_{c}%
)}{\lambda}}\right),\mbox{ if}\;-\lambda\leq s-s_{c}\leq\lambda,\\
1,\mbox{ if}\;s-s_{c}>\lambda.\end{array}\right.$$
(2.19)
In (2.19), $\lambda=const.$ is the small half-width of the transition region from $0$ to $1$ around
$s_{c}=const.\in(0,S)$. (2.18) and (2.19)
imply that the intervals of the interface $0\leq s<s_{c}-\lambda$ and $s_{c}-\lambda\leq s\leq s_{c}+\lambda$ evolve
with lesser degree of anisotropy
than the remaining interval $s_{c}+\lambda<s\leq S$. For instance, for the simulation shown in
Fig. 9, we chose $\epsilon_{\gamma}^{(0)}=0.01,s_{c}=0.08,\lambda=0.02$.
Of course,
such choice is difficult to justify physically, but nevertheless it allows to perform simulations and
obtain the right interface shape some (small) distance away from the contact point [33].
3 Results
Since the parameter space is very large, the overgrowth was computed
for the separate cases of evolution with the anisotropic mobility and crystal-vapor interfacial energy.
We studied the influence of parameters $r$ and $\gamma_{cm}$. Other parameters are fixed
at their values cited in Table LABEL:Table2.
3.1 Overgrowth with anisotropic mobility
It is assumed in this section that $\epsilon_{\gamma}=0$, thus $\hat{\gamma}_{cv}\equiv 1$ and
the contact angle $\phi$ is determined from the equation (2.9) where the third term identically
equals zero.
Fig. 4(a) shows the overgrowth over the blunt edge $r=0.5$ (the radius = 50 nm) of the energetically
inhomogeneous mask.
Also shown in Fig. 4(b)
is the actual contact angle $\phi$ vs. $\theta$. Value for $\gamma_{cm}$ is chosen 100 (500 erg/cm${}^{2}$), so that $\phi(\theta)>\pi/2\ \forall\ \theta$ (partial “wetting”).
One of accuracy checks is the comparison every
time step of the actual value of the contact angle with the imposed $\phi(\theta)$
given by (2.9). The difference is negligible for all runs performed.
Notice that the perfectly straight facet is formed in the direction 135${}^{\circ}$ to the substrate.
The feature (“bump”) is present at the junction of this facet and the horizontal, 90${}^{\circ}$ facet.
This bump is present on the surface even before the contact point reaches the mask edge, but the edge
makes it more pronounced. The bump gradually disappears after the edge is passed.
Fig. 5(a,b) compares the overgrowth over the sharper edge $r=0.0125$ (the radius = 1.25 nm)
of the energetically inhomogeneous
mask as in Fig. 4(b) with the overgrowth over
the edge of same sharpness of the energetically homogeneous mask. For the latter,
value for the constant $\phi$ is taken the average of the
smallest and largest values of $\phi$ in Fig. 4(b) ($\phi=1.956$). The overgrowth over the
energetically homogeneous mask is faster, and the bump is more pronounced in this case.
Nevertheless, the difference in the overgrowth of the mask edges of different energy is quite small.
Fig. 6(a,b) compares the overgrowth over the $r=0.0125$ edge
of the energetically inhomogeneous mask that supports different $\phi(\theta)$
(that is, the dynamics corresponding to $\phi(\theta)>\pi/2$ as in Fig. 4(b), and
$\phi(\theta)\leq\pi/2$ as in Fig. 6(c)).
Interestingly,
the bump and the crystal thickness are larger for the less wetting case (larger $\phi$’s)
when the contact point slides over the mask edge. However, immediately after the edge
the overgrowth of the more wetting crystal is faster, but it slows down as the overgrowth
on the mask progresses. This is expected since $\phi=\pi/2$ on the horizontal part of the mask.
Finally, the less wetting crystal takes over, but its
thickness in the $y$-direction is less than one of the more wetting crystal.
The distance from the contact point to the
foot point (0,0) of the mask vs. time is shown in Fig. 7(a,b) for
the overgrowth
over $r=0.5$ and $r=0.0125$ edge, respectively, and $\gamma_{cm}=100$.
For the blunt edge, the computational data is fitted well
by the cubic polynomial and thus the speed of the contact point vs. time
is the quadratic function that has
the minimum value $\approx 4\times 10^{-4}$ (4 nm/s) at $t=125$ (6.3 s) and the maximum value
$\approx 1.4\times 10^{-3}$ (14 nm/s) at $t=50,200$ (2.5,10 s). For the sharp edge,
the quadratic fit is better, and the speed of the contact point is linear function of time
(the min. value $\approx 3.25\times 10^{-4}$ (3.25 nm/s) at $t=100$ (5 s),
the max. value $\approx 5.5\times 10^{-4}$ (5.5 nm/s) at $t=86$ (4.3 s)).
This data,
as well as Figures 4-6 demonstrate
that de-pinning of the contact line of a crystal from the solid defect of finite size does
not have a threshold nature;
that is, the contact line is in constant motion over the defect, and the problem does not have a critical
eigenvalue such that the contact line is pinned when some parameter is smaller than this eigenvalue and
de-pinned (moving) when the parameter is larger.
In the language of [15], where the
analogy between perturbed (on a defect) liquid contact line and an elastic spring is used to phenomenologically
explain pinning (pages 836-38),
the total pinning force exerted by the defect on the line is such that graphs of this force and the restoring
linear elastic force (vs. position on a defect) are not intersecting, and thus there is no
equilibrium, pinning line position(s) on a defect.
Finally, the average nondimensional speed of the contact point’s motion over the edge
(naturally defined as the length of the circular segment of the mask, $\pi R/2$,
divided by the traversal time) is plotted in Fig. 8 vs. $r$.
There is an evident tendency for pinning of the contact point at the edge as $r\rightarrow 0$.
Data can be satisfactory fitted by a hyperbolic tangent curve (not shown).
In the limit $r=0$ (the mathematically sharp edge)
the true pinning is expected (the speed $=$ 0). The contact line starts to actually
“feel” the edge only when the latter has sufficiently small radius of curvature
($r\approx 0.1-0.15\;\Leftrightarrow\;$10-15 nm).
Note that despite such characteristic of the motion of the contact line over the edge as the speed
(instantaneous and average) is, of course, a function of all parameters of the system, the
pinning on the edge as its radius of curvature approaches zero is the universal tendency.
This was checked by means of multiple runs with different parameters. The computation fails
for $r<0.0075$ (the radius$<0.75$ nm) because as the number of discretizing nodes along the mask increases,
the nodes on the two straight segments of the mask come too close to each other and the
interpolation becomes inaccurate. In principle, the computation with smaller radii can be made successful
by using the nonuniform grid along the mask; we did not attempt to do that. It is also doubtful that
very sharp mask edges (having the radii of curvature of the order of a molecule size (1-2 nm))
are achievable by the lithography, etching and polishing used in the ELO/SAG practice
for the preparation of a mask. For comparison, the sapphire disks used for the late 1970s experiments with
the liquid drops by Oliver et al. [16] had the edge radius of curvature smaller than 50 nm,
and “much smaller” than that
value if the defects (chipped-off hollows) appear along the edge.
These disks were prepared by cutting from a fused sapphire boule.
The aluminum disks they used had very blunted edges with the radius of curvature 1 to 5 $\mu$m.
3.2 Overgrowth with strongly anisotropic $\gamma_{cv}$
It is assumed in this section that $\sigma=0$, thus $\hat{M}\equiv 1$.
The contact angle $\phi$ is determined from the equation (2.9), where the third term is nonzero
due to the anisotropy in the crystal-vapor surface energy.
Fig. 9 shows, as an example, the overgrowth over the blunt edge $r=0.5$
(the contact angle $\phi$ as in Fig. 4(b)). In Fig. 9(a),
the value of the anisotropy
parameter $\Upsilon$
(see (2.18)) is 0.07. This is just above the critical value 1/15. $\Upsilon=0.085$ in
Fig. 9(b). For both figures, the value of the phase angle $\beta_{\gamma}$ is chosen 1.67
(a little larger than $\pi/2$).
In Fig. 9(a), the corner forms slowly at the elevation, and it is followed by the
straight facet; other than that, the evolution resembles the isotropic evolution [8].
In Fig. 9(b), the interface instantly becomes unstable and the characteristic, coarsening
with time hill-and-valley structure
is formed [29, 30]. As the result of coarsening, the single elevated corner
(bump) followed by the $\alpha=45^{\circ}$
inclined facet are again formed. Thus this numerical experiment suggests that the strong
interfacial energy anisotropy may be one of the possible causes for the “bumpy” crystal shapes
routinely observed in ELO deposits [1, 4].
Fig. 10 shows the contact angle along the mask surface. In contrast to Figures 4(b) and
6(c), this function is not monotone. As $\epsilon_{\gamma}^{(0)}$ increases quite beyond value 0.01
chosen for computations in this section, the variations
in $\phi$ become more
pronounced and the computation becomes unstable.
Acknowledgements
The author thanks Professors Richard Braun (University of Delaware) and Brian Spencer (SUNY at Buffalo)
for the stimulating and useful discussions of this work. The reviewer is acknowledged for pointing out
the relevance of Herring’s equation to the problem considered in this work.
Appendix A On equation (1.1)
The equation (1.1) follows from the following equation derived by Herring [34]:
$$\gamma_{1}-\gamma_{2}\cos{\alpha_{2}}-\gamma_{3}\cos{\alpha_{3}}+\frac{%
\partial\gamma_{2}}{\partial\phi_{2}}\sin{\alpha_{2}}+\frac{\partial\gamma_{3}%
}{\partial\phi_{3}}\sin{\alpha_{3}}=0.$$
(A.1)
The equation (A.1) generalizes the Young-Dupré equation
$$\gamma_{2}\cos{\alpha_{2}}=\gamma_{1}-\gamma_{3}$$
(A.2)
to orientational dependence of surface energies (tensions) at the tri-junction (see Fig. 11).
To obtain (1.1) from (A.1), we identify the interface 1 in Fig. 11 with the mask-vapor interface,
the interface 2 with the crystal-vapor interface, the interface 3 with the crystal-mask interface and notice that
$\alpha_{2}=\phi$ and $\alpha_{3}=0$ since the slope of the mask surface is everywhere continuous (Fig. 2).
Appendix B Parameters of the model (characteristic of epitaxy of GaAs-like semiconductor at T$\approx$ 650${}^{\circ}$C
)
This appendix has been removed from online submission to arXiv.org in order to comply with
file size requirement; check the full version at
www.math.buffalo.edu/$\sim$mkhenner.
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TCDMATH 19-15
$T\overline{T}$, $\widetilde{J}J$, $JT$ and $\widetilde{J}T$ deformations
Sergey Frolov222 Correspondent fellow at
Steklov Mathematical Institute, Moscow.333email:
[email protected]
School of Mathematics and Hamilton Mathematics Institute,
Trinity College, Dublin 2,
Ireland
Abstract
The light-cone gauge approach to ${T\overline{T}}$ deformed models is generalised to models deformed by U(1) conserved currents $J^{\alpha}$, $\widetilde{J}^{\alpha}$, stress-energy tensor $T^{\alpha}{}_{\beta}$, and their various quadratic combinations of the form $\epsilon_{\alpha\beta}K_{1}^{\alpha}K_{2}^{\beta}$. It is then applied
to derive a ten-parameter deformed Hamiltonian for a system of scalars with an arbitrary potential, the flow equations for the Hamiltonian density, and the flow equations for the energy of the deformed model.
The flow equations disagree with the ones recently proposed
in arXiv:1903.07606.
The results obtained are applied to analyse a CFT with left- and right-moving conserved currents deformed by these operators. It is shown that with a proper choice of the parameter of the ${T\overline{T}}$ deformation the deformed CFT Hamiltonian density is independent of the parameters of the $J\Theta$ and $\bar{J}\overline{\Theta}$ deformations. This leads to the existence of two extra relations which generalise the
$J\Theta=0$ and $\bar{J}\overline{\Theta}=0$ relations of the undeformed CFT. The spectrum of the deformed CFT is found and shown to satisfy the flow equations.
Contents
1 Introduction
2 String model and the transformation
2.1 Extended string model
2.2 Parameter dependent canonical transformation
3 Deformed Hamiltonian and the flow equations
3.1 Deformed Hamiltonian
3.2 Flow equations for the density of the Hamiltonian
3.3 Flow equations for the energy
3.4 Homogeneous inviscid Burgers equation
4 Flow equations in CFT conventions
4.1 Flow equations for the Hamiltonian density
4.2 Flow equations for the energy in CFT conventions
4.3 Deformed CFT with left- and right-moving currents
5 Conclusions
A Canonical transformation matrices
B Virasoro constraints coefficients
C Examples of deformed Hamiltonians
D Examples of the deformed spectrum
1 Introduction
The ${T\overline{T}}$ deformation of 2d field theories introduced in [1] admits various generalisations if the model under consideration possesses additional conserved currents. Examples include the Lorentz invariance preserving higher-spin deformations of integrable models [2], and the ${\rm J}\overline{T}$ type deformations [3] which break Lorentz invariance of an undeformed model. Any such a deformation is obtained by adding to the Hamiltonian of an undeformed model an operator of the form
${\cal O}_{K_{1}K_{2}}\sim\epsilon_{\alpha\beta}K^{\alpha}_{1}K^{\beta}_{2}$ where $K_{1}$ and $K_{2}$ are conserved currents.
It is hoped that the spectrum of any model deformed by these operators is completely fixed by the spectrum of the undeformed model just as it is for the ${T\overline{T}}$ deformation [1]. Indeed, the spectrum of
a CFT with left- and right-moving conserved currents
deformed by ${\rm J}{\overline{T}}$ was derived in
[4], completing the results of [3], see also [5], and the spectrum of such a CFT deformed by ${T\overline{T}}$, ${\rm J}{\overline{T}}$, ${\bar{{\rm J}}T}$ was conjectured in [6] by relating the problem to deformations of holographically dual strings on $AdS_{3}$. A very general nine-parameter deformation of a CFT by the stress energy tensor $T$, the conserved currents ${\rm J}$ and $\bar{\rm J}$, and their various quadratic combinations was studied in [7] where a set of flow equations with respect to the deformation parameters was proposed, and then used to conjecture the spectrum of a deformed CFT.
In this paper we generalise the light-cone gauge approach [8, 9] to ${T\overline{T}}$ deformed models
to the case of the most general ten-parameter deformation
by conserved currents $J^{\alpha}$, $\widetilde{J}^{\alpha}$, the canonical Noether stress-energy tensor $T^{\alpha}{}_{\beta}$, and their various quadratic combinations of the form $\epsilon_{\alpha\beta}K^{\alpha}_{1}K^{\beta}_{2}$. The light-cone gauge approach was used in [9]
to derive the ${T\overline{T}}$ deformed action for a very general system of any number of scalars, fermions and chiral bosons with an arbitrary potential.
Basically all of the ${T\overline{T}}$ deformed models [10, 11, 12, 13, 8, 14, 15, 16, 17] studied before and after [9] are particular cases of this system which includes various Lorentz invariant systems of bosons and fermions, in particular supersymmetric sigma models, and some of non-Lorentz invariant systems, e.g. the chiral SYK model and the nonlinear matrix
Schrödinger model.
The light-cone gauge approach is based on the observation that
the homogeneous inviscid Burgers equation which determines the spectrum of a ${T\overline{T}}$ deformed model with zero momentum coincides with the gauge invariance condition of the target space-time energy and momentum of a non-critical string theory quantised in a parameter dependent light-cone gauge introduced in [18].
The light-cone gauge-fixed Hamiltonian can be thought of as the Hamiltonian of a deformed model, and the deformation parameter can be identified with the light-cone gauge parameter, see [9] for a detailed discussion.
The deformed Hamiltonian ${\cal H}_{\alpha}$ can be used to derive the flow equation,
$\partial_{\alpha}{\cal H}_{\alpha}=\,$${T\overline{T}}$, with respect to the parameter $\alpha$ of the ${T\overline{T}}$ deformation.
The flow equation is then used to get the inviscid Burgers equation which governs the spectrum of the deformed model.
In this paper we only consider a system of $n$ scalars with an arbitrary potential.
We assume that the bosonic model is invariant under shifts of one of its fields, say $x^{1}$, and $J^{\alpha}$ is its canonical Noether current due to the symmetry. The current
$\widetilde{J}^{\alpha}$ is a topological current associated to the field $x^{1}$: $\widetilde{J}^{\alpha}=\epsilon^{\alpha\beta}\partial_{\beta}x^{1}$. In the case of a CFT the left- and right-moving conserved currents are linear combinations of $J^{\alpha}$ and $\widetilde{J}^{\alpha}$.
We begin with the usual action for bosonic strings invariant under shifts of three isometry coordinates $x^{a}$, $a=+,-,1$. The undeformed model is obtained by imposing the standard light-cone gauge $x^{+}=\tau\,,\,p_{-}=1$ where $p_{-}$ is the momentum conjugate to $x^{-}$.
It appears that to describe the most general ten-parameter deformation we need to introduce pairs of auxiliary non-dynamical co-vectors and scalars $(V^{a}_{\alpha}\,,\,\tilde{X}_{a})$ and $({\cal U}_{a\alpha}\,,\,\Upsilon^{a})$ associated with the conserved currents $J_{a}^{\alpha}$ and $\widetilde{J}^{a\alpha}$, respectively. We then perform a ten-parameter canonical transformation $A$ which involves the three coordinates $x^{a}$ and their momenta $p_{a}$, and the coordinates and momenta of the auxiliary fields. Finally, we impose the light-cone gauge $x^{+}=\tau\,,\,p_{-}=1$ on the transformed coordinates and momenta, solve the Virasoro and Gauss-law constraints, and identify the light-cone gauge-fixed Hamiltonian ${\cal H}_{A}=-p_{+}$ with the deformed one.
The derivatives of the deformed Hamiltonian with respect to the ten parameters
give rise to flow equations (3.9-3.13) which are valid for any bosonic model of the type we study. We believe that the flow equations are universal, and apply
to any Lorentz and non-Lorentz invariant model of bosons and fermions.
The flow equations disagree with the ones recently proposed in [7].
The main difference is that some of the coefficients in front of the deforming operators on the r.h.s. of our flow equations depend on the deformation parameters of the quadratic operators while those of the flow equations of [7] do not.
The reason for the disagreement is simple. The authors of [7] only analysed the model of a single massless free scalar which is a CFT with left- and right-moving conserved currents. As we show, for such a model there is a choice of the ${T\overline{T}}$ deformation parameter such that the deformed Hamiltonian does not depend on the parameters of the ${\rm J}\Theta$ and $\bar{\rm J}\overline{\Theta}$ deformations.
The flow equations with respect to these parameters then become
extra constraints (4.11) which generalise the relations ${\rm J}\Theta=0$ and $\bar{\rm J}\overline{\Theta}=0$ of the undeformed CFT. Taking into account these constraints one can make a choice of the deformation parameters which removes the parameter dependent terms from the flow equations. Such a choice exists only for a CFT with left- and right-moving conserved currents, and for any other model one has to deal with the flow equations (3.9-3.13).
The flow equations for the deformed Hamiltonian can be used to find a system of flow equations (3.17, 3.18) for the spectrum of the deformed model. The system does not involve derivatives of the energy with respect to the parameters of the linear deformations by the space components of the currents $J^{\alpha}$ and $\tilde{J}^{\alpha}$. These derivatives were used in [7] to find the expectation values of $J^{1}$ and $\tilde{J}^{1}$. As we explain in the paper, these expectation values are given by the derivatives of the energy with respect to the charges $\widetilde{P}^{1}$ and $P_{1}$ of the currents $J^{\alpha}$ and $\tilde{J}^{\alpha}$. Thus, we do not really need the two parameters. The system of flow equations is complicated and at the moment we do not know how to find a proper solution of the system for a generic state. Nevertheless, we show that if a state satisfies level-matching conditions which we derive then the energy of such a state obeys a version of the
homogeneous inviscid Burgers equation.
The flow equations for the energy however can be solved for a deformed CFT with left- and right-moving conserved currents. The solution (4.18) we find reduces to the known ones in the particular cases of one- and two-parameter deformations. It seems to agree with the one proposed in [7]
at least if one switches off their parameters of the deformation by ${\rm Q}_{\pm}^{1}$, and parameters of the ${\rm J}\Theta$ and $\bar{\rm J}\overline{\Theta}$ deformations.
To get a precise agreement it might be necessary to perform an extra redefinition of the parameter of the ${T\overline{T}}$ deformation. We have not tried to do it.
In the case of the ${T\overline{T}}$, ${\rm J}{\overline{T}}$, ${\bar{{\rm J}}T}$ deformation our solution seems different from the one proposed in [6] but we believe that the two solutions may be equivalent after a proper redefinition of the deformation parameters.
The plan of the paper is as follows. In section 2.1 we introduce an extended action for bosonic strings propagating in a target manifold possessing (at least) three
abelian isometries and determine all the constraints and gauge conditions necessary to recover the undeformed model. In section 2.2 we generate a ten-parameter deformation of the model as a chain of one-parameter canonical transformations producing different deformations. In section 3.1 we impose the light-cone gauge on the transformed coordinates and momenta, solve all the constraints, and find the deformed Hamiltonian. In section 3.2 we use the Hamiltonian to derive the flow equations with respect to the deformation parameters. Then in section 3.3 the flow equations are converted into the flow equations for the spectrum of the deformed model.
In section 3.4 we determine level-matching conditions physical states must satisfy, and derive the
homogeneous inviscid Burgers equations for the energy of those states.
The flow equations for the Hamiltonian density and the spectrum are then rewritten in the CFT conventions in section 4 where we also consider a very special case of a CFT with left- and right-moving conserved currents, and show that for such a CFT there are extra constraints (4.11) replacing the relations
${\rm J}\Theta=0$ and $\bar{\rm J}\overline{\Theta}=0$ of the undeformed CFT. In section 4.3 we propose a formula for the spectrum of such a deformed CFT which satisfies the flow equations.
In Conclusions we discuss open questions and generalisations of the light-cone gauge approach. Finally, in Appendices we collect some explicit formulae, and consider examples of deformed Hamiltonians, and deformed spectrum of CFT with left- and right-moving conserved currents.
2 String model and the transformation
2.1 Extended string model
In this section we follow the same approach as in [9] but simplify a bit their consideration. We begin with bosonic strings propagating in a $n+2-$dimensional target Minkowski manifold ${\cal M}$ possessing
(at least) three abelian isometries realised by shifts of the isometry
coordinates.
We denote coordinates of ${\cal M}$ by $x^{M}$, $M=+,-,1,\ldots,n$,
and choose the isometry
coordinates to be $x^{+}$, $x^{-}$ and $x^{1}$.
The “transversal” coordinates $x^{\mu}$, $\mu=1,\ldots,n$ are the fields of the model we wish to deform.
Obviously, the target-space metric $G_{MN}$ of ${\cal M}$ does not depend on $x^{\pm}$ and $x^{1}$.
Assuming for simplicity that the $B$-field vanishes but making no assumption on the form of $G_{MN}$, we write
the initial string action in the standard form
$$\displaystyle S=\int_{0}^{R}\,{\rm d}\sigma{\rm d}\tau\,{\cal L}\,,\quad{\cal L%
}=-{1\over 2}\gamma^{\alpha\beta}\partial_{\alpha}x^{M}\partial_{\beta}x^{N}G_%
{MN}\,,$$
(2.1)
where $\gamma^{\alpha\beta}=h^{\alpha\beta}\sqrt{-h}$ is the Weyl-invariant
combination of the world-sheet
metric $h_{\alpha\beta}$ with $\det\gamma=-1$, and $\epsilon^{01}=-\epsilon_{01}=1$. The range $R$ of the world-sheet space coordinate $\sigma$ will be fixed by a generalised uniform light-cone gauge.
The string action invariance under the shifts of $x^{\pm}$ and $x^{1}$
leads to the existence of the three conserved currents
$$\displaystyle J_{a}^{\alpha}={\partial{\cal L}\over\partial\partial_{\alpha}x^%
{a}}\,,\quad\partial_{\alpha}J_{a}^{\alpha}=0\,,\quad a=+,-,1\,.$$
Then, to any coordinate $x^{M}$ we can associate a topological conserved current
$$\displaystyle\widetilde{J}^{M\alpha}=\epsilon^{\alpha\beta}\partial_{\beta}x^{%
M}\,,$$
and the deformations we discuss in this paper involve the topological current $\tilde{J}^{\alpha}\equiv\tilde{J}^{1\alpha}$
which for a free massless scalar field is dual to $J^{\alpha}\equiv J_{1}^{\alpha}$. We denote the corresponding conserved charges as
$$\displaystyle P_{a}=\int_{0}^{R}{\rm d}\sigma J_{a}^{0}\,,\quad\widetilde{P}^{%
a}=\int_{0}^{R}{\rm d}\sigma\widetilde{J}^{a0}\,.$$
It is clear that $\widetilde{P}^{a}$ does not vanish only if $x^{a}$ has a nonzero winding number.
To get the most general deformation we introduce pairs of non-dynamical co-vectors and scalars $(V^{a}_{\alpha}\,,\,\tilde{X}_{a})$ and $({\cal U}_{a\alpha}\,,\,\Upsilon^{a})$ associated with currents $J_{a}^{\alpha}$ and $\tilde{J}^{a\alpha}$, respectively. Then, the extended string action together with the new fields has the form
$$\displaystyle S=\int_{0}^{R}\,{\rm d}\sigma{\rm d}\tau\,\Big{(}-{1\over 2}%
\gamma^{\alpha\beta}\partial_{\alpha}x^{M}\partial_{\beta}x^{N}G_{MN}-\epsilon%
^{\alpha\beta}V_{\alpha}^{a}\partial_{\beta}\tilde{X}_{a}-\epsilon^{\alpha%
\beta}{\cal U}_{a\alpha}\partial_{\beta}\Upsilon^{a}\Big{)}\,.$$
(2.2)
The action (2.1) is obviously invariant under reparametrisations, the six U(1) gauge symmetries
$$\displaystyle V_{\alpha}^{a}\to V_{\alpha}^{a}+\partial_{\alpha}\xi^{a}\,,%
\quad\tilde{X}_{a}\to\tilde{X}_{a}\,,\qquad{\cal U}_{a\alpha}\to{\cal U}_{a%
\alpha}+\partial_{\alpha}\zeta_{a}\,,\quad\Upsilon^{a}\to\Upsilon^{a}\,,$$
(2.3)
and the constant shifts of the coordinates $\tilde{X}_{a}$ and $\Upsilon^{a}$
$$\displaystyle\tilde{X}_{a}\to\tilde{X}_{a}+c_{a}\,,\quad\Upsilon^{a}\to%
\Upsilon^{a}+\eta^{a}\,.$$
The last symmetry leads to the existence of six conserved currents
$$\displaystyle\partial_{\alpha}(\epsilon^{\alpha\beta}V_{\beta}^{a})=0\,,\quad%
\partial_{\alpha}(\epsilon^{\alpha\beta}{\cal U}_{a\beta})=0\,.$$
These are equations of motion for $V_{\alpha}^{a}$ and ${\cal U}_{a\alpha}$ which imply that
$$\displaystyle V_{\alpha}^{a}=\delta_{1\alpha}v^{a}+\partial_{\alpha}\xi^{a}\,,%
\quad{\cal U}_{a\alpha}=\delta_{1\alpha}u_{a}+\partial_{\alpha}\zeta_{a}\,,$$
where $v^{a}$ and $u_{a}$ are time-independent zero modes of $V_{1}^{a}$ and ${\cal U}_{a1}$, respectively. Thus, one gets a family of models parametrised by $v^{a}$ and $u_{a}$, and if they vanish then we get back to the original string model (2.1). We are going to impose a gauge condition which depends on ten parameters, and consider the resulting gauge-fixed action as a deformation of our favourite one.
The simplest way to impose such a gauge condition is to switch to the Hamiltonian formalism. Introducing the momenta canonically-conjugate to the coordinates $x^{M}$
$$\displaystyle p_{M}={\delta S\over\delta\dot{x}^{M}}=-\gamma^{0\beta}\partial_%
{\beta}x^{N}\,G_{MN}\,,\quad\dot{x}^{M}\equiv\partial_{0}x^{M}\,,$$
we bring the
string action (2.2) to the first-order form
$$\displaystyle S=\int_{0}^{R}\,{\rm d}\sigma{\rm d}\tau\,\left(p_{a}\dot{x}^{a}%
+V^{a}\dot{\tilde{X}}_{a}+{\cal U}_{a}\dot{\Upsilon}^{a}+p_{k}\dot{x}^{k}+{%
\gamma^{01}\over\gamma^{00}}C_{1}+{1\over\gamma^{00}}C_{2}-V_{0}^{a}C_{a}^{V}-%
{\cal U}_{a0}C^{a}_{\cal U}\right)\,.$$
(2.4)
Here $a=+,-,1$, $k=2,\ldots,n$, and $V^{a}\equiv V^{a}_{1}$, $\,{\cal U}_{a}\equiv{\cal U}_{a1}$. Then,
$$\displaystyle C_{1}$$
$$\displaystyle=p_{M}x^{\prime M}=p_{a}x^{\prime a}+p_{k}x^{\prime k}\,,\quad x^%
{\prime M}\equiv\partial_{1}x^{M}\,.$$
$$\displaystyle C_{2}$$
$$\displaystyle={1\over 2}G^{ab}p_{a}p_{b}+{1\over 2}x^{\prime a}x^{\prime b}G_{%
ab}+G^{ak}p_{a}p_{k}+x^{\prime a}x^{\prime k}G_{ak}+\widetilde{\cal H}_{x}\,,$$
are the Virasoro constraints,
and $\widetilde{\cal H}_{x}$
depends only on the transversal fields $x^{k}$ and $p_{k}$
$$\displaystyle\widetilde{\cal H}_{x}={1\over 2}G^{kl}p_{k}p_{l}+{1\over 2}x^{%
\prime k}x^{\prime l}\,G_{kl}\,,\quad k,l=2,\ldots,n\,.$$
(2.5)
Finally,
$$\displaystyle C_{a}^{V}=\tilde{X}_{a}^{\prime}\,,\quad C^{a}_{\cal U}=\Upsilon%
^{\prime a}\,,\quad a=+,-,1\,,$$
are the constraints which generate the U(1) gauge transformations (2.3). It is clear from (2.4) that $V^{a}$ and ${\cal U}_{a}$ are the momenta canonically-conjugate to the coordinates $\tilde{X}_{a}$ and $\Upsilon^{a}$. It is worthwhile to mention that on the constraints surface one can replace
$$\displaystyle C_{1}\to\tilde{C}_{1}=p_{a}x^{\prime a}+V^{a}\tilde{X}_{a}^{%
\prime}+{\cal U}_{a}\Upsilon^{\prime a}+p_{k}x^{\prime k}\,.$$
(2.6)
This simple observation will be useful later.
If we do nothing with the coordinates and momenta of the extended model, and just impose the light-cone gauge
conditions
$$\displaystyle x^{+}=\tau\,,\quad p_{-}=1\,,\quad V^{\prime a}=0\,,\quad{\cal U%
}^{\prime}_{a}=0\,,$$
(2.7)
then, solving the constraints $C_{a}^{V}=0$ and $C^{a}_{\cal U}=0$, we get that $V^{a}=v^{a}$, $\tilde{X}_{a}=\tilde{x}_{a}$, ${\cal U}_{a}=u_{a}$ and $\Upsilon^{a}=\mbox{\scriptsize$\Upsilon$}^{a}$ are $\sigma$-independent.
Using the Virasoro
constraint $C_{1}$ to find $x^{\prime-}$
$$\displaystyle x^{\prime-}=-p_{\mu}x^{\prime\mu}\equiv-px^{\prime}\,,$$
(2.8)
we bring $C_{2}$ to the form
$$\displaystyle C_{2}$$
$$\displaystyle={G^{++}\over 2}p_{+}^{2}+(G^{-+}+G^{\mu+}p_{\mu})p_{+}+{G^{--}%
\over 2}+G^{\mu-}p_{\mu}+{G_{--}\over 2}(px^{\prime})^{2}-G_{\mu-}x^{\prime\mu%
}px^{\prime}+{\cal H}_{x}\,,$$
(2.9)
$$\displaystyle{\cal H}_{x}$$
$$\displaystyle={1\over 2}G^{\mu\nu}p_{\mu}p_{\nu}+{1\over 2}x^{\prime\mu}x^{%
\prime\nu}\,G_{\mu\nu}\,.$$
Solving the resulting equation $C_{2}=0$ for $p_{+}$,
we find the gauge-fixed action of the extended model
$$\displaystyle S_{0}=\int{\rm d}\tau\,\left(v^{a}\dot{\tilde{x}}_{a}+u_{a}\dot{%
\mbox{\scriptsize$\Upsilon$}}^{a}\right)+\int_{0}^{R}\,{\rm d}\sigma{\rm d}%
\tau\,\left(p_{\mu}\dot{x}^{\mu}\,-\,{\cal H}_{0}\right)\,,\quad{\cal H}_{0}\,%
=\,-p_{+}(p_{\mu},x^{\mu},x^{\prime\mu})\,.$$
(2.10)
Here the integration range $R=P_{-}$ is found by integrating the gauge condition $p_{-}=1$ over $\sigma$, and
the density ${\cal H}_{0}$ of the world-sheet Hamiltonian depends on the periodic
transversal fields. Thus, the gauge-fixed string
action describes a two-dimensional model on a cylinder of
circumference $R=P_{-}$. Obviously, $v^{a}$ and $u_{a}$ are time-independent, and setting them to any constants, one gets the usual light-cone gauge string action.
The gauge-fixed model in general is not Lorentz invariant, and, as was shown in [9], to get a Lorentz invariant model one has to choose the target space metric of the form
$$\displaystyle ds^{2}=dx^{+}dx^{-}-2Vdx^{+}dx^{+}+G_{\mu\nu}dx^{\mu}dx^{\nu}\,,$$
(2.11)
where $V$ is an arbitrary function of the transversal coordinates. Then, ${\cal H}_{0}$ becomes the Hamiltonian density of a sigma-model of $n$ scalar fields with the
potential $V$
$$\displaystyle{\cal H}_{0}={\cal H}_{x}+V(x)\,.$$
(2.12)
We are going to perform a parameter dependent canonical transformation of the coordinates $x^{a}$, $\tilde{X}_{a}$, $\Upsilon^{a}$ and momenta $p_{a}$, $V^{a}$, ${\cal U}_{a}$, and impose the light-cone gauge conditions (2.7) on the transformed $x^{+}$, $p_{-}$, $V^{a}$ and ${\cal U}_{a}$.
Then, solving the constraints, one gets a parameter-dependent gauge-fixed Hamiltonian density ${\cal H}_{A}$ where $A$ labels the parameters.
We want to think about this gauge-fixed model as a deformation of the model with the Hamiltonian density ${\cal H}_{0}$, and consider the parameters of the canonical transformation as the deformation parameters.
The deformed two-dimensional model with the Hamiltonian density ${\cal H}_{A}$ is invariant under the shifts of
the world-sheet coordinates $\tau$ and $\sigma$, and its canonical stress-energy tensor is given by
$$\displaystyle T^{0}{}_{0}$$
$$\displaystyle={\cal H}_{A}\,,\quad T^{1}{}_{0}=-{\partial{\cal H}_{A}\over%
\partial x^{\prime\mu}}{\partial{\cal H}_{A}\over\partial p_{\mu}}\,,$$
$$\displaystyle T^{0}{}_{1}$$
$$\displaystyle=p_{\mu}x^{\prime\mu}\,,\quad T^{1}{}_{1}={\cal H}_{A}-{\partial{%
\cal H}_{A}\over\partial x^{\prime\mu}}x^{\prime\mu}-p_{\mu}{\partial{\cal H}_%
{A}\over\partial p_{\mu}}\,.$$
In particular, the
world-sheet energy and momentum are conserved
$$\displaystyle E=\int_{0}^{R}\,{\rm d}\sigma\,{\cal H}_{A}=-P_{+}\,,\quad P=-%
\int_{0}^{R}\,{\rm d}\sigma\,p_{\mu}x^{\prime\mu}\,.$$
Then, the Hamiltonian density is invariant under a shift of the coordinate $x^{1}$, and the corresponding conserved current is
$$\displaystyle J^{0}=p_{1}\,,\quad J^{1}=-{\partial{\cal H}_{A}\over\partial x^%
{\prime 1}}\,.$$
Finally, the topological current in the Hamiltonian formalism is given by
$$\displaystyle\widetilde{J}^{0}=x^{\prime 1}\,,\quad\widetilde{J}^{1}=-\dot{x}^%
{1}=-{\partial{\cal H}_{A}\over\partial p_{1}}\,.$$
2.2 Parameter dependent canonical transformation
The simplest way to find a canonical transformation generating a multi-parameter deformation of ${\cal H}_{0}$ is to realise it as a chain of one-parameter transformations producing different deformations.
Analysing infinitesimal transformations, one can identify suitable one-parameter ones. We find convenient to use the following consecutive ones
1. Transformation, ${\cal T}_{T\bar{T}}$, generating the ${T\overline{T}}$deformation [9]
$$\displaystyle{\cal T}_{T\overline{T}}:\quad x^{+}\to x^{+}-a_{+-}x^{-}\,,\quad
x%
^{-}\to x^{-}\,,\quad p_{+}\to p_{+}\,,\quad p_{-}\to p_{-}+p_{+}a_{+-}\,.$$
It is easy to check that for any target space metric $G_{MN}$ the gauge-fixed Hamiltonian density ${\cal H}_{a_{+-}}$ satisfies the flow equation
$$\displaystyle{\partial{\cal H}_{a_{+-}}\over\partial a_{+-}}={T\overline{T}}\,%
,\quad{T\overline{T}}\equiv T^{0}{}_{1}T^{1}{}_{0}-T^{1}{}_{1}T^{0}{}_{0}=-%
\epsilon_{\alpha\beta}T^{\alpha}{}_{1}T^{\beta}{}_{0}=T_{1}T_{0}\,.$$
2. Transformation, ${\cal T}_{\tilde{J}T_{0}}$, generating the $-\widetilde{J}T_{0}$ deformation
$$\displaystyle{\cal T}_{\tilde{J}T_{0}}:\quad x^{+}\to x^{+}-a_{+1}x^{1}\,,%
\quad x^{1}\to x^{1}\,,\quad p_{+}\to p_{+}\,,\quad p_{1}\to p_{1}+p_{+}a_{+1}\,,$$
$$\displaystyle{\partial{\cal H}_{a_{+1}}\over\partial a_{+1}}=-\widetilde{J}T_{%
0}\,,\quad\widetilde{J}T_{0}\equiv\widetilde{J}^{0}T^{1}{}_{0}-\widetilde{J}^{%
1}T^{0}{}_{0}=-\epsilon_{\alpha\beta}\widetilde{J}^{\alpha}T^{\beta}{}_{0}\,.$$
3. Transformation, ${\cal T}_{JT_{1}}$, generating the $JT_{1}$ deformation
$$\displaystyle{\cal T}_{JT_{1}}:\quad x^{1}\to x^{1}-a_{1-}x^{-}\,,\quad x^{-}%
\to x^{-}\,,\quad p_{1}\to p_{1}\,,\quad p_{-}\to p_{-}+p_{1}a_{1-}\,,$$
$$\displaystyle{\partial{\cal H}_{a_{1-}}\over\partial a_{1-}}=JT_{1}\,,\quad JT%
_{1}\equiv J^{0}T^{1}{}_{1}-J^{1}T^{0}{}_{1}=-\epsilon_{\alpha\beta}J^{\alpha}%
T^{\beta}{}_{1}\,.$$
The sequence ${\cal T}_{JT_{1}}{\cal T}_{\tilde{J}T_{0}}{\cal T}_{T\overline{T}}$ of these three transformations can be represented by a single canonical transformation
$$\displaystyle x^{a}\to(A^{-1})^{a}{}_{b}\,x^{b}\,,\quad p_{a}\to p_{b}\,A^{b}{%
}_{a}\,,\quad a,b=+,-,1\,.$$
Here $A$ is the following matrix
$$\displaystyle A=\left(\begin{array}[]{ccc}1&a_{+-}&a_{+1}\\
0&1&0\\
0&a_{1-}&1\\
\end{array}\right)\,,\ A^{-1}=\left(\begin{array}[]{ccc}1&-\tilde{a}_{+-}&{-a_%
{+1}}\\
&{1}&0\\
0&-a_{1-}&1\\
\end{array}\right)\,,$$
where
$$\displaystyle\tilde{a}_{+-}=a_{+-}-a_{+1}a_{1-}\,.$$
In general $A$ can be any nondegenerate real matrix with entries $A_{ij}=\delta_{ij}+a_{ij}$, and one can easily show that
(i) $A_{++}$ deforms by $-T^{0}{}_{0}$:
$A_{++}\partial_{A_{++}}{\cal H}_{A_{++}}=-T^{0}{}_{0}=-{\cal H}_{A_{++}}$;
(ii) $A_{--}$ deforms by $T^{1}{}_{1}$:
$A_{--}\partial_{A_{--}}{\cal H}_{A_{--}}=T^{1}{}_{1}$;
(iii) $A_{11}$ deforms by $\widetilde{J}J$:
$A_{11}{\partial{\cal H}_{A_{11}}\over\partial A_{11}}={\widetilde{J}J}$;
(iv) $a_{-+}$ deforms by $1$:
$\partial_{a_{-+}}{\cal H}_{a_{-+}}=1$;
(v) $a_{-1}$ deforms by $-\widetilde{J}^{1}$:
$\partial_{a_{-1}}{\cal H}_{a_{-1}}=-\widetilde{J}^{1}$;
(vi) $a_{1+}$ deforms by $J^{0}=p_{1}$:
$\partial_{a_{1+}}{\cal H}_{a_{1+}}=J^{0}$.
We do not need the deformations by $T^{0}{}_{\alpha}$, $J^{0}$ and $\widetilde{J}^{0}$ because they just change the energy by the corresponding charges.
We could have used the transformations with $A_{--}$ and $A_{11}$ at the next steps but this leads to complicated flow equations when all ten parameters are nonvanishing, and a careful analysis shows that it is better to use these transformations (which commute) at the very end. As to the transformation with $a_{-1}$, it appears to be easier to use the auxiliary non-dynamical fields to generate the deformation by $\widetilde{J}^{1}$ and the remaining four deformations.
The remaining transformations are
4. Transformation, ${\cal T}_{T^{1}{}_{0}}$, generating the $-T^{1}{}_{0}$ deformation
$$\displaystyle{\cal T}_{T^{1}{}_{0}}:\quad x^{\prime+}\to x^{\prime+}-V^{+}\,,%
\quad\tilde{X}^{\prime}_{+}\to\tilde{X}^{\prime}_{+}+p_{+}\,,\quad p_{+}\to p_%
{+}\,,\quad V^{+}\to V^{+}\,,$$
$$\displaystyle{\partial{\cal H}_{v^{+}}\over\partial v^{+}}=-T^{1}{}_{0}\,.$$
This transformation obviously represents gauging the shift symmetry of $x^{+}$, and due to the light-cone gauge condition $x^{+}=\tau$ it is natural that the zero mode of $V^{+}$ generates the deformation by $-T^{1}{}_{0}$.
5. Transformation, ${\cal T}_{J^{1}}$, generating the $J^{1}$ deformation
$$\displaystyle{\cal T}_{J^{1}}:\quad x^{\prime 1}\to x^{\prime 1}-V^{1}\,,\quad%
\tilde{X}^{\prime}_{1}\to\tilde{X}^{\prime}_{1}+p_{1}\,,\quad p_{1}\to p_{1}\,%
,\quad V^{1}\to V^{1}\,,$$
$$\displaystyle{\partial{\cal H}_{v^{1}}\over\partial v^{1}}=J^{1}\,.$$
Gauging the shift symmetry of $x^{1}$ leads to the dependence of ${\cal H}_{v^{1}}$ on $x^{\prime 1}-v^{1}$.
6. Transformation, ${\cal T}_{JT_{0}}$, generating the $JT_{0}$ deformation
$$\displaystyle{\cal T}_{JT_{0}}:\quad x^{+}\to x^{+}-b_{+1}\tilde{X}_{1}\,,%
\quad\tilde{X}_{1}\to\tilde{X}_{1}\,,\quad p_{+}\to p_{+}\,,\quad V^{1}\to V^{%
1}+p_{+}b_{+1}\,,$$
$$\displaystyle{\partial{\cal H}_{b_{+1}}\over\partial b_{+1}}=JT_{0}\,,\quad JT%
_{0}\equiv J^{0}T^{1}{}_{0}-J^{1}T^{0}{}_{0}=-\epsilon_{\alpha\beta}J^{\alpha}%
T^{\beta}{}_{0}\,.$$
This transformation must be performed after the transformation ${\cal T}_{J^{1}}$.
7. Transformation, ${\cal T}_{\tilde{J}^{1}}$, generating the $\widetilde{J}^{1}$ deformation
$$\displaystyle{\cal T}_{\widetilde{J}^{1}}:\quad x^{1}\to x^{1}\,,\quad\Upsilon%
^{1}\to\Upsilon^{1}+x^{1}\,,\quad p_{1}\to p_{1}-{\cal U}_{1}\,,\quad{\cal U}_%
{1}\to{\cal U}_{1}\,,$$
$$\displaystyle{\partial{\cal H}_{u_{1}}\over\partial u_{1}}=\widetilde{J}^{1}\,.$$
This transformation in fact represents gauging the symmetry generated by
$$\displaystyle\widetilde{Q}(\xi)=\int d\sigma\widetilde{J}^{0}\xi=\int d\sigma x%
^{\prime 1}\xi\,,$$
which is trivial unless $\xi$ depends on $\sigma$.
This leads to the dependence of ${\cal H}_{u_{1}}$ on $p_{1}-u_{1}$.
8. Transformation, ${\cal T}_{\widetilde{J}T_{1}}$, generating the $\widetilde{J}T_{1}$ deformation
$$\displaystyle{\cal T}_{\widetilde{J}T_{1}}:$$
$$\displaystyle\tilde{X}_{-}\to\tilde{X}_{-}+p_{-}-{b_{--}\over c_{1-}}(x^{-}-V^%
{-})+b_{--}(x^{\prime 1}-V^{1})\,,\quad\tilde{X}_{1}\to\tilde{X}_{1}+c_{1-}%
\tilde{X}_{-}\,,$$
$$\displaystyle x^{\prime-}\to x^{\prime-}-V^{-}-c_{1-}(x^{\prime 1}-V^{1})\,,$$
$$\displaystyle p_{-}\to p_{-}+V^{-}{b_{--}\over c_{1-}}+b_{--}(x^{\prime 1}-V^{%
1})\,,\quad p_{1}\to p_{1}-c_{1-}\tilde{X}^{\prime}_{-}-c_{1-}\,,$$
$$\displaystyle V^{-}\to V^{-}+c_{1-}(x^{\prime 1}-V^{1})\,,$$
$$\displaystyle{\partial{\cal H}_{b_{--},c_{1-}}\over\partial b_{--}}=\widetilde%
{J}T_{1}\,,\quad{\partial{\cal H}_{b_{--},c_{1-}}\over\partial c_{1-}}=0\,,%
\quad\widetilde{J}T_{1}\equiv\widetilde{J}^{0}T^{1}{}_{1}-\widetilde{J}^{1}T^{%
0}{}_{1}=-\epsilon_{\alpha\beta}\widetilde{J}^{\alpha}T^{\beta}{}_{1}\,,$$
where in the gauge-fixed Hamiltonian we set $v^{-}=0$.
Note that this transformation requires both $b_{--}$ and $c_{1-}$ to be nonvanishing. This is the most complicated transformation. It can be represented as a combination of three simpler transformations
$$\displaystyle{\cal T}_{\tilde{J}T_{1}}={\cal T}_{3}{\cal T}_{2}{\cal T}_{1}\,,$$
where ${\cal T}_{1}$ gauges the shift symmetry of $x^{-}$
$$\displaystyle{\cal T}_{1}:\quad x^{\prime-}\to x^{\prime-}-V^{-}\,,\quad\tilde%
{X}^{\prime}_{-}\to\tilde{X}^{\prime}_{-}+p_{-}\,,\quad p_{-}\to p_{-}\,,\quad
V%
^{-}\to V^{-}\,,$$
${\cal T}_{2}$ is a linear transformation
$$\displaystyle{\cal T}_{2}:\quad\tilde{X}^{\prime}_{-}\to\tilde{X}^{\prime}_{-}%
-{b_{--}\over c_{1-}}x^{-}\,,\quad p_{-}\to p_{-}+V^{-}{b_{--}\over c_{1-}}\,,$$
and ${\cal T}_{3}$ is a transformation we call twisting $V^{-}$ with $x^{\prime 1}$
$$\displaystyle{\cal T}_{3}:$$
$$\displaystyle\tilde{X}_{1}\to\tilde{X}_{1}+c_{1-}\tilde{X}_{-}\,,\quad p_{1}%
\to p_{1}-c_{1-}\tilde{X}^{\prime}_{-}-c_{1-}\,,\quad V^{-}\to V^{-}+c_{1-}(x^%
{\prime 1}-V^{1})\,.$$
We could not find a simpler way to generate the $\widetilde{J}T_{1}$ deformation.
However, as we will see soon, setting $v^{-}=0$, one can take the limit $c_{1-}\to 0$ in the gauge-fixed Hamiltonian density, and this simplifies the flow equations drastically.
9. Transformation, ${\cal T}_{\widetilde{J}J}$, generating the $\widetilde{J}J$ deformation
$$\displaystyle{\cal T}_{\widetilde{J}J}:$$
$$\displaystyle x^{1}\to{1\over A_{11}}x^{1}\,,\quad p_{1}\to A_{11}p_{1}\,,%
\quad\tilde{X}_{1}\to A_{11}\tilde{X}_{1}\,,\quad V^{1}\to{1\over A_{11}}V^{1}\,,$$
(2.13)
$$\displaystyle\Upsilon^{1}\to{1\over A_{11}}\Upsilon^{1}\,,\quad{\cal U}_{1}\to
A%
_{11}{\cal U}_{1}\,,$$
$$\displaystyle A_{11}{\partial{\cal H}_{A_{11}}\over\partial A_{11}}={%
\widetilde{J}J}\,,\quad\widetilde{J}J\equiv\widetilde{J}^{0}J^{1}-\widetilde{J%
}^{1}J^{0}=-\epsilon_{\alpha\beta}\widetilde{J}^{\alpha}J^{\beta}\,.$$
The rescaling of $\tilde{X}_{1}$, $V^{1}$ and $\Upsilon^{1}$, ${\cal U}^{1}$ is necessary for a gauge-fixed Hamiltonian density to depend on the differences $x^{\prime 1}-v^{1}$ and $p_{1}-u_{1}$.
10. Transformation, ${\cal T}_{T^{1}{}_{1}}$, generating the $T^{1}{}_{1}$ deformation
$$\displaystyle{\cal T}_{T^{1}{}_{1}}:\quad x^{-}\to{1\over A_{--}}x^{-}\,,\quad
p%
_{-}\to p_{-}A_{--}\,,$$
$$\displaystyle A_{--}{\partial{\cal H}_{A_{--}}\over\partial A_{--}}=T^{1}{}_{1%
}\,.$$
Due to the light-cone gauge condition $p_{-}=1$ the deformation with $A_{--}$ is equivalent to a change of the circumference of the cylinder the gauge-fixed model lives on.
Now, we can generate a ten-parameter transformation by using the following sequence of the one-parameter ones
$$\displaystyle{\cal T}_{A}={\cal T}_{T^{1}{}_{1}}{\cal T}_{\widetilde{J}J}{\cal
T%
}_{\widetilde{J}T_{1}}{\cal T}_{\widetilde{J}^{1}}{\cal T}_{JT_{0}}{\cal T}_{J%
^{1}}{\cal T}_{T^{1}{}_{0}}{\cal T}_{JT_{1}}{\cal T}_{\widetilde{J}T_{0}}{\cal
T%
}_{T\overline{T}}\,.$$
(2.14)
We see that to generate all these deformations we need all three pairs of the auxiliary fields
$(V^{a}_{\alpha}\,,\,\tilde{X}_{a})$ and the pair $({\cal U}_{1\alpha}\,,\,\Upsilon^{1})$. Thus, in what follows we set
$${\cal U}_{+\alpha}=0\,,\,\Upsilon^{+}=0\,,\quad{\cal U}_{-\alpha}=0\,,\,%
\Upsilon^{-}=0\,.$$
The transformation (2.14) can be represented by a single canonical transformation.
Introducing a column of coordinates
$$\displaystyle{\cal X}=\big{\{}{\cal X}^{\hat{1}}\,{\cal X}^{\hat{2}},\ldots,{%
\cal X}^{\hat{7}}\big{\}}=\big{\{}x^{+},x^{-},x^{1},\tilde{X}_{+},\tilde{X}_{-%
},\tilde{X}_{1},\Upsilon^{1}\big{\}}\,,$$
and a column of momenta
$$\displaystyle{\cal P}=\big{\{}{\cal P}_{\hat{1}},{\cal P}_{\hat{2}},\ldots,{%
\cal P}_{\hat{7}}\big{\}}=\big{\{}p_{+},p_{-},p_{1},V^{+},V^{-},V^{1},{\cal U}%
_{1}\big{\}}\,,$$
the transformation can be written in the form
$$\displaystyle{\cal X}^{\prime}\to A_{x}{\cal X}^{\prime}+B_{x}{\cal P}+C_{x}\,%
,\quad{\cal P}\to A_{p}^{T}{\cal P}+B_{p}^{T}{\cal X}^{\prime}+C_{p}\,.$$
(2.15)
Here the matrices $A_{x}$, $B_{x}$ , $A_{p}$ and $B_{p}$, and the columns $C_{x}$, $C_{p}$ satisfy the identities
$$\displaystyle A_{x}A_{p}+B_{x}B_{p}={\mathbb{I}}\,,\quad A_{x}B^{T}_{x}+B_{x}A%
_{x}^{T}=0\,,\quad A_{p}^{T}B_{p}+B_{p}^{T}A_{p}=0\,,$$
which are necessary for the transformation to be canonical, and additional identities
$$\displaystyle A_{p}A_{x}+B_{x}^{T}B_{p}^{T}$$
$$\displaystyle={\mathbb{I}}\,,\quad A_{p}B_{x}+(A_{p}B_{x})^{T}=0\,,\quad B_{p}%
A_{x}+(B_{p}A_{x})^{T}=0\,,$$
$$\displaystyle C_{p}^{T}A_{x}{\cal X}^{\prime}+{\cal X}^{\prime T}B_{p}C_{x}$$
$$\displaystyle=-{c_{1-}\over A_{11}}x^{\prime 1}\,,\quad C_{p}B_{x}{\cal P}+{%
\cal P}^{T}A_{p}C_{x}=0\,,\quad C_{p}^{T}C_{x}=0\,,$$
which lead to ${\cal P}^{T}{\cal X}^{\prime}\to{\cal P}^{T}{\cal X}^{\prime}-{c_{1-}\over A_{%
11}}x^{\prime 1}$.
The explicit form of the matrices can be found in appendix A.
3 Deformed Hamiltonian and the flow equations
3.1 Deformed Hamiltonian
Now, we can derive the gauge-fixed Hamiltonian density ${\cal H}_{A}$ which we consider as a deformation of ${\cal H}_{0}$.
First, we note that up to total derivatives the kinetic term in the action (2.4) does not change under the canonical transformation (2.15).
Then, as was mentioned before on the constraints surface the constraint $C_{1}$ is equivalent to $\tilde{C}_{1}$ (2.6). Clearly,
under (2.15), $\tilde{C}_{1}$ changes as
$$\displaystyle\tilde{C}_{1}\overset{{\cal T}_{A}}{\longrightarrow}\tilde{C}_{1}%
={\cal P}^{T}{\cal X}^{\prime}-{c_{1-}\over A_{11}}x^{\prime 1}+p_{k}x^{\prime
k%
}\,.$$
Then, we obviously have
$$\displaystyle C_{a}^{V}\overset{{\cal T}_{A}}{\longrightarrow}C_{a}^{V}=(A_{x}%
{\cal X}^{\prime}+B_{x}{\cal P}+C_{x})^{\hat{3}+\hat{a}}=(A_{x})^{\hat{3}+\hat%
{a}}{}_{r}{\cal X}^{\prime r}+(B_{x})^{\hat{3}+\hat{a},r}{\cal P}_{r}+C_{x}^{%
\hat{3}+\hat{a}}\,.$$
$$\displaystyle C^{1}_{\cal U}\overset{{\cal T}_{A}}{\longrightarrow}C^{1}_{\cal
U%
}=(A_{x}{\cal X}^{\prime}+B_{x}{\cal P})^{\hat{7}}=(A_{x})^{\hat{7}}{}_{r}{%
\cal X}^{\prime r}+(B_{x})^{\hat{7},r}{\cal P}_{r}\,,\quad r=\hat{1},\hat{2},%
\ldots,\hat{7}\,.$$
Finally, a simple calculation gives the transformed Virasoro constraint $C_{2}$
$$\displaystyle C_{2}\overset{{\cal T}_{A}}{\longrightarrow}C_{2}$$
$$\displaystyle={1\over 2}\tilde{G}^{rs}{\cal P}_{r}{\cal P}_{s}+{1\over 2}{\cal
X%
}^{\prime r}{\cal X}^{\prime s}\tilde{G}_{rs}+\tilde{G}^{rk}{\cal P}_{r}p_{k}+%
{\cal X}^{\prime r}x^{\prime k}\tilde{G}_{rk}$$
(3.1)
$$\displaystyle+\hat{G}^{r}{}_{s}{\cal P}_{r}{\cal X}^{\prime s}+\check{G}^{r}{}%
_{k}{\cal P}_{r}x^{\prime k}+{\cal X}^{\prime r}p_{k}\check{G}_{r}{}^{k}+%
\tilde{\cal H}_{x}\,.$$
The coefficients in this equation are listed in appendix B.
Now, we impose the light-cone gauge conditions (2.7) on the transformed coordinates and momenta. Then, from the analysis of
the canonical transformation (2.15) we know that the zero mode $v^{-}$ does not generate any of the deformations. Thus, in addition to (2.7) we can set it to 0 (and we have already set ${\cal U}_{\pm\alpha}=\Upsilon^{\pm}=0$)
$$\displaystyle v^{-}=0\,.$$
Next, we solve the constraints $\tilde{C}_{1}=0$, $C_{a}^{V}=0$ and $C^{1}_{\cal U}=0$ for
$$x^{\prime-}\,,\,\tilde{X}^{\prime}_{+}\,,\,\tilde{X}^{\prime}_{-}\,,\,\tilde{X%
}^{\prime}_{1}\,,\,\Upsilon^{\prime 1}\,.$$
The solution can be written in the form
$$\displaystyle{\cal X}^{\prime r}=\varkappa^{r}\,p_{+}+\chi^{r}\,,$$
(3.2)
where we also have ${\cal X}^{\prime\hat{1}}=x^{\prime+}=0$ and ${\cal X}^{\prime\hat{3}}=x^{\prime 1}$. The coefficients $\varkappa^{r}$ and $\chi^{r}$ can be read off from eqs.(B.1).
Using the solution,
we bring $C_{2}$ to the form $C_{2}={\cal G}_{2}\,p_{+}^{2}+{\cal G}_{1}\,p_{+}+{\cal G}_{0}$,
where the coefficients ${\cal G}_{i}$ are listed in appendix B.
One then checks that even though the transformation (2.15) is singular in the limit $c_{1-}\to 0$, the coefficients ${\cal G}_{i}$ are regular at $c_{1-}=0$. In fact it is necessary to have $v^{-}=0$ for the regularity. Thus, in what follows we set $c_{1-}=0$, and only discuss this case. If $c_{1-}\neq 0$ then there should exist a redefinition of the parameters which makes the gauge-fixed Hamiltonian independent of $c_{1-}$.
The solution of the quadratic equation
$$\displaystyle C_{2}$$
$$\displaystyle={\cal G}_{2}\,p_{+}^{2}+{\cal G}_{1}\,p_{+}+{\cal G}_{0}=0\,\,,$$
(3.3)
which reduces to $-{\cal H}_{0}$ when all the parameters vanish gives us the deformed Hamiltonian and action
$$\displaystyle S_{A}=\int_{0}^{R}\,{\rm d}\sigma{\rm d}\tau\,\left(p_{\mu}\dot{%
x}^{\mu}\,-\,{\cal H}_{A}\right)\,,\quad{\cal H}_{A}\,=\,-p_{+}(p_{\mu},x^{\mu%
},x^{\prime\mu})\,,\quad R={P_{-}}\,.$$
(3.4)
We consider some deformations of a model with the Hamiltonian
(2.12) in appendix C.
3.2 Flow equations for the density of the Hamiltonian
To simplify the understanding of the origin of various terms in the flow equations in this subsection we use the notations
$$\displaystyle\alpha_{\mbox{\tiny${T\overline{T}}$}}\equiv a_{+-}\,,\quad\alpha%
_{\mbox{\tiny${\widetilde{J}T_{0}}$}}\equiv-a_{+1}\,,\quad\alpha_{\mbox{\tiny$%
{JT_{1}}$}}\equiv a_{1-}\,,\quad\alpha_{\mbox{\tiny${JT_{0}}$}}\equiv b_{+1}\,%
,\quad\alpha_{\mbox{\tiny${\widetilde{J}T_{1}}$}}\equiv b_{--}\,,\quad v\equiv%
-v^{+}\,,$$
(3.5)
so that the flow equations at the leading order would have the form
$$\displaystyle{\partial{\cal H}_{A}\over\partial\alpha_{\cal O}}={\cal O}\,,%
\quad{\partial{\cal H}_{A}\over\partial v}=T^{1}{}_{0}\,.$$
The Hamiltonian density ${\cal H}_{A}$ (or, better to say the defining equation (3.3)) can be used to derive the flow equations with respect to the ten deformation parameters.
First of all, it is easy to check that the coefficients ${\cal G}_{i}$, and, therefore, ${\cal H}_{A}$, depend on the differences $x^{\prime 1}-v^{1}$ and $p_{1}-u_{1}$. Thus,
$$\displaystyle{\partial{\cal H}_{A}\over\partial v^{1}}=J^{1}\,,\quad{\partial{%
\cal H}_{A}\over\partial u_{1}}=\widetilde{J}^{1}\,.$$
(3.6)
Moreover, all the operators appearing on the right hand side of the flow equations also must depend on $x^{\prime 1}-v^{1}$ and $p_{1}-u_{1}$. Let us introduce the following improved currents ${\mathbb{J}}^{\alpha}$, $\widetilde{\mathbb{J}}^{\alpha}$ which depend on those differences
$$\displaystyle{\mathbb{J}}^{0}=p_{1}-u_{1}=J^{0}-u_{1}\,,\quad{\mathbb{J}}^{1}=%
J^{1}\,,$$
(3.7)
$$\displaystyle\widetilde{\mathbb{J}}^{0}=x^{\prime 1}-v^{1}=\widetilde{J}^{0}-v%
^{1}\,,\quad\widetilde{\mathbb{J}}^{1}=\widetilde{J}^{1}\,,$$
and the improved stress-energy tensor ${\mathbb{T}}^{\alpha}{}_{\beta}$
$$\displaystyle{\mathbb{T}}^{0}{}_{0}=T^{0}{}_{0}\,,\quad{\mathbb{T}}^{1}{}_{0}=%
T^{1}{}_{0}\,,$$
$$\displaystyle{\mathbb{T}}^{0}{}_{1}$$
$$\displaystyle=(p_{1}-u_{1})(x^{\prime 1}-v^{1})+p_{k}x^{\prime k}-v\,T^{0}{}_{%
0}=T^{0}{}_{1}-v\,T^{0}{}_{0}-v^{1}J^{0}-u_{1}\widetilde{J}^{0}+u_{1}v^{1}\,,$$
(3.8)
$$\displaystyle{\mathbb{T}}^{1}{}_{1}$$
$$\displaystyle={\cal H}_{A}-{\partial{\cal H}_{A}\over\partial x^{\prime k}}x^{%
\prime k}-p_{k}{\partial{\cal H}_{A}\over\partial p_{k}}-{\partial{\cal H}_{A}%
\over\partial x^{\prime 1}}(x^{\prime 1}-v^{1})-(p_{1}-u_{1}){\partial{\cal H}%
_{A}\over\partial p_{1}}-v\,T^{1}{}_{0}$$
$$\displaystyle=T^{1}{}_{1}-v\,T^{1}{}_{0}-v^{1}J^{1}-u_{1}\widetilde{J}^{1}\,,$$
whose components ${\mathbb{T}}^{\alpha}{}_{1}$ are also shifted by $v\,T^{\alpha}{}_{0}$.
Then, in terms of the improved currents (3.7, 3.8) the flow equations take the form (in the limit $c_{1-}\to 0$)
$$\displaystyle{\partial{\cal H}_{A}\over\partial v}={\mathbb{T}}^{1}{}_{0}=T^{1%
}{}_{0}\,,$$
(3.9)
$$\displaystyle A_{--}{\partial{\cal H}_{A}\over\partial A_{--}}={\mathbb{T}}^{1%
}{}_{1}=T^{1}{}_{1}-vT^{1}{}_{0}-v^{1}J^{1}-u_{1}\widetilde{J}^{1}\,,$$
(3.10)
$$\displaystyle A_{11}{\partial{\cal H}_{A}\over\partial A_{11}}$$
$$\displaystyle=\widetilde{\mathbb{J}}{\mathbb{J}}=\widetilde{J}J-u_{1}J^{1}+v^{%
1}\widetilde{J}^{1}\,,$$
(3.11)
$$\displaystyle A_{--}{\partial{\cal H}_{A}\over\partial\alpha_{\mbox{\tiny${T%
\overline{T}}$}}}={\mathbb{T}}\overline{\mathbb{T}}\,,$$
(3.12)
$$\displaystyle{A_{--}}{\partial{\cal H}_{A}\over\partial\alpha_{\mbox{\tiny${JT%
_{1}}$}}}$$
$$\displaystyle=A_{11}{\mathbb{J}}{\mathbb{T}}_{1}\,,$$
(3.13)
$$\displaystyle{\partial{\cal H}_{A}\over\partial\alpha_{\mbox{\tiny${JT_{0}}$}}}$$
$$\displaystyle=A_{11}{\mathbb{J}}{\mathbb{T}}_{0}+{\alpha_{\mbox{\tiny${%
\widetilde{J}T_{1}}$}}\over A_{--}}{\mathbb{T}}\overline{\mathbb{T}}\,,$$
$$\displaystyle A_{--}{\partial{\cal H}_{A}\over\partial\alpha_{\mbox{\tiny${%
\widetilde{J}T_{1}}$}}}$$
$$\displaystyle={1\over A_{11}}\widetilde{\mathbb{J}}{\mathbb{T}}_{1}\,,$$
$$\displaystyle{\partial{\cal H}_{A}\over\partial\alpha_{\mbox{\tiny${\widetilde%
{J}T_{0}}$}}}$$
$$\displaystyle={1\over A_{11}}\widetilde{\mathbb{J}}{\mathbb{T}}_{0}+{\alpha_{%
\mbox{\tiny${JT_{1}}$}}\over A_{--}}{\mathbb{T}}\overline{\mathbb{T}}\,.$$
A simple analysis of eqs. (3.6), (3.9) and (3.10) shows that the following rescaling of $v^{1},v,u_{1}$
$$\displaystyle v^{1}\to A_{--}v^{1}\,,\quad v\to A_{--}v\,,\quad u_{1}\to A_{--%
}u_{1}\,,$$
brings the equations to the form
$$\displaystyle A_{--}{\partial{\cal H}_{A}\over\partial v^{1}}=J^{1}\,,\quad A_%
{--}{\partial{\cal H}_{A}\over\partial u_{1}}=\widetilde{J}^{1}\,,\quad A_{--}%
{\partial{\cal H}_{A}\over\partial v}=T^{1}{}_{0}\,,\quad A_{--}{\partial{\cal
H%
}_{A}\over\partial A_{--}}=T^{1}{}_{1}\,.$$
(3.14)
This rescaling, however, breaks the dependence of ${\cal H}_{A}$ on $x^{\prime 1}-v^{1}$ and $p_{1}-u_{1}$. To restore the dependence we can rescale the world-sheet space coordinate $\sigma$
$$\displaystyle\sigma\to\sigma/A_{--}\ \Rightarrow\ S_{A}\to S_{A}=\int_{0}^{RA_%
{--}}\,{{\rm d}\sigma\over A_{--}}{\rm d}\tau\,\left(p_{\mu}\dot{x}^{\mu}\,-\,%
{\cal H}_{A}(p_{\mu},x^{\mu},A_{--}x^{\prime\mu})\right)\,.$$
We see that to restore the canonical Poisson structure we also need to rescale $p_{\mu}$
$$\displaystyle p_{\mu}\to A_{--}p_{\mu}\ \Rightarrow\ S_{A}\to S_{A}=\int_{0}^{%
RA_{--}}\,{\rm d}\sigma{\rm d}\tau\,\left(p_{\mu}\dot{x}^{\mu}\,-\,{1\over A_{%
--}}{\cal H}_{A}(A_{--}p_{\mu},x^{\mu},A_{--}x^{\prime\mu})\right)\,.$$
One can then check by using the coefficients ${\cal G}_{i}$ that
$$\displaystyle{1\over A_{--}}{\cal H}_{A}(A_{--}p_{\mu},x^{\mu},A_{--}x^{\prime%
\mu})={\cal H}_{A}(p_{\mu},x^{\mu},x^{\prime\mu})\big{|}_{A_{--}=1}\,.$$
Thus, without loss of generality one can set $A_{--}=1$, and consider $R$ as the tenth deformation parameter. In quantum theory the flow equation (3.10) with respect to $A_{--}$ would imply the following flow equation for the energy with respect to $R$
$$\displaystyle{dE_{n}\over dR}=\langle n|T^{1}{}_{1}|n\rangle\,,$$
where $|n\rangle$ is an eigenstate of $H_{A}$ with the energy $E_{n}$.
The $\widetilde{J}J$ deformation can be also easily understood. It is clear from the way it was generated,
see eq.(2.13), and can be checked explicitly by using the defining equation (3.3) that the deformed Hamiltonian depends on $A_{11}$ only through the combinations
$$\displaystyle A_{11}(p_{1}-u_{1})\,,\quad{1\over A_{11}}(x^{\prime 1}-v^{1})\,.$$
Thus, the energy of a deformed model with parameters $A_{11}$, $u_{1}$, $v^{1}$ is related to the energy of the deformed model with $A_{11}=1$ as
$$\displaystyle E_{n}(A_{11},P_{1}-Ru_{1},\widetilde{P}^{1}-Rv^{1})=E_{n}(1,A_{1%
1}(P_{1}-Ru_{1}),(\widetilde{P}^{1}-Rv^{1})/A_{11})\,,$$
where $\widetilde{P}^{1}=x^{1}(R)-x^{1}(0)$ is basically an effective range of $x^{1}$ which may depend on winding numbers of several scalars and target space-time metric of the model, see the discussion below eq.(4.2). This can be also seen from the flow equations. Indeed, rescaling $u_{1}$ and $v^{1}$ as $u_{1}\to u_{1}/A_{11}$, $v^{1}\to A_{11}v^{1}$, and then rescaling the currents as $J^{\alpha}\to J^{\alpha}/A_{11}$, $\widetilde{J}^{\alpha}\to A_{11}\widetilde{J}^{\alpha}$ removes completely the $A_{11}$ dependence from the flow equations. Thus, we may set $A_{11}=1$.
3.3 Flow equations for the energy
Since the Hamiltonian ${\bf H}_{A}$, the world-sheet momentum ${\bf P}$, the U(1) charge ${\bf P}_{1}$, and the dual charge $\widetilde{\bf P}^{1}$ are mutually commuting,
the energy $E_{n}$ of their common eigenstate $|n\rangle$ is a function of $R$, and the eigenvalues $P$, $P_{1}$, and $\widetilde{P}^{1}$ of ${\bf P}$, ${\bf P}_{1}$, and $\widetilde{\bf P}^{1}$, and the deformation parameters.
Then, the flow equations for the Hamiltonian density, and the Hellmann-Feynman
theorem lead to the following relations
$$\displaystyle\langle n|T^{0}{}_{0}|n\rangle$$
$$\displaystyle={E_{n}\over R}\,,\quad\langle n|T^{0}{}_{1}|n\rangle=-{P\over R}%
\,,\quad\langle n|J^{0}|n\rangle={P_{1}\over R}\,,\quad\langle n|\widetilde{J}%
^{0}|n\rangle={\widetilde{P}^{1}\over R}\,,$$
(3.15)
$$\displaystyle\langle n|T^{1}{}_{1}|n\rangle$$
$$\displaystyle={dE_{n}\over dR}\,,\quad\langle n|T^{1}{}_{0}|n\rangle={1\over R%
}{\partial E_{n}\over\partial v}\,,\quad\langle n|J^{1}|n\rangle={1\over R}{%
\partial E_{n}\over\partial v^{1}}\,,\quad\langle n|\widetilde{J}^{1}|n\rangle%
={1\over R}{\partial E_{n}\over\partial u_{1}}\,.$$
In addition, assuming factorisation, for an operator of the form
${\cal O}_{K_{1}K_{2}}=-\epsilon_{\alpha\beta}K^{\alpha}_{1}K^{\beta}_{2}$, one has
$$\displaystyle\langle n|{\cal O}_{K_{1}K_{2}}|n\rangle=\langle n|K^{0}_{1}|n%
\rangle\langle n|K^{1}_{2}|n\rangle-\langle n|K^{1}_{1}|n\rangle\langle n|K^{0%
}_{2}|n\rangle\,,$$
where $K^{\alpha}_{i}$ are conserved currents, and $|n\rangle$ is an eigenvector of their charges.
By using these relations, it is straightforward to derive flow equations for the energy of the state.
There is, however, a better form of flow equations for the energy.
The Hamiltonian density depends on $p_{1}-u_{1}=J^{0}-u_{1}$, and $x^{\prime 1}-v^{1}=\widetilde{J}^{0}-v^{1}$, and therefore it is more natural to consider the energy $E_{n}$ as a function of the improved charges ${\mathbb{P}}_{1}=P_{1}-Ru_{1}$, and $\widetilde{\mathbb{P}}^{1}=\widetilde{P}^{1}-Rv^{1}$.
To make sure that the energy would depend on $u_{1}$ and $v^{1}$ only through ${\mathbb{P}}_{1}$ and $\widetilde{\mathbb{P}}^{1}$, one also should consider $E_{n}$ as a function of
$$\displaystyle\check{P}\equiv P+{P_{1}\widetilde{P}^{1}\over R}\,.$$
Indeed, the world-sheet momentum can be represented as
$$\displaystyle P=-\int_{0}^{R}d\sigma\,p_{\mu}x^{\prime\mu}=-\int_{0}^{R}d%
\sigma\,\big{(}({P_{1}\over R}+\check{p}_{1})({\widetilde{P}^{1}\over R}+%
\check{x}^{\prime 1})+p_{k}x^{\prime k}\big{)}=-{P_{1}\widetilde{P}^{1}\over R%
}+\check{P}\,,$$
where $\check{p}_{1}$ and $\check{x}^{\prime 1}$ satisfy $\int_{0}^{R}\check{p}_{1}=0$, $\check{x}^{1}(R)=\check{x}^{1}(0)$. Clearly, this formula demonstrates the splitting of the world-sheet momentum into its intrinsic part $\check{P}$, and the part due to the rotation and winding in the $x^{1}$ direction.
Thus, we can consider the energy as a function of $R$, $\check{P}$, ${\mathbb{P}}_{1}$ and $\widetilde{\mathbb{P}}^{1}$, and the deformation parameters $\alpha_{\cal O}$, $v$, $A_{11}$
$$\displaystyle E_{n}=E_{n}(R\,,\check{P}\,,{\mathbb{P}}_{1}\,,\widetilde{%
\mathbb{P}}^{1}\,,\alpha_{\cal O}\,,v\,,A_{11})=E_{n}(R\,,\check{P}\,,P_{1}-Ru%
_{1}\,,\widetilde{P}^{1}-Rv^{1}\,,\alpha_{\cal O}\,,v\,,A_{11})\,.$$
It is now clear that the last two relations in (3.15) take the form
$$\displaystyle\langle n|J^{1}|n\rangle=-{\partial E_{n}\over\partial\widetilde{%
\mathbb{P}}^{1}}\,,\quad\langle n|\widetilde{J}^{1}|n\rangle=-{\partial E_{n}%
\over\partial{\mathbb{P}}_{1}}\,.$$
Then, the flow equations for the Hamiltonian density depend on the improved currents and stress-energy tensor (3.7, 3.8), and their expectation values are
$$\displaystyle\langle n|{\mathbb{T}}^{0}{}_{1}|n\rangle$$
$$\displaystyle=-{1\over R}\big{(}\check{P}-{{\mathbb{P}}_{1}\widetilde{\mathbb{%
P}}^{1}\over R}+vE_{n}\big{)}\,,\quad\langle n|{\mathbb{T}}^{1}{}_{1}|n\rangle%
={\partial E_{n}\over\partial R}-{v\over R}{\partial E_{n}\over\partial v}\,,$$
(3.16)
$$\displaystyle\langle n|{\mathbb{J}}^{0}|n\rangle$$
$$\displaystyle={{\mathbb{P}}_{1}\over R}\,,\quad\langle n|\widetilde{\mathbb{J}%
}^{0}|n\rangle={\widetilde{\mathbb{P}}^{1}\over R}\,.$$
By using these relations, one gets the following flow equations for the energy of the state $|n\rangle$ (with $A_{--}=1$)
$$\displaystyle A_{11}{\partial E_{n}\over\partial A_{11}}$$
$$\displaystyle={\mathbb{P}}_{1}{\partial E_{n}\over\partial{\mathbb{P}}_{1}}-%
\widetilde{\mathbb{P}}^{1}{\partial E_{n}\over\partial\widetilde{\mathbb{P}}^{%
1}}\,,$$
(3.17)
$$\displaystyle{\partial E_{n}\over\partial\alpha_{\mbox{\tiny${T\overline{T}}$}}}$$
$$\displaystyle=-E_{n}{\partial E_{n}\over\partial R}-{\check{P}-{{\mathbb{P}}_{%
1}\widetilde{\mathbb{P}}^{1}\over R}\over R}{\partial E_{n}\over\partial v}\,,$$
(3.18)
$$\displaystyle{1\over A_{11}}{\partial E_{n}\over\partial\alpha_{\mbox{\tiny${%
JT_{1}}$}}}$$
$$\displaystyle={\mathbb{P}}_{1}\big{(}{\partial E_{n}\over\partial R}-{v\over R%
}{\partial E_{n}\over\partial v}\big{)}-\big{(}\check{P}-{{\mathbb{P}}_{1}%
\widetilde{\mathbb{P}}^{1}\over R}+vE_{n}\big{)}{\partial E_{n}\over\partial%
\widetilde{\mathbb{P}}^{1}}\,,$$
$$\displaystyle{1\over A_{11}}{\partial E_{n}\over\partial\alpha_{\mbox{\tiny${%
JT_{0}}$}}}$$
$$\displaystyle={{\mathbb{P}}_{1}\over R}{\partial E_{n}\over\partial v}+E_{n}{%
\partial E_{n}\over\partial\widetilde{\mathbb{P}}^{1}}+{\alpha_{\mbox{\tiny${%
\widetilde{J}T_{1}}$}}\over A_{11}}{\partial E_{n}\over\partial\alpha_{\mbox{%
\tiny${T\overline{T}}$}}}\,,$$
$$\displaystyle A_{11}{\partial E_{n}\over\partial\alpha_{\mbox{\tiny${%
\widetilde{J}T_{1}}$}}}$$
$$\displaystyle=\widetilde{\mathbb{P}}^{1}\big{(}{\partial E_{n}\over\partial R}%
-{v\over R}{\partial E_{n}\over\partial v}\big{)}-\big{(}\check{P}-{{\mathbb{P%
}}_{1}\widetilde{\mathbb{P}}^{1}\over R}+vE_{n}\big{)}{\partial E_{n}\over%
\partial{\mathbb{P}}_{1}}\,,$$
$$\displaystyle A_{11}{\partial E_{n}\over\partial\alpha_{\mbox{\tiny${%
\widetilde{J}T_{0}}$}}}$$
$$\displaystyle={\widetilde{\mathbb{P}}^{1}\over R}{\partial E_{n}\over\partial v%
}+E_{n}{\partial E_{n}\over\partial{\mathbb{P}}_{1}}+A_{11}\alpha_{\mbox{\tiny%
${JT_{1}}$}}{\partial E_{n}\over\partial\alpha_{\mbox{\tiny${T\overline{T}}$}}%
}\,.$$
It is unclear how to solve this system of equations. However, the equation (3.17) with respect to $A_{11}$ for the $\widetilde{J}J$ deformation can be easily solved. Its solution is
$$\displaystyle E_{n}({\mathbb{P}}_{1}\,,\widetilde{\mathbb{P}}^{1}\,,A_{11})=E_%
{n}(A_{11}{\mathbb{P}}_{1}\,,{\widetilde{\mathbb{P}}^{1}\over A_{11}}\,,1)\,,$$
(3.19)
where we kept the energy dependence only on the essential variables. Thus, without loss of generality we can set $A_{11}=1$ in the remaining five equations.
3.4 Homogeneous inviscid Burgers equation
The deformed and undeformed models are obtained from the same extended string sigma model by applying different light-cone type gauge conditions.
The physical quantities, therefore, must be the same in both models. However, as usual in a light-cone gauge, physical states must satisfy additional conditions which follow from the requirement that the strings are closed. We will refer to these conditions as to generalised level-matching conditions. In this section we determine which states of the deformed model are physical, and discuss a relation between the energy of physical states in the deformed and undeformed models. We set $A_{--}=1$ and use $R$ as a parameter.
Let us denote $x^{a}_{0}$, $\tilde{X}_{a}^{0}$, $\Upsilon^{a}_{0}$, $p_{a}^{0}$, $V^{a}_{0}$ and ${\cal U}_{a}^{0}$ the coordinates and momenta of the extended string model (2.4) before one applies the canonical transformation (2.15). They are related to the transformed coordinates and momenta as
in eqs.(2.15). The transformed coordinates and momenta satisfy the light-cone gauge conditions and the constraints. Substituting the solution of the gauge conditions and the constraints into the relations (2.15), one finds for the coordinates
$$\displaystyle x^{\prime-}_{0}$$
$$\displaystyle=-{\mathbb{T}}^{0}{}_{1}\,,$$
(3.20)
$$\displaystyle x^{\prime 1}_{0}$$
$$\displaystyle=\alpha_{\mbox{\tiny${JT_{1}}$}}{\mathbb{T}}^{0}{}_{1}+{%
\widetilde{\mathbb{J}}^{0}\over A_{11}}+\alpha_{\mbox{\tiny${JT_{0}}$}}{\cal H%
}_{A}\,,$$
$$\displaystyle x^{\prime+}_{0}$$
$$\displaystyle=(\alpha_{\mbox{\tiny${T\overline{T}}$}}+\alpha_{\mbox{\tiny${JT_%
{1}}$}}\alpha_{\mbox{\tiny${\widetilde{J}T_{0}}$}}+\alpha_{\mbox{\tiny${%
\widetilde{J}T_{1}}$}}\alpha_{\mbox{\tiny${JT_{0}}$}}){\mathbb{T}}^{0}{}_{1}+%
\alpha_{\mbox{\tiny${\widetilde{J}T_{0}}$}}\big{(}{\widetilde{\mathbb{J}}^{0}%
\over A_{11}}+\alpha_{\mbox{\tiny${JT_{0}}$}}{\cal H}_{A}\big{)}+A_{11}\alpha_%
{\mbox{\tiny${JT_{0}}$}}{\mathbb{J}}^{0}+v\,,$$
$$\displaystyle\tilde{X}^{\prime}_{a}{}^{0}$$
$$\displaystyle=0\,,\quad\Upsilon^{\prime}{}^{a}_{0}=0\,,\quad a=+,-,1\,,$$
and for the momenta
$$\displaystyle p_{+}^{0}$$
$$\displaystyle=-{\cal H}_{A}\,,$$
(3.21)
$$\displaystyle p_{-}^{0}$$
$$\displaystyle=1-\alpha_{\mbox{\tiny${T\overline{T}}$}}{\cal H}_{A}+A_{11}%
\alpha_{\mbox{\tiny${JT_{1}}$}}{\mathbb{J}}^{0}+\alpha_{\mbox{\tiny${%
\widetilde{J}T_{1}}$}}{\widetilde{\mathbb{J}}^{0}\over A_{11}}+\alpha_{\mbox{%
\tiny${JT_{1}}$}}\alpha_{\mbox{\tiny${\widetilde{J}T_{1}}$}}{\mathbb{T}}^{0}{}%
_{1}\,,$$
$$\displaystyle p_{1}^{0}$$
$$\displaystyle=A_{11}{\mathbb{J}}^{0}+\alpha_{\mbox{\tiny${\widetilde{J}T_{0}}$%
}}{\cal H}_{A}+\alpha_{\mbox{\tiny${\widetilde{J}T_{1}}$}}{\mathbb{T}}^{0}{}_{%
1}\,,$$
$$\displaystyle V^{+}_{0}$$
$$\displaystyle=-v\,,$$
$$\displaystyle V^{1}_{0}$$
$$\displaystyle=v^{1}/A_{11}-\alpha_{\mbox{\tiny${JT_{0}}$}}{\cal H}_{A}\,,$$
$$\displaystyle{\cal U}_{1}^{0}$$
$$\displaystyle=v^{1}/A_{11}-\alpha_{\mbox{\tiny${JT_{0}}$}}A_{11}u_{1}{\cal H}_%
{A}\,,$$
where the improved currents and stress-energy tensor are given by (3.7, 3.8).
Eqs.(3.20) show that the generalised level-matching conditions are
$$\displaystyle x^{-}_{0}(R)-x^{-}_{0}(0)$$
$$\displaystyle=-\int_{0}^{R}d\sigma\,{\mathbb{T}}^{0}{}_{1}={\mathbb{P}}=0\,,$$
(3.22)
$$\displaystyle x^{1}_{0}(R)-x^{1}_{0}(0)$$
$$\displaystyle=\widetilde{P}_{0}^{1}=\int_{0}^{R}d\sigma\,({\widetilde{\mathbb{%
J}}^{0}\over A_{11}}+\alpha_{\mbox{\tiny${JT_{0}}$}}{\cal H}_{A})={\widetilde{%
\mathbb{P}}^{1}\over A_{11}}+\alpha_{\mbox{\tiny${JT_{0}}$}}E_{A}=w{\cal R}^{1%
}_{0}\,,$$
$$\displaystyle x^{+}_{0}(R)-x^{+}_{0}(0)$$
$$\displaystyle=\alpha_{\mbox{\tiny${\widetilde{J}T_{0}}$}}w{\cal R}^{1}_{0}+%
\int_{0}^{R}d\sigma\,(A_{11}\alpha_{\mbox{\tiny${JT_{0}}$}}{\mathbb{J}}^{0}+v)%
=\alpha_{\mbox{\tiny${\widetilde{J}T_{0}}$}}w{\cal R}^{1}_{0}+A_{11}\alpha_{%
\mbox{\tiny${JT_{0}}$}}{\mathbb{P}}_{1}+vR=0\,,$$
where $w\in{\mathbb{Z}}$ is the winding number of $x^{1}_{0}$ and $x^{1}$, and the improved world-sheet momentum ${\mathbb{P}}$, and charges ${\mathbb{P}}_{1}$ and $\widetilde{\mathbb{P}}^{1}$ are given by
$$\displaystyle{\mathbb{P}}$$
$$\displaystyle=P+v\,E+v^{1}P_{1}+u_{1}\widetilde{P}^{1}-u_{1}v^{1}R\,,$$
(3.23)
$$\displaystyle{\mathbb{P}}_{1}$$
$$\displaystyle=P_{1}-u_{1}R\,,$$
$$\displaystyle\widetilde{\mathbb{P}}^{1}$$
$$\displaystyle=\widetilde{P}^{1}-v^{1}R=w{\cal R}^{1}-v^{1}R\,.$$
Solving the generalised level-matching conditions for $P$, $v$ and $v^{1}$, one gets
$$\displaystyle P$$
$$\displaystyle=w\big{(}{{\cal R}_{0}^{1}\over R}\alpha_{\mbox{\tiny${\widetilde%
{J}T_{0}}$}}\,E_{A}+{A_{11}{\cal R}^{1}_{0}\over R}{\mathbb{P}}_{1}-{{\cal R}^%
{1}\over R}P_{1}\big{)}\,,$$
(3.24)
$$\displaystyle vR$$
$$\displaystyle=-\alpha_{\mbox{\tiny${\widetilde{J}T_{0}}$}}w{\cal R}^{1}_{0}-A_%
{11}\alpha_{\mbox{\tiny${JT_{0}}$}}{\mathbb{P}}_{1}\,,$$
$$\displaystyle v^{1}R$$
$$\displaystyle=A_{11}\alpha_{\mbox{\tiny${JT_{0}}$}}E_{A}+w({\cal R}^{1}-A_{11}%
{\cal R}^{1}_{0})\,.$$
Having found the generalised level-matching conditions, we can now relate the energies of physical states of the deformed and undeformed models.
Integrating the relations (3.21) over $\sigma$, one gets
$$\displaystyle E_{0}$$
$$\displaystyle=E_{A}\,,$$
(3.25)
$$\displaystyle R_{0}$$
$$\displaystyle=R-\alpha_{\mbox{\tiny${T\overline{T}}$}}E_{A}+\alpha_{\mbox{%
\tiny${\widetilde{J}T_{1}}$}}{\widetilde{\mathbb{P}}^{1}\over A_{11}}+\alpha_{%
\mbox{\tiny${JT_{1}}$}}(A_{11}{\mathbb{P}}_{1}-\alpha_{\mbox{\tiny${\widetilde%
{J}T_{1}}$}}{\mathbb{P}})\,,$$
$$\displaystyle P_{1}^{0}$$
$$\displaystyle=A_{11}{\mathbb{P}}_{1}-\alpha_{\mbox{\tiny${\widetilde{J}T_{1}}$%
}}{\mathbb{P}}+\alpha_{\mbox{\tiny${\widetilde{J}T_{0}}$}}E_{A}\,,$$
where $R_{0}\equiv P_{-}^{0}=\int_{0}^{R}p_{-}^{0}$ is the circumference of the cylinder, $P_{1}^{0}=\int_{0}^{R}p_{1}^{0}$ is the $U(1)$ charge, and $E_{0}$ is the energy
of the undeformed model which is equal to the energy of the deformed model for physical states.
These equations generalise the integrated form of the homogeneous inviscid Burgers equation which describes the spectrum of a ${T\overline{T}}$ deformed model for states with vanishing world-sheet momentum. For physical states one can set ${\mathbb{P}}=0$ and ${\widetilde{\mathbb{P}}^{1}}=w{\cal R}^{1}-A_{11}\alpha_{\mbox{\tiny${JT_{0}}$%
}}E_{A}$ but we prefer to keep them in (3.25) because they might be important for states which do not satisfy the generalised level-matching conditions. Due to the momentum $P_{1}$ quantisation
$$\displaystyle P_{1}^{0}={2\pi m\over{\cal R}^{1}_{0}}\,,\quad P_{1}={2\pi m%
\over{\cal R}^{1}}\,,\quad m\in{\mathbb{Z}}\,,$$
and the last equation in (3.25) can be considered as a relation between the ranges ${\cal R}_{0}^{1}$ and ${\cal R}^{1}$ of $x^{1}_{0}$ and $x^{1}$.
4 Flow equations in CFT conventions
To analyse the deformations of a CFT it is convenient to use the light-cone world-sheet coordinates
$$\displaystyle\sigma^{\pm}=\sigma\pm\tau\,,\quad\partial_{\pm}={1\over 2}(%
\partial_{\sigma}\pm\partial_{\tau})\,,\quad\partial_{\alpha}J^{\alpha}=%
\partial_{+}J^{+}+\partial_{-}J^{-}\,.$$
The light-cone components of a vector $V^{\alpha}$, a co-vector $U_{\alpha}$, and the stress-energy tensor $T^{\alpha}{}_{\beta}$ are related to the $0,1$ components as
$$\displaystyle V^{\pm}$$
$$\displaystyle=V^{1}\pm V^{0}\,,\quad V^{0}={1\over 2}(V^{+}-V^{-})\,,\quad V^{%
1}={1\over 2}(V^{+}+V^{-})\,,$$
$$\displaystyle U_{\pm}$$
$$\displaystyle={1\over 2}(U_{1}\pm U_{0})\,,\quad U_{0}=U_{+}-U_{-}\,,\quad U_{%
1}=U_{+}+U_{-}\,,$$
$$\displaystyle T^{+}{}_{+}$$
$$\displaystyle={1\over 2}(T^{0}{}_{0}+T^{1}{}_{1}+T^{0}{}_{1}+T^{1}{}_{0})\,,%
\quad T^{-}{}_{-}={1\over 2}(T^{0}{}_{0}+T^{1}{}_{1}-T^{0}{}_{1}-T^{1}{}_{0})\,,$$
$$\displaystyle T^{-}{}_{+}$$
$$\displaystyle={1\over 2}(T^{1}{}_{1}-T^{0}{}_{0}-T^{0}{}_{1}+T^{1}{}_{0})\,,%
\quad T^{+}{}_{-}={1\over 2}(T^{1}{}_{1}-T^{0}{}_{0}+T^{0}{}_{1}-T^{1}{}_{0})\,.$$
In a CFT one has $T^{0}{}_{0}+T^{1}{}_{1}=0$, $T^{0}{}_{1}+T^{1}{}_{0}=0$, and therefore $T^{+}{}_{+}=T^{-}{}_{-}=0$.
In terms of the light-cone components the deformation operators become
$$\displaystyle T\overline{T}$$
$$\displaystyle=-\epsilon_{\alpha\beta}T^{\alpha}{}_{1}T^{\beta}{}_{0}=-2%
\epsilon_{\alpha\beta}T^{\alpha}{}_{-}T^{\beta}{}_{+}=T^{+}{}_{-}T^{-}{}_{+}-T%
^{-}{}_{-}T^{+}{}_{+}\,,$$
$$\displaystyle VT_{1}$$
$$\displaystyle=-\epsilon_{\alpha\beta}V^{\alpha}T^{\beta}{}_{1}={1\over 2}\big{%
(}V^{+}(T^{-}{}_{+}+T^{-}{}_{-})-V^{-}(T^{+}{}_{+}+T^{+}{}_{-})\big{)}\,,$$
$$\displaystyle VT_{0}$$
$$\displaystyle=-\epsilon_{\alpha\beta}V^{\alpha}T^{\beta}{}_{0}={1\over 2}\big{%
(}V^{+}(T^{-}{}_{+}-T^{-}{}_{-})-V^{-}(T^{+}{}_{+}-T^{+}{}_{-})\big{)}\,,$$
$$\displaystyle VT_{\pm}$$
$$\displaystyle=-\epsilon_{\alpha\beta}V^{\alpha}T^{\beta}{}_{\pm}={1\over 2}%
\big{(}V^{+}T^{-}{}_{\pm}-V^{-}T^{+}{}_{\pm}\big{)}\,,$$
where $V^{\alpha}$ could be either $J^{\alpha}$ or $\widetilde{J}^{\alpha}$.
4.1 Flow equations for the Hamiltonian density
Introducing new deformation parameters (and setting $A_{--}=1$)
$$\displaystyle\alpha_{\pm}=\alpha_{\mbox{\tiny${JT_{1}}$}}\pm\alpha_{\mbox{%
\tiny${JT_{0}}$}}\,,\quad\tilde{\alpha}_{\pm}=\alpha_{\mbox{\tiny${\widetilde{%
J}T_{1}}$}}\pm\alpha_{\mbox{\tiny${\widetilde{J}T_{0}}$}}\,,$$
we rewrite the flow equations (3.13) in the form
$$\displaystyle{\partial{\cal H}_{A}\over\partial\alpha_{\pm}}$$
$$\displaystyle=A_{11}{\mathbb{J}}{\mathbb{T}}_{\pm}\pm\,{\tilde{\alpha}_{+}+%
\tilde{\alpha}_{-}\over 4}{\mathbb{T}}\overline{\mathbb{T}}\,,$$
(4.1)
$$\displaystyle{\partial{\cal H}_{A}\over\partial\tilde{\alpha}_{\pm}}$$
$$\displaystyle={1\over A_{11}}\widetilde{\mathbb{J}}{\mathbb{T}}_{\pm}\pm{%
\alpha_{+}+\alpha_{-}\over 4}{\mathbb{T}}\overline{\mathbb{T}}\,.$$
For a generic model (even a CFT) there is no reason to mix the currents $J$ and $\widetilde{J}$.
However, the situation is different if we want to deform a Lorentz-invariant model with the Lagrangian
$$\displaystyle{\cal L}_{0}=-{1\over 2}\partial_{\alpha}x^{\mu}\partial^{\alpha}%
x^{\nu}\,G_{\mu\nu}-V(x)\,,$$
(4.2)
which is invariant under shifts of $x^{1}$, and whose target-space metric components $G_{1\nu}$ are independent of $x$. Then, shifting and rescaling $x^{1}$
$$\displaystyle x^{1}\to{1\over\sqrt{G_{11}}}(x^{1}-G_{1k}x^{k})\,,$$
we (almost) decouple $x^{1}$ from the other fields
$$\displaystyle{\cal L}_{0}\to{\cal L}_{0}=-{1\over 2}\partial_{\alpha}x^{1}%
\partial^{\alpha}x^{1}-{1\over 2}\partial_{\alpha}x^{k}\partial^{\alpha}x^{l}%
\,\tilde{G}_{kl}-V(x)\,.$$
(4.3)
We see that $x^{1}$ is a free massless scalar whose only knowledge of the other fields is in its winding numbers. For such a scalar the topological current $\widetilde{J}$ is dual to $J^{\alpha}=-\partial^{\alpha}x^{1}$
$$\displaystyle\widetilde{J}^{0}=x^{\prime 1}=-J^{1}\,,\quad\widetilde{J}^{1}=-%
\dot{x}^{1}=p_{1}=-J^{0}\,,$$
and introducing
$$\displaystyle{\rm J}_{\pm}^{\alpha}=-{1\over 2}(J^{\alpha}\pm\widetilde{J}^{%
\alpha})\,,\quad{\rm J}_{\pm}^{0}=\mp\partial_{\pm}x^{1}\,,\quad{\rm J}_{\pm}^%
{1}=\partial_{\pm}x^{1}\,,$$
$$\displaystyle{\rm J}_{+}^{-}=2\partial_{+}x^{1}\,,\quad{\rm J}_{+}^{+}=0\,,%
\quad{\rm J}_{-}^{+}=2\partial_{-}x^{1}\,,\quad{\rm J}_{-}^{-}=0\,,$$
we see that the current ${\rm J}^{\alpha}\equiv{\rm J}_{+}^{\alpha}$ is purely left-moving (holomorphic) while $\bar{\rm J}^{\alpha}\equiv{\rm J}_{-}^{\alpha}$ is purely right-moving (anti-holomorphic). Clearly, a generic deformation would ruin these nice properties but one can still hope that these currents ${\rm J}^{\alpha}$ and $\bar{\rm J}^{\alpha}$ would suit better to describe the deformation. In what follows discussing the deformation of a model with Lagrangian (4.2), we first decouple the scalar $x^{1}$, and then deform the model.
Coming back to the flow equations (4.1), we see that in the presence of the $\tilde{J}J$-deformation the currents ${\rm J}^{\alpha}$ and
$\bar{\rm J}^{\alpha}$ should be defined as
$$\displaystyle{\rm J}_{\pm}^{\alpha}=-{1\over 2}({A_{11}}J^{\alpha}\pm{1\over A%
_{11}}\widetilde{J}^{\alpha})\,,\quad\tilde{J}J=-2\epsilon_{\alpha\beta}{\rm J%
}_{+}^{\alpha}{\rm J}_{-}^{\beta}\,.$$
(4.4)
The appearance of $A_{11}$ in this formula is expected because the $\widetilde{J}J$-deformation (2.13) rescales the coordinate $x^{1}$ and momentum $p_{1}$, and the factors of $A_{11}$ just reverse this rescaling.
Now, introducing the deformation parameters
$$\displaystyle\beta_{\pm}=-{1\over 2}(\alpha_{\pm}+\tilde{\alpha}_{\pm})\,,%
\quad\bar{\beta}_{\pm}=-{1\over 2}(\alpha_{\pm}-\tilde{\alpha}_{\pm})\,,$$
we bring the flow equations (4.1) to the form
$$\displaystyle{\partial{\cal H}_{A}\over\partial\beta_{\pm}}=2{\cal J}{\mathbb{%
T}}_{\pm}\pm\,{\beta_{+}+\beta_{-}\over 2}{\mathbb{T}}\overline{\mathbb{T}}\,,$$
(4.5)
$$\displaystyle{\partial{\cal H}_{A}\over\partial\bar{\beta}_{\pm}}=2\bar{\cal J%
}{\mathbb{T}}_{\pm}\mp{\bar{\beta}_{+}+\bar{\beta}_{-}\over 2}{\mathbb{T}}%
\overline{\mathbb{T}}\,.$$
Here ${\cal J}^{\alpha}\equiv{\cal J}^{\alpha}_{+}$ and $\bar{\cal J}\equiv{\cal J}^{\alpha}_{-}$ are the improved currents
$$\displaystyle{\cal J}_{\pm}^{\alpha}=-{1\over 2}({A_{11}}{\mathbb{J}}^{\alpha}%
\pm{1\over A_{11}}\tilde{\mathbb{J}}^{\alpha})\,,$$
(4.6)
and up to a normalisation in the Euclidean CFT terminology $JT_{+}\leftrightarrow J\Theta$, $JT_{-}\leftrightarrow J\overline{T}$,
$\bar{J}T_{+}\leftrightarrow\bar{J}T$, $\bar{J}T_{-}\leftrightarrow\bar{J}\overline{\Theta}$.
A nice feature of these equations is that in the absence of linear deformations ($v^{1}=v=u_{1}=0$) the flow equations for the left- and right-moving currents are coupled to each other only through the ${T\overline{T}}$ operator. Nevertheless, it still may not be the best form of the equations because they all involve the ${T\overline{T}}$ operator. Indeed, we have the flow equation for the ${T\overline{T}}$ deformation, and we can make nonlinear changes of the parameter $a_{+-}$. Performing
the following transformation
$$\displaystyle a_{+-}=\alpha+{\varkappa_{+-}\over 2}\beta_{+}\beta_{-}+{%
\varkappa_{+}\over 4}\beta_{+}^{2}+{\varkappa_{-}\over 4}\beta_{-}^{2}+{\bar{%
\varkappa}_{+-}\over 2}\bar{\beta}_{+}\bar{\beta}_{-}+{\bar{\varkappa}_{+}%
\over 4}\bar{\beta}_{+}^{2}+{\bar{\varkappa}_{-}\over 4}\bar{\beta}_{-}^{2}\,,$$
one finds that the equations (3.12, 4.5) take the form
$$\displaystyle{\partial{\cal H}_{A}\over\partial\alpha}={\mathbb{T}}\overline{%
\mathbb{T}}\,,$$
(4.7)
$$\displaystyle{\partial{\cal H}_{A}\over\partial\beta_{+}}=2{\cal J}{\mathbb{T}%
}_{+}\,+\,{\beta_{+}(1+\varkappa_{+})+\beta_{-}(1+\varkappa_{+-})\over 2}{%
\mathbb{T}}\overline{\mathbb{T}}\,,$$
(4.8)
$$\displaystyle{\partial{\cal H}_{A}\over\partial\beta_{-}}=2{\cal J}{\mathbb{T}%
}_{-}\,-\,{\beta_{+}(1-\varkappa_{+-})+\beta_{-}(1-\varkappa_{-})\over 2}{%
\mathbb{T}}\overline{\mathbb{T}}\,,$$
$$\displaystyle{\partial{\cal H}_{A}\over\partial\bar{\beta}_{+}}=2\bar{\cal J}{%
\mathbb{T}}_{+}-{\bar{\beta}_{+}(1-\bar{\varkappa}_{+})+\bar{\beta}_{-}(1-\bar%
{\varkappa}_{+-})\over 2}{\mathbb{T}}\overline{\mathbb{T}}\,,$$
$$\displaystyle{\partial{\cal H}_{A}\over\partial\bar{\beta}_{-}}=2\bar{\cal J}{%
\mathbb{T}}_{-}+{\bar{\beta}_{+}(1+\bar{\varkappa}_{+-})+\bar{\beta}_{-}(1+%
\bar{\varkappa}_{-})\over 2}{\mathbb{T}}\overline{\mathbb{T}}\,.$$
It is clear that natural values of the parameters $\varkappa$’s are $0,\pm 1$. A particular choice of $\varkappa$’s depends on the model under study. If the model does not have left- and right-moving conserved currents then the original choice with all $\varkappa$’s equal to 0 seems to be the best one.
However, if a CFT has left- and right-moving currents then there are better choices.
Since in such a CFT $JT_{+}=0$ and $\bar{J}T_{-}=0$,
it seems reasonable to choose $\varkappa$’s so that the flow equations for $JT_{-}$ and $\bar{J}T_{+}$ would take the simplest form.
Thus, one would choose $\varkappa_{-}=1$, $\varkappa_{+-}=1$, $\bar{\varkappa}_{+}=1$, $\bar{\varkappa}_{+-}=1$. Then, the remaining two flow equations take the simplest form if
one chooses $\varkappa_{+}=-1$ and $\bar{\varkappa}_{-}=-1$. With this choice
the flow equations with respect to $\beta_{\pm}$ and $\bar{\beta}_{\pm}$ are
$$\displaystyle\varkappa_{+}=-1\,,\quad\varkappa_{-}=1\,,\quad\bar{\varkappa}_{+%
}=1\,,\quad\bar{\varkappa}_{-}=-1\,,\quad\varkappa_{+-}=1\,,\quad\bar{%
\varkappa}_{+-}=1\,,$$
(4.9)
$$\displaystyle{\partial{\cal H}_{A}\over\partial\beta_{-}}$$
$$\displaystyle=2{\cal J}{\mathbb{T}}_{-}\,,\qquad{\partial{\cal H}_{A}\over%
\partial\beta_{+}}=2{\cal J}{\mathbb{T}}_{+}\,+\,\beta_{-}{\mathbb{T}}%
\overline{\mathbb{T}}\,,$$
(4.10)
$$\displaystyle{\partial{\cal H}_{A}\over\partial\bar{\beta}_{+}}$$
$$\displaystyle=2\bar{\cal J}{\mathbb{T}}_{+}\,,\qquad{\partial{\cal H}_{A}\over%
\partial\bar{\beta}_{-}}=2\bar{\cal J}{\mathbb{T}}_{-}\,+\,\bar{\beta}_{+}{%
\mathbb{T}}\overline{\mathbb{T}}\,.$$
Calculating the coefficients ${\cal G}_{i}$ with this choice of $\varkappa$’s for a deformed CFT which had left- and right-moving conserved currents before the deformation, see appendix C, one finds that they are independent of $\beta_{+}$ and $\bar{\beta}_{-}$ (in fact with all the ten parameters switched on), and, therefore, the Hamiltonian density ${\cal H}_{A}$ is independent of these two parameters too. Thus, the flow equations with respect to $\beta_{+}$ and $\bar{\beta}_{-}$ lead to the existence of the following two relations
$$\displaystyle 2{\cal J}{\mathbb{T}}_{+}\,+\,\beta_{-}{\mathbb{T}}\overline{%
\mathbb{T}}$$
$$\displaystyle=0\,,$$
(4.11)
$$\displaystyle 2\bar{\cal J}{\mathbb{T}}_{-}+\bar{\beta}_{+}{\mathbb{T}}%
\overline{\mathbb{T}}$$
$$\displaystyle=0\,,$$
in a deformed CFT. One might think that these relations imply the existence of improved left- and right-moving currents but it is so only in the case of a single $J\overline{T}$ or $\bar{J}T$ deformation.
4.2 Flow equations for the energy in CFT conventions
To write the flow equations for the energy in CFT conventions we introduce the charges ${\rm Q}_{\pm}$ and ${\cal Q}_{\pm}$ of the left- and right-moving currents (4.4) and (4.6)
$$\displaystyle{{\rm Q}_{\pm}\over R}=\langle n|{\rm J}_{\pm}^{0}|n\rangle=-{1%
\over 2R}(A_{11}P_{1}\pm{1\over A_{11}}\widetilde{P}^{1})\,,$$
and similar expressions for the improved charges ${\cal Q}_{\pm}$ of the currents (4.6).
Then, we get
$$\displaystyle{\partial E_{n}\over\partial{\rm Q}_{\pm}}={\partial E_{n}\over%
\partial{\cal Q}_{\pm}}=-\big{(}{1\over A_{11}}{\partial E_{n}\over\partial{%
\mathbb{P}}_{1}}\pm A_{11}{\partial E_{n}\over\partial\widetilde{\mathbb{P}}^{%
1}}\big{)}\,,$$
and therefore
$$\displaystyle\langle n|{\rm J}_{\pm}^{1}|n\rangle=\langle n|{\cal J}_{\pm}^{1}%
|n\rangle={1\over 2}\big{(}A_{11}{\partial E_{n}\over\partial\widetilde{P}^{1}%
}\pm{1\over A_{11}}{\partial E_{n}\over\partial P_{1}}\big{)}=\mp{1\over 2}{%
\partial E_{n}\over\partial{\cal Q}_{\pm}}\,.$$
We also need
$$\displaystyle\langle n|{\mathbb{T}}^{1}{}_{\pm}|n\rangle={1\over 2}\big{(}%
\langle n|{\mathbb{T}}^{1}{}_{1}|n\rangle\pm\langle n|{\mathbb{T}}^{1}{}_{0}|n%
\rangle\big{)}={1\over 2}\big{(}{\partial E_{n}\over\partial R}\pm{1\over R}(1%
\mp v){\partial E_{n}\over\partial v}\big{)}\,,$$
$$\displaystyle\langle n|{\mathbb{T}}^{0}{}_{\pm}|n\rangle={1\over 2}\big{(}%
\langle n|{\mathbb{T}}^{0}{}_{1}|n\rangle\pm\langle n|{\mathbb{T}}^{0}{}_{0}|n%
\rangle\big{)}={1\over 2R}\big{(}-\check{P}+{{\cal Q}_{+}^{2}-{\cal Q}_{-}^{2}%
\over R}\pm(1\mp v)E_{n}\big{)}\,.$$
By using these relations and eqs.(4.10), one gets the following flow equations for the energy of the state $|n\rangle$ (with $A_{--}=1$)
$$\displaystyle A_{11}{\partial E_{n}\over\partial A_{11}}$$
$$\displaystyle={\cal Q}_{+}{\partial E_{n}\over\partial{\cal Q}_{-}}+{\cal Q}_{%
-}{\partial E_{n}\over\partial{\cal Q}_{+}}\,,$$
(4.12)
$$\displaystyle{\partial E_{n}\over\partial\alpha}$$
$$\displaystyle=-E_{n}{\partial E_{n}\over\partial R}-{\check{P}-{{\cal Q}_{+}^{%
2}-{\cal Q}_{-}^{2}\over R}\over R}{\partial E_{n}\over\partial v}\,,$$
(4.13)
$$\displaystyle{\partial E_{n}\over\partial\beta_{-}}$$
$$\displaystyle={{\cal Q}_{+}}\big{(}{\partial E_{n}\over\partial R}-{1\over R}(%
1+v){\partial E_{n}\over\partial v}\big{)}-{1\over 2}{\partial E_{n}\over%
\partial{\cal Q}_{+}}\big{(}\check{P}-{{\cal Q}_{+}^{2}-{\cal Q}_{-}^{2}\over R%
}+(1+v)E_{n}\big{)}\,,$$
$$\displaystyle{\partial E_{n}\over\partial\bar{\beta}_{+}}$$
$$\displaystyle={{\cal Q}_{-}}\big{(}{\partial E_{n}\over\partial R}+{1\over R}(%
1-v){\partial E_{n}\over\partial v}\big{)}+{1\over 2}{\partial E_{n}\over%
\partial{\cal Q}_{-}}\big{(}\check{P}-{{\cal Q}_{+}^{2}-{\cal Q}_{-}^{2}\over R%
}-(1-v)E_{n}\big{)}\,,$$
$$\displaystyle{\partial E_{n}\over\partial\beta_{+}}$$
$$\displaystyle={{\cal Q}_{+}}\big{(}{\partial E_{n}\over\partial R}+{1\over R}(%
1-v){\partial E_{n}\over\partial v}\big{)}-{1\over 2}{\partial E_{n}\over%
\partial{\cal Q}_{+}}\big{(}\check{P}-{{\cal Q}_{+}^{2}-{\cal Q}_{-}^{2}\over R%
}-(1-v)E_{n}\big{)}+{\beta_{-}}{\partial E_{n}\over\partial\alpha}\,,$$
$$\displaystyle{\partial E_{n}\over\partial\bar{\beta}_{-}}$$
$$\displaystyle={{\cal Q}_{-}}\big{(}{\partial E_{n}\over\partial R}-{1\over R}(%
1+v){\partial E_{n}\over\partial v}\big{)}+{1\over 2}{\partial E_{n}\over%
\partial{\cal Q}_{-}}\big{(}\check{P}-{{\cal Q}_{+}^{2}-{\cal Q}_{-}^{2}\over R%
}+(1+v)E_{n}\big{)}+{\bar{\beta}_{+}}{\partial E_{n}\over\partial\alpha}\,.$$
As was discussed in the previous subsection, in the case of a CFT with left- and right-moving conserved currents ${\partial E_{n}\over\partial\beta_{+}}={\partial E_{n}\over\partial\bar{\beta}%
_{-}}=0$.
4.3 Deformed CFT with left- and right-moving currents
In this subsection we use the defining equation (3.3) for the Hamiltonian density to find a solution of the flow equations (4.13) for a deformed CFT with left- and right-moving conserved currents. The undeformed Lagrangian is given by (4.2), and before deforming the model we shift and rescale $x^{1}$ as discussed in section 4.1 to reduce (4.2) to the canonical form (4.3) with the undeformed Hamiltonian density
$$\displaystyle{\cal H}_{0}={1\over 2}p_{1}^{2}+{1\over 2}(x^{\prime 1})^{2}+{1%
\over 2}G^{kl}(x)p_{k}p_{l}+{1\over 2}x^{\prime k}x^{\prime l}\,G_{kl}(x)%
\equiv{\cal K}\,.$$
(4.14)
We set $A_{11}=1$ because the effect of the $\tilde{J}J$ deformation was discussed in section
3.3, and the energy of a $\tilde{J}J$ deformed model is related to the energy of the model with $A_{11}=1$ by eq.(3.19). To simplify the notations we also set the parameters $v^{1}$, $u_{1}$ to zero. The dependence of these parameters can be easily restored by shifting $p_{1}$ and $x^{\prime 1}$ in the Hamiltonian density, or by replacing the charges ${\rm Q}_{\pm}$ with the improved charges ${\cal Q}_{\pm}$ in the energy eigenvalues.
Let us assume that ${\cal H}_{A}$ and the coefficients ${\cal G}_{i}$ of the defining equation (3.3) are operators, and let us rewrite (3.3) in the following form
$$\displaystyle\langle n|\,{\cal G}_{2}\,{\cal H}_{A}^{2}-{\cal G}_{1}\,{\cal H}%
_{A}+{\cal G}_{0}\,|n\rangle=0\,\,.$$
(4.15)
Clearly, this equation does not really make sense, because the products of operators require careful definitions. In particular, ${\cal G}_{1}$ given by (C.9) depends on $px^{\prime}$, $p_{1}$ and $x^{\prime 1}$ which do not commute with ${\cal H}_{A}$. This can be cured because ${\cal G}_{2}$ does not depend on any fields, and therefore, shifting ${\cal H}_{A}$ by a linear combination of $px^{\prime}$, $p_{1}$ and $x^{\prime 1}$, one can remove all field-dependent terms from ${\cal G}_{1}$.
Performing this shift, we bring eq.(4.15) to the form111This is also the simplest form to check (C.12).
$$\displaystyle{\cal G}_{2}\,\langle n|({\cal H}_{A}-{\mathscr{S}}_{A})^{2}|n%
\rangle-\widetilde{\cal G}_{1}\,\langle n|({\cal H}_{A}-{\mathscr{S}}_{A})|n%
\rangle+\langle n|\widetilde{\cal G}_{0}\,|n\rangle=0\,\,,$$
where
$$\displaystyle{\cal G}_{2}$$
$$\displaystyle=\alpha(1-v^{2})+{1\over 2}\beta_{-}^{2}(1+v)^{2}+{1\over 2}\bar{%
\beta}_{+}^{2}(1-v)^{2}\,,$$
$$\displaystyle\widetilde{\cal G}_{1}$$
$$\displaystyle=1-v^{2}\,,$$
$$\displaystyle\widetilde{\cal G}_{0}={\cal K}+$$
$$\displaystyle{1\over{\cal G}_{2}}\Big{[}\frac{1}{2}px^{\prime}\left(\bar{\beta%
}_{+}^{2}(1-v)^{2}-\beta_{-}^{2}(1+v)^{2}\right)-(1-v^{2})\big{(}\beta_{-}{\rm
J%
}_{+}^{0}(1+v)+\bar{\beta}_{+}{\rm J}_{-}^{0}(1-v)\big{)}$$
$$\displaystyle+(px^{\prime})^{2}\big{(}\bar{\beta}_{+}^{2}\beta_{-}^{2}-\alpha^%
{2}\big{)}+2{\rm J}_{+}^{0}px^{\prime}\big{(}\alpha\beta_{-}(1+v)+\bar{\beta}_%
{+}^{2}\beta_{-}(1-v)\big{)}$$
$$\displaystyle-2{\rm J}_{-}^{0}px^{\prime}\big{(}\alpha\bar{\beta}_{+}(1-v)+%
\bar{\beta}_{+}\beta_{-}^{2}(1+v)\big{)}-\big{(}\beta_{-}{\rm J}_{+}^{0}(1+v)+%
\bar{\beta}_{+}{\rm J}_{-}^{0}(1-v)\big{)}^{2}\Big{]}\,,$$
$$\displaystyle{\mathscr{S}}_{A}$$
$$\displaystyle={1\over{\cal G}_{2}}\Big{(}\frac{1}{2}px^{\prime}\left(-2\alpha v%
-\bar{\beta}_{+}^{2}(1-v)+\beta_{-}^{2}(1+v)\right)+\beta_{-}{\rm J}_{+}^{0}(1%
+v)+\bar{\beta}_{+}{\rm J}_{-}^{0}(1-v)\Big{)}\,.$$
Now, to get an equation for the spectrum of the deformed CFT with left- and right-moving conserved currents we use the following relations
$$\displaystyle\langle n|{\cal H}_{A}|n\rangle={E_{n}\over R}\,,\quad\langle n|{%
\rm J}_{\pm}^{0}|n\rangle={{\rm Q}_{\pm}\over R}\,,$$
(4.16)
$$\displaystyle\langle n|px^{\prime}|n\rangle=-{P\over R}=-{\check{k}-{\rm Q}_{+%
}^{2}+{\rm Q}_{-}^{2}\over R^{2}}\,,\quad\langle n|{\cal K}|n\rangle={E_{n}^{(%
0)}\over R}={\check{{\cal E}}_{n}^{(0)}+{\rm Q}_{+}^{2}+{\rm Q}_{-}^{2}\over R%
^{2}}\,.$$
(4.17)
Here, as was discussed in section 3.3, we
split of the world-sheet momentum $P=k/R$ into its intrinsic part $\check{P}=\check{k}/R$, and the part due to the rotation and winding in the $x^{1}$ direction, and do the same with the eigenvalues $E_{n}^{(0)}={\cal E}_{n}^{(0)}/R$ of the Hamiltonian (4.14). It is worthwhile to mention that only $\check{{\cal E}}_{n}^{(0)}$ always coincides with the corresponding part of the spectrum of the undeformed model. As to ${\rm Q}_{\pm}$ dependent part, it depends on the point of view on the deformation. One may say that the range of $x^{1}$ does not change with the deformation and then ${\rm Q}_{\pm}$ do not change either, or one may use that the deformation was generated by the chain of transformations (2.14) which explicitly changed the range of $x^{1}$ and, therefore, ${\rm Q}_{\pm}$. In any case ${\rm Q}_{\pm}$ are the charges measured in the deformed theory.
Finally, we use the following replacement rule
$$\displaystyle\langle n|K_{1}K_{2}|n\rangle\to\langle n|K_{1}|n\rangle\langle n%
|K_{2}|n\rangle\,,$$
for products of any two operators in (4.16, 4.17). The only justification for this rule we have is that it leads to the deformed eigenvalues $E_{n}$ which satisfy the flow equations (4.13).
By using these rules we find that the spectrum of the deformed CFT with left- and right-moving conserved currents is given by
$$\displaystyle{\cal A}\big{(}{E_{n}\over R}-{\cal S}_{E}\big{)}^{2}-\big{(}{E_{%
n}\over R}-{\cal S}_{E}\big{)}+{\cal C}=0\,,$$
(4.18)
$$\displaystyle{E_{n}\over R}={\cal S}_{E}+{1-\sqrt{1-4{\cal A}{\cal C}}\over 2{%
\cal A}}\,.$$
Here
$$\displaystyle{\cal A}$$
$$\displaystyle=\alpha+{b_{-}^{2}\over 2}+{\bar{b}_{+}^{2}\over 2}\,,$$
(4.19)
$$\displaystyle{\cal A}{\cal C}$$
$$\displaystyle={\cal A}\,{{\cal E}_{n}^{(0)}\over{\cal R}^{2}}+\frac{k}{2{\cal R%
}^{2}}\left(b_{-}^{2}-\bar{b}_{+}^{2}\right)-\frac{1}{{\cal R}}\big{(}b_{-}{%
\rm Q}_{+}+\bar{b}_{+}{\rm Q}_{-}\big{)}+\frac{k^{2}}{{\cal R}^{4}}\big{(}\,{%
\bar{b}_{+}^{2}b_{-}^{2}}-\alpha^{2}\big{)}$$
$$\displaystyle-2{k\,{\rm Q}_{+}\over{\cal R}^{3}}b_{-}\big{(}\alpha+{\bar{b}_{+%
}^{2}}\big{)}+2{k\,{\rm Q}_{-}\over{\cal R}^{3}}\bar{b}_{+}\big{(}\alpha+{b_{-%
}^{2}}\big{)}-\frac{1}{{\cal R}^{2}}\big{(}b_{-}{\rm Q}_{+}+\bar{b}_{+}{\rm Q}%
_{-}\big{)}^{2}\,,$$
$$\displaystyle{\cal S}_{E}$$
$$\displaystyle={k\,v\over{\cal R}^{2}}+{1\over{\cal A}}\big{(}\frac{k}{2{\cal R%
}^{2}}\big{(}\bar{b}_{+}^{2}-b_{-}^{2}\big{)}+\frac{1}{{\cal R}}b_{-}{\rm Q}_{%
+}+\frac{1}{{\cal R}}\bar{b}_{+}{\rm Q}_{-}\big{)}\,.$$
where
$$\displaystyle b_{-}=\beta_{-}\,\sqrt{1+v\over 1-v}\,,\quad\bar{b}_{+}=\bar{%
\beta}_{+}\,\sqrt{1-v\over 1+v}\,,\quad{\cal R}=R\,\sqrt{1-v^{2}}\,.$$
(4.20)
We see that all the dependence of $v$ is absorbed in the rescaling of $\beta_{-}$, $\bar{\beta}_{+}$ and $R$, and a shift of the energy by $Pv/(1-v^{2})$. This is in agreement with the consideration in appendix C.
Our solution (4.18, 4.19)
seems to agree with the one proposed in [7]
at least if one switches off their parameters of the deformation by ${\rm J}_{\pm}^{1}$, and parameters of the ${\rm J}\Theta$ and $\bar{\rm J}\overline{\Theta}$ deformations. We see in particular that the parameter dependence of our coefficient ${\cal A}$ and their coefficient $A$ is similar. To check a precise agreement it
would be necessary to match the conventions, and it also
might be necessary to perform an extra redefinition of the parameter of the ${T\overline{T}}$ deformation. We have not tried to do it.
Then, at $v=0$ our solution is very similar to the one proposed in [6] for closely related single-trace deformations by using dual strings on deformed $AdS_{3}$. In the case of ${{\rm J}\overline{T}}$ deformation our formula agrees with theirs. Still, there are some differences. The easiest one to notice is the parameter dependence of our coefficient ${\cal A}$ and their coefficient $A$. To reproduce their dependence we would have to shift $\alpha$ by a term proportional to $\beta_{-}\bar{\beta}_{+}$. Such a shift would ruin the nice structure of the flow equations (4.10), and introduce ${T\overline{T}}$ terms on the r.h.s. of these equations. It is also unclear how their charges are related to ours. By these reasons we have not manage to find an exact map between our and their formulae but we believe they may agree. In appendix D we presents the spectrum deformed by one or two of the operators ${T\overline{T}}$ , ${{\rm J}\overline{T}}$ , and ${\bar{{\rm J}}T}$.
5 Conclusions
In this paper the light-cone gauge approach [8, 9] to the ${T\overline{T}}$ deformed two-dimensional models has been extended to include
the most general ten-parameter deformation by U(1) conserved currents, stress-energy tensor and their various quadratic combinations.
We have found an explicit ten-parameter deformed Hamiltonian for a rather general system of scalars with an arbitrary potential, and used it to derive the flow equations with respect to the parameters, and the spectrum of a deformed CFT with left- and right-moving conserved currents.
There are many open questions and generalisations of the approach. We mention just a few of them.
Clearly, the light-cone gauge approach can also be used to study multi-parameter deformations of the system of scalars, fermions and chiral bosons introduced in [9]. Since these deformations breaks Lorentz symmetry it is natural to analyse non-Lorentz invariant models such as the nonlinear matrix Schrödinger model.
The system of flow equations for the spectrum of a deformed model is complicated, and admits many solutions. We could not find its solution which would be completely determined by the spectrum of the undeformed model. It would be interesting to understand if such a solution exists for a generic model.
Even in the case of a CFT but without left- and right-moving conserved currents it is unclear how to find a unique solution.
If a model is integrable then there is a generalisation of the ${T\overline{T}}$ deformation to more general higher-spin deformations introduced in [2].
It is expected that they can be described in the light-cone gauge approach by coupling strings to W gravity, see [19] for a review.
Higher-spin conserved currents can be also used to study ${{\rm J}\overline{T}}$ type deformations which break Lorentz invariance of an undeformed model, see [20, 21] for a recent discussion. Clearly, as soon as the higher-spin deformations of [2] are understood in the light-cone gauge approach, one can consider multi-parameter deformations involving the higher-spin currents.
Acknowledgements
I would like to thank Tristan McLoughlin for useful discussions.
Appendix A Canonical transformation matrices
The matrices $A_{x}$, $B_{x}$, $A_{p}$ and $B_{p}$ appearing in the canonical transformation (2.15) are
$$\displaystyle A_{x}=\left(\begin{array}[]{ccccccc}1&-\frac{\tilde{a}_{+-}}{A_{%
--}}&\frac{c_{1-}\tilde{a}_{+-}-a_{+1}}{A_{11}}&0&-c_{1-}b_{+1}&-b_{+1}A_{11}&%
0\\
0&\frac{1}{A_{--}}&-\frac{c_{1-}}{A_{11}}&0&0&0&0\\
0&-\frac{a_{1-}}{A_{--}}&\frac{1+c_{1-}a_{1-}}{A_{11}}&0&0&0&0\\
0&0&0&1&0&0&0\\
0&-\frac{b_{--}}{c_{1-}A_{--}}&\frac{b_{--}}{A_{11}}&0&1&0&0\\
0&0&0&0&0&A_{11}&0\\
0&0&\frac{1}{A_{11}}&0&0&0&\frac{1}{A_{11}}\\
\end{array}\right)\,,$$
$$\displaystyle B_{x}=\left(\begin{array}[]{ccccccc}a_{+1}b_{+1}&0&0&-1&\tilde{a%
}_{+-}&\frac{a_{+1}-c_{1-}\tilde{a}_{+-}}{A_{11}}&0\\
0&0&0&0&-1&\frac{c_{1-}}{A_{11}}&0\\
-b_{+1}&0&0&0&a_{1-}&-\frac{1+c_{1-}a_{1-}}{A_{11}}&0\\
1&0&0&0&0&0&0\\
0&A_{--}&0&0&\frac{b_{--}}{c_{1-}}&-\frac{b_{--}}{A_{11}}&0\\
0&0&A_{11}&0&0&0&-A_{11}\\
0&0&0&0&0&0&0\\
\end{array}\right)\,,$$
$$\displaystyle A_{p}=\left(\begin{array}[]{ccccccc}1&a_{+-}&a_{+1}&0&0&b_{+1}&0%
\\
0&A_{--}&0&0&0&0&0\\
0&a_{1-}A_{11}&A_{11}&0&0&0&0\\
0&0&0&1&0&0&0\\
0&\frac{b_{--}}{c_{1-}}&0&0&1&0&0\\
0&-\frac{b_{--}}{A_{11}}&0&0&-\frac{c_{1-}}{A_{11}}&\frac{1}{A_{11}}&0\\
0&-a_{1-}A_{11}&-A_{11}&0&0&0&A_{11}\\
\end{array}\right)\,,$$
$$\displaystyle B_{p}=\left(\begin{array}[]{ccccccc}0&0&0&0&0&0&0\\
0&0&0&0&0&0&0\\
0&\frac{b_{--}}{A_{11}}&0&0&\frac{c_{1-}}{A_{11}}&0&0\\
0&0&0&0&0&0&0\\
0&-c_{1-}a_{1-}&-c_{1-}&0&0&0&0\\
0&0&0&0&0&0&0\\
0&0&0&0&0&0&0\\
\end{array}\right)\,,$$
$$\displaystyle C_{x}=\left(\begin{array}[]{c}0\\
0\\
0\\
0\\
0\\
-c_{1-}\\
0\\
\end{array}\right)\,,\quad C_{p}=\left(\begin{array}[]{c}0\\
-c_{1-}a_{1-}\\
-c_{1-}\\
0\\
0\\
0\\
0\\
\end{array}\right)$$
where
$$\displaystyle\tilde{a}_{+-}={a_{+-}-a_{+1}a_{1-}}\,.$$
Appendix B Virasoro constraints coefficients
The coefficients $\tilde{G}^{rs}$ and so on appearing in the transformed Virasoro constraint $C_{2}$
(3.1)
are given by
$$\displaystyle\tilde{G}^{rs}$$
$$\displaystyle=G^{ab}(A_{p})^{r}{}_{\hat{a}}(A_{p})^{s}{}_{\hat{b}}+G_{ab}(B_{x%
})^{{\hat{a}}r}(B_{x})^{{\hat{b}}s}\,,$$
$$\displaystyle\tilde{G}_{rs}$$
$$\displaystyle=G_{ab}(A_{x})^{\hat{a}}{}_{r}(A_{x})^{\hat{b}}{}_{s}+G^{ab}(B_{p%
})_{r{\hat{a}}}(B_{p})_{s{\hat{b}}}\,,$$
$$\displaystyle\hat{G}^{r}{}_{s}$$
$$\displaystyle=G^{ab}(A_{p})^{r}{}_{\hat{a}}(B_{p})_{s{\hat{b}}}+G_{ab}(B_{x})^%
{{\hat{a}}r}(A_{x})^{\hat{b}}{}_{s}\,,$$
$$\displaystyle\tilde{G}^{rk}$$
$$\displaystyle=G^{ak}(A_{p})^{r}{}_{\hat{a}}\,,\quad\tilde{G}_{rk}=G_{ak}(A_{x}%
)^{\hat{a}}{}_{r}\,,$$
$$\displaystyle\check{G}^{r}{}_{k}$$
$$\displaystyle=G_{ak}(B_{x})^{{\hat{a}}r}\,,\quad\check{G}_{r}{}^{k}=G^{ak}(B_{%
p})_{r{\hat{a}}}\,.$$
Here the indices $a,b=+,-,1$ are in one-to-one correspondence with ${\hat{a}},{\hat{b}}=\hat{1},\hat{2},\hat{3}$, and $r,s=\hat{1},\ldots,\hat{7}$.
The solution (3.2) to the constraints equations $\tilde{C}_{1}=0$, $C_{a}^{V}=0$ and $C^{1}_{\cal U}=0$, and the light-cone gauge conditions $x^{+}=\tau$, $p_{-}=1$, $v^{-}=0$ is given by
$$\displaystyle{\cal X}^{\prime\hat{2}}$$
$$\displaystyle=x^{\prime-}=v^{+}p_{+}-\widetilde{px}^{\prime}+{c_{1-}\over A_{1%
1}}(x^{\prime 1}-v^{1})\,,$$
(B.1)
$$\displaystyle{\cal X}^{\prime\hat{4}}$$
$$\displaystyle=\tilde{X}^{\prime}_{+}=-p_{+}\,,$$
$$\displaystyle{\cal X}^{\prime\hat{5}}$$
$$\displaystyle=\tilde{X}^{\prime}_{-}=\frac{b_{--}}{c_{1-}A_{--}}(v^{+}p_{+}-%
\widetilde{px}^{\prime})-A_{--}+({1\over A_{--}}-1){b_{--}\over A_{11}}(x^{%
\prime 1}-v^{1})\,,$$
$$\displaystyle{\cal X}^{\prime\hat{6}}$$
$$\displaystyle=\tilde{X}^{\prime}_{1}=-p_{1}+u_{1}+{c_{1-}\over A_{11}}\,,$$
$$\displaystyle{\cal X}^{\prime\hat{7}}$$
$$\displaystyle=\Upsilon^{\prime 1}=-x^{\prime 1}\,,$$
where
$$\displaystyle\widetilde{px}^{\prime}=(p_{1}-u_{1})(x^{\prime 1}-v^{1})+p_{k}x^%
{\prime k}\,.$$
The coefficients ${\cal G}_{i}$ appearing in (3.3)
are given by
$$\displaystyle{\cal G}_{2}$$
$$\displaystyle={1\over 2}\tilde{G}^{++}+{1\over 2}\varkappa^{r}\varkappa^{s}%
\tilde{G}_{rs}+\hat{G}^{+}{}_{s}\varkappa^{s}\,,$$
$$\displaystyle{\cal G}_{1}$$
$$\displaystyle=\tilde{G}^{+s^{\prime}}{\cal P}_{s^{\prime}}+{1\over 2}\varkappa%
^{r}\chi^{s}\tilde{G}_{rs}+\tilde{G}^{r^{\prime}k}{\cal P}_{r^{\prime}}p_{k}+%
\varkappa^{r}x^{\prime k}\tilde{G}_{rk}$$
$$\displaystyle+\hat{G}^{+}{}_{s}\chi^{s}+\hat{G}^{r^{\prime}}{}_{s}{\cal P}_{r^%
{\prime}}\varkappa^{s}+\check{G}^{+}{}_{k}x^{\prime k}+\varkappa^{r}p_{k}%
\check{G}_{r}{}^{k}\,.$$
$$\displaystyle{\cal G}_{0}$$
$$\displaystyle={1\over 2}\tilde{G}^{r^{\prime}s^{\prime}}{\cal P}_{r^{\prime}}{%
\cal P}_{s^{\prime}}+{1\over 2}\chi^{r}\chi^{s}\tilde{G}_{rs}+\tilde{G}^{r^{%
\prime}k}{\cal P}_{r^{\prime}}p_{k}+\chi^{r}x^{\prime k}\tilde{G}_{rk}$$
$$\displaystyle+\hat{G}^{r^{\prime}}{}_{s}{\cal P}_{r^{\prime}}\chi^{s}+\check{G%
}^{r^{\prime}}{}_{k}{\cal P}_{r^{\prime}}x^{\prime k}+\chi^{r}p_{k}\check{G}_{%
r}{}^{k}+\tilde{\cal H}_{x}\,,$$
where $k=2,\ldots,n$ are the indices of the remaining transversal coordinates and momenta, and $r^{\prime},s^{\prime}=\hat{2},\ldots,\hat{7}$.
Appendix C Examples of deformed Hamiltonians
In this appendix we consider deformations of a model with the Hamiltonian density
$$\displaystyle{\cal H}_{0}={1\over 2}G^{\mu\nu}(x)p_{\mu}p_{\nu}+{1\over 2}x^{%
\prime\mu}x^{\prime\nu}\,G_{\mu\nu}(x)+V(x)={\cal K}+V\,,$$
(C.1)
where ${\cal K}$ is the kinetic energy, and $G_{\mu\nu}$ and $V$ do not depend on $x^{1}$. The target space metric of the string sigma model is given by (2.11). Models of this type include also models with rotational symmetries where one should first use spherical coordinates to bring the model’s Hamiltonian to the form (C.1).
In all considered examples we switch off the deformations by $J^{1}$, $\widetilde{J}^{1}$, $T^{1}{}_{1}$ and $\widetilde{J}J$, that is we set the parameters $v^{1}$, $u_{1}$ to zero, and $A_{--}$, $A_{11}$ to one. The dependence of these parameters can be easily restored if desired. We use the notations (3.5)
$$\displaystyle\alpha_{\mbox{\tiny${T\overline{T}}$}}\equiv a_{+-}\,,\quad\alpha%
_{\mbox{\tiny${\widetilde{J}T_{0}}$}}\equiv-a_{+1}\,,\quad\alpha_{\mbox{\tiny$%
{JT_{1}}$}}\equiv a_{1-}\,,\quad\alpha_{\mbox{\tiny${JT_{0}}$}}\equiv b_{+1}\,%
,\quad\alpha_{\mbox{\tiny${\widetilde{J}T_{1}}$}}\equiv b_{--}\,,\quad v\equiv%
-v^{+}\,.$$
Deformation by ${T\overline{T}}$ and $T^{1}{}_{0}$
In this case the only nonvanishing parameters are $\alpha\equiv\alpha_{\mbox{\tiny${T\overline{T}}$}}$ and $v\equiv-v^{+}$. Calculating the coefficients ${\cal G}_{i}$, one gets
$$\displaystyle{\cal G}_{2}$$
$$\displaystyle=\tilde{\alpha}(1-v^{2})\,,\quad\tilde{\alpha}=\alpha(1+\alpha V)\,,$$
(C.2)
$$\displaystyle{\cal G}_{1}$$
$$\displaystyle=(1-v^{2})(1+2\alpha V)-2v\,\tilde{\alpha}\,px^{\prime}\,,\quad px%
^{\prime}\equiv p_{\mu}x^{\prime\mu}\,,$$
$$\displaystyle{\cal G}_{0}$$
$$\displaystyle={\cal K}+V(1-v^{2})-\tilde{\alpha}(px^{\prime})^{2}-v(1+2\alpha V%
)px^{\prime}\,.$$
Solving the defining equation (3.3), one gets the deformed Hamiltonian density
$$\displaystyle{\cal H}_{\alpha,v}={1\over 2\tilde{\alpha}}\left(1-\sqrt{1-{4%
\tilde{\alpha}{\cal K}\over 1-v^{2}}+\frac{4\tilde{\alpha}^{2}(px^{\prime})^{2%
}}{\left(1-v^{2}\right)^{2}}}\,\right)+{V\over 1+\alpha V}-\frac{v\,px^{\prime%
}}{1-v^{2}}\,.$$
(C.3)
The 2-parameter deformed model is in fact related to the ${T\overline{T}}$ deformed one. Indeed, if one performs the following rescaling of the momenta, and the world-sheet space coordinate $\sigma$
$$\displaystyle p_{\mu}\to\sqrt{1-v^{2}}\,p_{\mu}\,,\quad\sigma\to{\sigma\over%
\sqrt{1-v^{2}}}\,,$$
(C.4)
then the deformed action (3.4) transforms as
$$\displaystyle S_{A}\to S_{A}=\int_{0}^{R\sqrt{1-v^{2}}}\,{\rm d}\sigma{\rm d}%
\tau\,\big{(}p_{\mu}\dot{x}^{\mu}\,-\,{1\over\sqrt{1-v^{2}}}({\cal H}_{\alpha,%
0}-v\,px^{\prime})\big{)}\,.$$
(C.5)
The last term in the formula is just proportional to the world-sheet momentum $P$, and, therefore, the energy $E_{\alpha,v}(R,P_{v})$ of a state with momentum $P_{v}$ in the 2-parameter deformed model on a circle of circumference $R$ is related to the energy $E_{\alpha,0}(R\sqrt{1-v^{2}}\,,P_{0})$ of the corresponding state with momentum $P_{0}$ in the ${T\overline{T}}$ deformed model on a circle of circumference $R\sqrt{1-v^{2}}$ as
$$\displaystyle E_{\alpha,v}(R,P_{v})={1\over\sqrt{1-v^{2}}}(E_{\alpha,0}(R\sqrt%
{1-v^{2}}\,,P_{0})+v\,P_{0})\,.$$
(C.6)
Due to the momentum quantisation $P_{v}$ and $P_{0}$ are related as $P_{v}=\sqrt{1-v^{2}}\,P_{0}$ which also follows from the transformation (C.4).
Clearly, this simple relation exists only because of the Lorentz invariance of the ${T\overline{T}}$ model.
Deformation by ${T\overline{T}}$, $JT_{0}$, $JT_{1}$, $\tilde{J}T_{0}$ and $\tilde{J}T_{1}$
In this case we set the parameter $v$ of the $T^{1}{}_{0}$ to 0: $v=0$. Calculating the coefficients ${\cal G}_{i}$, one gets
$$\displaystyle{\cal G}_{2}$$
$$\displaystyle=\alpha_{\mbox{\tiny${T\overline{T}}$}}+{1\over 2}\alpha_{\mbox{%
\tiny${JT_{0}}$}}^{2}G_{11}+{1\over 2}\alpha_{\mbox{\tiny${\widetilde{J}T_{0}}%
$}}^{2}G^{11}+(\alpha_{\mbox{\tiny${T\overline{T}}$}}^{2}-\alpha_{\mbox{\tiny$%
{\widetilde{J}T_{0}}$}}^{2}\alpha_{\mbox{\tiny${JT_{0}}$}}^{2})V\,,$$
(C.7)
$$\displaystyle{\cal G}_{1}$$
$$\displaystyle=1+\alpha_{\mbox{\tiny${JT_{1}}$}}\big{(}p_{1}+(\alpha_{\mbox{%
\tiny${\widetilde{J}T_{1}}$}}-\alpha_{\mbox{\tiny${JT_{0}}$}}G_{11})px^{\prime%
}\big{)}$$
$$\displaystyle-\alpha_{\mbox{\tiny${\widetilde{J}T_{0}}$}}(p_{\mu}G^{\mu 1}-%
\alpha_{\mbox{\tiny${JT_{0}}$}}px^{\prime}+\alpha_{\mbox{\tiny${\widetilde{J}T%
_{1}}$}}G^{11}px^{\prime})+\alpha_{\mbox{\tiny${\widetilde{J}T_{1}}$}}x^{%
\prime 1}-\alpha_{\mbox{\tiny${JT_{0}}$}}G_{1\mu}x^{\prime\mu}\big{]}$$
$$\displaystyle+4V\big{[}\alpha_{\mbox{\tiny${T\overline{T}}$}}\big{(}1+\alpha_{%
\mbox{\tiny${\widetilde{J}T_{0}}$}}\alpha_{\mbox{\tiny${JT_{0}}$}}px^{\prime}+%
\alpha_{\mbox{\tiny${JT_{1}}$}}(p_{1}+\alpha_{\mbox{\tiny${\widetilde{J}T_{1}}%
$}}px^{\prime})+\alpha_{\mbox{\tiny${\widetilde{J}T_{1}}$}}x^{\prime 1}\big{)}$$
$$\displaystyle+\alpha_{\mbox{\tiny${\widetilde{J}T_{0}}$}}\alpha_{\mbox{\tiny${%
JT_{0}}$}}\big{(}\alpha_{\mbox{\tiny${JT_{0}}$}}(p_{1}+\alpha_{\mbox{\tiny${%
\widetilde{J}T_{1}}$}}px^{\prime})+\alpha_{\mbox{\tiny${\widetilde{J}T_{0}}$}}%
(\alpha_{\mbox{\tiny${JT_{1}}$}}px^{\prime}+x^{\prime 1})\big{)}\,,$$
$$\displaystyle{\cal G}_{0}$$
$$\displaystyle={\cal K}+{1\over 2}px^{\prime}\big{[}\alpha_{\mbox{\tiny${JT_{1}%
}$}}^{2}G_{11}px^{\prime}+\alpha_{\mbox{\tiny${\widetilde{J}T_{1}}$}}^{2}G^{11%
}px^{\prime}-2\alpha_{\mbox{\tiny${JT_{0}}$}}(p_{1}+\alpha_{\mbox{\tiny${%
\widetilde{J}T_{1}}$}}px^{\prime})$$
$$\displaystyle\qquad\quad-2(-\alpha_{\mbox{\tiny${\widetilde{J}T_{1}}$}}p_{\mu}%
G^{\mu 1}+\alpha_{\mbox{\tiny${JT_{1}}$}}\alpha_{\mbox{\tiny${\widetilde{J}T_{%
0}}$}}px^{\prime}+\alpha_{\mbox{\tiny${T\overline{T}}$}}px^{\prime}+\alpha_{%
\mbox{\tiny${\widetilde{J}T_{0}}$}}x^{\prime 1})+2\alpha_{\mbox{\tiny${JT_{1}}%
$}}G_{1\mu}x^{\prime\mu}\big{]}$$
$$\displaystyle\qquad\quad+V\big{[}\big{(}1+\alpha_{\mbox{\tiny${JT_{1}}$}}(p_{1%
}+\alpha_{\mbox{\tiny${\widetilde{J}T_{1}}$}}px^{\prime})+\alpha_{\mbox{\tiny$%
{\widetilde{J}T_{1}}$}}x^{\prime 1}\big{)}^{2}$$
$$\displaystyle\qquad\quad-\big{(}\alpha_{\mbox{\tiny${JT_{1}}$}}\alpha_{\mbox{%
\tiny${\widetilde{J}T_{0}}$}}px^{\prime}+\alpha_{\mbox{\tiny${T\overline{T}}$}%
}px^{\prime}+\alpha_{\mbox{\tiny${JT_{0}}$}}(p_{1}+\alpha_{\mbox{\tiny${%
\widetilde{J}T_{1}}$}}px^{\prime})+\alpha_{\mbox{\tiny${\widetilde{J}T_{0}}$}}%
x^{\prime 1}\big{)}^{2}\big{]}\,.$$
The Hamiltonian density ${\cal H}_{A}$ is then found by solving the defining equation (3.3). The resulting Hamiltonian is very complicated, and it is unclear how one could find its spectrum even in the CFT limit where $V=0$. The situation becomes better if $G_{11}=1$ (or a constant), and $G_{1k}=0\,,\,k=2,\ldots,n$. Then, the model has left- and right-moving conserved currents, and one can redefine the deformation parameters to simplify the Hamiltonian. This is done in the next subsection.
Deformed CFT with left- and right-moving currents
In this subsection we use parameters $\alpha\equiv\alpha_{\mbox{\tiny${T\overline{T}}$}}$, $v\equiv-v^{+}$ and $\beta_{\pm}$, $\bar{\beta}_{\pm}$, and consider deformations of a CFT with the Hamiltonian
$$\displaystyle{\cal H}_{0}={1\over 2}p_{1}^{2}+{1\over 2}(x^{\prime 1})^{2}+{1%
\over 2}G^{kl}(x)p_{k}p_{l}+{1\over 2}x^{\prime k}x^{\prime l}\,G_{kl}(x)%
\equiv{\cal K}\,,$$
(C.8)
by ${T\overline{T}}$, $JT_{+}\leftrightarrow J\Theta$, $JT_{-}\leftrightarrow J\overline{T}$,
$\bar{J}T_{+}\leftrightarrow\bar{J}T$, $\bar{J}T_{-}\leftrightarrow\bar{J}\overline{\Theta}$ and $T^{1}{}_{0}$.
Calculating the coefficients ${\cal G}_{i}$ by choosing $\varkappa$’s as in (4.9), one gets
$$\displaystyle{\cal G}_{2}$$
$$\displaystyle=\alpha(1-v^{2})+{1\over 2}\beta_{-}^{2}(1+v)^{2}+{1\over 2}\bar{%
\beta}_{+}^{2}(1-v)^{2}\,,$$
(C.9)
$$\displaystyle{\cal G}_{1}$$
$$\displaystyle=1-v^{2}-2v\alpha px^{\prime}+\big{(}\beta_{-}^{2}(1+v)-\bar{%
\beta}_{+}^{2}(1-v)\big{)}px^{\prime}$$
$$\displaystyle-\bar{\beta}_{+}(p_{1}-x^{\prime 1})(1-v)-\beta_{-}(p_{1}+x^{%
\prime 1})(1+v)\,,$$
$$\displaystyle{\cal G}_{0}$$
$$\displaystyle={\cal K}-\alpha(px^{\prime})^{2}+px^{\prime}\big{(}\bar{\beta}_{%
+}(p_{1}-x^{\prime 1})-\beta_{-}(p_{1}+x^{\prime 1})+{1\over 2}(\bar{\beta}_{+%
}^{2}+\beta_{-}^{2})px^{\prime}-v\big{)}\,.$$
We see that ${\cal G}_{i}$, and therefore the deformed Hamiltonian, do not depend on $\beta_{+}$ and $\bar{\beta}_{-}$.
Thus, for a CFT with left- and right-moving conserved currents there is no deformation by $JT_{+}\leftrightarrow J\Theta$ and $\bar{J}T_{-}\leftrightarrow\bar{J}\overline{\Theta}$ if one chooses correctly the parameter of the ${T\overline{T}}$ deformation. Let us also mention that if the potential $V$ does not vanish then ${\cal G}_{i}$ do depend on $\beta_{+}$ and $\bar{\beta}_{-}$. The Hamiltonian of the model is obviously given by
$$\displaystyle{\cal H}_{A}={\cal H}(R,\alpha,\beta_{-},\bar{\beta}_{+},v)={{%
\cal G}_{1}-\sqrt{{\cal G}_{1}^{2}-4{\cal G}_{2}{\cal G}_{0}}\over 2{\cal G}_{%
2}}\,,$$
(C.10)
but its surprising feature is that its $v$-dependence is very similar to the one in (C.5).
Indeed it is easy to check that the deformed action
$$\displaystyle S_{A}=\int_{0}^{R}\,{\rm d}\sigma{\rm d}\tau\,\big{(}p_{\mu}\dot%
{x}^{\mu}\,-\,{\cal H}(\alpha,\beta_{-},\bar{\beta}_{+},v)\big{)}\,,$$
(C.11)
satisfies
$$\displaystyle S_{A}=\int_{0}^{R\sqrt{1-v^{2}}}\,{\rm d}\sigma{\rm d}\tau\,\big%
{(}p_{\mu}\dot{x}^{\mu}\,-\,{1\over\sqrt{1-v^{2}}}(\,{\cal H}(\alpha,b_{-},%
\bar{b}_{+},0)-v\,px^{\prime}\,)\big{)}\,.$$
(C.12)
where
$$\displaystyle b_{-}=\beta_{-}\,\sqrt{1+v\over 1-v}\,,\quad\bar{b}_{+}=\bar{%
\beta}_{+}\,\sqrt{1-v\over 1+v}\,.$$
(C.13)
Thus, up to the rescaling of $\beta_{-}$, $\bar{\beta}_{+}$, the energy of the model with nonvanishing $v$ is related to its energy with $v=0$ as in
(C.6).
Let us also calculate the coefficients ${\cal G}_{i}$ with another interesting
choice of $\varkappa$’s
$$\displaystyle\varkappa_{+}=-1\,,\quad\varkappa_{-}=-1\,,\quad\bar{\varkappa}_{%
+}=-1\,,\quad\bar{\varkappa}_{-}=-1\,,\quad\varkappa_{+-}=1\,,\quad\bar{%
\varkappa}_{+-}=1\,.$$
(C.14)
Then, the flow equations with respect to $\beta_{-}$ and $\bar{\beta}_{+}$ take the form
$$\displaystyle{\partial{\cal H}_{A}\over\partial\beta_{-}}=2{\cal J}{\mathbb{T}%
}_{-}\,-\,{\beta_{-}}{\mathbb{T}}\overline{\mathbb{T}}\,,\qquad{\partial{\cal H%
}_{A}\over\partial\bar{\beta}_{+}}=2\bar{\cal J}{\mathbb{T}}_{+}-{\bar{\beta}_%
{+}}{\mathbb{T}}\overline{\mathbb{T}}\,.$$
(C.15)
The coefficients ${\cal G}_{i}$ with $\varkappa$’s as in (C.14) are given by
$$\displaystyle{\cal G}_{2}$$
$$\displaystyle=\alpha(1-v^{2})+\beta_{-}^{2}(1+v)v-\bar{\beta}_{+}^{2}(1-v)v\,,$$
(C.16)
$$\displaystyle{\cal G}_{1}$$
$$\displaystyle=1-v^{2}-2v\alpha px^{\prime}+\big{(}\beta_{-}^{2}(1+2v)-\bar{%
\beta}_{+}^{2}(1-2v)\big{)}px^{\prime}$$
$$\displaystyle-\bar{\beta}_{+}(p_{1}-x^{\prime 1})(1-v)-\beta_{-}(p_{1}+x^{%
\prime 1})(1+v)\,,$$
$$\displaystyle{\cal G}_{0}$$
$$\displaystyle={\cal K}-\alpha(px^{\prime})^{2}+px^{\prime}\big{(}\bar{\beta}_{%
+}(p_{1}-x^{\prime 1})-\beta_{-}(p_{1}+x^{\prime 1})+(\bar{\beta}_{+}^{2}+%
\beta_{-}^{2})px^{\prime}-v\big{)}\,.$$
Now, setting $v=0$, one gets
$$\displaystyle{\cal G}_{2}$$
$$\displaystyle=\alpha\,,$$
(C.17)
$$\displaystyle{\cal G}_{1}$$
$$\displaystyle=1+\big{(}\beta_{-}^{2}-\bar{\beta}_{+}^{2}\big{)}px^{\prime}-%
\bar{\beta}_{+}(p_{1}-x^{\prime 1})-\beta_{-}(p_{1}+x^{\prime 1})\,,$$
$$\displaystyle{\cal G}_{0}$$
$$\displaystyle={\cal K}-\alpha(px^{\prime})^{2}+px^{\prime}\big{(}\bar{\beta}_{%
+}(p_{1}-x^{\prime 1})-\beta_{-}(p_{1}+x^{\prime 1})+(\bar{\beta}_{+}^{2}+%
\beta_{-}^{2})px^{\prime}\big{)}\,.$$
Finally, setting $\alpha=0$, and solving the defining equation (3.3), one gets the deformed Hamiltonian
$$\displaystyle{\cal H}_{\beta_{-},\bar{\beta}_{+}}={{\cal G}_{0}\over{\cal G}_{%
1}}={{\cal K}-\alpha(px^{\prime})^{2}+px^{\prime}\big{(}\bar{\beta}_{+}(p_{1}-%
x^{\prime 1})-\beta_{-}(p_{1}+x^{\prime 1})+(\bar{\beta}_{+}^{2}+\beta_{-}^{2}%
)px^{\prime}\big{)}\over 1+\big{(}\beta_{-}^{2}-\bar{\beta}_{+}^{2}\big{)}px^{%
\prime}-\bar{\beta}_{+}(p_{1}-x^{\prime 1})-\beta_{-}(p_{1}+x^{\prime 1})}\,,$$
(C.18)
which satisfies the flow equations (C.15).
The flow equations are now more complicated but
a curious feature of this choice is that setting $v=\alpha=0$, one gets Hamiltonian (C.18) which is a rational function of the coordinates and momenta. It suggests that
it is the ${T\overline{T}}$ deformation which is responsible for the square root form of the deformed Hamiltonian.
Appendix D Examples of the deformed spectrum
In this appendix we specialise eqs.(4.18, 4.19)
to the one- and two-parameter deformations.
Deformation by ${T\overline{T}}$
Setting $v=0$, $\beta_{-}=0$, $\bar{\beta}_{+}=0$, one gets
$$\displaystyle{E_{n}\over R}$$
$$\displaystyle={1-\sqrt{1-{4\alpha\over R}E_{n}^{(0)}+\frac{4\alpha^{2}}{R^{2}}%
P^{2}}\over 2\alpha}\,,$$
(D.1)
which is a well-known result.
Deformation by ${{\rm J}\overline{T}}$ and $\bar{J}T$
Setting $v=0$, $\alpha=0$, $\bar{\beta}_{+}=0$, one gets
$$\displaystyle{E_{n}+P\over R}$$
$$\displaystyle=\frac{2{\rm Q}_{+}}{R\,\beta_{-}}+{1-\sqrt{\big{(}1+{2\beta_{-}%
\over R}{\rm Q}_{+}\big{)}^{2}-{2\beta_{-}^{2}\over R}(E_{n}^{(0)}+P)}\over%
\beta_{-}^{2}}\,,$$
(D.2)
which agrees with [4].
Let us also give the formula for the $\bar{J}T$ deformation
$$\displaystyle{E_{n}^{\bar{J}T}-P\over R}$$
$$\displaystyle=\frac{2{\rm Q}_{-}}{R\,\beta_{+}}+{1-\sqrt{\big{(}1+{2\beta_{+}%
\over R}{\rm Q}_{-}\big{)}^{2}-{2\beta_{+}^{2}\over R}(E_{n}^{(0)}-P)}\over%
\beta_{-}^{2}}\,.$$
(D.3)
Deformation by ${T\overline{T}}$+${{\rm J}\overline{T}}$
Setting $v=0$, $\bar{b}_{+}=0$, one gets
$$\displaystyle{E_{n}+P\over R}$$
$$\displaystyle=\frac{2\alpha P+2\beta_{-}{\rm Q}_{+}}{(2\alpha+\beta_{-}^{2})R}%
+{1-\sqrt{\big{(}1+{2\alpha\over R}P+{2\beta_{-}\over R}{\rm Q}_{+}\big{)}^{2}%
-{2(2\alpha+\beta_{-}^{2})\over R}(E_{n}^{(0)}+P)}\over 2\alpha+\beta_{-}^{2}}\,,$$
(D.4)
which agrees with [6, 7].
Deformation by ${{\rm J}\overline{T}}$+$\bar{J}T$
Setting $v=0$, $\bar{b}_{+}=0$, one gets
$$\displaystyle{E_{n}\over R}=\frac{(\bar{\beta}_{+}^{2}-\beta_{-}^{2})P+2\beta_%
{-}{\rm Q}_{+}+2\beta_{+}{\rm Q}_{-}}{(\beta_{-}^{2}+\bar{\beta}_{+}^{2})R}$$
(D.5)
$$\displaystyle+{1-\sqrt{\big{(}1+{2\beta_{-}{\rm Q}_{+}+2\beta_{+}{\rm Q}_{-}%
\over R}\big{)}^{2}-{2\beta_{-}^{2}(E_{n}^{(0)}+P)+2\beta_{+}^{2}(E_{n}^{(0)}-%
P)\over R}-{4\beta_{m}\bar{\beta}_{+}P(\beta_{m}\bar{\beta}_{+}P+2\beta_{-}{%
\rm Q}_{-}-2\bar{\beta}_{+}{\rm Q}_{+})\over R^{2}}}\over\beta_{-}^{2}+\bar{%
\beta}_{+}^{2}}\,.$$
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Hadronic $B$ Decay
T. E. Browder
Physics Department, University of Hawaii at Manoa,
Honolulu, HI, 96822, USA
Abstract
We review recent experimental
results from CLEO and LEP
experiments on hadronic decays of hadrons containing $b$
quarks[1]. We discuss charm counting and the
semileptonic branching fraction in $B$ decays and
the color suppressed amplitude in $B$ decay.
1 Charm counting and the semileptonic branching fraction
A complete picture of inclusive
$B$ decay is beginning to emerge from recent measurements
by CLEO II and the LEP experiments.[1]
These measurements can be used to address
the question of whether the hadronic decay of the $B$ meson is compatible with
its semileptonic branching fraction.
Three facts emerge from the experimental examination of inclusive $B$
decay at the $\Upsilon(4S)$:
$$n_{c}=1.10\pm 0.05$$
(1)
where $n_{c}$ is the number of charm quarks
produced per $B$ decay from recent CLEO II
results[3]
and using
${\cal B}(D^{0}\to K^{-}\pi^{+})=(3.91\pm 0.08\pm 0.17\%)$.[4]
$${\cal B}(B\to X\ell\nu)=10.23\pm 0.39\%.$$
(2)
This value is the
average of the CLEO and ARGUS model independent measurements using
dileptons.[2]
We note that the value used by the LEP Electroweak Working Group
for ${\cal B}(b\to X\ell\nu)=11.16\pm 0.20\%$ is only marginally
consistent with the $\Upsilon(4S)$ results.
The third quantity, ${\cal B}(b\to c\bar{c}s)$,
is calculated from the inclusive $B\to D_{s}$,
$B\to(c\bar{c})X$, and $B\to\Xi_{c}$ branching fractions, and is
$${\cal B}(b\to c\bar{c}s)=14.0\pm 2.8\%.$$
(3)
The above value is determined assuming no contribution from $B\to D$ decays,
an assumption which can be checked using data and is discussed in further
detail below.
In the parton model, it is
difficult to accomodate a low semileptonic branching fraction unless the
hadronic width of the $B$ meson is increased.[5]
The explanations for the semileptonic branching fraction
which have been proposed can be distinguished by expressing
the hadronic width of the $B$ meson in terms of three components:
$$\Gamma_{hadronic}(b)=\Gamma(b\to c\bar{c}s)+\Gamma(b\to c\bar{u}d)+\Gamma(b\to
s%
~{}g).$$
If the semileptonic
branching fraction is to be reduced to the observed level, then one of the
components must be enhanced.
A large number of explanations for the low semileptonic branching fraction
and charm yield have been proposed in the last few years.
These explanations can be logically classified as follows:
1.
An enhancement of $b\to c\bar{c}s$ due to large QCD corrections
or the breakdown of local duality. A variety of possible
experimental signatures
have been suggested.[6]${}^{,}$[7]${}^{,}$
[8]${}^{,}$[9]${}^{,}$[10]${}^{,}$[11]
2.
An enhancement of $b\to c\bar{u}d$ due to non-perturbative effects.
[12]${}^{,}$[13]${}^{,}$[14]${}^{,}$[15]
3.
An enhancement of $b\to s~{}g$ or $b\to d~{}g$
from New Physics.[16]${}^{,}$[17]${}^{,}$[18]
4.
The cocktail solution: For example,
if both the $b\to c\bar{c}s$ and the
$b\to c\bar{u}d$ mechanisms are increased,
this could suffice to explain the inclusive observations.
5.
There might also be a
systematic experimental problem in the determination of either $n_{c}$,
${\cal B}(b\to c\bar{c}s)$, or ${\cal B}(B\to X\ell\nu)$.[19]
Inclusive charm particle-lepton correlations can be used to probe
the $B$ decay mechanism and give further insight into this problem.
The correlation of the lepton charge and the charm particle flavor
distinguishes between different production mechanisms.
High momentum leptons, $p_{\ell}>1.4$ GeV, are used
to tag the flavor of the $B$. The angular correlation between the meson
and the lepton is then employed to select events in which the tagging lepton
and meson are from different $B$s. When the lepton and meson
originate from the same $B$ meson they tend to be back to back, whereas when
the meson and leptons come
from different $B$ mesons they are uncorrelated.
After this separation is performed, wrong sign
charge correlations from $B-\bar{B}$
mixing must be subtracted. Since the mixing rate is well measured,
this correction is straightforward and has little uncertainty.
This technique has been applied previously to several types of
correlations of charmed hadrons and leptons.
For example, the sign of $\Lambda_{c}$-lepton
correlations distinguishes between the
$b\to c\bar{u}d$ and the $b\to c\bar{c}s$ mechanisms.
It was found that the $b\to c\bar{c}s$ mechanism comprises
$19\pm 13\pm 4\%$ of $B\to\Lambda_{c}$ decays[20].
This observation effectively ruled out one proposed source of
additional $b\to c\bar{c}s$ decays.[6]
Similiarly, examination of the sign of
$D_{s}$-lepton correlations shows that
most $D_{s}$ mesons originate from $b\to c\bar{c}s$ rather than
from $b\to c\bar{u}d$ with $s\bar{s}$ quark popping at the lower vertex.
In this case, it was found that $17.2\pm 7.9\pm 2.6\%$ of $D_{s}$ mesons
originate from the latter mechanism[21].
The same experimental
technique has now been applied to $D$-lepton correlations.
The conventional $b\to c\bar{u}d$ mechanism which was previously assumed to be
responsible for all $D$ production in $B$ decay will give $D\ell^{-}$
correlations.
If a significant fraction of $D$ mesons
arise from $b\to c\bar{c}s$ with light quark popping at the
upper vertex as proposed by Buchalla, Dunietz,
and Yamamoto significant wrong sign $D\ell^{+}$ correlations will be
observed.[7]
Final results of this study have been presented by CLEO II which finds,
$\Gamma(B\to D~{}X)/\Gamma(B\to\bar{D}X)=0.100\pm 0.026\pm 0.016$.[22]
This implies a new contribution to the $b\to c\bar{c}s$ width
$${\cal B}(B\to DX)=7.9\pm 2.2\%$$
ALEPH finds evidence for semi-inclusive
$B\to D^{0}\bar{D^{0}}X+D^{0}D^{\mp}X$ decays with a somewhat larger branching
fraction of $12.8\pm 2.7\pm 2.6\%$.[23]
DELPHI reports the observation of
$B\to D^{*+}D^{*-}X$ decays with a branching fraction
of $1.0\pm 0.2\pm 0.3\%$.[29]
Additional and quite compelling evidence that these
signals are due to $B\to D^{(*)}\bar{D}^{(*)}K^{(*)}$ decays
has been presented by CLEO[24], which has observed
fully reconstructed signals in exclusive modes:
$${\cal B}(\bar{B}^{0}\to D^{*+}\bar{D}^{0}K^{-})=0.45^{+0.25}_{-0.19}\pm 0.08\%$$
$${\cal B}(B^{-}\to D^{*0}\bar{D}^{0}K^{-})=0.54^{+0.33}_{-0.24}\pm 0.12\%$$
$${\cal B}(\bar{B}^{0}\to D^{*+}\bar{D}^{*0}K^{-})=1.30^{+0.61}_{-0.47}\pm 0.27\%$$
$${\cal B}(B^{-}\to D^{*0}\bar{D}^{*0}K^{-})=1.45^{+0.78}_{-0.58}\pm 0.36\%$$
The rates observed by ALEPH and DELPHI are
consistent with the rate of wrong sign
$D$-lepton correlation reported by CLEO.[22]
It is possible that these channels are actually resonant modes
of the form $B\to D{D}_{s}^{**}$
decays, where the p-wave $D_{s}^{**}$ or radially excited $D_{s}^{{}^{\prime}}$ state
decays to $\bar{D}^{(*)}\bar{K}$.[30] A direct
search by CLEO has ruled
out the possibility of narrow $B\to D_{s_{1}}X$ decay:
${\cal B}(B\to D_{s1}^{+}X)<0.95\%$ at the 90% confidence level.[31]
There are other implications of these observations.
A $B$ decay mechanism with a ${\cal O}(10\%)$ branching
fraction has been found
which was not previously included in the CLEO or LEP
Monte Carlo simulations of $B$ decay. This may have consequences
for other analyses of particle-lepton correlations. For example,
CLEO has re-examined the model independent dilepton measurement
of ${\cal B}(B\to X\ell\nu)$. Due to the lepton threshold of 0.6 GeV
and the soft spectrum of leptons, the CLEO measurement
is fortuitously unchanged. It is also important to check the size
of this effect in LEP measurements of the $B$ semileptonic branching fraction
using dileptons.
We can now recalculate[22]
$${\cal B}(b\to c\bar{c}s)=21.9\pm 3.7\%$$
which would suggest a somewhat larger charm yield ($n_{c}\sim 1.22$).
This supports hypothesis (1), large QCD corrections in $b\to c\bar{c}s$
BUT the charm yield $n_{c}$ as computed in the usual way
is unchanged. Moreover, the contribution of
$B\to D\bar{D}KX$ decays was properly accounted for
in the computation of $n_{c}$. This suggests that the experimental
situation is still problematic.
One possibility that must be addressed is whether
there could be an error in
the normalization ${\cal B}(D^{0}\to K^{-}\pi^{+})$.[19]
This branching fraction calibrates the inclusive measurements of
$B\to D^{0}$, $B\to D^{+}$, and $B\to D_{s}$ rates as well as
$n_{c}$. Historically, a flaw in
${\cal B}(D^{0}\to K^{-}\pi^{+})$ has been the culprit in other consistency
problems with charm counting.
The most precise measurements of ${\cal B}(D^{0}\to K^{-}\pi^{+})$
are obtained by fitting the $p_{T}$ spectrum
of soft pions in charm jets. An examination of Table 1 shows
that these measurements are statistically precise but systematics
dominated. The Particle Data Group world average is currently dominated by
the ALEPH measurement.[25]
Using $B$ decay data it is also possible to make consistency checks on the
calibration branching fraction. For example, the ratio
$${{\cal B}(B\to D^{*}\ell\nu)_{\rm partial}}\over{{\cal B}(B\to D^{*}\ell\nu)_{%
\rm full}}$$
where the decay in the
numerator is observed without reconstructing the $D$ decay gives
a measurement of the calibration branching fraction with very different
systematic effects. In CLEO data, this method
gives
${\cal B}(D^{0}\to K^{-}\pi^{+})=3.81\pm 0.15\pm 0.16\%$.[26]
Another quantity which can be examined is
$${{\cal B}(B\to DX\ell\nu)}\over{{\cal B}(B\to X\ell\nu)}$$
which should be unity modulo small corrections for semileptonic $D_{s}$ and
baryon production as well as for processes
which do not produce charm. Applied
to CLEO data, this method gives
${\cal B}(D^{0}\to K^{-}\pi^{+})=3.69\pm 0.08\pm 0.17\%$.[22]
Neither method is sufficiently precise yet to
conclusively demonstrate that either the
$D$ branching fraction scale is correct or that it has a systematic flaw.
Another possibility is enhanced ${\cal B}(b\to cu\bar{d})$.
On the theoretical side, Bagan et al. find that at next to leading order,
$$r_{ud}={{{\cal B}(b\to c\bar{u}d)}\over{{\cal B}(b\to c\ell\nu)}}=4.0\pm 0.4$$
The value of ${\cal B}(b\to c\bar{u}d)$
can be checked using measurements
of inclusive $B$ decay from the $\Upsilon(4S)$ experiments:
$${\cal B}(b\to c\bar{u}d)_{exp}={\cal B}(B\to DX)+{\cal B}(B\to\Lambda_{c}X)$$
$$-{\cal B}(B\to DD_{s}X)-2{\cal B}(B\to D\bar{D}KX)-2.25{\cal B}(b\to c\ell\nu)$$
$$=(0.871\pm 0.035)+(0.036\pm 0.020)$$
$$-(0.10\pm 0.027)-2\times(0.079\pm 0.022)-(0.236\pm 0.010)$$
$${\cal B}(b\to c\bar{u}d)_{exp}=0.41\pm 0.07$$
In the above calculation, a small correction ($0.004$) has been applied
to the $B\to\Lambda_{c}X$ branching fraction to account
for $b\to c\bar{c}s$ production in baryonic $B$ decay.
The factor of $2.25$ accounts for phase space suppression in
$b\to c\tau\nu$ decay.
The experimental result is consistent with the theoretical
expectation,
$${\cal B}(b\to c\bar{u}d)_{theory}=0.42\pm 0.04$$
However, the present experimental accuracy is not quite sufficient
to completely rule out $b\to c\bar{u}d$ as the cause of the discrepancy.
We note that ALEPH and OPAL have recently reported a value for
$n_{c}$ in $Z\to b\bar{b}$ decay.[27],[28]
ALEPH finds $n_{c}^{Z}=1.230\pm 0.036\pm 0.038\pm 0.053$.
The rate of $D_{s}$ and $\Lambda_{c}$ production is significantly higher
than what is observed at the $\Upsilon(4S)$. It is not clear whether
the quantity being measured is the same as $n_{c}$ at the $\Upsilon(4S)$, which
would be the case if the spectator model holds and if the contribution
from the other $b$-hadrons, $B_{s}$ and $\Lambda_{b}$, could be neglected.
OPAL reports a somewhat lower value of $n_{c}=1.10\pm 0.045\pm 0.060\pm 0.037$
after correcting for unseen charmonium states.
OPAL assumes no contribution from $\Xi_{c}$ production while ALEPH includes
a very large contribution from this source. The
contribution of $B\to$baryon decays to charm counting
as well as the $\Lambda_{c}$, $\Xi_{c}$ branching fraction scales are still poorly
measured and definitely merit further investigation.
2 Exclusive Hadronic Decays
Recent progress has been made on partial reconstruction of hadronic
$B$ decays.[32]
For example, the decay chain
$$B\to D^{*}\pi_{f},D^{*}\to(D)\pi_{s}$$
can be measured without reconstructing the $D$ meson. In this reaction,
there are five particles ($B,D^{*},D,\pi_{s},\pi_{f}$) with five 4-momenta
give 20 unknowns. The 4-momenta of the
$\pi_{s},\pi_{f}$ are measured which gives 8 constraints.
The $B,D,D^{*}$ masses and beam energy are known and gives 4 constraints.
Then energy-momentum conservation in the $B\to D^{*}\pi_{f}$ and $D^{*}\to D\pi_{s}$
decay chains gives 8 additional constraints. Thus, one can
perform a $20-8-8-4=0$ C fit.
Two variables are used to extract the signal:
$\cos\Theta_{D}^{*}$, the angle between the $p_{\pi_{s}}$ and $p_{B}$
in the $D^{*}$ rest frame, and
$\cos\theta_{B}^{*}$, the angle between the $p_{\pi_{F}}$ and $P_{B}$
in the $B$ rest frame.
This method gives the most precise measurements of two exclusive branching
fractions:
$${\cal B}(\bar{B}^{0}\to D^{*+}\pi^{-})=(2.81\pm 0.11\pm 0.21\pm 0.05)\times 10%
^{-3}$$
$${\cal B}(B^{-}\to D^{*0}\pi^{-})=(4.81\pm 0.42\pm 0.40\pm 0.21)\times 10^{-3}.$$
The second systematic error is from the $D^{*}$ branching fractions.
A similiar partial reconstruction
analysis has been applied to the $B^{-}\to D^{**}(2420)^{0}\pi^{-}$
and $B^{-}\to D^{**}(2460)^{0}\pi^{-}$ decay modes.[33]
The event yields from fitting these distributions are substantial:
$281\pm 56$ $D^{**}(2420)$, $165\pm 61$ $D^{**}(2460)$, although
there are also large background subtractions. These correspond
to branching fractions,
$${\cal B}(B^{-}\to D_{1}(2420)\pi^{-})=(1.17\pm 0.24\pm 0.16\pm 0.03)\times 10^%
{-3}$$
$${\cal B}(B^{-}\to D_{2}^{*}(2460)\pi^{-})=(2.1\pm 0.8\pm 0.3\pm 0.05)\times 10%
^{-3}$$
The former mode was previously
observed using a similiar technique by ARGUS. The latter mode is
observed for the first time by CLEO. As noted by J. Gronberg and
H. Nelson, the partial reconstruction technique may also be useful for
observing a time dependent CP asymmetry in $\bar{B}^{0}\to D^{*+}\pi^{-}$.
2.1 The sign of the color suppressed amplitude and lifetimes
The sign
and magnitude of the color suppressed amplitude can be determined using
several classes of decay modes in charm and bottom mesons. The numerical
determination assumes factorization and uses form factors from various
phenemonological models.
For $D$ decay one
uses exclusive modes such as $D\to K\pi$, $D\to K\rho$ etc.,
and obtains
$$a_{1}=1.10\pm 0.03,~{}a_{2}=-0.50\pm 0.03$$
The destructive interference observed in two body $D^{+}$
decays leads to the $D^{+}$-$D^{0}$ lifetime difference.[1]
For $B$ decay, one can find the magnitude of $|a_{1}|$ from
the branching fractions for the decay modes
$\bar{B}^{0}\to D^{(*)+}\pi^{-}$, $\bar{B}^{0}\to D^{+(*)}\rho^{-}$.
This gives $|a_{1}|=1.06\pm 0.03\pm 0.06$.
One can also extract $|a_{1}|$
from measurements of branching fractions
$B\to D^{+,(0)}D_{s}^{(*)-}$[34].
The magnitude $|a_{2}|$ can be determined from the branching
fractions for $B\to\psi K^{(*)}$. This yields $|a_{2}|=0.23\pm 0.01\pm 0.01$.
The value of $a_{2}/a_{1}$ can be found by comparing
$B^{-}$ decays where both the external and spectator diagrams
contribute to $\bar{B}^{0}$ decays where only the external spectator
decays contribute. For example,
the model of Neubert et al. predicts the following ratios:
$$R_{1}={{\cal B}(B^{-}\to D^{0}\pi^{-})\over{\cal B}(\bar{B^{0}}\to D^{+}\pi^{-%
})}=(1+1.23a_{2}/a_{1})^{2}$$
(4)
$$R_{2}={{\cal B}(B^{-}\to D^{0}\rho^{-})\over{\cal B}(\bar{B^{0}}\to D^{+}\rho^%
{-})}=(1+0.66a_{2}/a_{1})^{2}$$
(5)
$$R_{3}={{\cal B}(B^{-}\to D^{*0}\pi^{-})\over{\cal B}(\bar{B^{0}}\to D^{*+}\pi^%
{-})}=(1+1.29a_{2}/a_{1})^{2}$$
(6)
$$R_{4}={{\cal B}(B^{-}\to D^{*0}\rho^{-})\over{\cal B}(\bar{B^{0}}\to D^{*+}%
\rho^{-})}\approx(1+0.75a_{2}/a_{1})^{2}$$
(7)
Improved measurements of these exclusive branching fractions
with better background subtraction
and additional data have recently been
presented by CLEO (see Fig. 2).[34]
Using the latest branching fractions,
$$a_{2}/a_{1}=0.21\pm 0.03\pm 0.03^{+0.13}_{-0.12},$$
where the third
error is a conservative estimte of the uncertainty ($\sim 20\%$)
in the relative production of $B^{+}$ and
$B^{0}$ mesons at the $\Upsilon(4S)$. There are a number of
additional theoretical uncertainties
which could significantly modify the magnitude of $a_{2}/a_{1}$ but
not its sign. For example,
the ratios of some heavy-to-heavy to heavy-to-light form factors is needed
(e.g. $B\to\pi/B\to D$).
Comparing the value of $a_{2}/a_{1}$ determined using
form factors from the model of Neubert et al. with the value obtained
using form factors from the
model of Deandrea et al. shows that this uncertainty is small.
The effect
of including the $B\to VV$ mode for which the form factors have somewhat
larger theoretical uncertainties is also small.
It is important to remember that
the determination of $a_{2}/a_{1}$ also assumes the factorization hypothesis.
The large error on the relative production of $B^{+}$ and $B^{0}$ mesons
is the most significant experimental
uncertainty in the determination of $a_{2}/a_{1}$.
The value of $a_{2}/a_{1}$ determined above
is consistent with the ratio $|a_{2}|$/$|a_{1}|$ where $|a_{2}|$ is computed
from $B\to\psi$ modes and $|a_{1}|$ is computed from $\bar{B}^{0}\to D^{(*)}\pi,D^{(*)}\rho$ modes. Although the result is surprisingly different
from what is observed in hadronic charm decay (where the interference
is destructive) and from what is expected in the $1/N_{c}$ expansion,
Buras claims that the result can be accomodated
in NLO QCD calculations.[35]
If the constructive interference which is observed in these
$B^{+}$ decays is present in all $B^{+}$ decays, then we expect
a significant $B^{+}$-$B^{0}$ lifetime difference ($\tau_{B}^{+}<\tau_{B^{0}}$),
of order $15-20\%$, in a direction opposite
to the $D^{+}-D^{0}$ lifetime difference. This
scenario is only marginally consistent
with experimental measurements of lifetimes;
the world average computed by the LEP lifetime working group
in August 1997 is
$$\tau_{B^{+}}/\tau_{B^{0}}=1.07\pm 0.04$$
It is possible
that the hadronic $B^{+}$ decays that have been observed to date are
atypical. The remaining higher multiplicity $B^{+}$ decays could
have destructive interference or no interference.[37]
Or perhaps there is a mechanism which also enhances the $\bar{B}^{0}$
width to compensate for the increase in the $B^{+}$ width
and which maintains the $B^{+}/B^{0}$ lifetime ratio near unity.
Such a mechanism would be relevant to the charm counting and
semileptonic branching fraction problem.
In either case, there will be experimental consequences in the
pattern of hadronic $B$ branching fractions.[38]
Experimentally one can compare other $B^{-}$ and $B^{0}$ decays including
$D^{**}\pi^{-}$ and $D^{**}\rho^{-}$ as well
decays to $D^{(*)}a_{1}^{-}$, $a_{1}^{-}\to\rho^{0}\pi^{-}$
and $D^{(*)}b_{1}^{-}$, $b_{1}^{-}\to\omega\pi^{-}$ to check the first
possibility.
3 Conclusion
The charm counting and semileptonic $B$ branching
fraction problem persists. Three possible
solutions are still experimentially viable.
These are (1) a systematic problem in the $D$
branching fraction, (2) an enhancement of $b\to c\bar{u}d$ or
(3) an enhancement of $b\to sg$. Any proposed solution must also satisfy
the experimental constraints on ${\cal B}(b\to c\bar{c}s)$ and
${\cal B}(b\to c\bar{u}d)$.
The sign of $a_{2}/a_{1}$ is found to be positive in the low multiplicity hadronic
$B$ decays that have been observed so far. This indicates
constructive interference in hadronic $B^{+}$ decay.
It will be interesting to see whether this pattern persists
as higher multiplicity $B$ decay modes are measured.
References
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A Survey of Binary Code Similarity
Irfan Ul Haq
IMDEA Software Institute &
Universidad Politécnica de Madrid
[email protected]
Juan Caballero
IMDEA Software Institute
[email protected]
Abstract
Binary code similarity approaches compare two or more pieces of binary code to identify their similarities and differences.
The ability to compare binary code enables many real-world applications
on scenarios where source code may not be available
such as patch analysis, bug search, and malware detection and analysis.
Over the past 20 years numerous binary code similarity approaches have been proposed,
but the research area has not yet been systematically analyzed.
This paper presents a first survey of binary code similarity.
It analyzes 61 binary code similarity approaches,
which are systematized on four aspects:
(1) the applications they enable,
(2) their approach characteristics,
(3) how the approaches are implemented, and
(4) the benchmarks and methodologies used to evaluate them.
In addition, the survey discusses the scope and origins of the area,
its evolution over the past two decades, and the challenges that lie ahead.
Code diffing, Code search, Cross-architecture, Program executables
I Introduction
Binary code similarity approaches compare two or more pieces of binary code e.g., basic blocks, functions, or whole programs,
to identify their similarities and differences.
Comparing binary code is fundamental in scenarios
where the program source code is not available, which happens with
commercial-of-the-shelf (COTS) programs, legacy programs, and malware.
Binary code similarity has a wide list of real-world applications such as
bug search [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15],
malware clustering [16, 17, 18],
malware detection [19, 20, 21],
malware lineage [22, 23, 24],
patch generation [25],
patch analysis [26, 27, 28, 29, 8, 30, 31],
porting information across program
versions [32, 26, 27], and
software theft detection [33].
Identifying binary code similarity is challenging because
much program semantics are lost due to the compilation process including
function names, variable names, source comments, and
data structure definitions.
Additionally, even when the program source code does not change,
the binary code may change if the source is recompiled,
due to secondary changes introduced by the compilation process.
For example, the resulting binary code
can significantly change when
using different compilers,
changing compiler optimizations,
and selecting different target operating systems and CPU architectures.
Furthermore, obfuscation transformations can be applied on
both the source code and the generated binary code,
hiding the original code.
Given its applications and challenges, over the past 20 years numerous binary code similarity approaches have been proposed.
However, as far as we know there does not exist a systematic survey of this
research area.
Previous surveys deal with
binary code obfuscation techniques in packer tools [34],
binary code type inference [35], and
dynamic malware analysis techniques [36].
Those topics are related because binary code similarity may need to tackle obfuscation,
binary code type inference may leverage similar binary analysis platforms,
and malware is often a target of binary code similarity approaches.
But, binary code similarity is well-separated from those topics
as shown by previous surveys having
no overlap with this paper on the set of papers analyzed.
Other surveys have explored similarity detection on any binary data,
i.e., not specific to code,
such as hashing for similarity search [37] and
similarity metrics on numerical and binary feature
vectors [38, 39].
In contrast, this survey focuses on approaches that compare
binary code, i.e., that disassemble the executable byte stream.
This paper presents a first survey of binary code similarity.
It first identifies 61 binary code similarity approaches through a
systematic selection process that examines over a hundred papers
published in research venues
from different computer science areas such as computer security,
software engineering, programming languages, and machine learning.
Then, it systematizes four aspects of those 61 approaches:
(1) the applications they enable,
(2) their approach characteristics,
(3) how the approaches have been implemented, and
(4) the benchmarks and methodologies used to evaluate them.
In addition, it discusses the scope and origin of binary code similarity,
its evolution over the past two decades, and the challenges that lie ahead.
Binary code similarity approaches widely vary in their approach,
implementation, and evaluation.
This survey systematizes each of those aspects, and
summarizes the results in easy to access
tables that compare the 61 approaches across multiple dimensions,
allowing beginners and experts
to quickly understand their similarities and differences.
For example, the approach systematization includes, among others,
the number of input pieces of binary code being compared
(e.g., one-to-one, one-to-many, many-to-many);
the granularity of the pieces of binary code analyzed
(e.g., basic blocks, functions, programs);
whether the comparison happens at the syntactical representation,
the graph structure, or the code semantics;
the type of analysis used
(e.g., static, dynamic, symbolic), and
the techniques used for scalability
(e.g., hashing, embedding, indexing).
The implementation systematization includes
the binary analysis platforms used to build the approach,
the programming languages used to code it,
the supported architectures for the input pieces of binary code being compared,
and whether the approach is publicly released.
The evaluation systematization covers the datasets on which the approaches
are evaluated and the evaluation methodology including
the type of evaluation (e.g., accuracy, comparison wih prior works, performance) and how the robustness of the approach is evaluated in face of
common code transformations such as compiler and compilation option changes,
different architectures, and obfuscation.
Beyond the systematization,
this survey also discusses how binary code similarity has evolved from binary code diffing to
binary code search and how the focus has moved from a single architecture to
cross-architecture approaches.
It shows that the present of the field is vibrant as many new approaches are
still being proposed.
It discusses technical challenges that remain open, but
concludes that the future of the area is bright with important applications
scenarios under way such as those related to binary code search engines and the
Internet-of-Things.
Paper structure.
This paper is organized as follows.
Section II provides an overview of binary code similarity.
Section III details the scope of the survey and
the paper selection process.
Section IV summarizes applications of binary code similarity and
Section §V the evolution of the field over
the last two decades.
Section VI systematizes the characteristics of the
61 binary code similarity approaches,
Section VII their implementation, and
Section VIII their evaluation.
Finally, we discuss future research directions in Section IX,
and conclude in Section X.
II Overview
In this section, we first provide background on
the compilation process (§II-A).
Then, we present an overview of the binary code similarity problem (§II-B).
II-A Compilation Process
Binary code refers to the machine code that is produced by the
compilation process and that can be run directly by a CPU.
The standard compilation process takes as input the
source code files of a program.
It compiles them with a chosen compiler and optimization level and
for a specific platform (defined by the architecture, word size, and OS)
producing object files.
Those object files are then linked into a binary program,
either a stand-alone executable or a library.
Binary code similarity approaches typically deal with an
extended compilation process, illustrated in Figure 1,
which adds two optional steps to the standard compilation process:
source code and binary code transformations.
Both types of transformations are typically semantics-preserving
(i.e., do not change the program functionality) and are
most commonly used for obfuscation,
i.e., to hamper reverse-engineering of the distributed binary programs.
Source code transformations happen pre-compilation.
Thus, their input and output are both source code.
They can be applied regardless of the target platform, but may be
specific to the programming language used to write the program.
On the other hand, binary code transformations happen post-compilation.
Thus, their input and output are binary code.
They are independent of the programming language used,
but may be specific to the target platform.
Obfuscation is a fundamental step in malware, but can also be applied
to benign programs, e.g., to protect their intellectual property.
There exist off-the-shelf obfuscation tools that use
source code transformations (e.g., Tigress [40]),
as well as binary code transformations (e.g., packers [41]).
Packing is a binary code transformation widely used by malware.
Once a new version of a malware family is ready, the malware authors
pack the resulting executable to hide its functionality and
thus bypass detection by commercial malware detectors. The packing process
takes as input an executable and produces another executable with the
same functionality, but with the original code hidden
(e.g., encrypted as data and unpacked at runtime).
The packing process is typically applied many times
to the same input executable, creating polymorphic variants
of exactly the same source code,
which look different to malware detectors.
Nowadays, the majority of malware is packed
and malware often uses custom packers for which off-the-shelf unpackers
are not available [41].
A main challenge in binary code similarity is that the compilation process
can produce different binary code representations for the same source code.
An author can modify any of the grey boxes in
Figure 1 to produce a different,
but semantically-equivalent, binary program
from the same source code.
Some of these modifications may be due to the standard compilation process.
For example, to improve program efficiency an author may vary the
compiler’s optimization level, or change the compiler altogether.
Both changes will transform the produced binary code,
despite the source code remaining unchanged.
An author may also change the target platform to obtain a version of the
program suitable for a different architecture.
In this case, the produced binary code may radically differ if the
new target architecture uses a different instruction set.
An author may also deliberately apply obfuscation transformations to
produce polymorphic variants of the same source code.
The produced variants will typically have the same functionality defined by
the original source code.
A desirable goal for binary code similarity approaches is that they are able to identify
the similarity of binary code that corresponds to the same source code
having undergone different transformations.
The robustness of a binary code similarity approach captures
the compilation and obfuscation transformations that it can handle,
i.e., the transformations despite which it can still detect similarity.
II-B Binary Code Similarity Overview
Binary code similarity approaches compare pieces of binary code.
The three main characteristics of binary code similarity approaches are:
(1) the type of the comparison
(identical, similar, equivalent),
(2) the granularity of the pieces of binary code being compared
(e.g., instructions, basic blocks, functions), and
(3) the number of input pieces being compared
(one-to-one, one-to-many, many-to-many).
We detail these three characteristics next.
For simplicity, we describe the comparison type and comparison granularity
for two inputs and then generalize to multiple inputs.
Comparison type.
Two (or more) pieces of binary code are identical if they have
the same syntax, i.e., the same representation.
The binary code can be represented in different ways such as
an hexadecimal string of raw bytes,
a sequence of disassembled instructions, or
a control-flow graph.
Determining if several pieces of binary code are identical is
a Boolean decision (either they are identical or not)
that it is easy to check:
simply apply a cryptographic hash
(e.g., SHA256)
to the contents of each piece.
If the hash is the same, the pieces are identical.
However, such straightforward approach fails to detect similarity
in many cases.
For example, compiling the same program source code twice,
using the same compilation parameters
(i.e., same compiler version, same optimization level,
same target platform)
produces two executables with different file hash.
This happens because the executable may include metadata
that differs in both compilations such as the
compilation date, which is automatically computed and included into the
header of the generated executable.
Two pieces of binary code are equivalent if they have the
same semantics,
i.e., if they offer exactly the same functionality.
Equivalence does not care about the syntax of the binary code.
Clearly, two identical pieces of binary code will have the same semantics,
but different pieces of binary code may as well.
For example, mov %eax,$0 and xor %eax,%eax
are semantically equivalent x86 instructions
because both set the value of register EAX to zero.
Similarly, the same source code compiled for two different target
architectures should produce equivalent executables, whose syntax
may be completely different if the architectures use different instruction sets.
Proving that two arbitrary programs are functionally equivalent is an
undecidable problem that reduces to solving the halting problem [42].
In practice, determining binary code equivalence is a very
expensive process that can only be performed for small pieces of binary code.
Two pieces of binary code can be considered similar if their
syntax, structure, or semantics are similar.
Syntactic similarity compares the code representation.
For example, clone detection approaches consider that a target piece of binary code is a clone of some source binary code if their syntax are similar.
Structural similarity compares graph representations of binary code
(e.g., control flow graphs, call-graphs).
It sits between syntactic and semantic similarity.
The intuition is that the control flow of the binary code captures
to some extent its semantics, e.g., the decisions taken on the data.
Furthermore, the graph representation captures
multiple syntactic representations of the same functionality.
However, it is possible to modify the graph structure without affecting
the semantics, e.g., by inlining functions.
Semantic similarity compares the code functionality.
A simple approach to semantic similarity compares
the interaction between the program and its environment through OS APIs or
system calls.
But, two programs with similar system calls can perform significantly
different processing on their output,
so more fine-grained semantic similarity approaches focus on a
syntax-independent comparison of the code.
Generally speaking, the more robust an approach is,
i.e., the more transformations it can capture,
the more expensive it also is.
Syntactic similarity approaches are cheapest to compute,
but least robust.
They are sensitive to simple changes in the binary code,
e.g., register reallocation, instruction reordering,
replacing instructions with semantically equivalent ones.
Structural similarity sits in the middle.
It is robust against multiple syntactical transformations,
but sensitive to transformations that change code structure such as
code inlining or removal of unused function parameters.
Semantic similarity is robust against semantics-preserving transformations,
despite changes to the code syntax and structure,
but it is very expensive to compute for large pieces of binary code.
Comparison granularity.
Binary code similarity approaches can be applied at different granularities.
Common granularities are
instructions;
basic blocks;
functions; and
whole programs.
To perform a comparison at a coarser granularity,
some approaches use a different comparison at a finer granularity,
and then combine the finer granularity results.
For example, to compare whether two programs are similar,
an approach could determine the fraction of identical functions
between both programs.
Thus, we differentiate between the input granularity,
i.e., the granularity of the input pieces of binary code that the approach compares,
and the approach granularities,
i.e., the granularities of the different comparisons in the approach.
Applying a specific comparison at a finer granularity may restrict the type of
comparison that can be performed at a coarser granularity,
as illustrated in Figure 2.
The figure shows that computing whether two pieces of binary code are
identical at a finer granularity (e.g., basic block) can be used to
compute that the coarser granularity pieces that encompass them
(e.g., their functions) are identical, equivalent, or similar.
However, similarity at a finer granularity cannot be used to
infer that the coarser granularity code is equivalent or identical.
For example, when comparing two functions,
just because all their basic blocks are similar,
it cannot be concluded that the functions are identical or equivalent.
On the other hand, similarity is the most general type of comparison
and any finer granularity comparison type can be used to infer it.
Number of inputs.
Binary code similarity approaches can compare two or more pieces of binary code.
Those that compare more than two pieces can further be split into comparing one piece to the rest or
comparing each piece to all other pieces.
Thus, we identify three types of approaches based on the number of inputs and
how they are compared:
one-to-one ($OO$), one-to-many ($OM$), and many-to-many ($MM$).
The source of the input pieces is application-dependent.
They may come
from the same program version
(e.g., two functions of the same executable),
from two versions of the same program,
and from two different programs.
One-to-one approaches compare an original piece of binary code (also called source, old, plaintiff, or reference) to a target piece of binary code (also called new, patched, or upgrade).
Most $OO$ approaches perform binary code diffing,
i.e., they diff two consecutive, or close, versions of the
same program to identify what was added, removed, or modified in the
target (patched) version.
The granularity of binary code diffing is most often functions and the
diffing tries to obtain a mapping between a function in the original
program version and another function in the target program version.
Added functions are original functions that cannot be mapped to a
target function;
removed functions are target functions that cannot be mapped to an
original function; and
modified functions are mapped functions that are not identical.
One-to-many approaches compare a query piece of binary code to many target pieces of binary code.
Most $OM$ approaches perform binary code search,
i.e., they search if the query piece is
similar to any of the target pieces and
return the top $k$ most similar target pieces of binary code.
The target pieces may come from multiple versions of the same program
(different than the version the query piece comes from),
from different programs compiled for the same architecture, or
from programs compiled for different architectures.
In contrast to $OO$ and $OM$ approaches,
many-to-many approaches do not distinguish between source and target pieces.
All input pieces are considered equal and compared against each other.
These approaches typically perform binary code clustering,
i.e., they output groups of similar pieces of binary code called clusters.
III Scope & Paper Selection
To keep our survey of the state-of-the-art focused and manageable
it is important to define what is, and what is not, within scope.
Overall, the main restriction is that we focus on works that compare
binary code.
This restriction, in turn, introduces the following four constraints:
1.
We exclude approaches that require access to the source code,
namely source-to-source (e.g., [43]) and
source-to-binary (e.g., [44]) similarity approaches.
2.
We exclude approaches that operate on bytecode
(e.g., [45, 46]).
3.
We exclude behavioral approaches that compare similarity exclusively
on the interaction of a program with its environment through
system calls or OS API calls
(e.g., [47, 48, 49, 50]).
4.
We exclude approaches that consider binary code as a sequence of
raw bytes with no structure such as
file hashes (e.g., [51]),
fuzzy hashes (e.g., [52, 53]), and
signature-based approaches (e.g., [54, 55]).
Approaches need to disassemble raw bytes into instructions to be considered.
While we do not include the papers describing byte-level approaches,
we do examine the use of some of those techniques (e.g., fuzzy hashing)
by the analyzed approaches.
In addition, we introduce the following constraints to keep the
scope of the survey manageable:
5.
We limit the survey to papers published on peer-reviewed venues
and technical reports from academic institutions.
Thus, we do not analyze tools,
but rather the research works describing their approach
(e.g., [26, 27] for BinDiff).
6.
We exclude papers that do not propose a new binary code similarity approach or technique,
but simply apply off-the-shelf binary code similarity tools as a step
towards their goal.
Paper selection.
To identify candidate papers, we first systematically
examined all papers published in the last 20 years in 14 top venues
for computer security and software engineering:
IEEE S&P, ACM CCS, USENIX Security, NDSS, ACSAC, RAID, ESORICS, ASIACCS, DIMVA, ICSE, FSE, ISSTA, ASE, and MSR. Not all relevant binary code similarity approaches have been published in those venues,
which is especially true for early approaches.
To identify candidate papers in other venues,
we extensively queried specialized search engines such as Google Scholar
using terms related to binary code similarity and its applications,
e.g., code search, binary diffing, bug search.
We also carefully examined the references of the candidate papers
for any further papers we may have missed.
This exploration identified over a hundred candidate papers.
We then read each candidate paper to determine if they
proposed a binary code similarity approach that satisfied the above scope constraints.
In the end, we identified the 61 binary code similarity research works in Table I,
whose approaches are systematized.
The first three columns of Table I capture
the name of the approach,
the year of publication, and
the venue where the work was published.
The research works are sorted by publication date
creating a timeline of the development of the field.
Papers are identified by their system name, if available, otherwise
by the initials of each author’s last name and the year of publication.
The 61 papers have been published in 37 venues.
Binary code similarity is quite multidisciplinary;
while most papers appear in computer security venues (36 papers in 20 venues),
there are works in software engineering (13 papers in 8 venues),
systems (6 papers in 4 venues), and
machine learning (2 papers in 2 venues).
The venues with most binary code similarity papers are:
DIMVA (6), ASE (4), CCS (3), USENIX Security (3), and PLDI (3).
IV Applications
This section motivates the importance of binary code similarity by
describing the applications it enables.
Of the 61 papers analyzed, 36 demonstrate an application,
i.e., present a quantitative evaluation, or case studies,
of at least one application.
The other 23 papers present generic binary code similarity capabilities
that can be used for multiple applications such as
binary diffing tools (e.g., [84, 85, 86]),
binary code search platforms (e.g., [72, 62, 71]), and
binary clone detection approaches
(e.g., [74, 73, 56, 65]).
Table II summarizes the eight applications identified.
Most of the 36 papers demonstrate a single application, although a few
(e.g., F2004, BinSequence)
demonstrate multiple.
One property of an application is whether the application compares
different versions of the same program
(patch analysis, patch generation, porting information, malware lineage),
different programs (malware clustering, malware detection,
software theft detection),
or can be applied to both cases (bug search).
Next, we detail those applications.
1.
Bug search –
Arguably the most popular application of binary code similarity is finding a known bug in a large repository of target
pieces of binary code [2, 3, 4, 6, 11, 12, 9, 8, 7, 15, 5, 10, 13, 14, 1].
Due to code reuse, the same code may appear in multiple programs,
or even in multiple parts of the same program.
Thus, when a bug is found, it is important to identify similar code that
may have reused the buggy code and contain the same, or a similar, bug.
Bug search approaches take as input a query buggy piece of binary code and search for
similar pieces of binary code in a repository.
A variant of this problem is cross-platform bug search,
where the target pieces of binary code in the repository may be compiled for
different platforms (e.g., x86, ARM, MIPS) [3, 4, 6, 12, 9, 7, 15, 10, 13].
2.
Malware detection –
Binary code similarity can be used to detect malware by comparing a given
executable to a set of previously known malware samples.
If similarity is high then the input sample is likely a variant of a previously known malware family.
Many malware detection approaches are purely behavioral,
comparing system or API call behaviors
(e.g., [47, 48]).
However, as described in Section III,
we focus on approaches that use
binary code similarity [19, 20, 21].
3.
Malware clustering –
An evolution of malware detection is
clustering similar, known malicious, executables into families.
Each family cluster should contain executables
from the same malicious program,
which can be different versions of the malicious program,
as well as polymorphic (e.g., packed) variants of a version.
Similar to malware detection,
we focus on approaches that compare binary code [16]
and exclude purely behavioral approaches
based on system calls and network traffic
(e.g., [50, 87, 88]).
4.
Malware lineage –
Given a set of executables known to belong to the same program,
lineage approaches build a graph where nodes are program versions and edges
capture the evolution of the program across versions.
Lineage approaches are most useful with malware because version information is
typically not available [22, 23, 24, 89].
Since input samples should belong to the same family,
malware lineage often builds on the results of malware clustering.
5.
Patch generation and analysis –
The earliest binary code similarity application, and one of the most popular,
is to diff two consecutive, or close, versions of the
same program to identify what was patched in the newer version.
This is most useful with proprietary programs where the vendor
does not disclose patch details.
The diffing produces small binary code patches that can be efficiently
shipped to update the program [25].
It can also be used to automatically identify security patches that
fix vulnerabilities [30],
analyze those security patches [26, 27, 28, 29, 31, 8], and
generate an exploit for the old vulnerable version [90].
6.
Porting information –
Binary code similarity can be used for porting information between two
close versions of the same program.
Since close versions typically share a large amount of code,
the analysis done for one version may be largely reusable for a newer version.
For example, early binary code similarity approaches ported
profiling data [32, 91] and analysis results obtained during
malware reverse engineering [26, 27].
7.
Software theft detection –
Binary code similarity can be used for identifying unauthorized reuse of code from
a plantiff’s program such as
the source code being stolen, its binary code reused,
a patented algorithm being reimplemented without license,
or the license being violated (e.g., GPL code in a commercial application).
Early approaches for detecting such infringements used software birthmarks,
i.e., signatures that capture inherent functionality of the plaintiff
program [92, 93].
However, as described in Section III,
we exclude signature-based approaches and focus on approaches using
binary code similarity [33].
V Binary Code Similarity Evolution
This section describes the origins of binary code similarity and its evolution over
the last 20 years, highlighting some noteworthy approaches.
The origins.
The origins of binary code similarity are in the problem of generating a patch (or delta)
that captures the differences between
two consecutive (or close) versions of the same program.
Text diffing tools had existed since the 1970’s
(the popular UNIX diff was released in 1974) and
had been integrated in early source code versioning systems such as
SCCS [94] (1975) and RCS [95] (1985).
The increasing popularity of low bandwidth communication networks
(e.g., wireless)
and the limited resources in some devices,
raised interest in techniques that would increase efficiency by transmitting
a small patch that captured the differences
between two versions of a binary, i.e., non-text, file,
instead of transmitting the whole file.
In 1991, Reichenberger proposed a diffing technique
for generating patches between arbitrary binary files
without any knowledge about the file structure [96].
The approach worked at the byte-level,
instead of the line-level granularity of text diffing,
and efficiently identified the
byte sequences that should appear in the patch because they only appeared
in the updated version and thus were not available in the original file
to be patched.
Several tools for generating and applying binary patches soon appeared
such as RTPatch [97],
BDiff95 [98], and
xDelta [99].
Those tools worked at byte-level and could diff any type of file.
The first binary code similarity approaches are from 1999.
That year, Baker et al. proposed an approach for compressing differences
of executable code and built a prototype diffing tool
called Exediff [25],
which generated patches for DEC Alpha executables.
Their intuition was that many of the changes when diffing two executable
versions of the same program represent secondary changes due to
the compilation process, as opposed to direct changes in the source code.
One example they mentioned is register allocation
that may change at recompilation.
Another example they mentioned is that code added in the newer version
would displace parts of the old code,
and thus the compilation process would have to adjust
pointer values in the displaced code to point to the correct addresses.
Their idea was to reconstruct secondary changes at patch time,
so that they would not need to be included in the patch,
reducing the patch size.
Exediff is the earliest approach we have identified
that focused on computing similarity between binary code,
taking advantage of the code structure by disassembling the raw bytes
into instructions.
Also in 1999, Wang et al. presented BMAT [32], a tool that aligned two versions of a Windows DLL library executable
to propagate profile information from the older (extensively profiled) version
to a newer version, thus reducing the need for re-profiling.
Their approach is the first to compare functions and basic blocks
(Exediff compared two sequences of instructions).
It first matched functions in the two executables and then
matched similar blocks within each matched function.
It used a hashing technique to compare blocks.
The hashing removed relocation information to handle pointer changes,
but was order-sensitive.
The first decade.
After the initial works of Exediff and BMAT,
we only identify 7 binary code similarity approaches in the next decade (2000-2009).
However, some of these are highly influential as
they extend binary code similarity from purely syntactical to also include semantics;
they widen the scope from binary code diffing (OO) to also include
binary code clustering (MM) and binary code search (OM);
and they apply binary code similarity to malware.
In 2004, Thomas Dullien (alias Halvar Flake) proposed a
graph-based binary code diffing approach that focused on the structural properties
of the code by heuristically constructing a callgraph isomorphism
that aligns functions in two versions of the same
binary program [26].
This is the first approach to handle instruction reordering inside a
function introduced by some compiler optimizations.
A followup work [27]
extended the approach to also match basic blocks inside matched functions
(as done in BMAT)
and introduced the Small Primes Product (SPP) hash
to identify similar basic blocks despite instruction reordering.
These two works are the basis for the
popular BinDiff binary code diffing plugin for the Ida disassembler [84].
In 2005, Kruegel et al. proposed a graph coloring technique to detect
polymorphic variants of a malware.
This is the first approach that performed semantic similarity
and MM comparison.
They categorized instructions with similar functionality into 14
semantic classes.
Then, they colored the inter-procedural control-flow graph (ICFG)
using those classes.
The graph coloring is robust against syntactical obfuscations such as
junk insertion, instruction reordering, and instruction replacement.
In 2008, Gao et al. proposed BinHunt [28]
to identify semantic differences between two versions of the same program.
This is the first approach that checks code equivalence.
It uses symbolic execution and a constraint solver to check if two basic blocks
provide the same functionality.
In 2009, Xin et al. presented Smit [16],
an approach that given a malware sample finds similar malware in
a repository. Smit is the first OM and binary code search approach.
It indexes malware callgraphs in a database and uses
graph edit distance to find malware with similar callgraphs.
The last decade.
The last decade (2010-2019) has seen a huge increase in the popularity of
binary code similarity, with 52 approaches identified.
The focus on this decade has been on binary code search approaches,
with an emphasis since 2015 on its cross-architecture version
(16 approaches),
and in recent years on machine learning approaches. In 2013, Wei et al. proposed Rendezvous,
a binary code search engine that given the binary code of a query function,
finds other functions in a repository with similar syntax and
structural properties.
Reducing the search granularity from whole programs (Smit) to
smaller pieces of binary code such as functions
enables an array of applications such as clone detection and bug search.
Most binary code search approaches target the bug search application.
This application was first addressed on source code in 2012 by
Jang et al. [100].
In 2014, David et al. proposed Tracy [1],
the first binary code search approach focused on bug search.
Tracy used the concept of tracelets,
an execution path in a CFG,
to find functions similar to a vulnerable function.
In 2015, Pewny et al. presented Multi-MH [3],
the first cross-architecture binary code search approach.
Multi-MH indexed functions by their input-output semantics.
Given a function compiled for one CPU architecture (e.g., x86)
Multi-MH can find similar functions compiled for other
architectures (e.g., MIPS).
This problem quickly gained traction due to the popularity of embedded devices.
In 2016, Lageman et al. [72] trained a neural network to decide
if two functions were compiled from the same source code.
The use of deep learning has picked up in the last two years,
e.g., $\alpha$Diff (2018), InnerEye (2019), and Asm2Vec (2019).
VI Approaches
In this section, we systematize the approaches of binary code similarity,
describing the Approach Characteristics columns in
Table I.
We recommend the reader to print Table I in
a separate page to have it in hand while reading this section.
VI-A Comparison Type
Columns: Input Comparison; Approach Comparison
This section discusses the type of comparison between the approach inputs,
as well as the finer granularity comparisons that may be used by the approach.
Input comparison.
All 61 works analyzed compare their inputs to identify similarity.
That is, no approach identifies identical inputs
(since a hash suffices for that) or input equivalence
since it is an undecidable problem [101]
that can only be solved efficiently for small pieces of binary code.
Thus, we classify binary code similarity approaches based on their input
comparison as:
one-to-one (OO, 21 approaches), one-to-many (OM, 30 approaches), and
many-to-many (MM, 10 approaches).
The dominance of $OM$ approaches is due to the high interest in binary code search in the last decade.
It is always possible to build an $OM$ or $MM$
approach from an $OO$ approach.
For example, a simple implementation of an $OM$ approach
is to compare the given query piece of binary code with each of the $n$ targets
using an $OO$ approach that returns the similarity between both inputs.
Then, simply rank the $n$ targets by decreasing similarity
and return the top $k$ entries or the entries above a similarity threshold.
However, most $OM$ approaches avoid this simple implementation
since it is inefficient.
The two main solutions to improve performance are
extracting a feature vector from each input and
storing the target pieces of binary code in a repository with indices.
Obtaining a feature vector for each input
allows to perform the feature extraction only once per input.
This offers significant benefits when the feature extraction is expensive,
e.g., in the case of Blex whose feature extraction requires executing
a piece of binary code multiple times with different inputs.
Once the feature vectors have been extracted,
a similarity metric between two feature vectors is used.
This similarity metric is typically cheap to compute as feature vectors
are often numerical or Boolean.
A common source of confusion is that some approaches propose similarity
metrics, while others propose distance metrics.
It is important to keep in mind that when the metrics are normalized
between zero and one, the distance is simply one minus the similarity.
The other solution used by $OM$ approaches is adding indices
on a subset of the features in the feature vector.
The indices can be used to reduce the number of comparisons by
applying the similarity metric only between the feature vector of
the input piece of binary code and the feature vectors of selected targets
more likely to be similar.
Approach comparison.
Most approaches use a single type of comparison:
similarity (42 approaches), equivalence (5), and identical (2).
Note that even if only one type of comparison is used in the approach,
it may differ from the input comparison.
For example, Exediff looks for identical instructions in the process of
diffing two programs.
There are 12 approaches that use multiple comparison types at
different granularities.
Of those, six use identical comparison at finer granularities
to quickly identify the same pieces of binary code (BMAT, DR2005, SWPQS2006, BinClone)
or to reduce expensive comparisons such as graph isomorphism (Smit, Spain).
The other six use equivalence comparisons at finer granularities
to capture semantic similarity.
VI-B Granularity
Columns: Input Granularity; Approach Granularities
We separate the input granularity from
the granularities of the pieces of binary code compared in the approach
(i.e., approach granularities)
since it is common to use finer granularities (e.g., functions) to
compare coarser input granularity (e.g., whole programs).
We have identified 8 comparison granularities:
instruction (I),
set of related instructions (I*),
basic block (B),
set of related basic blocks (B*),
function (F),
set of related functions (F*),
trace (T), and
whole program (P).
Instruction, basic block, function, and whole program are
standard granularities that require no explanation.
Related instructions (I*) are either consecutive (e.g., n-gram) or
share a property (e.g., data dependent).
They may belong to different basic blocks,
and even to different functions.
For example, Tracy groups instructions that collectively
impact an output variable.
Related basic blocks (B*)
share structural properties (e.g., graphlets in a CFG)
or belong to the same execution path.
Basic blocks in a set may belong to the same or multiple functions.
Related functions (F*) implement a program component such as
a library, a class, or a module.
Trace granularity compares the execution trace of two binary programs
on the same input.
The most common input granularity is
function (26 approaches) followed by whole program (25) and
related basic blocks (4).
Whole program is the preferred input granularity for $OO$ approaches
(16/21 )
since most binary code diffing approaches try to establish a one-to-one mapping
between all functions in the input programs,
and also for $MM$ (7/10 ) approaches that tend to cluster input programs.
On the other hand, function is the preferred granularity
for binary code search approaches (21/30 ).
Another four binary code search approaches use B* to
identify code reuse that covers only a subset of a function or
crosses function boundaries.
The most common approach granularity is function (30 approaches)
followed by basic block (20).
The majority of approaches (47/61) use different input and
approach granularities,
i.e., use finer approach granularities to compare coarser input granularity.
Most approaches with the same input and approach granularity perform
function searches (12/14).
The 11 approaches that perform equivalence comparisons
do so at fine granularities due to its low efficiency:
six have B granularity,
five I*, and one F. Some approaches accumulate features at a fine granularity that
are never directly compared and thus do not show in the
approach granularities column.
For instance, Gemini accumulates basic block features
to generate a numerical vector at function granularity.
Thus, only functions are compared.
VI-C Syntactic Similarity
Column: Syntactic similarity
Syntactic approaches capture similarity of the code representation,
more especifically they compare sequences of instructions.
Most commonly, the instructions in a sequence are consecutive
in the virtual address space and belong to the same function.
The instructions in the sequence may first be normalized,
e.g., considering only the mnemonic, only the opcode, or
normalizing the operands into classes.
We detail instruction normalization in Section VI-J
and simply refer to instruction sequences in the rest of this subsection.
The instruction sequences may be of fixed or variable length.
Fixed-size sequences are obtained by sliding a window
over the instruction stream,
e.g., over the linearly-ordered instructions in a function.
This process is characterized by the window size,
i.e., the number of instructions in the sequence,
and the stride, i.e., the number of instructions to slide the
start of the window to produce the next sequence.
When the stride is smaller than the window size,
consecutive sequences overlap.
When the stride is one, the resulting sequence is called an n-gram.
For example, given the sequence of instruction mnemonics
$\{mov,push,add\}$ two 2-grams will be extracted:
$\{mov,push\}$ and $\{push,add\}$.
There are 7 works that use n-grams:
Idea, MBC, Rendezvous, MutantX-S, Exposé, iLine, and Kam1n0.
Fixed-size sequences are also used by SWPQS2006 with a configurable stride larger than one.
Rendezvous, in addition to n-grams, also uses n-perms,
unordered n-grams that capture instruction reordering within the sequence.
An n-perm may capture multiple n-grams,
e.g., 2-perm $\{mov,push\}$ captures 2-grams $\{mov,push\}$ and $\{push,mov\}$.
The most common methods to compare instruction sequences are
hashing, embedding, and alignment.
Hashing is used by 6 approaches
(BMAT, DR2005, SWPQS2006, Smit, BinClone, Spain)
to obtain a fixed-length value out of a variable-length instruction sequence.
If the hash values are the same, the sequences are similar.
Five approaches generate an embedding from n-gram sequences
(Idea, MBC, MutantX-S, Exposé, Kam1n0).
Three approaches (Exediff, Tracy, BinSequence) align two sequences to
produce a mapping between them by inserting gaps in either sequence
to account for inserted, removed, and modified instructions.
These approaches define a similarity score when instructions are aligned,
and a gap score when an instruction aligns with a gap.
Other less common comparison methods are using
vectors of Boolean features (iLine) and encoding sequences as strings for indexing (Rendezvous).
VI-D Semantic Similarity
Column: Semantic similarity
Semantic similarity captures if the code being compared has similar
effects, as opposed to syntactic similarity that captures
similarity in the code representation.
The semantics of a piece of binary code can be described by the changes it produces in the
process state, i.e., updates to the content of registers and memory.
We identify 26 approaches computing semantic similarity.
Most of them capture semantics at basic block granularity
because basic blocks are straight-line code without control flow.
Three methods are used to capture semantics:
instruction classification,
input-output pairs, and
symbolic formulas.
Instruction classification.
The first approach to introduce semantics into binary code similarity was KKMRV2005,
which classified instructions into 14 classes
(e.g., arithmetic, logic, data transfer) and used a 14-bit value to
capture the classes of the instructions in a basic block.
This semantic $color$ bitvector captures the effects of the basic block.
This approach was later adopted by
BMM2006, Beagle, Fossil, and Sigma.
Instruction classification can be used to compute semantic similarity,
but cannot determine if two pieces of binary code are, or are not, equivalent.
Input-output pairs.
Intuitively, two pieces of binary code are functionally equivalent if given the same input
they produce the same output, for all possible inputs.
Such equivalence is independent of the code representation and
compares the final state after the code is executed,
ignoring intermediate program states.
This approach was proposed by Jiang et al.
on source code [102] and later used by numerous binary code similarity approaches:
BinHash, MXW2015, Blex, Multi-MH, BinGo, CACompare, Spain, KS2017 and IMF-sim.
It involves executing both pieces of binary code with the same input and
comparing their output, repeating the process many times.
If the output differs for any input, then the two pieces of binary code are not equivalent.
Unfortunately, to determine that both pieces are equivalent, the approach
would require testing all possible inputs,
which is not realistic for any non-trivial piece of binary code.
Thus, in practice, this approach can only determine
that two pieces of binary code are likely equivalent, with a confidence proportional
to the fraction of inputs that have been tested, or
that they are not equivalent (with certainty).
The tested inputs are typically selected randomly,
although it is possible to use other selection rules,
e.g., taking values from the program data section (CACompare).
It is generally a dynamic approach,
but some approaches (e.g., Multi-MH) evaluate concrete inputs on
statically-extracted symbolic formulas.
Symbolic formula.
A symbolic formula is an assignment statement in which the
left side is an output variable and the
right side is a logical expression of input variables and literals
that captures how to derive the output variable.
For instance, the instruction add %eax,%ebx
can be represented with the symbolic formula
EBX2 = EAX + EBX1
where EBX2 and EBX1 are symbols representing the values of
the EBX register before and after executing the instruction.
Eleven approaches use symbolic formulas:
BinHunt, iBinHunt, BinHash, Exposé, Tracy, RMKNHLLP2014, Tedem,
CoP, Multi-MH, Esh, and Xmatch.
Of those, eight approaches extract symbolic formulas at basic block granularity,
Xmatch and Exposé extract formulas for the return values of a function, and
BinSim extracts symbolic formulas from an execution trace that capture how
the arguments of a system call were derived.
Three methods are used to compare symbolic formulas:
using a theorem prover to check for equivalence,
comparing the semantic hash of the formulas to check for equivalence,
and computing the similarity of the graph representation of the formulas.
Theorem prover –
BinHunt introduced the idea of using
theorem provers such as STP [103] or Z3 [104]
to check if two symbolic formulas are equivalent,
i.e., whether the output variable always contains the same value
after the execution of both formulas,
assuming that the input variables share the same values.
The main limitation of this approach
is that it is computationally expensive because
it can only perform pairwise equivalence queries,
the solving time quickly increases with formula sizes, and
the solver may fail to return an answer for some queries.
Note that a piece of binary code may have multiple outputs
(registers and variables in memory),
each represented by its own symbolic formula.
These approaches need to try all pair-wise comparisons
and check if there exists a permutation of variables
such that all matched variables contain the same value.
Semantic hashes –
An alternative to using a theorem prover is to check if two symbolic formulas
have the same hash, after normalizing the formulas
(e.g., using common register names) and
simplifying them (e.g., applying constant propagation).
The intuition is that if the two symbolic formulas have the same hash
they should be equivalent.
Three approaches use semantic hashes: BinHash, BinJuice, and GitZ.
Semantic hashes are efficient, but are limited in that
it is possible for two equivalent formulas to have different hashes
even after normalization and simplification.
For example, reordering of symbolic terms in one of the formulas
(e.g., due to instruction reordering) results in a different hash.
Graph distance –
Xmatch and Tedem represent the symbolic formula of a basic block as a tree,
and compute their similarity by applying graph/tree edit distance.
Computing the graph/tree edit distance is more expensive than
comparing semantic hashes, but the graph representation has the advantage
over semantic hashes that it can handle term reordering.
VI-E Structural Similarity
Column: Structural similarity
Structural similarity computes similarity on graph representations of
binary code.
It sits between syntactic and semantic similarity since
a graph may capture multiple syntactic representations of the same code and
may be annotated with semantic information.
Structural similarity can be computed on different graphs.
The three most common are the intra-procedural control flow graph (CFG),
the inter-procedural control flow graph (ICFG),
and the callgraph (CG).
All three are directed graphs.
In the CFG and ICFG, nodes are basic blocks and
an edge indicates a control flow transition (e.g., branch, jump).
Basic blocks in a CFG belong to a single function;
each function has its own CFG.
Basic blocks in the ICFG belong to any program function;
there is one ICFG per program.
In the CG, nodes are functions and
edges capture caller-callee relationships.
The intuition behind structural approaches is that CG, ICFG, and CFGs
are fairly stable representations whose structure varies little
for similar code.
Approaches that operate on the CG or ICFG have a
whole program input granularity,
while those that operate on CFGs may have function granularity,
or use function similarity as a step towards whole program similarity.
Structural similarity approaches may use labeled graphs.
For example, F2004 and DR2005 use node labels in the CG to capture
structural information about a function’s CFG
(e.g., number of instructions and edges).
Other approaches label basic blocks in the CFG/ICFG with a feature vector that
captures the semantics of the basic clock (e.g., KKMRV2005, BinJuice)
or its embedding (Genius, Gemini, see §VI-F).
Edge labels can be used
to capture the type of control flow transfer (BMM2006)
or to aggregate the semantic labels of source and destination nodes (Fossil).
Structural similarity is used by 27 approaches.
14 approaches operate only on CFGs;
five on both CFGs and the CG;
four only on the ICFG; and
two only on the CG.
There are also three approaches that use non-standard graphs:
Sigma proposes a semantic integrated graph
that combines information from the CFG, CG, and register flow graph,
while QSM2015 and libv use the
execution dependence graph [105].
The remainder of this subsection discusses different approaches
used to compute graph similarity.
(Sub)Graph isomorphism –
Most structural similarity approaches check for variations of graph isomorphism.
An isomorphism of two graphs $G$ and $H$ is an edge-preserving bijection $f$
between their node sets such that if any two nodes $u$, $v$ are adjacent in $G$,
then $f(u)$ and $f(v)$ are also adjacent in $H$.
Graph isomorphism requires that the node set cardinality
is the same in both graphs, which is too strict for binary code similarity.
Thus, approaches instead check for subgraph isomorphism,
which determines if $G$ contains a subgraph isomorphic to $H$.
Subgraph isomorphism is a known NP-complete problem.
Other approaches check for the maximum common subgraph isomorphism (MCS),
which finds the largest subgraph isomorphic to two graphs and is also
NP-Complete.
Given the high complexity of both problems, approaches try to reduce the
number of graph pairs that need to be compared,
as well as the size of the compared graphs.
For example, DR2005 avoids comparing CFGs with the same hash
(match) and CFGs with very different number of nodes and edges
(unlikely match).
iBinHunt reduces the number of nodes to consider
by assigning taint labels to basic blocks.
Only nodes with the same taint label are considered in the
subgraph isomorphism.
For candidate graph pairs that pass the filtering,
approximate algorithms are used that
can be grouped into greedy and backtracking.
Greedy –
These approaches perform neighborhood exploration.
An initial set of matching nodes is first identified.
Then, the matching is recursively expanded by checking only the neighbors
(i.e., parents or children) of already matched nodes.
BMAT, F2004, DR2005, LKI2013, Tedem, Multi-MH, KLKI2016,
Kam1n0, BinSequence, and BinArm use this approach.
A limitation of greedy algorithms is that early errors propagate,
significantly reducing the accuracy.
Backtracking –
Backtracking algorithms fix a wrong matching by revisiting the solution,
and if the new matching does not improve the overall matching it is reverted
(BMM2006, BinHunt, iBinHunt, MXW2015, QSM2015, discovRE).
Backtracking is more expensive,
but can improve accuracy by avoiding local optimal matching.
Optimization.
An alternative used by four approaches
(Smit, BinSlayer, CXZ2014, Genius)
is to model graph similarity as an optimization problem.
Given two CFGs and a cost function between two basic blocks,
they find a bijective mapping between the two CFGs with minimum cost.
Such bipartite matching ignores graph structure,
i.e., does not use edge information.
To address this, Smit and BinSlayer assign lower cost to connected
basic blocks.
To perform the matching,
Smit, BinSlayer, and CXZ2014 use the $O(n^{3})$ Hungarian algorithm,
while Genius uses a genetic algorithm.
K-subgraph matching.
KKMRV2005 proposed to divide a graph into $k$-subgraphs,
where each subgraph contains only $k$ connected nodes.
Then, generate a fingerprint for each k-subgraph and the
similarity of two graphs corresponds to the maximum number
of $k$-subgraphs matched.
Four other approaches later leveraged this approach:
Beagle, CXZ2014, Rendezvous, and Fossil.
Path similarity.
There are three approaches (CoP, Sigma, BinSequence)
that convert function similarity into a path similarity comparison.
First, they extract a set of executions paths from a CFG,
then define a path similarity metric between execution paths, and
finally combine the path similarity into a function similarity.
Graph embedding.
Another method used by Genius, VulSeeker, and Gemini,
detailed in §VI-F,
is to extract a real-valued feature vector from each graph and then
compute the similarity of the feature vectors.
VI-F Feature-Based Similarity
Column: Feature-based, Machine learning
A common method (28 approaches) to compute similarity is to
represent a piece of binary code as a vector or a set of features
such that similar pieces of binary code have similar
feature vectors or feature sets.
A feature captures a syntactic, semantic, or structural property
of the binary code.
Features can be Boolean, numeric, or categorical.
Categorical features have discrete values,
e.g., the mnemonic of an instruction.
A feature vector typically has all numeric or all Boolean features,
the latter is called a bitvector.
Categorical features are typically first encoded into Boolean features using
one-hot encoding or into real-valued features using an embedding.
Of the 28 approaches, 21 use numeric feature vectors,
six use feature sets, and
BinClone uses bitvectors.
Once the features have been extracted,
a similarity metric between feature vectors or feature sets
is used to compute the similarity. Common similarity metrics are
the Jaccard index for feature sets,
dot product for bitvectors,
and the Euclidean or cosine distance for numeric vectors.
Figure 3 shows two alternative methods for
feature-based similarity.
The top method comprises of two steps:
feature selection and feature encoding.
Feature selection is a manual process
where an analyst uses domain knowledge to identify representative features.
The alternative approach showed below is learning to automatically
generate real-valued feature vectors,
called embeddings, from training data.
Embeddings are used by eight recent approaches
(Genius, Gemini, $\alpha$Diff, VulSeeker, RLZ2019, InnerEye, Asm2Vec, SAFE).
Embeddings are used in natural language processing (NLP)
to encode categorical features using real-valued numbers,
which helps deep learning algorithms by reducing the dimensionality
and increasing the density of feature vectors compared to one-hot encoding.
Embeddings enable automatic feature extraction
and efficient similarity computation.
But, features in embeddings do not provide information about what
has been learnt.
Binary code similarity embeddings can be classified by the properties they captured
and their granularity.
The first binary code similarity approach using embeddings was Genius,
later followed by VulSeeker and Gemini.
All the three approaches build a graph embedding for the ACFG of a
function, i.e., a CFG with nodes annotated with selected basic block features.
While Genius uses clustering and graph edit distance to compute the embedding,
VulSeeker and Gemini improve efficiency by training a neural network that avoids
expensive graph operations.
Later approaches
($\alpha$Diff, RLZ2019, InnerEye, Asm2Vec, SAFE)
avoid manually selected features by focusing on instruction,
or raw byte, co-ocurrencence.
In NLP, it is common to extract a word embedding that captures word co-occurrence
(e.g., word2vec) and then build a sentence embedding that builds upon it.
RLZ2019 and InnerEye use an analogous approach by considering
instructions as words and basic blocks as sentences.
SAFE uses similar approach to create functions embedding than basic blocks.
Also related is Asm2Vec that obtains a function embedding by
combining path embeddings capturing instruction co-occurrence
along different execution paths in the function.
Instead of using instruction co-ocurrence, $\alpha$Diff computes a function embedding directly from the sequence of raw bytes
of a function using a convolutional network.
Machine learning.
We identify three uses of machine learning in binary code similarity approaches:
(1) to generate an embedding as explained above,
(2) to cluster similar pieces of binary code using unsupervised learning (BinHash, MutantX-S, iLine, RMKNHLLP2014, KLKI2016, Genius), and
(3) to classify with a probability if the pieces of binary code are being compiled from the same source code
(BinDnn, IMF-sim).
BinDnn is the first use of neural networks for binary code similarity.
Instead of generating an embedding,
BinDnn directly uses a neural network classifier to determine if two functions
are compiled from the same source code.
Surprisingly, BinDnn is not cited
by later binary code similarity approaches including those using neural networks to build embeddings.
VI-G Hashing
Column: Locality sensitive hashing
A hash is a function that maps data of arbitrary size to a fixed-size value.
Hash values are compact to store, efficient to compute, and
efficient to compare, which makes them great for indexing.
Hashes operate at the raw-byte level.
They are not especifically designed for binary code,
but rather for arbitrary binary data.
However, three classes of hashes have been used for binary code similarity:
cryptographic hashes, locality-sensitive hashes, and executable file hashes.
Cryptographic hashes capture identical, rather than similar, inputs.
They are used by some binary code similarity approaches to quickly identify duplicates at
fine granularity (e.g., basic block).
Locality-sensitive hashes produce similar hash values for similar inputs,
as oppossed to cryptographic hashes where a small difference in the input
generates a completely different hash value.
Executable file hashes take as input an executable file,
but the hash computation only considers parts of the executable
such as the import table or selected fields of the executable’s header.
Their goal is to output the same hash value for polymorphic variants
of the same malware.
Locality sensitive hashing (LSH).
LSH produces similar hash values for similar inputs,
efficiently approximating a nearest neighbor search.
LSH algorithms typically apply multiple hash functions on the given input
and produce the same hash value for similar inputs,
i.e., they increase collision probability for similar inputs.
LSH is used in binary code similarity to boost performance.
For example, it is used by $OM$ approaches for indexing pieces of binary code,
enabling efficient binary code search (Kam1n0, Gemini, InnerEye).
Of the 11 approaches that use LSH, seven use MinHash [106]
(BinHash, Multi-MH, Genius, BinSequence, BinShape, CACompare, BinSign),
two do not specify the algorithm used (Gemini, InnerEye),
SWPQS2006 uses the algorithm by Gionis et al. [107],
and Kam1n0 proposes its own Adaptive Locality Sensitive Hashing (ALSH).
Fuzzy hashing is a popular type of LSH used
to compute similarity of arbitrary files.
For example, the VirusTotal file analysis service [108]
reports ssdeep [52] hashes for submitted files.
Other fuzzy hashes include tlsh [53],
sdhash [109], and
mrsh-v2 [110].
None of the 61 approaches use them,
but we briefly discuss them because
they are often applied to executable files.
When applied on executables, fuzzy hashes may capture similarity
of the binary code, but also similarity of data present in the executables.
This issue has been recently examined by Pagani et al. [111].
Their results show that when applied only on the code of the
.text section they work significantly worse than
when used for whole program similarity.
In fact, they show that a byte change at the right position,
extra nop instructions, and instruction swapping
can degrade similarity significantly (in some cases bring it down to zero).
They observe that when compiling the same source code with
different optimizations,
the data sections of the executables remain the same,
which seems to be a key reason fuzzy hashes work better on whole executables.
Executable file hashes.
This class of hashes are especifically designed for executable files.
They are designed to output the same hash value for
polymorphic variants of the same malware.
The hash computation only considers parts of the executable that are less
likely to change when simply repacking, or resigning, the same executable.
They are not used by any of the 61 approaches,
but we briefly describe three popular hashes for completeness.
peHash [51] hashes selected fields of a PE executable
that are less susceptible to changes during compilation and packing,
e.g., initial stack size, heap size.
ImpHash hashes the import table of an executable.
Since the functionality of packed variants is the same, their imported
functions should be the same as well.
Unfortunately, it gives false positives with unrelated executables packed
with the same packer, if the packer reconstructs the original import table
at runtime.
Authentihash is the hash of a PE executable ignoring its
Windows Authenticode code signing data.
It enables identifying the same executable signed by different publishers.
It hashes the whole PE executable except three pieces:
the Authenticode data, pointers to the Authenticode data, and
the file checksum.
VI-H Supported Architectures
Column: Cross-architecture
A cross-architecture approach can compare pieces of binary code for
different CPU architectures, e.g., x86, ARM, and MIPS.
This differs from architecture-independent approaches (e.g., F2004)
that support different architectures,
but cannot cross-compare among them,
i.e., they can compare two x86 inputs and two MIPS inputs,
but cannot compare an x86 input with a MIPS input.
There are 16 cross-architecture approaches, all proposed since 2015.
A common application is given a buggy piece of binary code,
to search for similar pieces of binary code, compiled for other architectures,
which may also contain the bug.
For example, to search for programs in firmware images
where a version of OpenSSL vulnerable to Heartbleed
has been statically compiled.
The code syntax for different architectures may significantly differ as
they may use separate instruction sets with different instruction mnemonics,
sets of registers, and default calling conventions.
Thus, cross-architecture approaches compute semantic similarity.
Cross-architecture approaches employ one of two techniques.
Seven approaches lift the
binary code to an architecture-independent intermediate representation (IR):
Multi-MH, MockingBird, BinGo, Xmatch, CACompare, GitZ, FirmUp.
Then, identical analysis can be peformeed on the IR,
regardless of the origin architecture.
The advantage is that the analysis only depends on the IR and the IR design
can be outsourced to a separate group.
Section VII details the specific architectures
supported by each approach and the IRs they use.
An alternative approach used by 9 approaches is to
use feature-based similarity (discussed in §VI-F).
These approaches use a separate module for each architecture to obtain
a feature vector that captures the semantics of the binary code
(discovRE, BinDnn, Genius, Gemini, $\alpha$Diff, VulSeeker, RLZ2019,
InnerEye, SAFE).
VI-I Type of Analysis
Column: Static analysis; Dynamic analysis; Dataflow analysis
Binary code similarity approaches can use static analysis, dynamic analysis, or both.
Static analysis examines the disassembled binary code, without executing it.
Instead, dynamic analysis examines code executions by running the
code on selected inputs.
Dynamic analysis can be performed online,
as the code executes in a controlled environment,
or offline on traces of the execution.
Fifty one approaches use only static analysis,
four use only dynamic analysis, and six combine both.
The dominance of static analysis for binary code similarity is due to most applications
requiring all the input code to be compared.
This is easier with static analysis as it provides complete code coverage.
Dynamic analysis examines one execution at a time and
can only determine similarity of the code run in that execution.
To increase the coverage of dynamic analysis,
three approaches (iBinHunt, Blex, IMF-sim) run the code on multiple inputs
covering different execution paths and combine the results.
However, it is infeasible to run any non-trivial piece of binary code on inputs that cover all possible execution paths.
One advantage of dynamic analysis is simplicity,
as memory addresses, operand values, and control-flow targets
are known at runtime, which sidesteps static analysis challenges such as
memory aliasing and indirect jumps.
Another advantage is that it can handle some obfuscations and does not
require disassembly of the code,
also difficult with obfuscated code [112].
Section VIII details which approaches have evaluated its
robustness on obfuscated code.
Overall, dynamic analysis has been used in binary code similarity for
malware unpacking (Smit, Beagle, MutantX-S, CXZ2014, BinSim),
for operating on trace granularity (BinSim, KS2017), and for
collecting runtime values for semantic similarity (Blex, BinSim, IMF-sim).
Dataflow analysis is a common type of analysis that
examines how values propagate through the code.
It comprises of data sources to be tracked
(e.g., registers or memory locations holding specific variables),
propagation rules defining how values are propagated by
different instructions or IR statements, and sinks,
i.e., program points where to check the values reaching them.
Of the 19 approaches that use dataflow analysis,
16 use symbolic execution to extract a set of symbolic formulas
to compute semantic similarity (§VI-D).
Spain uses taint analysis to summarize the patterns of
vulnerabilities and their security patches.
And, iBinHunt uses both taint analysis and symbolic execution.
It first uses taint analysis as a filter to find pieces of binary code that process the
same user input, restricting the expensive subgraph isomorphism computation
to those with the same taint label.
And, it computes basic block similarity using symbolic formulas.
IMF-sim uses backward taint analysis to infer pointer
arguments of a function from dereference errors.
VI-J Normalization
Column: Normalization
Syntactic similarity approaches often normalize instructions,
so that two instructions that are normalized to the same form
are considered similar despite some syntactic differences,
e.g., different registers being used.
Overall, there are 33 approaches that use instruction normalization.
They apply the following three types of instruction normalization:
•
Operand removal –
A normalization used by nine approaches is to abstract an
instruction only by its mnemonic or opcode, ignoring all operands.
For example, $add\,\%eax,\%ebx$ and $add\,[\%ecx],\%edx$ would be both
represented by $add$ and considered similar, despite both using
different operands.
•
Operand normalization –
A normalization used by 17 approaches is to replace
instruction operands with symbols that capture the operand type
such as REG for register, MEM for memory, and
IMM for immediate values. For example, $add\,\%eax,\%ebx$ and $add\,\%ecx,\%edx$ would be both
represented as $add\,REG,REG$,
matching the instructions despite different register allocations
used by the compiler. Operand normalization abstracts less than operand removal.
For example, $add\,[\%ecx],\%edx$ would be represented
as $add\,MEM,REG$ and thus be considered different from the above.
Some approaches also use different symbols for
general purpose registers and segment registers, and
for operands of different sizes (e.g., RegGen8 and RegGen32 in BinClone).
•
Mnemonic normalization –
A normalization used by 3 approches (BMAT, Exposé and iLine) is to
represent multiple mnemonics by the same symbol.
For example, both BMAT and iLine represent all conditional jumps
(e.g., je, jne) with the same symbol to account for the compiler
modifying jump conditions.
Another type of normalization is to ignore code that should not
affect the semantics.
For example, some instruction sets contain no-op instructions
that do not change the process state.
And, compilers often use no-op equivalent instructions for padding
such as instructions that move a register to itself,
e.g., $mov\,\%eax,\%eax$.
No-op equivalent instructions do not matter for semantic similarity and
structural similarity approaches,
but since they change the syntax they may affect syntactic
similarity approaches.
Three approaches remove no-op instructions (iLine, MXW2015, QSM2015).
A few approaches also remove unreachable dead code (BMM2006, FirmUp),
which may be introduced by obfuscations;
function epilogue and prologue (Exposé) instructions,
which may make small unrelated functions to look similar; and
functions added by the compiler to load the program,
not present in the source code (BinGo, Sigma).
VII Implementations
This section systematizes the implementation of the 61 approaches.
For each approach, Table III shows
the static and dynamic platforms it builds on,
the programming language used to implement the approach,
whether the implementation supports distributing the analysis,
the supported target program architectures and operating systems, and
how the approach is released.
A dash (-) indicates that we could not find the information (i.e., unknown),
while a cross ( ✗) means unsupported/unused.
Building a binary code similarity approach from scratch requires significant effort.
Thus, all approaches build on top of previously
available binary analysis platforms or tools, which provide functionality
such as disassembly and control flow graphs for static analysis, or
instruction-level monitoring for dynamic analysis.
However, the implementation of an approach may only use
a subset of the functionality offered by the underlying platform.
The most popular static platform is Ida (42 approaches),
followed by BAP (4), DynInst (3), and McSema (2).
Ida main functionalities are disassembly and building control flow graphs.
Its popularity comes from supporting a large number of architectures.
Some binary analysis platforms already support using Ida as their
disassembler, so it is not uncommon to combine Ida and another platform
(6 approaches).
Among dynamic approaches,
PIN is the most popular with 5 approaches,
followed by Temu and Anubis with 3 approaches each.
Previous work has analyzed binary analysis platforms used by binary code
type inferece approaches [35].
Since most platforms overlap,
we refer the reader to that work for platform details,
but provide an extended version of their table in the Appendix
(Table V) with six extra platforms.
A few approaches build on top of previous binary code similarity approaches.
One case is that both approaches have overlapping authors.
For example, RMKNHLLP2014 is based on BinJuice,
and MXW2015 extends iBinHunt, which already shared components with BinHunt.
The other case is using a previously released approach.
For example, BinSlayer and Spain use BinDiff as a first stage filter to
find matched functions.
The most popular programming language is Python (20 approaches),
followed by C++ (14).
One reason for this is that the majority of approaches use Ida for disassembly
and Ida supports both C++ and Python plugins.
Moreover, two approaches (Kam1n0, BinSign) have used distributed
platforms, such as Hadoop, to distribute their analysis.
Binary code analysis can operate directly on a particular instruction set
(e.g., x86, ARM) or convert the instruction set into an
intermediate representation (IR).
Using an IR has two main advantages.
First, it is easier to reuse the analysis built on top of an IR.
For example, supporting a new architecture only requires adding a new front-end
to translate into the IR, but the analysis code can be reused.
Second, complex instruction sets (e.g., x86/x86-64) can be converted
into a smaller set of IR statements and expressions,
which make explicit any side-effects such as implicit operands or
conditional flags.
There are 15 approaches that use an IR and most use the IR
provided by the underlying platform.
Out of 15 approaches, 5 use VINE provided by BitBlaze (BinHunt, iBinHunt, CoP, MXW2015, BinSim),
another 5 use VEX provided with Valgrind (Multi-MH, MockingBird, CACompare, GitZ, FirmUp), and
two use LLVM-IR (Esh, Xmatch).
The remaining three approaches use
SAGE III (SWPQS2006), METASM (Tedem) and REIL (BinGo).
It is surprising that only 6 of the 16 cross-architecture approaches use an IR
(Multi-MH, Xmatch, MockingBird, CACompare, GitZ, FirmUp).
The remaining approaches provide separate analysis modules for each architecture
that typically output a feature vector with a common format.
The most supported architectures for the target programs to be analyzed
are x86/x86-64 (59 approaches),
followed by ARM (16), and MIPS (10).
The only two approaches that do not support x86/x86-64 are Exediff that targets Digital Alpha, and BinArm that targets ARM.
There are 16 cross-architecture approaches and 9
approaches that support firmware.
The first cross-architecture approach was Multi-MH,
which added support for ARM in 2015.
Since then, ARM support has become very prevalent due to the popularity
of mobile and IoT devices.
It is worth noting that even if there are 42 approaches that use Ida,
which supports more than 60 processor families,
most approaches built on top of Ida only analyze x86/x86-64 programs.
The most supported OS is Linux (41 approaches)
followed by Windows (35).
Only 4 approaches support MacOS.
Early approaches that focused on x86/x86-64 often used Ida to obtain support
for both PE/Windows and ELF/Linux executables.
Most recently, all approaches that leverage ARM support Linux,
which is used by Android and also by many IoT devices.
Of the 61 approaches, only 12 are open source
(BinSlayer, Tracy, QSM2015, Esh, Genius, Kam1n0, Gemini, Asm2Vec,
VulSeeker, RLZ2019, InnerEye, SAFE).
DR2005 was implemented in the BinDiff commercial tool,
which is now available as a free binary.
The remaining approaches have not been released in any form,
although the platforms they build on may be open source.
Moreover, 4 of the 12 open-source approaches (Esh, Genius, RLZ2019, InnerEye)
have partially released their source code and machine learning models.
VIII Evaluations
This section systematizes the evaluation of the 61 binary code similarity approaches.
For each approach, Table IV summarizes the
datasets used (§VIII-A) and
the evaluation methodology (§VIII-B).
VIII-A Datasets
The left side of Table IV describes the datasets used
by each approach.
It first shows
the total number of executables used in the evaluation and
their split into benign and malicious executables.
Executables may come from different programs or correspond to
multiple versions of the same program,
e.g., with varying compiler and compilation options.
Then, for approaches that have function granularity, it captures the total
number of functions evaluated,
and for approaches that analyze firmware,
the number of images from where the executables are obtained.
A dash (–) in a column means that we could not find the number in the paper.
For example, SWPQS2006 evaluates on system library files in Windows XP,
but the total number of executables is not indicated.
Most approaches use custom datasets,
the one popular benchmark is Coreutils used by 18 approaches.
In addition, approaches that evaluate on firmware use
two openly available firmwares
(ReadyNAS [113] and DD-WRT [114]).
Over half of the approaches evaluate on
less than 100 executables,
7 on less than 1K,
8 on less than 10K, and
only 8 on over 10K.
Out of 61 approaches,
37 have evaluated only on benign programs,
8 only on malware, and 16 on both.
This indicates that binary code similarity is also popular for malware analysis.
However, of the 24 approaches evaluated on malware,
only five use packed malware samples
(Smit, Beagle, MutantX-S, CXZ2014, BinSim).
These approaches first unpack the malware using a
custom unpacker (Smit) or a generic (write-and-execute) unpacker
(Beagle, MutantX-S, CXZ2014, BinSim), and then compute binary code similarity.
The rest have access to the malware’s source code or to
unpacked samples.
For binary code search approaches that use function granularity,
the number of functions in the repository better captures the dataset size.
The largest dataset is by Genius,
which evaluates on 420M functions extracted from 8,126 firmwares.
Prior approaches had evaluated on at most 0.5M functions,
which demonstrates the scalability gains from its embedding approach.
Five other approaches have evaluated on over 1M functions:
Fossil (1.5M),
BinSequence (3.2M),
BinArm (3.2M),
FirmUp (40M), and
Gemini (420M), which uses the same dataset as Genius.
In addition, InnerEye has evaluated on a repository of 1.2M basic blocks.
VIII-B Methodology
The right side of Table IV describes four aspects of
the evaluation methodology used by each approach:
robustness,
accuracy,
performance,
and comparison with prior approaches.
Robustness.
The first 8 columns on the right side of Table IV
capture how authors evaluate the robustness of their approaches,
i.e., their ability to capture similarity despite transformations
applied to the input programs.
First, it shows whether they use each of the four compilers we have
observed being employed to compile the programs in the dataset:
GCC, ICC, Visual Studio (MSVS), and Clang.
Then, it captures whether the authors evaluate similarity between
programs compiled with different compilation options (cross-optimization),
between programs compiled with different compilers (cross-compiler), and
between programs compiled for different architectures (cross-architecture).
Finally, it captures whether the authors evaluate similarity when
obfuscation transformations are applied to the input programs.
There are 34 approaches that evaluate robustness
(at least one ✓in the last four robustness columns) and 27 that do not.
Many early works did not evaluate robustness.
This evaluation has become increasingly popular as approaches mature.
The most popular robustness evaluation is
cross-optimization (23 approaches), followed by
cross-compiler (19), cross-architecture (16), and obfuscation (13).
There are 9 approaches that have evaluated cross-optimization, cross-compiler, and
cross-architecture.
Approaches that evaluate cross-compiler also typically evaluate
cross-optimization, as it is a simpler case.
Simlarly, approaches that evaluate cross-architecture
typically also evaluate cross-compiler,
as cross-architecture programs may be produced using different compilers.
Note that it is possible for approaches to compile programs with multiple
compilers, but not perform cross-compiler evaluation,
i.e., not compare similarity between programs compiled with different
compilers.
There are 13 approaches that have evaluated on obfuscated programs.
Of those, two use only source code transformations (CoP, Asm2Vec),
three use only binary code transformations (KKMRV2005, Tpm, BinShape),
five use packed malware (Smit, Beagle, MutantX-S, CXZ2014, BinSim), and
three evaluate both source code and binary code transformations
(BinSim, IMF-sim, Fossil).
Accuracy evaluation and comparison.
There are 49 approaches that perform a quantitative evaluation
of their accuracy using some ground truth (✓), and
12 that perform qualitative accuracy evaluation through case studies ( ✗).
Quantitative evaluation most often uses standard accuracy metrics such as
true positives, false positives, precision, and recall.
However, two approaches propose novel application-specific accuracy
metrics (iLine, KS2017).
There are 33 approaches that compare with prior approaches.
All of them compare accuracy and six also compare runtime.
The top target for comparison is BinDiff (13 approaches compare with it),
followed by Tracy (5), discovRE (4), and Multi-MH (4).
Comparing accuracy across binary code similarity approaches is challenging for
multiple reasons.
First, only a small fraction of the proposed approaches have publicly released
their code (Section VII).
Since most approaches are not publicly available,
comparison is often performed by re-implementing previous approaches,
which may require significant effort.
One advantage of reimplementation is that approaches can be compared
on new datasets.
The alternative to re-implementation is to evaluate the new approach on
the same dataset used by a prior approach,
and compare with the reported results.
This method is only used by 6 approaches
(Genius, BinGo, Xmatch, CACompare, Gemini, and BinArm)
likely because most datasets are custom and not publicly available.
Fortunately, we observe that public code release has become more
common in recent approaches.
Second, the input comparison and input granularity may differ among approaches
making it nearly impossible to perform a fair comparison.
For instance, it is hard to compare in a fair manner an approach that
identifies program similarity using callgraphs (e.g., Smit)
with an approach comparing basic blocks (e.g., InnerEye).
Third, even when the input comparison and input granularity are the same,
the evaluation metrics and methodology may differ,
significantly impacting the measured accuracy.
The latter challenge is best illustrated on binary code search approaches operating at
function granularity.
These approaches find the most similar functions in a repository to a
given function.
They return multiple entries ranked in descending similarity order and
count a true positive if one of the top-k most
similar entries is a true match.
Unfortunately, the values of $k$ vary across approaches and significantly
impact the accuracy,
e.g., a 98% precision on top 10 is significantly worse than a
98% precision on top 3.
Thus, it becomes hard to compare accuracy numbers obtained with different
$k$ values and it becomes tempting to raise $k$ until a sufficiently
high accuracy number is achieved.
Furthermore, many approaches do not describe the similarity threshold
used to determine that no similar entry exists in the repository.
This means that they always find some similar entry in the repository,
even if the similarity may be really low.
Performance.
It is common (49/61) to measure the runtime performance of an approach.
Runtime is typically measured end-to-end,
but a few approaches report it for each approach component (e.g., BinSim).
Four approaches report their asymptotic complexity
(BMM2006, SWPQS2006, iLine, MockingBird).
IX Discussion
This section discusses open challenges and possible future research directions.
Small pieces of binary code.
Many binary code similarity approaches ignore small pieces of binary code,
setting a threshold on the minimum number of instructions or basic blocks
to be considered.
Oftentimes, only pieces of binary code with a handful of instructions are ignored,
e.g., functions with less than 5 instructions,
but some approaches use large thresholds like 100 basic blocks in Tracy.
Small pieces of binary code are challenging because they are common,
may comprise of a single basic block that prevents structural analysis, and
may have identical syntax despite different semantics.
For example, setter functions that update the value of a field in an object
have nearly identical syntax and structure,
simply setting a memory variable with the value of a parameter.
But, they may have very different semantics,
e.g., setting a security level or updating a performance counter,
Furthermore, semantic similarity techniques like instruction classification,
symbolic formulas, and input-output pairs may fail to capture their
differences, e.g., if they do not distinguish different memory variables.
Similarity of small pieces of binary code remains an open challenge.
One potential avenue would be to further incorporate context.
Some structural approaches already consider the callgraph to match functions
(e.g., F2004), but this does not always suffice.
We believe that it may be possible to further incorporate other context
like locality (e.g., how close the binary code is in the program structure) or
data references (e.g., whether they use equivalent variables).
Source-to-binary similarity.
Some applications like plagiarism detection
may require source code to be compared with binary code.
Early approaches for source-to-binary similarity used software birthmarks
that capture inherent functionality of the
source code [92, 93].
Recently, source-to-binary similarity has been applied for
searching if a known bug in open source code exists in
some target binary code [44].
The availability of source code provides more semantics
(e.g., variable and function names, types, comments)
potentially improving accuracy compared to simply
compiling the source code and performing binary code similarity.
We believe other applications remain that require
determining if a target piece of binary code has been compiled, or has evolved,
from some given source code.
For example, there may be programs for which source code is only available
for an old version and there is a need to understand how newer binary versions
have evolved from the original source code.
Data similarity.
This survey has focused on binary code similarity,
but programs comprise both code and data.
There may be situations where the data used by the code is as important as
the code, e.g., when the only change between versions of a program is in the
data used such as changing parameters for a machine learning classifier.
Furthermore, data may be stored in complex data structures that may be
key to the functionality.
There exists a long history of type inference techniques on
binary code [35],
which we believe could be combined with binary code similarity to compare the data structures
used by different pieces of binary code, or how two pieces of binary code use the same data structure.
Semantic relationships.
A different aspect of semantic similarity is to identify binary code
with related functionality,
e.g., cryptographic or networking functions.
This is challenging as code that is related in its functionality
may not have the same semantics.
For example, a decryption function is clearly related to its
encryption function, but performs opposite operations.
So far, most work has focused on domain specific techniques such as those
for identifying cryptographic functions
(e.g., [115, 116]).
But, recently some approaches have started exploring domain-agnostic
techniques [117].
We believe further work is needed to better define such semantic relationships
and their identification.
Scalability.
The largest binary code similarity evaluation so far is by Genius and Gemini on 420M functions
from 8,126 firmware images (500K functions per image).
While that is a significant step from prior approaches,
if we consider instead 100K unique firmware,
a conservative number since it is expected that there will be
20 Billion IoT devices connected to the Internet by 2020 [118],
we need a binary code similarity approach that can handle 50 Billion functions.
Thus, further improvements on scalability will be needed to realize the
vision of binary code search engines.
Obfuscation.
Many challenges still remain for binary code similarity on obfuscated code.
For example, a recent study has shown that state of the art
unpacking techniques can miss 20%–60% of the original code,
e.g., due to incomplete function detection [89].
And, no binary code similarity approach currently handles virtualization-based packers
such as Themida [119] and VMProtect [120],
which generate a random bytecode instruction set,
transform an input piece of binary code into that bytecode instruction set, and
attach an interpreter (or virtual machine)
from the generated bytecode to native code.
Furthermore, obfuscation is best addressed with semantic similarity,
which has a challenge with obfuscation transformations that do not
respect the original semantics of the code, but still perform its main goals.
Approach comparison.
The variety of datasets and methodologies used to evaluate the approaches
(e.g., top $k$ evaluation for OM approaches),
together with the absence of source code for many of them,
makes it hard to perform a fair comparison to understand
their benefits and limitations.
We believe the field would greatly benefit from open datasets for benchmarking,
and independent validation under the same experimental conditions.
Furthermore, there is a need to improve the comparison of approaches
that handle obfuscation, beyond changes of compiler and compiler options.
Building a dataset that covers real-world obfuscations is fundamental
given the huge space of possible obfuscation transformations and that
each approach supports a different subset.
X Conclusion
During the last 20 years,
researchers have proposed many approaches to perform binary code similarity and
have applied them to address important problems such as
patch analysis, bug search, and malware detection and analysis.
The field has evolved
from binary code diffing to binary code search;
from syntactic similarity to incorporate structural and semantic similarity;
to cover multiple code granularities,
to strengthen the robustness of the comparison
(i.e., cross-optimization, cross-compiler, cross-OS, cross-architecture,
obfuscation);
to scale the number of inputs compared
(e.g., through hashing, embedding, indexing); and
to automatically learn similarity features.
Despite its popularity, the area had not yet been systematically analyzed.
This paper has presented a first survey of binary code similarity.
The core of the paper has systematized 61 binary code similarity approaches on four dimensions:
the applications they enable,
their approach characteristics,
how the approaches are implemented, and
the benchmarks and evaluation methodologies used to evaluate them.
It has discussed the advantages and limitations of
different approaches, their implementation, and their evaluation.
It has summarized the results into easy to access tables useful
for both beginners and experts.
Furthermore, it has discussed the evolution of the area and
outlined the challenges that remain open,
showing that the area has a bright future.
Acknowledgments.
This research was partially supported
by the Regional Government of Madrid through the
BLOQUES-CM (S2018/TCS-4339) grant; and
by the Spanish Government through the
SCUM (RTI2018-102043-B-I00) grant.
This research is also supported by the EIE Funds of the European Union.
All opinions, findings and conclusions, or recommendations expressed herein
are those of the authors and do not necessarily reflect the views
of the sponsors.
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Appendix
Implementation platforms.
Table V shows
the intermediate representation (IR) used by the platform,
whether it supports static and dynamic analysis,
the target architectures and operating systems it supports, and
how it is released (open source or free binary).
Among the 17 platforms, 12 support static analysis and
7 dynamic analysis.
The functionality provided by static analysis platforms widely varies.
Ida is a disassembler,
Boomerang is a decompiler, and the rest
offer diverse static analysis functionality such as disassembly,
building control flow graphs and call graphs,
IR simplifications, and data flow propagation.
All dynamic analysis platforms can run unmodified binaries
(i.e., no need to recompile or relink).
QEMU is a whole system emulator that can run a full guest OS (e.g., Windows)
on a potentially different host OS (e.g., Linux).
Dyninst, PIN, and Valgrind execute an unmodified target program with
customizable instrumentation (e.g., through binary rewriting)
on the CPU of the host system.
BAP and BitBlaze build their dynamic analysis components on top of PIN and QEMU. |
On the evaluation of the ac-Stark shift and Kramers-Heisenberg dispersion formula
R. Radhakrishnan
[email protected]
Department of Theoretical Physics,
Guindy Campus, University of Madras, Chennai-25, India.
Ramesh Babu Thayyullathil
Department of physics, Cochin university of Science and
Technology, Cochin 682022, India.
(November 21, 2020)
Abstract
We describe a unified approach for the determination of ac-Stark
correction and Kramers-Heisenberg dispersion formula. In both cases
the contribution from infinite intermediate summation appearing in
the expression for the corresponding matrix elements are evaluated
exactly in the dipole approximation for the ground state of hydrogen
atom using a variation of the Dalgarno-Lewis method. The analytical
expressions obtained can be efficiently used for the numerical
evaluation of matrix element for all values of incident photon
energy and comparison is made with results obtained by different
methods.
pacs: 32.80.Rm, 42.50.Hz
I Introduction
In the presence of intense laser field, atomic or molecular levels
are displaced (shifted or broadened) and these stimulated radiative
corrections are alternatively known as ac-Stark shift, dynamic Stark
effect or light shift cohen . Being a fundamental process
ac-stark effect is very well studied theoretically as well as
experimentally in wide area of atomic and molecular physics. This
effect is also very important in the area of laser trapping and
cooling and is a basic mechanism behind many nonlinear optical
effects bec . It is a promising tool for the field of optical
communications (Optical switching methods) and plays a very
decisive role in many current research areas
dyn ; schu ; glib ; shin ; zou ; sch ; rud ; bach ; apa ; ago ; chen ; yak . In this
work we have described an efficient method to obtain a closed form
analytical expression for two of the basic process viz. ac stark
shift and elastic scattering cross section of photons for the atomic
hydrogen in the ground state.
Our aim is to get an analytical expression for dipole dynamical
polarizability $\tau^{(2)}(\omega)$ from which one can calculate
the level shift $\delta E^{(2)}$ and level width $\Gamma^{(2)}$.
The hydrogen atom being the simplest of all atomic systems, plays
a very special role in this respect because it has closed-form
wavefunctions. Since these wave functions are analytically known,
both the dipole dynamic polarizability and Kramer’s-Heisenberg
dispersion formula for hydrogen atom can be written in a closed form.
The level shift for atomic hydrogen was previously
calculated by both perturbative and nonperturbative methods
pan ; maq:83 ; kar:04 ; bak ; chu:coo ; gon:89 ; maq:90 ; sha:tan ; ben:kri ; flo:pat .
In the perturbative method, evaluation of the higher order
matrix element is the key problem and it is achieved through
Greens function formalism pan ; maq:83 ; kar:04 or by
solving a system of second-order differential equations
gon:89 .
The Kramers-Heisenberg formula alternatively known as
dispersion formula plays a very important role in the determination of
scattering cross-sections hei .
Considering its importance as an age-old relation
there were only very few attempts to evaluate it
analytically mit ; gav . There were some generalization
of it and alternate forms are also available in the
literature hea:woo ; hol ; bre ; tul .
Recently a low-energy expansion of the Kramers-Heisenberg
formula for Rayleigh scattering process in atomic hydrogen was reported lee .
Similar to the ac Stark shift this also involves the
evaluation of the second order matrix element.
The exact calculation of higher order process
in the perturbative formulation is nontrivial because of the presence
of infinite summation over intermediate states in the expression
for the higher order matrix element. In our approach the matrix element
containing infinite summation over the whole hydrogenic
spectrum (discrete and continuum) is performed by using an
implicit summation technique due to Dalgarno Lewis dar:lew ,
which reduces the evaluation of the infinite summation
to finding a solution of some inhomogeneous differential
equation. The closed form expression which we have obtained
is very simple and also very convenient for analytical
continuation. Thus with this method, we can very easily obtain
the relevant matrix element for radiation with energy larger
than ionization energy (above threshold ionization),
while other methods need some kind of
approximation like Pade approximation in the
case of Coulomb Greens function formalism.
II Radiative correction in the ground state
The level shift depends on the intensity $I$ and the frequency $\omega$ of the radiation and the complex second order shift
$\Delta^{(2)}$ is given by maq:83 ; kar:04 ; chu:coo ; gon:89
$$\Delta^{(2)}(\omega)=\delta E^{(2)}-i\Gamma^{(2)}=-\frac{I}{I_{0}}\tau^{(2)}(\omega)$$
(1)
where the real part $\delta E^{(2)}$ is the energy shift and the
imaginary part $\Gamma^{(2)}$ gives the level width and
$I_{0}=7.016\times 10^{16}$ W/cm${}^{2}$ is the characteristic atomic field
strength intensity.
Here $\tau^{(2)}(\omega)$ is the dipole dynamic polarizability
and in the atomic unit it can be written as
$$\tau^{(2)}=\sum_{n}\left\{\frac{\langle g\mid\epsilon^{*}\cdot{\bf r}\mid n%
\rangle\langle n\mid\epsilon\cdot{\bf r}\mid g\rangle}{(E_{g}-E_{n}+\omega)}+%
\frac{\langle g\mid\epsilon\cdot{\bf r}\mid n\rangle\langle n\mid\epsilon^{*}%
\cdot{\bf r}\mid g\rangle}{(E_{g}-E_{n}-\omega)}\right\}$$
(2)
where $E_{g}$ is the atomic ground state energy
and $\epsilon$ is
the polarization of the radiation. First term in the bracket is
the absorption-emission term and the second is the
emission-absorption term. The infinite summation over the complete
set of intermediate state in equation (2) can be performed
exactly by defining a set of operators $F$ and $\widetilde{F}$
such that
$$\mathbf{\epsilon}\cdot{\bf r}\left|g\right\rangle=\left(FH_{0}-H_{0}F+\omega F%
\right)\left|g\right\rangle$$
(3)
$$\mathbf{\epsilon}^{*}\cdot{\bf r}\left|g\right\rangle=\left(\widetilde{F}H_{0}%
-H_{0}\widetilde{F}-\omega\widetilde{F}\right)\left|g\right\rangle$$
(4)
where $H_{0}=-\nabla^{2}/2-1/r$ is the unperturbed Hamiltonian for
atomic hydrogen and $|g\rangle=e^{-r}/\sqrt{\pi}$ is the ground
state wave function in atomic units. With these definitions and the closure relation
$\sum_{n}\mid n\rangle\langle n\mid\ =\hat{I}$ the expression in
equation (2) for dipole dynamical polarizability will be reduced to
$$\tau^{(2)}(\omega)=\langle g\mid\mathbf{\epsilon}^{*}\cdot{\bf r}F\mid g%
\rangle+\langle g\mid\mathbf{\epsilon}\cdot{\bf r}\widetilde{F}\mid g\rangle.$$
(5)
Thus the infinite summation over the intermediate states are
reduced to the determination of the operator $F$ and
$\widetilde{F}$. Using co-ordinate space representation and writing
$F=\mathbf{\epsilon}\cdot\mathbf{r}f(r)$ and
$\widetilde{F}=\mathbf{\epsilon}^{*}\cdot\mathbf{r}\widetilde{f}(r)$,
the equations (3) and (4) become
$$\displaystyle r\frac{d^{2}}{dr^{2}}f(r)+(4-2r)\frac{d}{dr}f(r)+(2\omega r-2)f(%
r)=2r$$
(6)
$$\displaystyle r\frac{d^{2}}{dr^{2}}\widetilde{f}(r)+(4-2r)\frac{d}{dr}%
\widetilde{f}(r)-(2\omega r+2)\widetilde{f}(r)=2r$$
(7)
By the method of Laplace transform inc for the
solution of differential equation we can obtain the solutions to
the above differential equations as
$$f(r)=\frac{1}{\omega}-\frac{1}{2\omega^{3}}\Phi(1,1,\lambda,r)$$
(8)
and
$$\widetilde{f}(r)=\frac{1}{\omega}+\frac{1}{2\omega^{3}}\widetilde{\Phi}(1,1,%
\widetilde{\lambda},r)$$
(9)
with
$$\Phi(p,q,\lambda,r)=\int\limits_{\lambda}^{1}ds\,e^{-r(s-1)}K(p,q,\lambda,s)$$
(10)
$$\widetilde{\Phi}(p,q,\lambda,r)=\int\limits_{1}^{\lambda}ds\,e^{-r(s-1)}%
\widetilde{K}(p,q,\lambda,s)$$
(11)
$$K(p,q,\lambda,s)=\left(\frac{1-\lambda}{1+\lambda}\right)^{\frac{1}{\lambda}}%
\left(s+\lambda\right)^{p+\frac{1}{\lambda}}\left(s-\lambda\right)^{q-\frac{1}%
{\lambda}}$$
(12)
$$\widetilde{K}(p,q,\lambda,s)=\left(\frac{\lambda-1}{\lambda+1}\right)^{\frac{1%
}{\lambda}}\left(\lambda+s\right)^{p+\frac{1}{\lambda}}\left(\lambda-s\right)^%
{q-\frac{1}{\lambda}}$$
(13)
where $\lambda=\sqrt{1-2\omega}$ and
$\widetilde{\lambda}=\sqrt{1+2\omega}$. In this work we obtain the solutions
with integer values of $p$ and $q$. But in general for the purpose of analytic
continuation it can be complex and $\lambda$ also can become complex depending
on the value of the frequency $\omega$.
Now using equations (10) and (11)
in equation (5) the final form of the dipolar polarizability
becomes
$$\tau^{(2)}(\omega)=\frac{2}{3\omega^{3}}\int\limits_{0}^{\infty}dr\,e^{-2r}\,r%
^{4}\{\widetilde{\Phi}(1,1,\widetilde{\lambda},r)-\Phi(1,1,\lambda,r)\}$$
(14)
The limit $\omega\rightarrow 0$ of the expression for $\tau^{(2)}(\omega)$
can be shown to approach the value corresponding to the static dipolar
polarizability which is $9/2$.
III Elastic scattering of photons
The differential scattering cross section for low energy
elastic scattering of photons with frequency $\omega$,
by atoms is given by the dispersion formula hei ; mit ; gav
$$\frac{d\sigma}{d\Omega}=a_{0}^{2}\left(\epsilon\cdot\epsilon^{\prime}\right)^{%
2}\mid M(\omega)\mid^{2}=a_{0}^{2}\left(\epsilon\cdot\epsilon^{\prime}\right)^%
{2}\mid 1-P(\omega)-P(-\omega)\mid^{2}~{}$$
(15)
where $a_{0}$ is the Bohr radius, $M$ is the Kramers-Heisenberg
matrix element, $\epsilon$ and $\epsilon^{\prime}$ respectively are the
initial and final polarization of photons and $P(\omega)$ in atomic
units is given as gav
$$P(\omega)=-\frac{2}{3}\sum_{n}\frac{\langle g\mid\mathbf{p}\mid n\rangle\cdot%
\langle n\mid\mathbf{p}\mid g\rangle}{E_{g}-E_{n}+\omega}.$$
(16)
Here $\mathbf{p}$ is the momentum
operator and the summation is over the complete set of states including continuum
states. It is useful to note from equation (15) that the
differential cross section for coherent scattering of photons is
just the Thompson cross section modified by the dynamic polarizability.
In a similar fashion we can consider Raman scattering where
initial and final states are different. The analytical expressions
for $P(\omega)$ were derived earlier using Schwartz and Teinman method
mit and Coulomb Greens function (CGF) formalism gav .
Using a slight variation of the formalism in the previous section
we can easily calculate $P(\omega)$. In this case the infinite
summation over the intermediate state in equation (16) is
performed by defining a set of operators $\mathbf{U}$ and
${\widetilde{\mathbf{U}}}$ such that,
$$\displaystyle\mathbf{p}\left|g\right\rangle=\left(\mathbf{U}H_{0}-H_{0}\mathbf%
{U}+\omega\mathbf{U}\right)\left|g\right\rangle$$
(17)
$$\displaystyle\mathbf{p}\left|g\right\rangle=\left(\widetilde{\mathbf{U}}H_{0}-%
H_{0}\widetilde{\mathbf{U}}-\omega\mathbf{\widetilde{U}}\right)\left|g\right\rangle$$
(18)
and the expression for $M$ will become
$$M(\omega)=1+\frac{2}{3}\left(\langle g\mid\mathbf{p}\cdot\mathbf{U}\mid g%
\rangle+\langle g\mid\mathbf{p}\cdot\mathbf{\widetilde{U}}\mid g\rangle\right)$$
(19)
Now the evaluation of the infinite summation is reduced to the
evaluation of the operators $\mathbf{U}$ and
$\widetilde{\mathbf{U}}$. To obtain a similar expression as in
the previous section, instead of going to momentum space
representation mit ; gav , we use the coordinate space
representation of $\mathbf{U}$ and $\widetilde{\mathbf{U}}$.
This is done by taking
$\mathbf{U}=\mathbf{r}u(r)$ and
$\widetilde{\mathbf{U}}=\mathbf{r}\widetilde{u}(r)$ and the
equations (17) and (18) become
$$\displaystyle r\frac{d^{2}}{dr^{2}}u(r)+(4-2r)\frac{d}{dr}u(r)+(2\omega r-2)u(%
r)=2i$$
(20)
$$\displaystyle r\frac{d^{2}}{dr^{2}}\widetilde{u}(r)+(4-2r)\frac{d}{dr}%
\widetilde{u}(r)-(2\omega r+2)\widetilde{u}(r)=2i$$
(21)
These differential equations has the same form which appeared in the
previous section and the solutions can be written as
$$u(r)=-\frac{i}{2\omega^{2}}\Phi(1,1,\lambda,r)$$
(22)
and
$$\widetilde{u}(r)=\frac{i}{2\omega^{2}}\widetilde{\Phi}(1,1,\widetilde{\lambda}%
,r)$$
(23)
Using this in the definition of $U$ and $\widetilde{U}$ and substituting it
in equation (19)
we get a closed form expression for the Kramers-Heisenberg matrix
element. It now takes the form
$$M(\omega)=1-\frac{4}{3\omega^{2}}\int\limits_{0}^{\infty}dr\,e^{-2r}\,r^{3}%
\left\{\Phi(1,1,\lambda,r)-\widetilde{\Phi}(1,1,\widetilde{\lambda},r)\right\}$$
(24)
If the incident photons are unpolarized and the polarization of
the scattered photons are not observed the differential scattering
cross section will take the standard form
$$d\sigma=r_{0}^{2}\frac{1}{2}(1+cos^{2}\theta)|M(\omega)|^{2}d\Omega$$
(25)
IV Discussion and Conclusion
The radial integrals in equations (14) and (24) can be
done exactly isg . For the numerical evaluation of various
integrals it is very convenient to define the following
$$\displaystyle I(p,q,\lambda,n)$$
$$\displaystyle=$$
$$\displaystyle\int\limits_{0}^{\infty}dr\,e^{-2r}r^{n}\,\Phi(p,q,\lambda,r)$$
$$\displaystyle=$$
$$\displaystyle n!\left(\frac{1-\lambda}{1+\lambda}\right)^{\frac{1}{\lambda}}%
\int\limits_{\lambda}^{1}ds\frac{\left(s+\lambda\right)^{p+\frac{1}{\lambda}}%
\left(s-\lambda\right)^{q-\frac{1}{\lambda}}}{(1+s)^{n+1}}$$
and
$$\displaystyle\widetilde{I}(p,q,\lambda,n)$$
$$\displaystyle=$$
$$\displaystyle\int\limits_{0}^{\infty}dr\,e^{-2r}r^{n}\,\widetilde{\Phi}(p,q,%
\lambda,r)$$
$$\displaystyle=$$
$$\displaystyle n!\left(\frac{\lambda-1}{\lambda+1}\right)^{\frac{1}{\lambda}}%
\int\limits_{1}^{\lambda}ds\frac{\left(\lambda+s\right)^{p+\frac{1}{\lambda}}%
\left(\lambda-s\right)^{q-\frac{1}{\lambda}}}{(1+s)^{n+1}}.$$
With these, the Kramers-Heisenberg matrix element and the dipole
dynamic polarizability respectively will take the following forms
$$M(\omega)=1-\frac{4}{3\omega^{2}}\left\{I(1,1,\lambda,3)-\widetilde{I}(1,1,%
\widetilde{\lambda},3)\right\}$$
(28)
$$\tau^{(2)}(\omega)=\frac{2}{3\omega^{3}}\left\{\widetilde{I}(1,1,\widetilde{%
\lambda},4)-I(1,1,\lambda,4)\right\}$$
(29)
This is the final form of the analytic expression for the matrix elements.
We did not attempt to rewrite these in terms of well known special
functions because the remaining integrals in equations (IV) and (IV)
can be done very efficiently by a simple numerical integration.
The integrals in
equation (10) is defined only for Re$(q-1/\lambda)>-1$. But in order to
calculate the matrix elements in equations (28) and (29) for
all values of $\omega$, we require an analytical continuation of the
definition of $\Phi(.)$ given in equation (10).
It can be achieved by performing a partial integration in the definition given
in equation (10) and this gives
$$\displaystyle\Phi(p,q,\lambda,r)$$
$$\displaystyle=$$
$$\displaystyle\frac{1}{q+1-\frac{1}{\lambda}}\left[K(p,q+1,\lambda)+r\Phi(p,q+1%
,\lambda,r)\right.$$
$$\displaystyle\left.+\left(p+\frac{1}{\lambda}\right)\Phi(p-1,q+1,\lambda,r)%
\right].$$
This is a very important recurrence relation because (IV) can be used to analytically
continue the definition of these function for Re$(q-1/\lambda)>-2$.
We can also repeat this procedure to analytically continue the definition
to other values of Re$(q-1/\lambda)$. And for the
numerical evaluation of various integrals as a function of
radiation frequency $\omega$ we have extensively used this
relation. It is also very important to note the appearance of
simple poles in the definition of $\Phi(.)$ when $q=-1+1/\lambda$.
Since $\lambda=\sqrt{1-2\omega}$, this is due to the presence of
intermediate resonances (when $2\omega=1-1/n^{2}$, where
$n=2,3,\ldots$) in the expression for second order matrix element
in the perturbation theory. This also leads to a strong frequency
dependence of ac stark effect and scattering cross section.
Similarly for above one photon ionization threshold i.e. $\omega>1/2$, a simple analytic continuation makes $\lambda$
to be a purely imaginary number. Thus by this simple method, with the same
analytic expression for the matrix element, we can numerically
evaluate it for the whole physical range of photon frequency $\omega$.
Displayed in table 1 are
$\tau^{(2)}(\omega)$ as a function of photon frequency in atomic units and are
compared with values obtained by other methods.
It is purely real for $\omega<1/2$.
For $\omega>1/2$ one photon ionization is possible and this makes the
intermediate virtual state to lie in the continuum and $\tau^{(2)}(\omega)$
becomes complex.
The dipole dynamic polarizability for photon
energy below and above one photon ionization threshold is given in table 1. Our values are compared
with values obtained by Arnous et al. maq:83 using Coulomb Greens function
method, but with an additional factor of $\omega^{-2}$
in their result. It is very easy to see that this factor is
missing from their result by taking the $\omega\rightarrow 0$ limit.
In this limit our result approaches the dc polarizability, which is 9/2.
The values given in maq:83 also approaches to the same limit
provided their results are multiplied by an overall factor of $\omega^{-2}$.
Using the analytic continuation described in equation (IV) we can calculate
$\tau^{(2)}(\omega)$ for $\omega$ very close to intermediate resonance also.
Our result also agrees with the second order
level shift and width reported by Pan et al. pan .
From the values given in tables 1 and 2
it is useful to note the change
in sign of the level shift when the photon frequency cross these resonance
values. The values of Kramers-Heisenberg matrix element are given in tables
2 and 3. They are in good agreement
with the results of Gavrila gav .
Acknowledgements.Authors acknowledge the support from UGC
through DSA-COSIST Scheme
References
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Abstract
The analytical representation of the longitudinal hadronic shower development
from the face of a calorimeter is presented and compared with
experimental data.
The suggested formula is particularly useful at designing,
testing and calibration of huge calorimeter complex like in ATLAS at LHC.
Analytical Representation
of the Longitudinal Hadronic Shower Development
Y.A. Kulchitsky
Institute of Physics, National Academy of Sciences, Minsk, Belarus
&
JINR, Dubna, Russia
V.B. Vinogradov
JINR, Dubna, Russia
One of the important questions of hadron calorimetry is the question
of the longitudinal development of hadronic showers
[1].
For analysing of energy depositions in various depths of calorimeters
it is necessary to have the analytical representation of longitudinal shower
development from the face of calorimeter.
Such formula is particularly useful at designing, testing and calibration
of huge calorimeter complex like in ATLAS [2] at LHC.
There is well known parameterisation of longitudinal shower development
from shower origin suggested by Bock et al. [3]:
$$\frac{dE_{s}(x)}{dx}=k\ \Biggl{\{}w\ \biggl{(}\frac{x}{X_{o}}\biggr{)}^{a-1}\ %
e^{-b\frac{x}{X_{o}}}\ +\ (1-w)\ \biggl{(}\frac{x}{\lambda_{I}}\biggr{)}^{a-1}%
\ e^{-d\frac{x}{\lambda_{I}}}\Biggr{\}},$$
(1)
where
$X_{o}$ is the radiation length,
$\lambda_{I}$ is the interaction length,
$a,\ b,\ d,\ w$ are parameters,
$k$ is the normalisation factor.
In this work we use the values of parameters obtained in
[4]:
$a=0.786+0.427\cdot lnE$,
$b=0.29$,
$d=0.978$,
$\omega=1.032-0.159\cdot lnE$.
In practice, the calorimeter longitudinal segmentation is unsufficient
for precise determination of shower vertex position.
Because of it is necessary to integrate the longitudinal profile from shower
origin over the shower position:
$$\frac{dE(x)}{dx}=\int\limits_{0}^{x}\frac{dE_{s}(x-x_{v})}{dx}\ e^{-\frac{x_{v%
}}{\lambda_{I}}}\ dx_{v},$$
(2)
where $x_{v}$ is a coordinate of the shower vertex.
In given work the analytical representation of the hadronic shower
longitudinal development from the calorimeter face has been derived.
It turns out that
$$\displaystyle\frac{dE(x)}{dx}$$
$$\displaystyle=$$
$$\displaystyle N\ \Biggl{\{}\frac{wX_{o}}{a}\biggl{(}\frac{x}{X_{o}}\biggr{)}^{%
a}e^{-b\frac{x}{X_{o}}}{}_{1}F_{1}\biggl{(}1,a+1,\biggl{(}b-\frac{X_{o}}{%
\lambda_{I}}\biggr{)}\frac{x}{X_{o}}\biggr{)}$$
(3)
$$\displaystyle+\ \frac{(1-w)\lambda_{I}}{a}\biggl{(}\frac{x}{\lambda_{I}}\biggr%
{)}^{a}e^{-d\frac{x}{\lambda_{I}}}{}_{1}F_{1}\biggl{(}1,a+1,-\bigl{(}1-d\bigr{%
)}\frac{x}{\lambda_{I}}\biggr{)}\Biggr{\}},$$
where
${}_{1}F_{1}(\alpha,\beta,z)$
is the confluent hypergeometric function [5] and
$N$ is the normalisation factor.
The confluent hypergeometric function in the first term
of formula (3) can be calculated by using the following
relation:
$${}_{1}F_{1}(1,a+1,z)=az^{-a}e^{z}\ \gamma(a,z),$$
(4)
where $\gamma(a,z)$ is the incomplete gamma function [5].
This function can be calculated by using the entry G106 in the program library
[6].
The confluent hypergeometric function in the second term
can be calculated by using the series development:
$${}_{1}F_{1}(1,a+1,-z)=1-\frac{z}{a+1}+\frac{z^{2}}{(a+1)(a+2)}+\dots.$$
(5)
This series has the remarkable property: at $a=0$ it corresponds to the
function $e^{-z}$.
For real calorimeters with longitudinal size usually less then 10 $\lambda_{I}$
the value of $z=(1-d)\frac{x}{\lambda_{I}}$ does not exceed $0.2$.
Therefore it is sufficient only 3 terms in order to calculate the
function ${}_{1}F_{1}(1,a+1,-z)$ with the precision of 0.1%.
The normalisation factor is
$$N=E_{beam}\biggr{/}\int\limits_{0}^{\infty}\frac{dE(x)}{dx}dx,$$
(6)
where $E_{beam}$ is the beam energy.
The integral in the denominator can be calculated by using the following
relation [7]:
$$\int\limits_{0}^{\infty}z^{\nu}e^{-\lambda z}{}_{1}F_{1}(\alpha,\gamma,kz)\ dz%
=\Gamma(\nu+1){\lambda}^{-\nu-1}{}_{2}F_{1}\biggl{(}\alpha,\nu+1,\gamma,\frac{%
k}{\lambda}\biggr{)},$$
(7)
where ${}_{2}F_{1}(\alpha,\beta,\gamma,z)$
is the hypergeometric function [5],
which in our case have the very simple form:
$${}_{2}F_{1}(1,a+1,a+1,z)=\frac{1}{1-z}.$$
(8)
As a result we obtain
$$N=\frac{E_{beam}}{\lambda_{I}\ \Gamma(a)\ \bigl{(}w\ X_{o}\ b^{-a}+(1-w)\ %
\lambda_{I}\ d^{-a}\bigr{)}}.$$
(9)
In Fig. 1 the calculations by formula (3) are shown
and compared with the experimental data at 20 – 140 GeV obtained by
using the conventional iron-scintillator calorimeter [4] and
the one with innovative feature — the longitudinal orientation of the
scintillating tiles [8].
The good agreement is observed.
So, now there is useful analytical formula which gives the possibility
immediately to obtain the longitudinal shower energy deposition
from calorimeter face.
References
[1]
R.K. Bock and A. Vasilescu, The Particle Detector Briefbook,
http://www.cern.ch/Physics/ParticleDetector/BriefBook.
[2]
ATLAS Collaboration, ATLAS Technical Proposal for a
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CERN/LHCC/ 94-93,
CERN, Geneva, Switzerland.
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ATLAS Collaboration,
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p. 525, FNAL, Batavia, 1990.
[5]
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[6]
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[7]
I.S. Gradshteyn and I.M. Ryzhik,
Table of Integrals, Series, and Products, 5th ed., Academic Press, 1994.
[8]
J.A. Budagov, Y.A. Kulchitsky, V.B. Vinogradov et al.,
Preprint JINR, E1-97-318, 1997, Dubna, Russia;
ATLAS Internal note, TILECAL-No-127, 1997, CERN, Geneva, Switzerland. |
The Evolutionary Sequence of Post-Starburst Galaxies
C. L. Wilkinson,${}^{1}$
K. A. Pimbblet,${}^{1,2}$
J. P. Stott${}^{3,4}$
${}^{1}$E.A. Milne Centre for Astrophysics, University of Hull, Cottingham Road, Kingston Upon Hull, HU6 7RX, UK
${}^{2}$School of Physics and Astronomy, Monash University, Clayton, VIC 3800, Australia
${}^{3}$Department of Physics, Lancaster University, Lancaster LA1 4YB, UK
${}^{4}$Department of Physics, University of Oxford, Keble Road, Oxford, OX1 3RH, UK
E-mail: [email protected] (CLW)
(Accepted XXX. Received YYY; in original form ZZZ)
Abstract
There are multiple ways in which to select post-starburst galaxies in the literature.
In this work, we present a study into how two well-used selection techniques have consequences on observable post-starburst galaxy parameters, such as colour, morphology and environment and how this affects interpretations of their role in the galaxy duty cycle. We identify a master sample of H$\delta$ strong (EW${}_{H\delta}$ > 3Å) post-starburst galaxies from the value-added catalogue in the 7th data release of the Sloan Digital Sky Survey (SDSS DR7) over a redshift range 0.01 < $z$ < 0.1. From this sample we select two E+A subsets, both having a very little [OII] emission (EW${}_{[OII]}$ $>-2.5$Å) but one having an additional cut on EW${}_{H\alpha}$ ($>-3$Å). We examine the differences in observables and AGN fractions to see what effect the H$\alpha$ cut has on the properties of post-starburst galaxies and what these differing samples can tell us about the duty cycle of post-starburst galaxies. We find that H$\delta$ strong galaxies peak in the ‘blue cloud’, E+As in the ‘green valley’ and pure E+As in the ‘red sequence’. We also find that pure E+As have a more early-type morphology and a higher fraction in denser environments compared with the H$\delta$ strong and E+A galaxies. These results suggest that there is an evolutionary sequence in the post-starburst phase from blue disky galaxies with residual star formation to passive red early-types.
keywords:
galaxies: evolution - galaxies: interactions - galaxies: clusters: general
††pubyear: 2017††pagerange: The Evolutionary Sequence of Post-Starburst Galaxies–The Evolutionary Sequence of Post-Starburst Galaxies\savesymbol
iint
\restoresymbolTXFiint
1 Introduction
Post-starburst galaxies are a vital link between star-forming spirals and quiescent E/S0 galaxies. E+A galaxies, also known as k+a galaxies, are specific type of post-starburst galaxy that were identified by Dressler &
Gunn (1983) as having an elliptical morphology but rich in A-class stars. Dressler &
Gunn (1983) find that 26 E/S0 galaxies display spectra showing no [OII]${}_{\lambda 3727}$ or H$\alpha$ emission but have deep Balmer absorption lines. The absence of [OII] indicates that there is no ongoing star formation whilst the presence of H$\delta$ absorption is a typical sign of young and recently formed A-class stars. This mix of spectral lines and morphology suggests a recent starburst approximately 1-3 Gyr ago, that has been quenched by an ‘unknown’ process (Dressler &
Gunn 1982; Couch &
Sharples 1987).
E+A galaxies are thought to be a transitionary phase between star-forming disks and quiescent early-type galaxies. Since their discovery, there have been many studies on E+As, but there is a large variance on how they are defined. Most observational work adopts a lack of [OII] emission and strong absorption in the H$\delta$ Balmer line to classify E+As (Poggianti
et al. 2009; Vergani
et al. 2010; Rodríguez Del Pino et al. 2014). Some have combined the Balmer absorption lines such as H$\delta$ with H$\gamma$ and/or H$\beta$ (Zabludoff et al. 1996; Norton et al. 2001; Chang et al. 2001; Blake et al. 2004; Tran et al. 2004; Yang et al. 2004) in order to maximise their E+A samples.
However, there are a number of studies that use a lack of H$\alpha$ emission (an indicator of star-formation) in their selection, such as Quintero
et al. (2004), Hogg et al. (2006), Goto (2007a) and Goto (2007b). While this method eliminates dusty star-formers it simultaneously excludes those E+As that contain H$\alpha$ caused by an active galactic nucleus. With different lines being used to select post-starburst galaxies, we ask if this has an impact on their characteristics and what can this tell us about their star formation duty cycle?
In earlier works E+A galaxies have been observed to be blue (Dressler &
Gunn 1983; Couch &
Sharples 1987). This disagrees with the originally belief that galaxies turn red immediately after a starburst (Larson
et al. 1980). Poggianti et al. (1999); Poggianti
et al. (2009) select E+As on the basis of having EW${}_{[OII]}$ emission < 5Å and EW${}_{H\delta}$ absorption > 3Å and find that there are two populations of E+As; blue and red, although without a discussion of their origin. Further works by Tran et al. (2004) and Vergani
et al. (2010), who select E+A primarily on [OII] emission and Balmer absorption lines, find that their location on a colour magnitude diagram is towards the redder end of the blue cloud and into the green valley. However, Quintero
et al. (2004) and Hogg et al. (2006), who select E+As without H$\alpha$ emission, find that E+As are located towards the bluer end of the red sequence and into the green valley on the colour-magnitude diagram. This suggests that the lack of H$\alpha$ emission could be the cause of the red E+A galaxies found in Poggianti et al. (1999).
Work on the morphology of E+A galaxies reveals them to be generally bulge dominated and in some cases having an underlying disk component (Tran et al. 2004, Quintero
et al. 2004). This combination of bulge and disk components replicates the S0 type morphology and reinforces the link between late- and early-type galaxies. This may suggest that the starburst triggering the E+A phase is centrally located, and not in the disc. The latter study opts for different selection methods yet their results on morphology are unanimous: H$\alpha$ has no effect on the morphology of post-starburst galaxies. Poggianti et al. (1999) find that the majority of E+A galaxies contain signs of spiral morphology; this indicates that the time-scale or process responsible for quenching star-formation is different from the process responsible for a morphological transformation. In turn, this could indicate multiple processes involved in triggering the E+A phase. Some have been shown to have a disturbed morphology, which is an indicator of a recent merger (Yang et al. 2004). By identifying E+A galaxies we are essentially catching them in the act of transition. Not all starbursts are strong enough to trigger the E+A phase (McIntosh
et al. 2014), but it is estimated that 30% of elliptical galaxies have passed through the E+A phase at some point during their evolutionary duty cycle (Tran et al. 2003; Goto 2005) making the E+A phase an important and significant part of galaxy evolution.
The properties of E+A galaxies are found to vary with different environments. For example, cluster E+A galaxies contain a more prominent disk component than those outside a cluster (Tran et al. 2003) and are found to be 1.5$\sigma$ bluer than those in the field at $z$ $\sim$ 0.1. This finding goes against the typical colour-density relation that suggests galaxies in clusters should be redder, not bluer (Hogg et al. 2003). This difference in colour is attributed to differing galaxy masses in both field and cluster E+A galaxies. Clusters have been found to contain much smaller E+As than in the field (Tran et al. 2003).
E+A galaxies are found both in a cluster environment as well as a field environment, however, different studies show different environmental preferences. From a sample of 21 nearby E+A galaxies (selected by [OII] and Balmer lines) from the Las Campanas Redshift Survey, Zabludoff et al. (1996) finds 75% of the E+As are in the field. They find 5 E+A galaxies within their sample display tidal features, a typical sign of galaxy-galaxy interactions and mergers. This suggests that mergers and galaxy-galaxy interactions could be the trigger for the E+A phase. The type of mergers found to dominate in low density environments are gas-rich (Bekki
et al. 2001; Lin et al. 2010; Sánchez-Blázquez
et al. 2009) which are most likely to trigger a starburst strong enough of producing an E+A signature.
When H$\alpha$ is included into the selection criteria different results are found; Mahajan (2013) find that E+A galaxies prefer a weak-group environment ranging from 4 to 10 galaxies. For local X-ray bright galaxy clusters at 0.02$<z<$0.06, they find 86% of local clusters contained some form of substructure on a weak group-scale (4-10 members) and 91.4% of these weak-groups contained E+A galaxies. This result suggests that star-formation is quenched due to a galaxy being pre-processed in a weak-group environment before being accreted into a cluster environment.
Opposite results are found by Poggianti
et al. (2009): at higher redshifts ($0.4<z<0.8$) E+A galaxies are predominantly in clusters where the star-formation has been quenched on a short time scale of $\ll$ 1 Gyr. Poggianti
et al. (2009) find only a few E+A galaxies are located in other environments such as the field and in weak-groups and that the variation in E+A fraction is found to be dependant on environment. Mahajan (2013). Regardless of whether H$\alpha$ has been used in the selection criteria of E+As there is a discrepancy in what different works report with regards to environmental preference. Lemaux et al. (2016) suggests that ‘true’ E+As (those with no H$\alpha$) have a higher fraction in a cluster environment compared to those selected by ‘traditional’ methods (using only Balmer absorption lines and the absence of [OII]). Could environment be driving changes in H$\alpha$ emission or are we seeing two different stages of the E+A duty cycle?
A typical sign that star-formation has ceased is a lack of H$\alpha$ emission and as E+As galaxies are no longer producing stars, one would expect to see no significant H$\alpha$ emission. Although not commonly used as a criteria for E+A selection, this work aims to test whether including a cut in H$\alpha$ could explain some of the differences found in other studies and connect up these potentially different phases of post-starburst evolution. Here we present a study of the properties of low redshift post-starburst galaxies using different selection criteria from the Sloan Digital Sky Survey Data Release 7 (SDSS DR7; Abazajian
et al. 2009) with the broad aim of testing the star formation duty cycle of post-starburst galaxies.
In section 2 we describe the data used in this study and the different selection cuts we impose. In section 3 we analyse and discuss the results. Our conclusions are presented in in section 4. We assume the following cosmological parameters for a flat Universe: $H_{0}=\rm 69.3\,kms^{-1}Mpc^{-1},\Omega_{M}=0.238\ and\ \Omega_{\Lambda}=0.762$ (Spergel
et al. 2007).
2 Sample Selection
Using the MPA/JHU value-added catalogue from SDSS DR7 (Abazajian
et al. 2009), we create a master sample of 234 H$\delta$ strong (EW${}_{H\delta}$ > 3Å; note that positive/negative values for equivalent widths refer to absorption/emission respectively) post-starburst galaxies with 0.01 $<z<$ 0.10. This limit on H$\delta$ is comparable to studies such as Poggianti et al. (1999), Bekki
et al. (2001), Vergani
et al. (2010) and Rodríguez Del Pino et al. (2014). The lower limit on the redshift range ensures we exclude any stars from our sample. Although the SDSS spectroscopy aims to be 100% complete at r=17.77 (Strauss
et al. 2002), we find that our sample is $\sim$80% complete at this level, as shown in Fig.1.
Most studies of post-starbursts use a typical S/N of 3-5 in the continuum, as evidenced by Pracy et al. (2005) (S/N$>$5); Couch &
Sharples (1987) (S/N$>$3) and Brown et al. (2009) (S/N$>$4), hence we adopt a minimum S/N of 5.
Our complete S/N $>$ 5 H$\delta$ strong sample contains 192 galaxies. Using this sample we apply additional cuts to create a further two sub-samples; E+As and ‘pure’ E+As (E+As with low H$\alpha$ emission) in order to test whether different criteria for post-starburst galaxies can explain the differences found in literature and to gain a better insight into the post-starburst duty cycle.
Our first sub-sample is based on the Poggianti et al. (1999) definition of E+A galaxies. With the primary selection of EW${}_{H\delta}$ > 3Å, we apply an additional cut of $>-2.5$Å in the equivalent width of [OII]${}_{\lambda 3727}$ (Zabludoff et al. (1996); Yang et al. 2004). This sample of E+As contains 67 galaxies.
To account for this, our second sub-sample ensures there is no ongoing (potentially dusty) star-formation in the E+A sample by applying a further cut in H$\alpha$ emission (EW${}_{H\alpha}$ > -3Å) (Goto 2007b). Doing this we obtain 25 pure E+A galaxies. We list the selection criteria for our three post-starburst sub-samples below:
H$\delta$ Strong:
1.
EW${}_{\delta}$ > 3Å
E+A:
1.
EW${}_{\delta}$ > 3Å
2.
EW${}_{[OII]}$ > -2.5Å
Pure E+A:
1.
EW${}_{H\delta}$ > 3Å
2.
EW${}_{[OII]}$ > -2.5Å
3.
EW${}_{H\alpha}$ > -3Å
We test whether or not we would expect to see H$\alpha$ given the limit on [OII] emission stated above. We use a simple approach by plotting EW${}_{[OII]}$ against EW${}_{H\alpha}$ as shown in Fig. 2. In this plot we witness a significant amount of scatter (regression analysis gives R${}^{2}$ to be 0.3), however, there is an underlying trend present and we are able to fit a regression line using a robust linear model (Huber 1981). From this we see at the EW${}_{[OII]}$ limit we would expect EW${}_{H\alpha}$ to be -2.1Å in emission. On average the [OII] cut should remove any significant star formation indicated by H$\alpha$. However, the large scatter means that there is a population of galaxies with up to EW=-15Å of H$\alpha$ emission at the [OII] limit (See fig. 2). We also calculate the star formation rate derived from the H$\alpha$ flux, SFR${}_{H\alpha}$ from Kennicutt (1998) and apply an extinction correction (10${}^{A/-2.5}$) from Treyer
et al. (2007). Assuming $Av=0.5$ we find the average SFR${}_{H\alpha}$ for H$\delta$ strong galaxies that didn’t make the E+A cut to be 0.07M${}_{\odot}$/yr${}^{-1}$. For the E+A that are not classed as pure we find SFR${}_{H\alpha}$ to be 0.06M${}_{\odot}$/yr${}^{-1}$ and pure E+A have an average SFR${}_{H\alpha}$ of 0.01M${}_{\odot}$/yr${}^{-1}$. Stacked spectra for our three sub-samples can be found in Fig. 3. We note here that emission-filling could potentially affect our H$\delta$ measurements. We believe this would be a small effect (Goto et al. 2003; Blake et al. 2004). However, the usual method of measuring this in conjunction with, e.g., H$\alpha$ to determine the strength of its effect is unable to be achieved due to our sample selection approach.
3 Colour
In this section we investigate whether the differences in selection criteria, based on spectral lines, has an impact on the colours of the post-starburst population. This will allow us to determine why Poggianti et al. (1999); Poggianti
et al. (2009) finds two populations of E+A galaxies that peak in the red sequence and blue cloud.
To study the differences in colour between our samples we first plot a colour magnitude diagram, shown in Fig. 4. Using the galaxies from the SDSS DR7 catalogue, we are able to locate the red sequence and blue cloud regions and over plot our samples to see which regions they reside (Fig.4).
We include a colour histogram (Fig. 5) that shows the bi-modality of the colour distribution. Using Fig. 4 and Fig. 5 we find that the H$\delta$ strong sample peaks in the blue cloud with a median colour of 1.36$\pm 0.04$. The E+A sample peaks in the green valley with a median colour of 1.46$\pm 0.04$. We find that the pure E+As predominantly reside in the red sequence with a median colour of 1.66$\pm 0.04$.
Hogg et al. (2006) select E+A galaxies using a cut in H$\alpha$ and find that E+As are located in the red sequence towards the green valley. While E+As have a significant connection to the green valley (Vergani
et al. 2010), we find that by selecting E+As using H$\alpha$ emission we are predominantly selecting those towards the red sequence as star formation has ceased and the galaxies are truly passive.
To test whether the E+A and pure E+A samples are distinguishable we apply a two sample Kolmogorov-Smirnov statistical (abbreviated to KS onwards) test. When comparing the E+A sample with the pure E+As the KS test returns a p-value of 0.007 which demonstrated that E+As and pure E+As are unlikely to be drawn from the same parent distribution. In this study we consider any p-value less than 0.01 to be statistically different, whilst p-value over 0.1 we consider to be the same. P-values between these limits we consider to have a weak difference.
To test the distinguishability between the E+A sample and the H$\delta$ strong sample, we apply the same test and output a p-value of 0.110. These distributions are likely to be from the same parent distribution. When we take H$\delta$ strong galaxies that are not defined as E+As and compare to the E+A sample we get a p-value of 0.005. This means there is a statistical difference between non-E+A H$\delta$ strong galaxies and E+A galaxies. Fig. 5 clearly shows that the pure E+A sample and the H$\delta$ strong sample are statistically different. To quantify the distinguishability of the H$\delta$ strong sample and pure E+As we include a p-value which is found to be 10${}^{-6}$.
4 Morphology and Morphological Parameters
To explore the differences in findings on morphology we test the morphological fractions of each sample using the Galaxy Zoo (GZ) catalogue (Lintott
et al. 2008, 2011). The GZ project provides an extensive catalogue of galaxy morphologies and is based on the data from SDSS. GZ is a citizen science project in which volunteers view images of galaxies and cast their vote as to whether the galaxy is elliptical, spiral, merger or unknown. We assign a classification of elliptical, spiral or merger to our galaxies if that classification takes 50% or more of the votes. The minimum number of votes per galaxy in our sample is 16 and the mean number or counts our sample has is 39.5. We note here that S0 galaxies are not easily distinguishable from the elliptical and spiral morphologies in the GZ catalogue. Whilst the majority of S0 galaxies are within the elliptical classification there will certainly be S0 contamination in the spiral classifications (Bamford
et al. 2009).
We find that from our H$\delta$ strong sample we are able to match 185 (96.4%) galaxies to the GZ catalogue. This in turn means we matched 62 E+As (92.5%) and 21 pure E+As (84.0%) to the catalogue. The morphology fractions for each of the samples are shown in Table 1. We find that there is no statistically significant merger fraction in either of the three samples, this is due to the very low numbers that were identified. We find that $\sim$55% of the H$\delta$ strong sample have a spiral morphology and $\sim$30% have an elliptical morphology. There is a clear majority here for a preference of disc morphologies. This result somewhat changes in the E+A sample in which there is a $\approx$50/50 (42%/44%) split between the elliptical and spiral morphologies for those matched to GZ. However, in the pure E+A sample we see $\sim$80% of the galaxies are classed as ellipticals and no significant spiral fraction. This is in agreement with the findings that pure E+As are redder than the H$\delta$ strong and E+A galaxies and occupy the red sequence with the majority of early-type. Interestingly we note that mergers make up less than 5% of any of the post-starburst categories and therefore do not seem to be important for this phase of galaxy evolution.
As well as using the GZ catalogue to test morphology fractions we also look at the $fracDeV$ parameter, which corresponds to the Sérsic index (a generalization of the de Vaucouleur profile, de Vaucouleurs 1948; Sersic 1968) and links together galaxy shape with surface brightness profiles (Fig. 6). $fracDeV$ is the fraction of light fit by the de Vaucouleur profile versus an exponential profile. When $fracDeV$=1, this represents a pure de Vaucouleur profile and is appropriate for an elliptical morphology. When $fracDev$=0, this represents a pure exponential profile and is appropriate for a disk morphology as seen in spiral galaxies. The $fracDev$ parameter is listed in the SDSS database for all $u,g,r,i,z$ filters (Abazajian
et al. 2009).
Fig. 6 shows the normalised distribution of the $fracDeV$ parameter and broadly agrees with the morphological classifications in Table 1. For the pure E+A the $fracDeV$ distribution tends to one meaning an elliptical morphology. For the H$\delta$ strong sample and the E+A sample we visually see little difference in distribution and that both distributions are dominated by a disk-like profile. We calculate their median $fracDeV$ values to be 0.086$\pm 0.025$ for H$\delta$ strong galaxies, 0.238$\pm 0.046$ for E+As and 0.779$\pm 0.063$ for pure E+As.To test this statistically we apply a KS test to all samples so determine their distinguishability. We find that when comparing the H$\delta$ strong sample to the E+A sample we obtain a p-value of 0.338; we conclude that these two samples are not statistically different. To determine whether the E+A sample and the pure E+As emission are distinguishable we apply the same test and obtain a p-value of 0.003. We can therefore claim that there is a significant difference between E+As and pure E+As. To complete these tests, a p-value of $\sim$10${}^{-6}$ was achieved when comparing the H$\delta$ strong sample to the pure E+As.
We examine the radii of our samples to further quantify our morphology results and to compare to the trends found in Shen et al. (2003). Here we use the effective radii based on the deVaucouleur profile in the SDSS r-band. We note here we have included a first order seeing correction to the radii, as shown in Equation 1 assuming the average SDSS seeing of 2".
$$r_{corrected}=(r_{observed}^{2}-seeing^{2})^{1/2}$$
(1)
Fig. 7 shows the normalised radius distributions of the three samples. The majority of the pure E+A galaxies are mainly under 10kpc in radius and are therefore relatively small galaxies. We find the following median radii for the H$\delta$ strong, E+A and pure E+A galaxies respectively: 11.4kpc$\pm$0.5, 9.8kpc$\pm$0.8 and 4.1kpc$\pm$1.2. When a KS test is applied to the H$\delta$ strong sample and the E+A sample we obtain a p-value of 0.176. When comparing the E+As and H$\delta$ strong sample to the pure E+As we obtain the p-values 0.006 and $\sim$10${}^{-6}$ respectively. This results shows that our third sample are statistically smaller in radii than the other two samples. This is also shown in Fig. 8 which shows the relation between radius and absolute magnitude with the corresponding trends from Shen et al. (2003). We look at the median separation between R${}_{50}$ and the trends from Shen et al. (2003) to quantify which line fits which population. From the H$\delta$ strong sample we find that the median separation from the early-type trend is 1.21kpc and 0.03kpc from the late-type trend. Looking at the E+A sample we find that the median separation is 0.48kpc from both the early-type trend and late-type trend, this demonstrate the finds from Table 1. Finally, we find that the median separation of the pure E+A sample to the early-type trend is 0.39kpc and 0.85kpc to the late-type trend. This shows the H$\delta$ strong galaxies follow the trend for late-type galaxies, the E+As an even mix of early- and late-type and the pure E+As resemble an early-type relation. This again confirms our findings from the colour and morphology analysis.
5 Environment
Whilst there is little evidence in the literature that selection criteria affect the environmental results, we aim to test whether there are any environmental differences between our sub-samples.
In order to explore the environments of the galaxies in our samples we use the Yang group and cluster catalogue (Yang et al. 2007). We directly match 176 galaxies in our sample (91.2%); this corresponds to 64 E+As (94.0%) and 22 pure E+As (88.0%). The Yang catalogue is based on the 4th data release of SDSS and selects galaxies with a redshift completeness $C>0.7$, so therefore may not match totally to the 7th data release.
First, we examine whether our galaxies are in groups/clusters, if so, we look at the membership of that group or cluster to determine the size. Our results are displayed in Table 2. We split our environment classifications into field galaxies, pairs, weak groups (3$<$N$<$10), rich groups (10$<$N$<$50) and clusters (N$>$50).
Table 2 shows that $\sim$70% and $\sim$60% of the H$\delta$ strong galaxies and E+A galaxies respectively are field galaxies with a strong preference for this type of environment; this supports the findings of Zabludoff et al. (1996) who find that 75% of E+A galaxies to be in a field environment. We find that only$\sim$40% of the pure E+As are in the field environment whilst there is a significant amount of pure E+A galaxies in the group environments. When combining these environments together into all field (field and pair, where a pair is two galaxies that are gravitationally bound) and all cluster (weak groups, rich groups and clusters) environments we see the following splits in preference (field/cluster); 80/20, 70/30 and 50/50 for H$\delta$ strong, E+A and pure E+As respectively. This suggests that pure E+As are more likely to be in denser environments than the other sub-samples of post-starburst galaxies.
6 AGN Connection
Hydrodynamical simulations show that major mergers of spirals contribute heavily to the populations of E+A galaxies (Snyder et al. 2011). When these gas-rich spirals merge, high levels of star-formation occur and to quench this star-formation a feedback mechanism is required, potentially AGN driven feedback.
During a merger gas is funnelled into the nucleus of a merging galaxy by tidal torques and and can trigger a nuclear starburst (Mihos &
Hernquist 1996; Hopkins et al. 2008a; Hopkins et al. 2008b) and therefore a potential progenitor of the E+A phase. With an excess of gas in the nucleus of the galaxy, this provides fuel for the supermassive black hole in the centre and can trigger an active galactic nuclei (AGN). AGN can produce jets, winds or radiation that heat or remove cold gas from galaxies, removing the fuel for further star formation.
De Propris &
Melnick (2014) show in their study of 10 K+A galaxies that at any age there is no presence of AGN. They suggest, however, that a quasar phase immediately follows the quenching mechanism and this leads onto the K+A phase.
In this section we investigate the E+A/AGN link further by examining the AGN fractions within our samples and comparing to the fractions found in SDSS. We start by plotting a BPT diagram (Baldwin
et al. 1981; Veilleux &
Osterbrock 1987) shown in Fig. 9 which includes the Kauffman (Kauffmann
et al. 2003), Kewley (Kewley
et al. 2001) and Schawinski lines (Schawinski
et al. 2014). Using these lines we locate the seyfert, LINER, composite and star formation regions. Starting with a control sample derived from SDSS we find that seyfert galaxies make up 10.8% of the general galaxy population, LINERs 11.5%, composites 20.7% and star formers 57.0%.
We plot the H$\delta$ strong and E+A samples in Fig. 9 on the BPT diagram and find that the majority these two samples are within the star forming region (65.4%$\pm$6.4 and 57.1%$\pm$11.7 respectively). We find that composites makes up 23.0%$\pm$3.8 of the H$\delta$ strong sample and 35.7%$\pm$9.2 of the E+A sample. Seyferts make up 8.2%$\pm$2.3 of the H$\delta$ strong sample and 4.8%$\pm$3.4 of the E+As whilst LINERs make up 3.1%$\pm$1.4 of the H$\delta$ strong sample and 2.4%$\pm$2.4 of the E+As. These results show similar fraction in both samples of star formers compared to the general galaxy population, however we witness an enhancement in E+A composite galaxies and decline in E+A AGN.
We also look at AGN fractions using a WHAN diagram (Fig. 10), described by Cid Fernandes
et al. (2010); Cid Fernandes et al. (2011) which unlike the BPT can identify ‘fake’ AGN (i.e. retired galaxies) from those with weakly active nuclei from the BPT LINER region. Using the SDSS control sample we find that 46.0% of the general galaxy population is made up of star forming galaxies, 27.2% passive galaxies, 19.3% strong AGN and 7.4% weak AGN.
We find that the H$\delta$ strong sample is comprised of 59.7%$\pm$6.1 star forming galaxies, 15.1%$\pm$3.1 passive galaxies, 16.4%$\pm$3.2 sAGN and 8.8%$\pm$2.3 wAGN. The E+A sample is found to be comprised of 54.2%$\pm$10.6 star forming galaxies, 27.1%$\pm$7.5 passive galaxies, 12.5%$\pm$5.1 sAGN and 8.3%$\pm$4.2 wAGN. These fractions are similar to those found in the SDSS control sample, however, we witness a decline in passive galaxies and slight enhancement of star formers within the H$\delta$ strong sample compared to SDSS galaxies, which is to be expected.
7 Discussion
7.1 Links to Previous Studies
We have discussed in section 1 how different studies have used different selection methods to select post-starburst galaxies. There are two main methods for selection; the most common selecting on the lack of [OII] emission and the presence Balmer absorption whilst the other selects on H$\alpha$ emission, as well as [OII] emission and Balmer absorption.
Poggianti et al. (1999); Poggianti
et al. (2009) find that there are two populations of E+As when they select using [OII] emission and H$\delta$ Balmer absorption; these two populations can be classed as red and blue. We have seen from Fig. 4 and Fig. 5 that the H$\delta$ galaxies peak in the blue cloud, the E+As peak in the green valley and the pure E+As peak in the red sequence. This result matches with that of Hogg et al. (2006) who place H$\alpha$ selected E+As within the red sequence. This means the cut in H$\alpha$ is removing galaxies with residual star formation, therefore, we are intrinsically selecting the red E+As mentioned in Poggianti et al. (1999) and Poggianti
et al. (2009).
Along with colour we have also tested the morphologies of our three samples. The literature tells us E+As are bulge dominated with an underlying disc component (Tran et al. 2004; Quintero
et al. 2004), similar to the S0 morphology. This was not seen to change with different selection methods mentioned above. We find in Table 1 and Fig. 6 that there are significant fractions in both the elliptical and spiral morphologies for H$\delta$ strong galaxies and E+As. However, we see that pure E+As are predominantly elliptical.
Regardless of what selection method is used in the literature there is a great deal of discrepancy in the environmental preferences of post-starburst galaxies. Table 2 shows that $\sim$70% and $\sim$60% of H$\delta$ strong galaxies and E+As are within the field, this result agrees with the findings of Zabludoff et al. (1996) at low redshifts $z<0.1$. Lemaux et al. (2016) states that pure E+As have a higher fraction in clusters than E+As selected only on [OII] emission and Balmer absorption, we find the same trend when examining the fractions of post-starburst galaxies in various environments. Mahajan (2013) find that E+As prefer a weak-group environment, whilst we do see a systematically higher fraction in the weak-group environment compared with rich environments this result is not statistically significant.
7.2 Duty Cycle of Post-Starburst Galaxies
In section 3 we find that the colours of H$\delta$ strong galaxies peak in the blue cloud, the E+As in the green valley and the pure E+As in the red sequence (Fig. 4 and Fig. 5). This result is backed up morphologically in section 4. We find that the majority of H$\delta$ strong galaxies have a spiral morphology whereas the E+As have a mix of spiral and elliptical morphologies . The majority of pure E+As display an elliptical morphology (Table 1). This is confirmed again by the light profiles of our samples (Fig. 6), we find that H$\delta$ strong and E+A galaxies have a disky light profile whereas the pure E+As have a de Vaucouleur light profile, typical of an early-type morphology. This difference in morphology is seen in the radii of our galaxies (Fig. 7 and Fig.8), in which the H$\delta$ strong and E+A galaxies follow the late-type relationship found by Shen et al. (2003) and the pure E+As follow the early-type relationship. The lack of merging morphologies suggests that major mergers do not play an important role in the post-starburst phase.
We speculate that the H$\delta$ strong galaxies, E+A and pure E+As are part of an evolutionary sequence in which star forming galaxies are quenched into blue disky H$\delta$ strong galaxies. They are then quenched further into green valley E+As before, finally, quenching fully into red elliptical pure E+As. If this is true, how does the morphology transition from disk to elliptical? This could be due to a fading disk with time as young stars die or possibly due to any mergers. We add here that we witness no evidence for mergers in the morphologies (Table 1) so the trigger would have to be a minor merger or a mergerless interaction.
Another factor we consider is environment, although we have low-number statistics, we see a higher proportion of pure E+As in denser environments than the H$\delta$ strong and E+A galaxies. Whilst in the H$\delta$ strong and E+A galaxies we see a strong preference for a field environment. We note this is not a migration of galaxies from the field to a cluster environment as the migration time scale is much longer than that of the evolution of our galaxies. Instead, this may point to an accelerated evolution from H$\delta$ strong galaxies to pure E+As in dense environments (Hatch et al. 2011). A dense environment would enhance processes such as interactions and minor mergers. Also the hot atmosphere of the intracluster medium would help quench and fade the disk by starving the galaxy of the fuel to replenish its cold gas supply.
We see from looking at AGN fractions using BPT and WHAN diagnostics that there is a higher fraction of AGN in the H$\delta$ strong sample compared with the E+As. This could suggest that E+As are the product of H$\delta$ strong galaxies that have been quenched by AGN feedback, however, as the E+A cut selects against line emission this is purely speculative and we have no solid evidence that AGN are important in quenching star formation after a burst.
8 Conclusion
The post-starburst phase in galaxy evolution is an important and significant link between star-forming discs and quiescent early-type galaxies. Throughout the literature there are several different selection techniques. We split these different techniques into two categories; those that select on the presence of Balmer absorption and the absence of [OII] emission and those which also select against H$\alpha$ emission. There are advantages and disadvantages in using either method, one includes those galaxies that have H$\alpha$ due to the presence of AGN and the other excludes dusty starburst galaxies. In this work we have explored how the different selection techniques can affect the results found in colour, morphology, environment and AGN fraction. Here we list our principal findings and suggest what we feel is an ideal selection method:
1.
By locating our samples on a colour magnitude diagram and colour histogram, we have been able to find where each population peaks. We find that H$\delta$ strong galaxies peak in the blue cloud, E+A galaxies in the green valley and pure E+As in the red sequence.
2.
Using the GZ catalogue we find that H$\delta$ strong galaxies are predominantly spiral in morphology with a fraction of 54%. We find that E+As have approximately an even split between the elliptical and spiral morphologies and pure E+As have an elliptical fraction of 81%. This results is confirmed when we look at the $fracdev$ parameter which shows the H$\delta$ strong galaxies and E+As have a disk morphology and the pure E+As have an early-type morphology. This is again demonstrated when we look at the radii of our galaxies, we see that the H$\delta$ strong galaxies and E+As follow the late-type trend found in Shen et al. (2003), whilst the pure E+As follow the early-type trend.
3.
We find that the majority of H$\delta$ strong galaxies and E+A galaxies have a strong preference for the low density environment, however, there is an even split between the low and high density environments within the pure E+A population. We see that there is a higher fraction of pure E+As in denser environments than the other two populations.
4.
In this study we have also tested the AGN fractions of our samples by using BPT and WHAN diagrams. Both diagrams suggest that the emission lines of the H$\delta$ strong and E+A populations are excited by star formation and not AGN activity. The AGN fraction of the H$\delta$ strong galaxies is typical of the general galaxy population, so provides no evidence that AGN play a role in quenching star-formation in post-starburst galaxies. However, we see a significant decrease in AGN fraction compared with the general population which could be a consequence of AGN fading into the E+A phase assuming AGN feedback leads to the E+A phase. However, we note that the E+A sample is selected to not have strong emission lines (albeit in [OII]) and so it is not possible to draw strong conclusions.
We postulate that different selection criteria allow us to probe different stages of post-starburst galaxies. After the initial starburst has ended the galaxies are seen as blue spirals/disks with a young stellar spectra, indicated by deep Balmer absorption features. However, $\sim$65% of these disks are still forming stars as seen by their [OII] and/or H$\alpha$ emission lines. If we cut on [OII] emission (and therefore star formation) we see a population of green valley disk galaxies suggestive of the next stage in the post-starburst evolution as galaxies quench towards the red sequence. If we make a further cut on star formation using a lack of H$\alpha$ emission then we find a population of early-type red sequence galaxies. These are galaxies that still show deep Balmer absorption but with no ongoing star formation and represent the final post starburst phase before the underlying stellar population becomes dominated by old stars.
The lack of any significant enhancement in the merger or AGN fractions for the post-starbursts indicates that, while we cannot rule out a scenario whereby the initial starburst was triggered by a major merger and then initially quenched by an AGN, it is not these processes that continue to quench the galaxies down to full quiescence. One puzzle in our proposed evolutionary sequence is how the morphologies have changed from disky to early-type without evidence of major merging. We speculate that this morphological evolution could be due to the secular fading of the star-forming disk leading to a bulge dominated system or to a series of minor interactions not detected in the imaging. There also seems to be a suggestion that the passive, pure E+As are more prevalent in denser environments, which perhaps suggests that the quenching that follows a starburst proceeds more quickly in dense environments perhaps due to the enhancement of minor interactions in groups or the removal of a cold gas supply by the hot intracluster medium of the most massive systems.
Acknowledgements
JPS gratefully acknowledges support through the Hintze Research Fellowship.
Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/.
The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington
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High-resolution transmission spectroscopy of four hot inflated gas giant exoplanets
Jiří Žák
Astronomical Institute, Czech Academy of Sciences,
Fričova 298, 25165, Ondřejov, Czech Republic
Department of Theoretical Physics and Astrophysics, Masaryk University, Kotlářská 2, 61137 Brno, Czech Republic
Petr Kabath
Astronomical Institute, Czech Academy of Sciences,
Fričova 298, 25165, Ondřejov, Czech Republic
Henri M.J. Boffin
ESO, Karl-Schwarzschild-str. 2, 85748 Garching, Germany
Valentin D. Ivanov
ESO, Karl-Schwarzschild-str. 2, 85748 Garching, Germany
Marek Skarka
Astronomical Institute, Czech Academy of Sciences,
Fričova 298, 25165, Ondřejov, Czech Republic
Department of Theoretical Physics and Astrophysics, Masaryk University, Kotlářská 2, 61137 Brno, Czech Republic
(Received: January 3, 2019; Revised: ; Accepted: July 15,2019)
Abstract
The technique of transmission spectroscopy allows us to constrain the chemical composition of the atmospheres of transiting exoplanets. It relies on very high signal-to-noise spectroscopic (or spectrophotometric) observations and is thus most suited for bright exoplanet host stars. In the era of TESS, NGST and PLATO, more and more suitable targets, even for mid-sized telescopes, are discovered. Furthermore, a wealth of archival data is available that could become a basis for long-term monitoring of exo-atmospheres.
We analyzed archival HARPS spectroscopic time series of four host stars to transiting bloated gas exoplanets, namely WASP-76b, WASP-127b, WASP-166b and KELT-11b, searching for traces of sodium (sodium doublet), hydrogen (H$\alpha$, H$\beta$), and lithium (670.8 nm).
The archival data sets include spectroscopic time series taken during transits. Comparing in- and out-of-transit spectra we can filter out the stellar lines and investigate the absorption from the planet. Simultaneously, the stellar activity is monitored using the Mg I and Ca I lines.
We independentely detect sodium in the atmosphere of WASP-76b at a 7–9 $\sigma$ level. Furthermore, we report also at 4–8 $\sigma$ level of significance the detection of sodium in the atmosphere of WASP-127b, confirming earlier result based on low-resolution spectroscopy. The data show no sodium nor any other atom at high confidence levels for WASP-166b nor KELT-11b, hinting at the presence of thick high clouds.
planetary systems — techniques: spectroscopic — planets and satellites: atmospheres —
planets and satellites: individual: WASP-76b, WASP-127b, WASP-166b, KELT-11b
††journal: AJ††software: HARPS Data Reduction Software, iSpec (Blanco-Cuaresma et al. 2014), IRAF (Tody 1986)
1 Introduction
Transmission spectroscopy has the power to probe the physical conditions in
the atmospheres of exoplanets, as it allows detecting the various absorption features that are imprinted on the
stellar light that passes through their atmospheres. Not surprisingly, the
first detections of features in exo-atmospheres were made from space
(Charbonneau et al., 2002; Vidal-Madjar et al., 2003), while ground-based transmission spectroscopy delivered its
first secure detection of sodium in extrasolar atmospheres only about a decade ago
(Redfield et al., 2008; Snellen et al., 2008). If from space, only low-resolution spectroscopy can be used, from the ground, both low-resolution spectrophotometry and high-resolution spectroscopy can and have been used to detect atoms and molecules in the atmospheres of several exoplanets. In particular, in most recent years, high-resolution spectroscopy has been more and more successful. Moreover, Wyttenbach et al. (2015) used HARPS at the ESO 3.6-m telescope to be able to detect
sodium in HD 189733b, demonstrating that spectrographs attached to moderate-size telescopes
can also be used for this purpose. In the past only a handful of suitable, bright enough targets were
available for such kind of study, but this is now changing rapidly with the discoveries from
the TESS mission (Ricker et al., 2015; Kabath et al., 2019).
The high resolving power, i.e.,
R$\geq$40 000–60 000, removes the need for a reference star that is at the same
time close on sky to the target, and of similar brightness – two rarely met
conditions that are limiting the low-resolution spectrophotometry studies. In
contrast, the high resolving power solely requires accurate telluric correction and
a careful analysis of the radial velocities of the star-planet system (e.g., Astudillo-Defru & Rojo, 2013; Wyttenbach et al., 2017).
Another advantage of high-resolution spectroscopy is its ability to resolve and characterize individual planetary absorption lines. In particular, the
Na D lines are well suited for transmission spectroscopy due to the strong expected signal and the fact that they are very narrow features, in comparison to the broad molecular absorption bands of, e.g., $\rm{CH_{4},\ H_{2}O,\ TiO\ and\ VO.}$
If sodium, and in some cases, potassium measurements
based on low-resolution spectra were reported
(e.g., for some of the most recent ones, Sedaghati et al., 2016; Chen et al., 2017; Pallé et al., 2017; Murgas et al., 2017; Lendl et al., 2017; Chen et al., 2018; Pearson et al., 2019), only a few studies detected sodium using high-resolution transmission spectroscopy:
Wyttenbach et al. (2017) registered strong absorption signal in WASP-49b.
Using HARPS-N mounted on the TNG, Casasayas-Barris et al. (2017) and Casasayas-Barris et al. (2018) concluded to the presence of sodium in the atmospheres of WASP-69b and MASCARA-2b. Khalafinejad et al. (2018) tentatively detected sodium in WASP-17b using the MIKE instrument on the 6.5m Magellan Telescope. Finally, Jensen et al. (2018) detected sodium and hydrogen in WASP-12b using the Hobby-Eberly Telescope and Deibert et al. (2019) found sodium in HAT-P-12b using the Subaru Telescope.
These are not the only elements to have been discovered in the atmospheres of exoplanets, as there are published detection of hydrogen (Ehrenreich et al., 2008; Lecavelier Des Etangs et al., 2010), helium
(Nortmann et al., 2018; Spake et al., 2018), calcium, scandium (Astudillo-Defru & Rojo, 2013), TiO and VO
(Désert et al., 2008; Sedaghati et al., 2017), CO (Snellen et al., 2010), H${}_{2}$O and CH${}_{3}$
(Tinetti et al., 2007; Swain et al., 2008; Birkby et al., 2013). Some of these detections were disputed by
Gibson et al. (2011) who discusses in great detail the complex systematics that affect
the results, even those obtained from space. This issue was addressed in recent
years by paying closer attention to the systematic effects (e.g., Sedaghati et al., 2015) and by
introducing new stable instruments.
It is also important to note that there is a tremendous amount of archival spectra – obtained during transits for other purposes, such as measuring the spin-orbit alignment measurements via the Rossiter-McLaughlin effect (Queloz et al., 2000) – that are available for further analysis. They can set the foundation of long-term exoclimate studies. Therefore, the method of high-resolution transmission spectroscopy from the ground is likely to become even more widely used than today, and to gain further importance for the characterization of exoatmospheres.
Here we investigate the atmospheric chemistry of two inflated super-Neptunes and two inflated hot-Jupiters with archival HARPS data111Data were obtained under Programme IDs: 090.C-0540(F), 098.C-0304(A) and 0100.C-0750(A). We study the instrument systematics that can influence the results and the effects of the stellar activity (see Klocová et al., 2017, for further discussion) using the calcium lines as activity indicators (Khalafinejad et al., 2017). This work is part of a larger effort to characterize systematically and uniformly a large number of exoplanetary atmospheres, and to shed more light onto their chemical composition. We also aim to identify interesting targets for CHEOPS and JWST follow-up.
2 Data sets and their analysis
2.1 Data summary
Table 2 shows the various datasets we used, while Table 1 presents the main properties of our targets. For illustration purpose we show part of the spectrum around the sodium Na D region of each of our targets, along with the telluric spectrum in Fig. 1.
WASP-76b:
The HARPS data are covering 3 transit events, on 11/12 November 2012, 24/25 October 2017 and 22/23 November 2017. The total number of frames was $176$ with $105$ in-transit, and the exposure times were between 300-600 seconds.
WASP-127b:
The HARPS data are covering 2 transit events during the nights of 27/28 February and 19/20 March 2017. The total number of frames was $82$, of which 52 were in-transit.
WASP-166b:
The HARPS data are covering three transits during the nights of 13/14 January, 03/04 March and 14/15 March 2017. The total number of frames was $188$, of which 112 were in-transit.
KELT-11b:
The HARPS data are covering 3 transit events and 4 nights of out-of-transit on 01/02 February (transit), 14/15 Februray, 15/16 February (transit), 16/17 February, 05/06 March, 06/07 March (transit) and 07/08 March 2017. The total number of frames was 342, of which 144 were in-transit.
2.2 Data Reduction
We retrieved the publicly available HARPS (Mayor et al., 2003) reduced spectroscopic data sets from the ESO Science data archive for the four planets WASP-76b, WASP-127b, WASP-166b and KELT-11b (see Table 1). The retrieved data are fully reduced products obtained with the HARPS Data Reduction Software (DRS version 3.5). In more detail, each spectrum is provided as a merged 1D spectrum with 0.01 $\AA$ wavelength step. The reduced spectrum covers the wavelength range between 380 nm and 690 nm, and has a spectral resolution R $\approx$ 115 000 corresponding to 2.7 km s${}^{-1}$ per resolving element. Spectra are corrected to the barycentric frame of reference. The sodium doublet is well-centered on the 16th order of HARPS.
2.3 Normalization
We obtained the radial velocities of the star (see below) from the FITS header of the spectra and subsequently shifted both during transit $f(\lambda,t_{\rm{in}})$ and out of transit $f(\lambda,t_{\rm{out}})$ frames to the stellar rest frame. We have made a cut of the region of interest and performed a normalization for the obtained spectra $f(\lambda,t_{\rm{in}})\rightarrow\tilde{f}(\lambda,t_{\rm{in}})$ and $f(\lambda,t_{\rm{out}})\rightarrow\tilde{f}(\lambda,t_{\rm{out}})$, where the tilde represents normalization. Normalization has been carried out using the standard IRAF routine continuum. More precisely, a Legendre polynomial has been used to perform the normalization of continuum regions without strong lines within the regions centred around the Na D lines (583 to 596 nm), H$\alpha$ (645 to 670 nm), H$\beta$ (475 to 495 nm) and lithium (666 to 674 nm).
2.4 Telluric correction
As the next step, the Earth’s atmospheric signature needs to be accounted for. There are several ways to remove telluric lines from the spectra. One of the methods is to simultaneously observe an early type rapidly rotating star to obtain the telluric spectrum. However, this method has a disadvantage of reducing the observing time spent on the primary target. Another method is described in Astudillo-Defru & Rojo (2013) and Wyttenbach et al. (2015) where they use the fact that the variation of the telluric lines depth follows the airmass variation linearly. We decided to follow the method used in Casasayas-Barris et al. (2017). We took the one-dimensional telluric spectrum constructed from the line list of HITRAN (Rothman et al., 2013) and matched its resolution to the resolution of the data. Then we corrected the reference telluric spectrum for the radial velocity to the same frame as the observed spectrum. The telluric lines absorption depth is changing with the airmass. Thus we used a Least Square Method on the unblended telluric lines to scale the telluric model to the same airmass (depth) as the observed spectrum. Afterwards, we divided each observed spectrum by the telluric spectrum scaled for the given airmass using standard IRAF commands resulting in the removal of the telluric lines. As discussed in Casasayas-Barris et al. (2017), the scaling of the telluric spectra is not perfect as line depth variation is not equal for all the telluric lines, hence possibly introducing small residuals into the final transmission spectrum. As an illustration,
the RMS of a single-night transmission spectrum in the telluric region (589.9 to 590.4 nm) before applying the telluric correction is 0.00339. The RMS of the same region was decreased to 0.00196 after applying the correction. For comparison, the RMS of the spectral region without telluric features (587 to 587.5 nm) is 0.00193.
2.5 Transmission spectrum
After the telluric correction, we averaged all out of transit spectra to create a “master out” frame $\tilde{F}_{\rm out}(\lambda)=\sum_{\rm{out}}\tilde{f}_{\rm{out}}(\lambda,t_{%
\rm{out}})$. Next, we divided each in-transit frame $\tilde{f}_{\rm in}(\lambda,t_{\rm{in}})$ by $\tilde{F}_{\rm out}(\lambda)$ to create the individual transmission spectra. Each of these frames was subsequently corrected for the planetary motion as the planetary signal is shifted from the stellar reference frame. We created a model of radial velocity taking only out-of-transit data222As the planetary signal is shifted by as much as tens of kilometers from the rest frame, using the observed radial velocities would imply correcting for the possible Rossiter-McLaughlin effect and would produce systematic effects in the final transmission spectrum Cegla et al. (2017); Wyttenbach et al. (2017).. Finally, we summed up all the individual transmission spectra and normalized this sum to unity and then subtracted unity to retrieve the final transmission spectrum, $\tilde{\mathcal{R}}(\lambda)$:
$$\tilde{\mathcal{R}}(\lambda)=\sum_{\rm{in}}\left.\frac{\tilde{f}_{\rm{in}}(%
\lambda,t_{\rm{in}})}{\tilde{F}_{\rm{out}}(\lambda)}\right|_{\rm{Planet\ RV\ %
shift}}-1$$
(1)
2.6 Error estimates
The uncertainties of the exoatmospheric Na D lines measurements
were estimated with a simple simulation from the data itself. This
approach has the advantage of preserving as much as possible the
systematic errors of the data.
The starting point of the simulation is our actual measurement –
the fit that reproduces the observed spectrum with a flat line and
two Gaussians (see Sec. 2.5). In the
following we only consider a region of the spectrum containing 190
pixel elements, centred on the Na doublet. We converted the fit
within this region into a noiseless spectrum with the same
wavelength sampling as the original observational data.
The first step of our error analysis is to calculate the residuals
of the observational data with respect to the fit. Second, the
residuals were shifted by one pixel and then added to the best fit
model spectrum. Thus, the residual from the first pixel of the data
was added to the second pixel of the fit, and the residual from
the last pixel of the data was added to the first pixel of the best
fit. This created a new realization of artificial data that have
the same Na D signal and a different noise for each pixel. However,
the overall statistical properties of the noise are preserved: for
example if there is a systematic trend of increasing noise toward
the end of the observations, because of increasing airmass, it will
be preserved. The third step is to remeasure the Na absorptions in
the new artificial spectrum.
The last two steps were repeated 189 times, creating 189 unique new
realizations – each time the shift of the residual pattern was
increased by one, so for the second realization the first deviation
was added to the third pixel and the last deviation was added to
the second pixel. For the third realization the first deviation was
added to the fourth pixel and the last deviation – to the third
pixel. In general, to create the i${}^{\rm th}$ realization we added
the first deviation to the (i+1)${}^{\rm th}$ pixel of the noiseless
model spectrum and the last realization – to the i${}^{\rm th}$
pixel of the noiseless model spectrum. Summarizing, this exercise
is a simple shift of the residuals in a circular manner. It has
been used before, e.g. by Cáceres et al. (2009, 2011).
The measurement errors were derived from the distributions (over
all realizations) of the fitting parameters determined for each
realization in the third step of the process described above.
These distributions are shown in Fig. 2. We adopt as
final measurements and as their errors the means and the FWHMs
values of these distributions.
2.7 Stellar activity
We also checked for possible false-positive detections. As suggested in Barnes et al. (2017), stellar activity could produce signatures in lines originating mainly from the stellar chromosphere. To rule out any such fake signals we investigated the vicinity of Mg I (581.36 nm) and Ca I (612.22 and 616.22 nm) lines. The lack of any features in these regions shows that the increase of absorption in the sodium Na D lines is not caused by the chromospheric activity or by systematic effects in our analysis.
2.8 Rossiter–McLaughlin effect
When looking at the presence of any atomic or molecular species in the spectrum of a planet, it is important to take into account the Rossiter–McLaughlin effect (R-M; Queloz et al., 2000; Ohta et al., 2005), as shown by Wyttenbach et al. (2017). In Fig. 4, we show the measured radial velocities of our targets as obtained with HARPS as well as the expected orbital motion of the star due to the presence of the planet. Any deviation from this could be due to the R-M effect. This is indeed the case for WASP-166, where the R-M effect is clearly visible (as also shown by Hellier et al. (2018)).
For the three other targets, the R-M effect is not measurable.
It is possible to have an order of magnitude estimate of the R-M effect (Triaud, 2018):
$$\Delta v_{r}\simeq\frac{2}{3}\gamma^{2}V\sin i\sqrt{1-b^{2}},$$
(2)
where $V\sin i$ is the rotational velocity of the star, $b$ is the impact parameter and $\gamma=\frac{R_{p}}{R_{s}}$, with $R_{p}$ being the radius of the planet and $R_{s}$ that of the star.
Estimating these values for the 4 targets we get the following values for $\Delta v_{r}$: 2.3 m s${}^{-1}$ (WASP-76), 2 m s${}^{-1}$ (WASP-127), 8 m s${}^{-1}$ (WASP-166), and 4 m s${}^{-1}$ (KELT-11). These are in agreement with what we see in Fig. 4.
Moreover, in all cases, the R-M effect is too small to have any real impact on the depth of the Na D lines (Nortmann et al., 2018). We thus conclude here that the R-M effect is not important in our analysis.
3 Results
Each data set was analyzed for Na D, H${}_{\rm\alpha}$, H${}_{\rm\beta}$ to see the absorption in the exo-atmosphere and for Ca I (612.2217 nm and 616.2173 nm) and Mg I (518.3604 nm) to monitor the stellar activity. The results are described for each star separately.
WASP-76b:
The scale height of the atmosphere of WASP-76b is estimated to be about 1212 km, making it an excellent system for the analysis by transmission spectroscopy (Kabath et al., 2019). This exoplanet was also suggested as one of the primary targets for JWST observations (Mollière et al., 2017). Fig.5 presents the final divided in/out transit spectra corrected for planetary radial velocity, displaying clearly visible planetary sodium (Na D lines) absorption. The RMS of the Na D region is 0.0027 and the RMS of the binned spectrum by 20 is 0.0013. The sodium line depths are $0.65\,\%$ and $0.57\,\%$, respectively, therefore, corresponding to 9.5 $\sigma$ and 7.1 $\sigma$ significance. This is the first detection of sodium in the upper atmosphere of WASP-76b. Details of the detected peaks are summarized in Table 3.
Figure 3 shows the non-significant detections in Ca I and Mg I lines (RMS of the data points is 0.0007), showing that the stellar activity is low and any planetary atmosphere sodium (Na D lines) detection should be significant and not influenced by the star. Furthermore, the star WASP-76 is a slow rotator as reported by Brown et al. (2017).
The D1 line is apparently stronger than the D2 line, which would imply different mixing ratios and compositions of the atmospheres of different planets because all other detections reported were with stronger D2 line. The explanation for the strength of the lines can be found in the literature for other Na lines in redder regions (Civiš et al., 2012), but such a measurement is missing in the optical. Civiš et al. (2012) present a laboratory spectrum of sodium lines demonstrating
that different atomic states are shaping the spectral lines and their strengths.
Furthermore, Slanger et al. (2005) measured the D2/D1 ratio in the Earth’s
atmosphere and found it to be varying between 1.2-1.8. They explained that the variability of the
ratio is originating from a competition between Oxygen reacting with
NaO($A^{3}\Sigma^{+}$), produced from the reaction of sodium with Oxygen. In other words, the line strength and the mixing ratio depend on the composition of the surrounding
atmosphere and it can even vary over time.
In the course of the review process of this paper, an independent detection of the sodium in the atmosphere of WASP-76b planet by Seidel et al. (2019) was reported for the same HARPS data set. Their results are in agreement with ours, confirming the methodology. We note, however, that our FWHM of the D2 line is about 30 % smaller compared to their values.
WASP-127b:
WASP-127b is a heavily bloated super-Neptune. Its scale height is $2500\pm 400$ km, making it another ideal target for transmission spectroscopy. Indeed, a cloudless sky was observed with many features such as sodium, potassium and lithium detected and hints of TiO absorption were reported with GTC instrumentation (Pallé et al., 2017; Chen et al., 2018).
Fig. 5 presents the final divided in/out transit spectrum. The RMS of the data is 0.0042 and by binning by 20 points we reach a final RMS of 0.0019. We were able to confirm only sodium (Na D lines) with a high significance of 4.2-$\sigma$ (D2) and 8.33-$\sigma$ (D1), respectively. We did not detect lithium (670.8 nm), which is most likely due to aperture differences between the GTC (10 m) and the 3.6 m telescope that hosts HARPS, while as mentioned earlier, potassium (766.5 & 769.9 nm) is out of range of HARPS. However, our confirmation of sodium is unique because this is one of the first proofs for observations using different instruments as well as two different methods.
The stellar activity of WASP-127 star can be monitored with the Ca I and Mg I lines. We were not able to see any activity in these lines, as shown in Fig. 3.
WASP-166b:
WASP-166b is a bloated super-Neptune with a scale height of 1103 km. Fig. 5 presents the final divided in/out transit spectrum of all regions of interest. The RMS of the data is 0.0025 and, after binning by 20 points, 0.0014. However, we are not able to detect any feature in the sodium (Na D) lines. Therefore, we can only put an upper limit of 0.14 $\%$ on the detection.
Fig. 3 displays no signatures in the Mg I and Ca I lines. Therefore, we can assume that the star is not active.
KELT-11b:
The scale height of KELT-11b is about 2500 km and since the planetary atmosphere is bloated, this is another excellent system for detection of exo-atmosphere (Kabath et al., 2019).
Fig. 5 presents the final divided in/out transit spectrum of all regions of interest. The RMS of the data is 0.0012 (unbinned) and 0.0006 (binned by 20 points). The upper limit for a sodium (Na D lines) detection is therefore 0.06 $\%$. This is a very stringent constraint on the presence of sodium in the atmosphere and possibly hints at the presence of clouds in the atmosphere of KELT-11b.
We checked the stellar activity on Ca I and Mg I lines. There was no significant detection of activity observed in our data (see Fig. 3).
3.1 Previous Na D detections
We have investigated all the previous Na D line detections in exoplanets based on high-resolution spectroscopy. We have summarized them in Table 4. As can be seen, WASP-76b is the first reported planet having potentially an absorption ratio of D2 to D1 lines less than one333One should note, however, that the deviation from unity is less than 1-$\sigma$, and we therefore do not claim much about this yet..
The unusual reported values make this planet an ideal target for future JWST follow-up to further clarify this discovery.
4 Conclusions
We investigated a group of four inflated gas giant planets, among which two inflated hot-Jupiters and two super-Neptunes. All four targets are ideal targets for transmission spectroscopy, given their magnitudes and their large scale heights. We reported for the first time the detection of sodium in WASP-76b with a 7–9 $\sigma$ significance.
We have been able to fully resolve the Na D lines. We report for the first time an absorption ratio of D2 to D1 lines less than one, although this is within 1-$\sigma$ of unity. If confirmed, this possibly hints at an unusual composition of the exo-atmosphere.
There was no hydrogen (H$\alpha$ and H$\beta$ lines) detected for WASP-76b. Another planet, WASP-127b was already characterized in previous studies and we could confirm the presence of sodium in the atmosphere. This is to our knowledge the first time such a detection is confirmed, and provides further credence to the transmission spectroscopy method given that two different instruments on two very different aperture telescopes (3.6-m vs. 10-m GTC) provide similar results. For WASP-166b and KELT-11b, we were not able to detect any feature, despite having extremely precise data, thus hinting at cloudy skies or simply the absence of sodium and other elements. We note that non-detections are usually not reported, although they provide useful information as well.
The data presented here are available in the ESO archive under ESO Prog.IDs: 090.C-0540(F), 098.C-0304(A) and 0100.C-0750(A).
PK and JZ would like to acknowledge the support from the GACR international grant 17-01752J. MS acknowledges financial support of Postdoc@MUNI project CZ$.02.2.69/0.0/0.0/16\_027/$ 0008360. This work benefited from support for mobility by the DAAD-18-08 programme, Project-ID: 57388159. We acknowledge the usage of IRAF and iSpec. We would like to
thank the anonymous referee for providing comments that helped to improve the manuscript.
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Exploring Linguistic Style Matching in Online Communities: The Role of Social Context and Conversation Dynamics
Aparna Ananthasubramaniam, Hong Chen11footnotemark: 1, Jason Yan11footnotemark: 1, Kenan Alkiek11footnotemark: 1, Jiaxin Pei11footnotemark: 1,
Agrima Seth11footnotemark: 1, Lavinia Dunagan11footnotemark: 1, Minje Choi11footnotemark: 1, Benjamin Litterer11footnotemark: 1 and David Jurgens
School of Information, University of Michigan
{akananth,hongcc,jasonyan,kalkiek,pedropei,
agrima,laviniad,minje,blitt,jurgens}@umich.edu
denotes equal contribution
Abstract
Linguistic style matching (LSM) in conversations can be reflective of several aspects of social influence such as power or persuasion. However, how LSM relates to the outcomes of online communication on platforms such as Reddit is an unknown question.
In this study, we analyze a large corpus of two-party conversation threads in Reddit where we identify all occurrences of LSM using two types of style: the use of function words and formality. Using this framework, we examine how levels of LSM differ in conversations depending on several social factors within Reddit: post and subreddit features, conversation depth, user tenure, and the controversiality of a comment. Finally, we measure the change of LSM following loss of status after community banning.
Our findings reveal the interplay of LSM in Reddit conversations with several community metrics, suggesting the importance of understanding conversation engagement when understanding community dynamics.
1 Introduction
Social influence can be subtle. When two persons converse, their interpersonal dynamics can lead to one person adopting the language of the other. For example, in settings where one person has higher status or power, the lower-status person may unconsciously begin mirroring the language of the other (Danescu-Niculescu-Mizil et al., 2012). This process has been described as accommodation Giles and Ogay (2007) or linguistic style matching (LSM) Niederhoffer and Pennebaker (2002) and can reflect the underlying influence that individuals have on each other Chartrand and Bargh (1999). Past work has primarily focused on how linguistic influence changes relative to the identities of the speakers. However, the larger social context in which a conversation happens also plays a role in determining whether an individual may be influential. Here, we perform a large-scale study of linguistic influence to test how specific types of social context influence the level of accommodation.
Past work in the social sciences has studied accommodation to understand the influence and social power dynamics in specific settings, like job interviews (applicants and interviewers) Willemyns et al. (1997) and academic context (students and faculty)Jones et al. (1999). Also, LSM has been studied to understand group dynamics (Gonzales et al., 2010) and negotiations (Ireland and Henderson, 2014). Work in NLP has operationalized these theories to test accommodation theory in new domains. Typically, these works adopt some tests for measuring influence in language and have shown these measures correlate with known social differences. However, it is yet unknown how LSM occurs in conversations in online community platforms and differs by community dynamics.
Our work examines the larger context in which linguistic influence occurs. Using a large sample of 2.3 million conversations from Reddit and two measures of linguistic influence, we test how the level of linguistic influence correlates with conversational outcomes, such as conversation length and even the continued presence of a person in a community. Further, we examine how specific social and contextual factors influence the rates of linguistic influence. For instance, we discover that the controversy level of the parent comment can lead to different dynamics of style matching in the conversation threads.
This paper offers the following three contributions. First, we systematically compare complementary measures of accommodation, showing clear evidence of style accommodation in Reddit conversations. Second, we draw the relationships of several social factors that affect LSM, including levels of engagement, the popularity of the content, and tenure within a subreddit.
Third, we demonstrate the use of LSM to measure the loss of status through the banning of subreddits. We have released all code and data for full reproducibility.111https://github.com/davidjurgens/style-influence
2 Accommodation and its Measurement
In this section, we discuss communication accommodation theory and associated sociolinguistic research to outline the accommodation of communicative behavior based on perceived social power dynamics. Subsequently, we explore the concept of linguistic style matching and methods adopted by researchers to quantify this phenomenon. We also investigate various factors that contribute to LSM variations and their strategic uses.
2.1 Accommodation Theory as Social Influence
When two individuals engage in social interaction, they may either converge or diverge in their communicative behavior. The Communication Accommodation Theory (CAT) suggests that the degree of convergence or divergence is affected by the relative social power between the interlocutors Xu et al. (2018). Asymmetric convergence is more likely to occur in situations where there is a power imbalance between the interlocutors. Individuals with lower social power or status are more likely to adapt their communication style to align with those in higher or dominant positions Muir et al. (2016). For instance, Puerto Ricans in New York City during the 1970s, who were perceived to hold less power than African Americans, adopted the dialect of African Americans to converge with their more powerful counterparts Wolfram (1974).
Social power has been often found to be an important determinant of degrees of accommodation Giles et al. (1991); Ng and Bradac (1993) and interactants of differential social power or social status can act in a complementary fashion Street (1991).
2.2 Linguistic Style Matching
Linguistic alignment is a pervasive phenomenon that occurs in human communication where interactants unconsciously coordinate their language usage. This coordination, described as convergence in the psycholinguistic theory of communication accommodation, involves aspects such as word choice, syntax, utterance length, pitch, and gestures Giles et al. (1991). Linguistic style matching (LSM) is a specific manifestation of linguistic alignment, wherein individuals unconsciously match their speaking or writing styles during conversations Ireland et al. (2011). Unlike content accommodation, LSM focuses on stylistic accommodation, examining how things are communicated rather than what they communicate.
Individuals strategically negotiate their language style to decrease social distance, seek approval, and accommodate each other. LSM can also reflect the level of common understanding and conceptualization of the conversation topic between speakers. The degree of LSM can indicate social power dynamics as indicated by Giles and Ogay (2007). Empirical evidence from recent studies Danescu-Niculescu-Mizil et al. (2012) showed that participants with less power (such as lawyers or non-administrative roles in Wikipedia) exhibit greater coordination in conversational behavior than participants with high power (such as justices or administrators). Additionally, Noble and Fernández (2015) identified a positive correlation between linguistic accommodation and social network centrality, which effect can be greater than the effect of power status distinction. Studies by Muir et al. (2016, 2017) further show that individuals in a lower position of power tend to accommodate their linguistic style to match that of their higher-power counterparts during face-to-face communication as well as computer-mediated communication.
The variance in LSM can be attributed to various social and psychological factors and can be triggered for different purposes. Linguistic alignment may signal likability and agreement, relate to seeking approval or arise from social desirability. Higher levels of accommodation in social behaviors are found to be associated with increased feelings of affiliation, liking, and successful interpersonal relationships Bayram and Ta (2019).
Thus, linguistic alignment can be strategically employed to establish relationship initiation and stability Ireland et al. (2011), increase group cohesion, and task performance Gonzales et al. (2010), and assist in negotiations Taylor and Thomas (2008). Furthermore, alignment has been found to enhance persuasiveness, motivating listeners to adopt healthier practices Cialdini (2001) while in some cases like presidential debates, it has been perceived as more aggressive Romero et al. (2015). The degree of matching may differ based on context and individual factors.
3 Data
Reddit is a popular social media platform with a forum-based interface. It allows users to interact with dispersed individuals who share similar experiences or topics of interest. Our dataset to study LSM spans from July 2019 to December 2022 and includes 35M users and 500K subreddits.
Using the Pushshift Reddit Dataset which contains the full history of comments aggregated on a monthly basis (Baumgartner et al., 2020), we construct conversation threads from the comments and filter those that satisfy the following conditions: (1) the conversation chain consists of exactly two users; (2) the beginning of the conversation chain must be a root comment which does not have a parent comment; and (3) the lengths of a conversation chain must between 3 and 100.
These conditions allow us to capture conversation dynamics between exactly two users without any interference. Our resulting dataset contains 16,893,013 conversation turns (or comments) across 2,305,775 conversation chains from 68,788 subreddits.
4 How should we measure linguistic influence?
Computational work has proposed multiple approaches for both what to measure and how to measure linguistic influence. In this section, we aim to build intuition for what the two measures of accommodation—using function words and formality—are operationalizing.
4.1 Linguistic Style Markers
Our study measures linguistic influence with two complementary style markers. We use the notation $m$ to refer to a marker throughout.
Marker 1: Function Words
Function words (e.g. pronouns, prepositions, articles, and auxiliary words) are primarily employed unconsciously and frequently and incorporate social knowledge for comprehension and usage Meyer and Bock (1999); Ireland and Pennebaker (2010).
Prior computational studies of linguistic accommodation have measured linguistic influence by tracking the relative frequencies of function words across conversation turns (Danescu-Niculescu-Mizil et al., 2011; Babcock et al., 2014; Gonzales et al., 2010).
Function words reflect how content is expressed, rather than what specific content is expressed (e.g., content words) and are thought to be a better proxy for unconscious language processing Tausczik and Pennebaker (2010). Here, we use the function words defined by the Linguistic Inquiry and Word Count (LIWC) lexicon Pennebaker et al. (2001); Pennebaker and Chung (2007).
Marker 2: Formality
Individuals adopt a specific register that is appropriate to their position in the social context, real or desired Niederhoffer and Pennebaker (2002). A commonly varied register is the level of formality used when speaking to another. The level of formality shown by a speaker is known to reflect the speaker’s opinion towards a topic or their closeness to the listener (Hovy, 1987).
Unlike function words, variation in formality often requires conscious processing to select the appropriate phrasing in a given circumstance. As a result, it offers a complementary view into how a speaker influences another through shifting the conversation towards a more formal or informal register.
Here, we measure formality using a supervised classification model. The model is a fine-tuned RoBERTa-based classifier (Liu et al., 2019) trained on the GYAFC (Rao and Tetreault, 2018) and Online Formality Corpus (Pavlick and Tetreault, 2016) datasets; we use the model available from the Hugging Face API222https://huggingface.co/s-nlp/roberta-base-formality-ranker. Both datasets contain social media text and the reported model performance is high for both blogs and Q&A text (Spearman’s $\rho$>0.7).
Using this classifier, each comment’s formality is measured on a continuous scale in [0,1].
Importantly, these style variables are related; function word frequency also changes in more formal contexts, where articles and prepositions typically become more common while pronouns and interjections become less common (Heylighen and Dewaele, 1999).
Content word-based measures of style and function word counts are thought to capture the same latent style variables, i.e., they are interchangeable at a stylometric level (Grieve, 2023).
4.2 Measuring Linguistic Influence
At a high-level, linguistic influence (also referred to as LSM or accommodation in this paper) is measured by testing whether the value for some measure $m$ of a comment made by user $a$ is predictive of the value of $m$ in the reply to that comment by user $b$.
Therefore, one straightforward way to measure accommodation is with linear regression: $m_{b}\sim\beta_{0}+\beta_{1}m_{a}$ where $\beta_{0}$ reflects the baseline level of the measure (e.g., the average formality) and $\beta_{1}$ measures the level of accommodation (e.g., the average increase in formality associated with a 1-unit increase in the formality of the parent comment).
However, as Xu et al. (2018) note, the characteristics of a comment are likely influenced by other unrelated factors such as the length of the comment or the number of turns in the conversation. Indeed, they show that unless one controls for such factors, linguistic influence may be overestimated.
Therefore, we used a mixed-effects regression to control for comment $a$ and $b$’s length in tokens (fixed effects $L_{a}$, $L_{b}$), the number of replies $r_{b\rightarrow a}$ that $b$ has made to $a$ so far in the conversation.
To capture individual and community-level variation, we include random effects to control for the effect of the subreddit $s$; these random effects let us control for differences in the norms of communities (e.g., some communities are more/less formal) to test for relative changes in $m$. Linguistic accommodation is modeled as
$$\displaystyle m_{b}\sim$$
$$\displaystyle\beta_{0}+\beta_{1}m_{a}+$$
$$\displaystyle\beta_{2}L_{a}+\beta_{3}L_{b}+\beta_{4}r_{b\rightarrow a}+(1|s)$$
where $\beta_{1}$ measures the level of accommodation.
4.3 Results
We first observe clear evidence of accommodation in both style markers: parent comments with more function words receive replies with more function words (Figure 0(a)), and more formal parent comments receive more formal replies (Figure 0(b)).
For comments where we have the text of the original post, we observe accommodation even after controlling for the author and original post’s style markers, suggesting that users may accommodate to the style of the person they are interacting with in the comment thread. However, this effect plateaus when the parent comment has above-average levels of a style marker, suggesting a potential threshold for the impact of parent comment style on reply style. This attenuation of effect may be the result of several mechanisms, including regression to the mean or an author modulating their replies according to their own personal style (i.e., a more extreme parent comment may trigger greater modulation).
Second, the two style markers are almost perfectly uncorrelated, suggesting that they measure distinct constructs. In order to calculate the correlation between these two measures, we randomly sample 1,000 subsets of the conversation turns and calculate the extent of accommodation in function words and formality in that subset.
The correlation between the function-word- and formality-based accommodation scores is -0.00171.
Third, accommodation in the two style markers seems to occur via fundamentally distinct psychological processes. Accommodation can occur either 1) through a subconscious priming mechanism, where the speaker instinctively repeats what they hear; or 2) through a more conscious, strategic act with communicative intent (Doyle and Frank, 2016).
Figure 2 suggests that function-word-accommodation seems to be an unconscious form of relating to the audience, while formality-accommodation seems to be more intentional and strategic. Commenters exhibit greater accommodation in function words when they take less time to reply to the prior comment (1(a)) and greater accommodation in formality when they reply more slowly (1(b)). These results are consistent with prior work, suggesting that accommodation of function words occurs subconsciously (reflexively, takes less time) and builds on this work to show that accommodation in other style markers, like formality, occurs strategically (intentionally, takes more time).
Fourth, there is little variation in accommodation across subreddit characteristics. Figure 3 shows the levels of accommodation across ten different types of subreddits, using an existing taxonomy of popular subreddits.333https://www.reddit.com/r/ListOfSubreddits/wiki/listofsubreddits/ While certain types of subreddits (e.g., lifestyle) tend to have higher levels of accommodation than others (e.g., technology, entertainment), most differences are only weakly significant ($p>0.01$) with a small effect size. Moreover, Figure 4 shows the relationship between subreddit size and variation in linguistic style, for 300 subreddits sampled based on their number of subscribers. To calculate variation in linguistic style, we use Biber (1988)’s comprehensive set of linguistic features. Linguistic variation within each subreddit is estimated as the mean Shannon Entropy of each Biber tag frequency at the subreddit level.
Despite expectations that larger communities may exhibit greater diversity in language use (Kocab et al., 2019), we find no relationship between community size and linguistic variation.
Overall, these findings point to the nuanced dynamics of LSM in online interactions, indicating that factors such as function word usage and formality in the parent comment are associated with the linguistic style and tone of replies.
5 What factors about a comment influence the degree of accommodation?
LSM can be affected by many factors and existing studies have pointed out the roles of not only linguistic characteristics but also the contextual factors affecting LSM (Niederhoffer and Pennebaker, 2002). In this section, we study the connection between LSM and a series of contextual factors where the comment is posted (i.e., comment depth) and the “success” of a comment (i.e., comment Karma and parent comment Karma).
5.1 Experimental Setup
To test for heterogeneity in the level of accommodation with respect to several covariates (e.g., depth, Karma), we run a mixed effects regression similar to Section 4.2, but include an interaction term to test whether accommodation changes significantly with respect to some covariate (say, Karma $K$):
$$\displaystyle m_{b}\sim$$
$$\displaystyle\beta_{0}+\beta_{1}m_{a}+\beta_{2}K+\beta_{3}m_{a}*K+$$
$$\displaystyle\beta_{4}L_{a}+\beta_{5}L_{b}+\beta_{6}r_{b\rightarrow a}+(1|b)+(1|s)$$
Here, $\beta_{1}$ measures the level of accommodation when $K=0$ and $\beta_{3}$ measures the increase in accommodation when $K$ increases by one point; if $\beta_{3}$ is significantly different from $0$, then we have evidence that accommodation is heterogeneous with respect to Karma.
In order to visualize these effects, we fit the model in the above equation to estimate accommodation at different values of Karma. In order to appropriately represent uncertainty in this model, we sample 100,000 conversation turns at each value of Karma 10 times and use this to obtain 10 different estimates of accommodation for each value of the covariate. To visualize the association between Karma and accommodation, we plot Karma on the x-axis and the LSM estimates on the y-axis.
5.2 Results
As shown in Figure 5, various factors of comments are related to LSM.
Comment depth
Comment depth reflects the position of a comment in the conversation tree. Deeper comments are usually posted in longer conversations and when the users are more engaged in the dialogue. As shown in Figure 4(a) and Figure 4(c), comment depth is positively correlated with LSM. However, accommodation in formality drops off for very deep comments. LSM happen more when the comment is deeper in the conversation tree, suggesting that users tend to match not only the content but also the structural aspects of their language in response to their interlocutor. Such a trend could be due to greater investment in the conversation. When two users are involved in longer and deeper conversations, they are more likely to be engaged in the conversation, which may lead to higher subconscious but lower conscious LSM.
Comment Karma
A key feature of Reddit is the ability for users to upvote or downvote comments, which determines the comment’s karma - a measure of its popularity within the community. In figure 5, we observe several non-linear associations between karma, comment characteristics, and LSM. In terms of comment karma, users’ LSM tends to remain relatively constant, except for cases where the comment has very high karma, which is associated with an increase in LSM. This finding implies that highly popular comments may foster greater linguistic alignment between users.
We also see that comments with low karma have lower levels of LSM than comments with high karma (Figure 4(d)), which makes sense since we’d expect users to respond better to comments whether the author is mirroring their interlocutor. Notably, this upward trend reverses in comments with very high karma – which have lower levels of LSM than comments with lower levels of karma. The reversal of the LSM trend in comments with high karma warrants further exploration. One possible explanation for this phenomenon is that highly upvoted comments may exhibit unconventional linguistic styles that deviate from the norm, which could be seen as novel by the Reddit community. Another explanation may be that comments with high karma are more likely to be popular in larger, diverse communities where users may have a wider range of linguistic styles. Additionally, it is possible that comments with high karma receive a higher volume of comments and interactions, which may dilute the overall LSM score due to the presence of diverse linguistic styles from multiple interlocutors.
6 What effect does accommodation have on the conversation itself?
Linguistic accommodation is usually associated with positive social benefits Taylor and Thomas (2008); Gonzales et al. (2010). Here, we test whether linguistic accommodation is associated with two positive behaviors in social media: sustained conversation and length of participation in a subreddit.
6.1 Experimental Setup
We fit a linear regression on conversational dyads following the LSM measure in Section 4.2. Following the procedure from the prior section, we estimate the level of accommodation for comments around a particular covariate by sampling 100,000 conversation turns at or near the respective value of the covariate. Once again, we verify that differences between covariates are significant, by introducing interaction terms in the regression and testing for a statistically significance effect.
6.2 Results
Figures 6a and 6b compare the effect of alignment when conditioned on the total length of the conversation thread. For both functions words and formality, we observe from the fitted lines that accommodation is more likely to happen from longer conversations, but only up to a certain length of approximately 30-40. This suggests the possibility of LSM being an earlier indicator of how engaged the users will be in a conversation. On the other hand, the likelihood of accommodation in formality decreases when the conversation becomes longer than a certain threshold, which suggests that speakers may stop consciously trying to accommodate once the conversation becomes sufficiently long.
Figures 6c and 6d compare accommodation likelihoods at a given turn within a conversation. Interestingly, we can observe that LSM starts off highest at the beginning of a conversation and decreases as the number of turns increases. Combining the two results, we can conjecture that while the degree of LSM generally decreases within a conversation thread, the initial levels of LSM observed at the early stages of a conversation can indicate how engaged the speakers will be, which one can use to estimate the overall conversation length.
How does LSM differ by tenure and number of subsequent posts in a subreddit? Figure 7 shows that, for both style markers, users who have a longer tenure in the subreddit or who post more in the subreddit in the next month tend to display higher subconscious and lower conscious LSM. We consider these results as evidence of the “lifespan” of a user’s engagement toward conversations held within that subreddit, and ultimately engagement toward the subreddit itself, which has been noted in prior work (Danescu-Niculescu-Mizil et al., 2013).
7 What effect does the social context have on accommodation: Controversiality?
In this section, we examine whether LSM differs by social contexts that arise during conversations. Specifically, we focus on the controversy level of the parent comment.
In contrast to non-controversial issues, controversial issues lead to competitive disagreement, where the goal of the groups involved in argumentation is to convince the opponent group(s) of the validity of one’s point of view Ilie (2021). The arguments on controversial issues tend to invite strong emotions with negative affect Mejova et al. (2014) and deteriorate the deliberation in the public sphere because interactions often turn uncivil Doxtader (1991).
7.1 Experimental Setup
Following the procedure from the prior section, we estimate the level of accommodation for comments at each covariate, separately for controversial and non-controversial comments.
When a comment or post receives a substantial number of upvotes and downvotes, Reddit automatically designates it as controversial. The exact method used by Reddit to determine controversy remains private. However, the Reddit API offers a binary label indicating whether a comment is controversial or non-controversial Koncar et al. (2021). Approximately 1.30% (n=218,899) of the comments in our sample are labeled as controversial.
We test that differences between conditions are significant with a three-way interaction term in the regression between the parent-comment style, the comment’s Karma (or other covariates) and the comment’s controversiality: $m_{a}\times K\times C$.
7.2 Results
Figure 8 reveals that LSM occurs differently in controversial and non-controversial comments. For both function words and formality, LSM is less likely to occur in controversial rather than non-controversial comments when the conversation length is below a certain threshold (12-14). Interestingly, we see that this trend is strengthened as the conversation length increases. One possible explanation is that controversial comments generate more initial interest that promotes users to engage more in conversations. However, this initial effect is washed away as the conversation takes further turns, and the conversation is less likely to continue due to reasons such as incivility. Non-controversial comments, on the other hand, enjoy less of this initial boost and is more likely to carry on if the users have accommodated each other’s language during their conversation.
With the addition of Karma, we can observe a more complex trend that plays out differently for each style marker. For function words, conversations in controversial comments have a nonlinear relationship that drops as the parent comment’s Karma increases, whereas a weak positive correlation can be observed for non-controversial comments and levels of Karma. In contrast, for formality, LSM occurs most at comments with about 0-5 Karma and decreases for higher Karma for both controversial and non-controversial comments.
Overall, we observe that social contexts that are defined by the community platform such as Karma or controversy have complex, nonlinear effects on how LSM occurs in conversations.
8 Loss of Status via Community Banning
Reddit bans specific subreddit communities as a result of policy violations, such as repeated posting of highly offensive content or lack of moderator oversight Chandrasekharan et al. (2017). When users are highly active in such communities, the ban potentially results in a loss of status, as they are forced to find new communities to participate in.
Here, we test the extent to which users change how they are linguistically influenced by others after such a ban. While prior work has studied how users change after gaining status (e.g., Danescu-Niculescu-Mizil et al., 2012), our unique setting allows us to perform a novel study of the potentially humbling effects of status loss. In addition, a study of the subreddit r/changemyview suggests that formality is (weakly) associated with more effective persuasion on Reddit (Dayter and Messerli, 2022); we hypothesize that users who recently experienced a ban may have multiple pragmatic reasons to accommodate more.
8.1 Experimental Setup
We test for changes to linguistic influence using a pseudo-causal difference-in-difference analysis Lechner et al. (2011). Subreddit ban dates were determined by identifying all banned subreddits and then using the last date of a post in that subreddit. Our sample includes 1,024 subreddits banned between July 2019 and December 2022. We identify 16,686 users in our sample who made at least one comment in these subreddits in the 30 days before their ban. Each user from a banned subreddit is considered as treated and matched with a control user who did not participate in that subreddit.
Three analyses of the effect of the ban are performed, controlling for user-level and temporal factors. First, we estimate the effect of commenting in a banned subreddit, by comparing posts made in banned subreddits $t$ months before the ban to posts made by the same users at the same time, in other subreddits. Second, using a difference-in-differences approach, we estimate the effect of banning a subreddit on authors’ use of accommodation in (unbanned) subreddits they were active in for $t$ months before and after the ban. This second analysis measures the spill-over effects of the ban on users’ behaviors in other subreddits; the difference-in-differences estimator uses users active in these subreddits at the same time, but not in a banned subreddit, as a control for temporal and subreddit-level effects. Third, we calculate the effect of the ban on commenting behavior in subreddits users migrated to (i.e., newly joined) after the ban was enacted. The difference-in-differences estimator compares accommodation in comments in the banned subreddits to comments in the subreddits these users migrated to; to isolate the effect of migration, the difference between the comments in the migrated and banned subreddits are compared against the spill-over effects in other subreddits that users were a part of during this time.
8.2 Results
Our results suggest that policy actions on Reddit, such as banning, have an effect on the level of accommodation by users. First, the level of subconscious accommodation tends to be lower in banned subreddits than other subreddits the users comment in during the 30 days before the ban (the effects are all below 0 in Figure 8(c) ($p<2e-16$).
Second, following the banning of a subreddit, users tend to change their LSM levels in other subreddits: Figure 9 shows that function-word-mirroring (Figure 8(a)) and formality-mirroring (Figure 8(b)) increase after a subreddit is banned. Our results suggest that users who had previously been active in banned subreddits may have been making an effort to index agreeableness by accommodating (e.g., to avoid losing status in another community).
Third, changes in accommodation are initially amplified in subreddits that these users migrate to after their original community was banned. The comments left by these users in banned subreddits exhibit higher levels of accommodation than would be expected immediately before the ban and maintain higher subconscious accommodation in subreddits they migrated (Figures 8(c) and 8(d)$p<2e-16$). Since function-word mirroring is likely subconscious and formality-mirroring strategic (Section 4), our results suggest that users who had previously been active in banned subreddits may have, intrinsically, indexed agreeableness by accommodating (e.g., to gain status in their new community) but without making a conscious effort (e.g., because they were upset about the loss of a status). These users also increased LSM in the subreddit immediately before it was banned (e.g., perhaps to index agreeableness when warnings about the ban were issued).
9 Discussion and Conclusion
In this study, we performed a large-scale computational analysis on Reddit conversations to understand when LSM occurs and its effect on platform engagement. Overall, do our findings indicate that LSM frequently occurs in online conversations within Reddit, and that it exhibits complex nonlinear relationships with conversation metrics such as Karma, conversation lengths, or controversy scores, which suggests linguistic influence can affect conversation dynamics. Furthermore, we show that the degree of accommodation in conversations is related to greater levels of engagement both at conversation and platform levels. Our findings highlight the possibility of identifying LSM as an indicator of engagement and civil conversations and suggest ideas for building and maintaining online communities that promote constructive discourse.
In our experiments, we have assumed LSM as a unidirectional concept by measuring the exhibition of a particular style conditioned on the previous turn. However, LSM can occur in several different directions, such as the two speakers converging into a single style or even diverging to separate styles. While not in the scope of this study, the existence of such types of LSM in Reddit conversation threads can be studies in future research.
10 Ethical Considerations
This study was conducted only on observational data and did not require any human intervention. We did not use any information that could identify individuals or specific demographic groups, and all of our presented results were obtained through aggregation from millions of users and comments.
Acknowledgments
This material is based in part upon work supported by the National Science Foundation under Grant No IIS-2143529.
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Acoustic radiation force and torque on an absorbing particle
Glauber T. Silva
Abstract
Exact formulas of the radiation force and torque exerted on an absorbing particle in the Rayleigh scattering limit
caused by an arbitrary harmonic acoustic wave are presented.
They are used to analyze the trapping conditions by a single-beam
acoustical tweezer based on a spherically focused ultrasound beam.
Results reveal that particle’s absorption has a pivotal role in single-beam trapping.
Furthermore, we obtain the radiation torque induced by a Bessel beam in an on-axis particle.
Physical Acoustics Group, Instituto de Física,
Universidade Federal de Alagoas, Maceió, AL 57072-970, Brazil.
[email protected]
PACS numbers: 43.25.Qp, 43.20.Fn, 43.55.Ev
1 Introduction
An increasing interest on acoustic radiation force has arisen after
the concept of acoustical tweezers was developed 1.
An acoustical tweezer can be accomplished by means of an ultrasound
standing wave 1 or a focused beam 2.
Understanding on how Rayleigh particles (i.e. particles much smaller than the incident wavelength) are
trapped is relevant, for instance, in lab-on-a-chip tweezers design 3.
Commonly, particle trapping is described through Gor’kov’s radiation force theory 4.
In turn, this theory is valid when the particle’s viscous boundary layer $\delta$
is much smaller than its radius $a$ 5, where $\delta=\sqrt{2\nu/\omega}$,
where $\nu$ is the kinematic viscosity of the host fluid and $\omega$ is the wave frequency.
Another important effect in radiation force generation is the ultrasound absorption inside the particle.
This effect has been analyzed in the radiation force exerted on a particle by
a plane wave 6.
The purpose of this letter is to extend Gor’kov’s radiation force theory with
absorption effects beyond plane wave results.
Ultrasound absorption is assumed to obey a frequency power-law model 7.
Shear viscosity inside the particle is not taken into consideration and $\delta\ll a$ is assumed .
Apart from radiation force, we also obtain the radiation torque induced
by an arbitrary ultrasound beam on the particle.
Exact formulas for the radiation force and torque are derived from the partial-wave expansion
up to the quadrupole moment 8, 9.
The developed theory is applied in the analysis of the radiation force due to a spherically focused beam.
Results show that transverse trapping depends on particle’s compressibility and density with respect to
the host fluid, while axial trapping also reckon on the particle’s absorption.
In addition, we study the radiation torque produced by a Bessel vortex beam in an on-axis particle.
It is demonstrated that only the first-order Bessel vortex beam can generate radiation torque in this configuration.
2 Rayleigh scattering
Consider an incident acoustic wave of angular frequency $\omega$ propagating
in a nonviscous fluid of density $\rho_{0}$ and speed of sound $c_{0}$.
The wave is scattered by a compressional fluid particle of radius $a$, density $\rho_{1}$,
and speed of sound $c_{1}$.
The incident wave can be represented by the pressure $p$ and the velocity particle ${\bf v}$,
while
the scattered pressure $p_{\text{s}}$ is described in spherical coordinates $(r,\theta,\varphi)$ with respect to
the particle’s center
through the partial-wave expansion method.
In the Rayleigh scattering limit, i.e. the particle is much smaller that the incident wavelength,
only monopole and dipole terms are relevant in the partial-wave expansion.
Accordingly, 10
$$p_{\text{s}}=p_{0}\left\{s_{0}a_{0}^{0}h_{0}^{(1)}(kr)Y_{0}^{0}(\theta,\varphi%
)+s_{1}\left[a_{1}^{-1}Y_{1}^{-1}(\theta,\varphi)+a_{1}^{0}Y_{1}^{0}(\theta,%
\varphi)+a_{1}^{1}Y_{1}^{1}(\theta,\varphi)\right]h_{1}^{(1)}(kr)\right\},$$
(1)
where $p_{0}$ is incident peak pressure magnitude, $k=\omega/c_{0}$ is the wavenumber,
$h_{n}^{(1)}$ the $n$th-order
spherical Hankel function of first-type,
and $Y_{n}^{m}$ is the $n$th-order and $m$th-degree
spherical harmonic.
The quantities $a_{n}^{m}$ and $s_{n}$ are the beam-shape and scattering coefficients, respectively.
The beam-shape coefficient is explained in the Appendix A.
The scattering coefficient is obtained by applying the continuity condition of pressure and particle
velocity across the particle’s surface.
These conditions lead to following scattering coefficient
$$s_{n}=-\det\left[\begin{matrix}\gamma j_{n}(ka)&j_{n}(k_{1}a)\\
j_{n}^{\prime}(ka)&j_{n}^{\prime}(k_{1}a)\end{matrix}\right]\det\left[\begin{%
matrix}\gamma h_{n}^{(1)}(ka)&j_{n}(k_{1}a)\\
{h_{n}^{(1)}}^{\prime}(ka)&j_{n}^{\prime}(k_{1}a)\end{matrix}\right]^{-1},$$
(2)
where
$k_{1}$ in the wavenumber inside the particle, $\gamma=\rho_{0}k_{1}/(\rho_{1}k)$, and
the prime symbol indicates differentiation.
It is assumed that the ultrasound absorption within the particle obeys a frequency power-law
as follows 7
$\alpha=\alpha_{\upsilon}\omega^{\upsilon},$
where $\alpha_{\upsilon}>0$, and $0<\upsilon\leq 2$.
Thus, the inner wavenumber of the particle is expressed as
$k_{1}=\omega_{1}/c_{1}+i\alpha_{\upsilon}\omega^{\upsilon}.$
Our analysis is limited to weak-absorption, thus, $\alpha_{\upsilon}\omega^{\upsilon}\ll\omega/c_{1}$ is taken.
Now, we may expand the scattering coefficients in Eq. (2) for $a\rightarrow 0$, $\rho_{1}/\rho_{0}=O(1)$,
and $c_{0}/c_{1}=O(1)$.
Hence, we find the first relevant terms of this expansion as
$$\displaystyle s_{0}$$
$$\displaystyle=-\left(\frac{2\alpha_{\upsilon}\omega^{\upsilon-1}\rho_{0}c_{0}^%
{2}}{\rho_{1}c_{1}}+if_{0}\right)\frac{(ka)^{3}}{3}-f_{0}^{2}\frac{(ka)^{6}}{9},$$
(3)
$$\displaystyle s_{1}$$
$$\displaystyle=-if_{1}\frac{(ka)^{3}}{6}-\frac{2\rho_{0}\rho_{1}c_{1}\alpha_{%
\upsilon}\omega^{\upsilon-1}}{5(\rho_{0}+2\rho_{1})^{2}}(ka)^{5}-f_{1}^{2}%
\frac{(ka)^{6}}{36},$$
(4)
where $f_{0}=1-\rho_{0}c_{0}^{2}/\rho_{1}c_{1}^{2}$ and $f_{1}=2(\rho_{1}-\rho_{0})/(2\rho_{1}+\rho_{0})$ are the monopole and the dipole scattering factors, respectively.
3 Radiation force
The linear momentum of the incident wave might be transferred
to the suspended particle generating the so-called acoustic radiation force.
This force can be expressed as
${\bf f}=\pi a^{2}(I_{0}/c_{0}){\bf Y}$, where
$I_{0}=p_{0}^{2}/2\rho_{0}c_{0}$ is the averaged incident intensity and
${\bf Y}$ is the dimensionless radiation force.
In the Rayleigh scattering limit, the radiation force vector has the following Cartesian components 8, 11
$$\displaystyle Y_{x}+iY_{y}$$
$$\displaystyle=\frac{i}{2\pi(ka)^{2}}\biggl{[}\sqrt{\frac{2}{3}}\bigl{[}(s_{0}+%
s_{1}^{*}+2s_{0}s_{1}^{*})a_{0}^{0}a_{1}^{1*}+(s_{0}^{*}+s_{1}+2s_{0}^{*}s_{1}%
)a_{0}^{0*}a_{1}^{-1}\bigr{]}$$
$$\displaystyle+\sum_{m=-1}^{1}\sqrt{\frac{(2+m)(3+m)}{15}}\bigl{(}s_{1}a_{1}^{m%
}a_{2}^{m+1*}+s_{1}^{*}a_{1}^{-m*}a_{2}^{-m-1}\bigr{)}\biggr{]},$$
(5)
$$\displaystyle Y_{z}$$
$$\displaystyle=\frac{1}{\pi(ka)^{2}}\textrm{Im}\left[\sqrt{\frac{1}{3}}(s_{0}+s%
_{1}^{*}+2s_{0}s_{1}^{*})a_{0}^{0}a_{1}^{0*}+\sum_{m=-1}^{1}\sqrt{\frac{(2-m)(%
2+m)}{15}}s_{1}a_{1}^{m}a_{2}^{m*}\right].$$
(6)
where the symbol ${}^{*}$ means complex conjugation and ‘Im’ signifies the imaginary-part of.
Note that the quadrupole mode of the incident wave $a_{2}^{m}$ is necessary to compute the radiation force.
By substituting the equations in (24) into
Eqs. (5) and (6), one obtains after some algebra
$${\bf f}=-\frac{2\pi}{k^{2}c_{0}}\text{Re}\bigg{[}\frac{3is_{1}}{k}c_{0}\nabla%
\left(\frac{\rho_{0}\|{\bf v}(0)\|^{2}}{2}-\frac{|p(0)|^{2}}{2\rho_{0}c_{0}^{2%
}}\right)+(s_{0}+3s_{1}+2s_{0}s_{1}^{*})p(0){\bf v}^{*}(0)\bigg{]}$$
(7)
where ‘Re’ means the real-part of.
We remark that the acoustic fields are evaluated in the particle’s center ${\bf r}=0$.
Equation (7) can be further developed by using Eqs. (3) and (4).
In doing so, we find that
the radiation force can be represented as the sum of a gradient, a scattering,
and a absorption component.
The gradient radiation force is given by
$${\bf f}_{\text{grad}}=-\nabla U(0),$$
(8)
where
$$U=\pi a^{3}\left(f_{0}\frac{|p|^{2}}{3\rho_{0}c_{0}^{2}}-f_{1}\frac{\rho_{0}\|%
{\bf v}\|^{2}}{2}\right)$$
(9)
is the radiation force potential function as obtained by Gor’kov 4.
This force does not depend on the ultrasound absorption by the particle.
Moreover, it might be important whenever the incident beam has a stationary spatial distribution
such as standing waves or focused beams.
On the other hand, the scattering radiation force reads
$${\bf f}_{\text{sca}}=\frac{4\pi a^{2}}{9c_{0}}\left(f_{0}^{2}+f_{0}f_{1}+\frac%
{3f_{1}^{2}}{4}\right)(ka)^{4}\overline{\bf I}(0),$$
(10)
where $\overline{\bf I}=(1/2)\text{Re}\{p{\bf v}^{*}\}$ is the
incident intensity averaged in time.
Equation (10) generalizes the previous result
obtained by Gor’kov for the radiation force produced by a plane traveling wave on a
non-absorbing particle 4.
Finally, the absorption radiation force is expressed as
$${\bf f}_{\text{abs}}=\frac{8\pi a^{3}}{3}\frac{\rho_{0}c_{0}^{\upsilon}}{\rho_%
{1}c_{1}}\alpha_{\upsilon}k^{\upsilon}\overline{\bf I}(0).$$
(11)
This force varies with frequency like the absorption itself.
We see that both the scattering and the absorbing radiation forces depend on the averaged incident intensity.
Therefore, they will be relevant whenever the incident wave has a traveling part, such as a plane wave, for instance.
For a standing wave, the scattering and the absorption radiation forces are much smaller than their gradient counterpart.
According to Eqs. (10) and (11), the magnitude ratio of the absorbing force to the scattering force
is given by
$$\left\|\frac{{\bf f}_{\text{abs}}}{{\bf f}_{\text{sca}}}\right\|=\frac{6\rho_{%
0}c_{0}^{\upsilon+1}\alpha_{\upsilon}k^{\upsilon-4}}{a^{3}\rho_{1}c_{1}(f_{0}^%
{2}+f_{0}f_{1}+3f_{1}^{2}/4)}.$$
(12)
Therefore, the importance of the absorbing radiation force decreases with frequency as $\omega^{\upsilon-4}$.
Though, this force increases with the particle’s radius to the power-third as it approaches to zero.
When $\upsilon=2$ and the incident beam is a plane wave, we recover the result given in Eq. (52) in Ref. 6,
which in turn has a typo (an extra factor of 1/3).
4 Radiation torque
An acoustic beam may set the particle to spin around its axis.
This happens because the beam’s angular momentum with respect
to the particle’s center might be transferred to the particle giving rise to the acoustic radiation torque.
In the Rayleigh scattering limit, the radiation torque is given in terms
of the monopole and the dipole scattering modes only.
Accordingly, we have 9
${\bf N}=\pi a^{3}(I_{0}/c_{0})\bm{\tau}$, where $\bm{\tau}$ is the
dimensionless radiation torque whose Cartesian components are
$$\displaystyle\tau_{x}+i\tau_{y}$$
$$\displaystyle=-\frac{\sqrt{2}}{\pi(ka)^{3}}\left(\frac{s_{1}+s_{1}^{*}}{2}+|s_%
{1}|^{2}\right)\left(a_{1}^{-1}a_{1}^{0*}+a_{1}^{0}a_{1}^{1*}\right),$$
(13)
$$\displaystyle\tau_{z}$$
$$\displaystyle=-\frac{1}{\pi(ka)^{3}}\left(\frac{s_{1}+s_{1}^{*}}{2}+|s_{1}|^{2%
}\right)\left(\left|a_{1}^{1}\right|^{2}-\left|a_{1}^{-1}\right|^{2}\right).$$
(14)
Substituting the beam-shape coefficients given in (24) into these equations results
$$\bm{\tau}=-\frac{12\rho_{0}^{3}\rho_{1}c_{0}^{\upsilon+1}c_{1}\alpha_{\upsilon%
}k^{\upsilon+1}a^{2}}{5p_{0}^{2}(\rho_{0}+2\rho_{1})^{2}}[i{\bf v}(0)\times{%
\bf v}^{*}(0)].$$
(15)
It is clear that no torque is produced in a non-absorbing particle suspended in an inviscid fluid.
Furthermore,
the radiation torque increases with frequency as $\omega^{\upsilon+1}$.
5 Spherically focused beam
Consider that a spherically focused transducer
with diameter $2b=50\,\mathrm{mm}$ and curvature radius $z_{0}=70\,\mathrm{mm}$ ($f$-number of 1.4) generates
an ultrasound beam of frequency $1\,\mathrm{MHz}$ in water at room temperature ($\rho_{0}=1000\,\mathrm{kg/m^{3}}$ and $c_{0}=1480\,\mathrm{m/s}$). The peak pressure at the transducer is $p_{0}=10\,\mathrm{kPa}$.
We chose these parameters because such transducer could be readily manufactured for experiments.
Moreover, the generated ultrasound beam can be described in the paraxial approximation.
The scattering particle has radius $a=10\,\mathrm{\mu m}$ and
hence $ka=0.04$.
It is worth to notice that the viscous boundary layer is $\delta=0.6\,\mathrm{\mu m}$; thus, $\delta\ll a$.
The pressure amplitude of the beam in the focal plane and along the axial direction are given
in cylindrical coordinates $(\varrho,z)$, respectively, by 12
$$\displaystyle p(\varrho,z_{0})$$
$$\displaystyle=\frac{ip_{0}b}{\varrho}\exp\left[ikz_{0}\left(\frac{\varrho^{2}}%
{z_{0}^{2}}+1\right)\right]J_{1}\left(\frac{k\varrho b}{z_{0}}\right),$$
(16)
$$\displaystyle p(0,z)$$
$$\displaystyle=\frac{p_{0}e^{ikz}}{z/z_{0}-1}\left[1-\exp\left(-ikb^{2}\frac{z-%
z_{0}}{2zz_{0}}\right)\right].$$
(17)
where $J_{n}$ is the $n$th-order Bessel function.
The particle might be transversely trapped if the radiation force potential has
a minimum at the beam’s axis, i.e. at $\varrho=0$.
Hence, substituting Eq. (16) into Eq. (9), one encounters
$$\lim_{\varrho\rightarrow 0}U(\varrho,z_{0})=\frac{\pi a^{3}I_{0}k^{2}b^{4}f_{0%
}}{6c_{0}z_{0}^{2}}.$$
(18)
Therefore, the particle will be transversely trapped if $f_{0}<0$, otherwise it
will be pushed away from the beam’s axis.
The situation is different in the axial direction.
Consider the limiting case $b=z_{0}/2$ ($f$-number equal to 1).
Then
and at the focus point, the radiation force potential becomes
$$\lim_{z\rightarrow z_{0}}U(0,z)=\frac{\pi a^{3}I_{0}k^{2}b^{4}}{2c_{0}z_{0}^{2%
}}\left(\frac{f_{0}}{3}-\frac{f_{1}}{2}\right),\quad\frac{z_{0}}{2b}\geq 1.$$
(19)
Thus, the radiation force potential has a minimum only when one of the following conditions is met:
$f_{0}<0$ and $f_{1}>0$, $f_{0}<0$ and $f_{1}<2f_{0}/3$, and
$f_{0}>0$ and $f_{1}>2f_{0}/3$.
Note that the last condition allows the particle to be axially trapped if $f_{0}>0$,
but this implies, according to Eq. (18), that no transverse trap occurs.
Another important issue is even if the radiation force potential has a minimum, the particle
may not be trapped.
This happens because
the absorption and the scattering radiation forces push the particle
away from the potential minimum.
In Fig. 1, we depict the radiation force due to the focused transducer
on a polyurethane (PU) particle with
$\rho_{1}=1130\,\mathrm{kg/m}$, $c_{1}=1468\,\mathrm{m/s}$, and
$\alpha_{\nu}/2\pi=1.43\,\mathrm{Np\cdot MHz/m}$; and a red blood cell (RBC) with
$\rho_{1}=1576\,\mathrm{kg/m}$, $c_{1}=1047\,\mathrm{m/s}$, and
$\alpha_{\nu}/2\pi=0.72\,\mathrm{Np\cdot MHz^{1.47}/m}$.
The PU particle has $f_{0}=0.10$, so it is not transversely trapped.
In axial direction, the absorption radiation force is dominant and pushes
the PU particle away from the focus point $kz_{0}=297$.
Note that the axial radiation force on this particle is rather different when absorption is neglected.
With $f_{0}=-0.26$ and $f_{1}=0.27$, the RBC is three-dimensionally trapped on the
transducer’s focus point.
Furthermore, trapping may also occur in a series of points (depicted by vertical arrows) prior to the focal region.
It is also noticeable that the absorption radiation force on the RBC can be neglected and the scattering radiation force is smaller
than the backward gradient force.
6 Bessel vortex beam
We derive here the radiation torque induced by a Bessel vortex beam on a particle
located in the beam’s axis.
Assume that the beam propagates along the $z$-axis.
Hence, the pressure amplitude of the beam is given, in cylindrical coordinates, by
$$p=p_{0}J_{\ell}(k\varrho\sin\beta)e^{i(\ell{\varphi}+kz\cos\beta)},\quad\ell=0%
,\pm 1,\pm 2,\dots,$$
(20)
where $\beta$ is the beam’s half-cone angle.
The index $\ell$ is known as the orbital angular momentum of the beam.
After calculating the particle velocity of the Bessel beam,
we find that only the axial radiation
torque remains.
Therefore, we have
$$\tau_{z}=\frac{6\ell\rho_{0}\rho_{1}c_{1}c_{0}^{\upsilon-1}\alpha_{\upsilon}k^%
{\upsilon+1}a^{2}}{5(\rho_{0}+2\rho_{1})^{2}}\sin^{2}\beta,\quad|\ell|=1.$$
(21)
When $|\ell|\neq 1$ no radiation torque is produced on the particle.
This happens because radiation torque is induced by
the dipole mode only as shown in Eqs. (13) and (14).
This mode with respect to beam’s axis is not present in all but the first-order Bessel vortex beam.
Equation (21) can also be obtained directly from Eq. (14) in Ref. 13.
7 Conclusions
An extension of Gor’kov’s radiation force theory to include absorption effects inside
the particle was presented.
With this theory, the stability of axial and transverse particle trapping by
a spherically focused ultrasound beam was analyzed.
Moreover, an exact expression for the radiation torque on a suspended particle was derived
and was applied to obtain the torque caused by a Bessel vortex beam in an on-axis particle.
In conclusion, we believe that the closed-form expressions i the limit of non-viscous fluids developed here might be useful
in analyzing the trapping stability of single-beam acoustical tweezers.
Appendix A Beam-shape coefficients
The beam-shape coefficients are given by 10
$$a_{n}^{m}=\frac{1}{p_{0}j_{n}(kR)}\int_{\Omega}p_{\text{i}}(kR,\theta,\varphi)%
Y_{n}^{m*}(\theta,\varphi)d\Omega,$$
(22)
where $\Omega$ is the unit sphere, $d\Omega$ is the differential solid angle, and $R$ is the radius of a control spherical
region in which the incident beam propagates.
Expanding the incident pressure in the origin up to the second-order approximation,
yields
$$p({\bf r})=p(0)+ik{\bf r}\cdot{\bf v}(0)+\frac{ik}{2}{\bf r}\cdot\left[{\bf r}%
\cdot\nabla{\bf v}(0)\right]\,$$
(23)
where ${\bf v}=(v_{x},v_{y},v_{z})$.
Substituting Eq. (23) into Eq. (22) along with
${\bf r}=r(\sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\theta)$,
we obtain the beam-shape coefficients up to the quadrupole approximation as
$$\displaystyle a_{0}^{0}$$
$$\displaystyle=\frac{\sqrt{4\pi}}{p_{0}}p(0),$$
$$\displaystyle a_{1}^{-1}$$
$$\displaystyle=\frac{\rho_{0}c_{0}}{p_{0}}\sqrt{6\pi}[iv_{x}(0)+v_{y}(0)],$$
$$\displaystyle a_{1}^{0}$$
$$\displaystyle=\frac{2i\rho_{0}c_{0}}{p_{0}}\sqrt{3\pi}v_{z}(0),$$
$$\displaystyle a_{1}^{1}$$
$$\displaystyle=\frac{\rho_{0}c_{0}}{p_{0}}\sqrt{6\pi}[-iv_{x}(0)+v_{y}(0)],$$
$$\displaystyle a_{2}^{-2}$$
$$\displaystyle=\frac{\rho_{0}c_{0}}{kp_{0}}\sqrt{30\pi}\left[i\left(\frac{%
\partial v_{x}(0)}{\partial x}-\frac{\partial v_{y}(0)}{\partial y}\right)+2%
\frac{\partial v_{x}(0)}{\partial y}\right],$$
(24)
$$\displaystyle a_{2}^{-1}$$
$$\displaystyle=\frac{2\rho_{0}c_{0}}{kp_{0}}\sqrt{30\pi}\left[i\frac{\partial v%
_{x}(0)}{\partial z}+\frac{\partial v_{y}(0)}{\partial z}\right],$$
$$\displaystyle a_{2}^{0}$$
$$\displaystyle=\frac{2i\rho_{0}c_{0}}{kp_{0}}\sqrt{5\pi}\left[i\left(\frac{%
\partial v_{x}(0)}{\partial x}+\frac{\partial v_{y}(0)}{\partial y}\right)-2%
\frac{\partial v_{z}(0)}{\partial z}\right],$$
$$\displaystyle a_{2}^{1}$$
$$\displaystyle=\frac{2\rho_{0}c_{0}}{kp_{0}}\sqrt{30\pi}\left[-i\frac{\partial v%
_{x}(0)}{\partial z}+\frac{\partial v_{y}(0)}{\partial z}\right],$$
$$\displaystyle a_{2}^{2}$$
$$\displaystyle=\frac{\rho_{0}c_{0}}{kp_{0}}\sqrt{30\pi}\left[i\left(\frac{%
\partial v_{x}(0)}{\partial x}-\frac{\partial v_{y}(0)}{\partial y}\right)-2%
\frac{\partial v_{x}(0)}{\partial y}\right].$$
References and links
1
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(2012). |
Semi-Supervised Classification and Segmentation on High Resolution Aerial Images
Sahil Khose${}^{*}$ Abhiraj Tiwari${}^{*}$ Ankita Ghosh${}^{*}$
Manipal Institute of Technology
Manipal
{sahil.khose, abhiraj.tiwari1, ankita.ghosh1}@learner.manipal.edu
Abstract
FloodNet is a high-resolution image dataset acquired by a small UAV platform, DJI Mavic Pro quadcopters, after Hurricane Harvey. The dataset presents a unique challenge of advancing the damage assessment process for post-disaster scenarios using unlabeled and limited labeled dataset. We propose a solution to address their classification and semantic segmentation challenge. We approach this problem by generating pseudo labels for both classification and segmentation during training and slowly incrementing the amount by which the pseudo label loss affects the final loss. Using this semi-supervised method of training helped us improve our baseline supervised loss by a huge margin for classification, allowing the model to generalize and perform better on the validation and test splits of the dataset. In this paper, we compare and contrast the various methods and models for image classification and semantic segmentation on the FloodNet dataset.
††
${}^{*}$Authors have contributed equally to this work and share first authorship
1 Introduction
The frequency and severity of natural disasters threaten human health, infrastructure and natural systems. It is extremely crucial to have accurate, timely and understandable information to improve our disaster management systems. Rapid data collection from remote areas can be easily facilitated using small unmanned aerial systems which provide high-resolution images. Visual scene understanding of these collected images is vital for quick response and large scale recovery post-natural disaster. Classification and segmentation tasks are fitting in such situations as they can provide scene information to help the task force make decisions.
One of the major challenges with generating a vision dataset is the cost of labeling the data, especially for semantic segmentation. This often leads to labels only for a small percentage of the data which gives rise to the need for semi-supervised methods that can produce results that are at par with supervised methods. Another challenge that we face apart from the lack of labeled dataset is the heavy class imbalance. Lack of labeled data coupled with class imbalance makes it a very challenging task to solve. Through our approach, we try to tackle these problems and produce creditable results.
Our contribution in this paper is two folds: semi-supervised classification and semi-supervised semantic segmentation. We discuss the existing literature in these fields based on which we have crafted our approach in Section 2. The subsequent sections throw light upon the pipeline of classification 3 and segmentation 4 respectively. Our paper describes the preprocessing and augmentation techniques used on the dataset, the methodology followed, the models and hyperparameters we have experimented with and the results obtained. Finally, we conclude by summarizing our results and discussing the scope of improvement in future 5.
2 Related Works
Supervised Classification:
Classification has been one of the earliest undertakings in the deep learning domain. Over the years, several architectures and methods have emerged which have leveraged extensive datasets like ImageNet [5] to produce state of the art results using supervised learning. The architectures that our paper makes use of are ResNet [6] and EfficientNet [14].
ResNet proposes residual connection architecture which makes it feasible to train networks with a large number of layers without escalating the training error percentage. Using the technique of skip connections, it resolves the issue of vanishing gradient.
EfficientNet proposes a simple but highly effective scaling method that can be used to scale up any model architecture to any target resource constraints while maintaining model efficiency. They observed the effects of model scaling and identified that carefully balancing network depth, width and resolution can lead to better performance.
Supervised Semantic Segmentation: Segmentation aids in extracting the maximum amount of information from an image. Instance segmentation treats every occurrence of a class as a unique object. Semantic segmentation is more coherent and associates every pixel of an image with a class label. Deep learning models like UNet[13], PSPNet[15] and DeepLabV3+[3] have provided exceptional results for this task.
The architecture of UNet is divided into two parts: contracting path and expansive path. The contracting path follows the generic framework of a convolutional network while the expansive path undergoes deconvolution to reconstruct the segmented image.
PSPNet exploits the capability of global context information using different region-based context aggregation by introducing a pyramid pooling module along with the proposed pyramid scene parsing.
DeepLabV3+ is a refinement of DeepLabV3 which uses atrous convolution. Atrous convolution is a powerful tool to explicitly adjust the filter’s field-of-view as well as control the resolution of feature responses computed by Deep Convolution Neural Network.
Semi-supervised Approach:
Pseudo-Label [9] proposes a simple semi-supervised learning approach. The idea is to train the neural network in a supervised fashion with both labeled and unlabeled data simultaneously. For unlabeled data, pseudo labels are generated by selecting the class which has maximum predicted probability. This is in effect equivalent to Entropy Regularization [2]. It favors a low-density separation between classes, a commonly assumed prior for semi-supervised learning.
3 Classification
The dataset has 2343 images of dimensions $3000\times 4000\times 3$ divided into training (1445), validation (450) and test (448) splits. Out of the 1445 training images 398 are labeled and 1047 are unlabeled. The labels for the classification task are Flooded and Non-Flooded.
In this section we describe our approach for classifying the FloodNet dataset [11] into 2 classes, Flooded and Non-Flooded.
3.1 Data and Preprocessing
In the given dataset 398 sample images were labeled out of which 51 samples were flooded and 347 were non-flooded. The large class imbalance prevents the model from achieving a good F1 score while training with the labeled dataset. To prevent this we used a weighted sampling strategy while loading the data in the model as inspired from [10]. Both the classes were sampled equally during batch generation.
The labeled dataset was heavily augmented to get more images for training the model under supervision. The image samples were randomly cropped, shifted, resized and flipped along the horizontal and vertical axes.
We downsized the image from $3000\times 4000$ to $300\times 400$ dimensions to strike a balance between processing efficiency gained by the lower dimensional images and information retrieval of the high-resolution images.
3.2 Methodology
ResNet18 with a binary classification head was used for semi-supervised training on the dataset. The model was trained for $E$ epochs out of which only the labeled samples were used for $E^{\alpha}_{i}$ epochs after which pseudo labels were used to further train the model. $\alpha$ has an initial value of $\alpha_{i}$ that increases up to $\alpha_{f}$ from epoch $E^{\alpha}_{i}$ to $E^{\alpha}_{f}$ as described in Algorithm 1.
A modified form of Binary Cross-Entropy (BCE) was used as the loss function as shown in line 10 in Algorithm 1 where $l$ is the label of a sample, $\hat{l}$ is the predicted class for labeled sample and $u_{epoch}$ is the predicted class for an unlabeled sample in the current epoch. This loss function was optimized using Stochastic Gradient Descent (SGD) [12].
3.3 Experiments
We used ResNet18 as it is computationally efficient. We experimented with Adam [8] optimizer and SGD. Optimizing using SGD was much more stable and the optimizer was less susceptible to overshooting. Different values of $\alpha$ were experimented with and it was found that a slow and gradual increase in alpha was better for training the model. Our best performing model uses $\alpha_{i}=0$ and $\alpha_{f}=1$. The value of $\alpha$ increases from epoch $E^{\alpha}_{i}=10$ to $E^{\alpha}_{f}=135$. The model was trained on batch size of 64.
3.4 Results
Our system performed significantly better than all the classification baseline results mentioned in the FloodNet paper while having a considerably smaller architecture (half the number of parameters) as shown in Table 1. Our best model achieves 98.10% F1 and 96.70% accuracy on the test set.
4 Segmentation
In this section, we detail our approach for training a model which generates multi-class segmentation masks for given images. The semantic labels for the task is a 10 pixel-level class segmentation mask consisting of Background, Building-flooded, Building non-flooded, Road-flooded, Road non-flooded, Water, Tree, Vehicle, Pool and Grass classes. They are mapped from 0 to 9 respectively.
4.1 Data and Preprocessing
To expedite the process of feature extraction for the deep learning model, we apply bilateral filter to the image, followed by two iterations of dilation and one iteration of erosion. For image augmentation we perform shuffling, rotation, scaling, shifting and brightness contrast. The images and masks are resized to $512\times 512$ dimensions while training since high-resolution images preserve useful information.
4.2 Methodology
The dataset contains labeled masks of dimension $3000\times 4000\times 3$ with pixel values ranging from 0 to 9, each denoting a particular semantic label. These are one-hot encoded to generate labels with 10 channels, where $i^{th}$ channel contains information about $i^{th}$ class.
We experiment with various encoder-decoder and pyramid pooling based architectures to train our model, the details of which are mentioned in Section 4.3. The loss function used is a weighted combination of Binary Cross-Entropy loss (BCE) and Dice loss as it provides visually cleaner results.
We apply a semi-supervised approach and generate pseudo masks for the unlabeled images. While training the model for $E$ epochs, the labeled samples were used for $E^{\alpha}_{i}$ epochs where Adam is used as an optimizer. After that pseudo masks were used to further train the model as described in Algorithm 1. $\alpha$ has an initial value of $\alpha_{i}$ that increases upto $\alpha_{f}$ from epoch $E^{\alpha}_{i}$ to $E^{\alpha}_{f}$. SGD optimizer with 0.01 LR is used when pseudo masks are introduced to the model.
4.3 Experiments
We apply three state-of-the-art semantic segmentation models on the FloodNet dataset. We adopt one encoder-decoder based network named UNet, one pyramid pooling module based network PSPNet and the last network model DeepLabV3+ employs both encoder-decoder and pyramid pooling based module. We train all of them in a supervised fashion.
For UNet, PSPNet and DeepLabV3+ the backbones used were ResNet34 , ResNet101 and EfficientNet-B3 respectively.
For UNet the learning rate was $0.01$ with step LR scheduler set at intervals [10,30,50] and decay factor $\gamma$ set to $0.1$. For PSPNet the learning rate was $0.001$ without any LR decay. For DeepLabV3+ the learning rate was 0.001 with step LR scheduler set at intervals [7,20] and $\gamma$ set to $0.1$.
Adam optimizer and batch size of 24 was used for all the models with Mean Intersection over Union (MIoU) as the evaluation metric. We observed the best results when we weighed the BCE loss and Dice loss equally.
Once we recognized the best performing model on the task, we trained a DeepLabV3+ model with EfficientNet-B3 as the backbone and used SGD optimizer instead of Adam optimizer in a semi-supervised fashion. Due to computation and memory constraints we randomly sampled unlabeled data with the ratio of $1:10$ for generating the pseudo masks.
4.4 Results
Table 2 showcases the comparison of the best models we achieved for each of the 3 architectures. The best test set result was achieved by a DeepLabV3+ architecture with EfficientNet-B3 backbone. A few example images of the predictions of the model against the ground truth are provided in Figure 1. It is evident from the table that small objects like vehicles and pools are the most difficult tasks for our models.
5 Conclusion
In this work, we have explored methods to approach semi-supervised classification and segmentation along with handling the class imbalance problem on high-resolution images. We have conducted a range of experiments to obtain the best possible technique and models to optimize for the tasks.
Our classification framework achieves laudable results with just 398 labeled images and also utilizes the entirety of the unlabeled data. Our segmentation framework shows an increase of 0.19% on using the unlabeled data as pseudo labels. This provides a wide scope of improvement. The amount of unlabeled data is approximately three times the amount of labeled data which if employed efficiently can produce superior results.
We foresee multiple opportunities for future research. Training the model in an unsupervised fashion and fine-tuning it with the labeled data followed by distillation as presented in SimCLRv2 [4] is a very promising method. Training with a contrastive loss [7] has been incremental at times. With the emergence of Visual Transformers, self-supervised Vision Transformers [1] could also be explored for this task.
6 Acknowledgement
We would like to thank Research Society Manipal for their valuable inputs and research guidance.
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On the waiting time distribution for continuous stochastic systems
Robert Gernert${}^{1}$
[email protected]
Clive Emary${}^{2}$
[email protected]
Sabine H. L. Klapp${}^{1}$
[email protected]
${}^{1}$ Institut für Theoretische Physik, Sekr. EW 7–1,
Technische Universität Berlin, Hardenbergstrasse 36,
D-10623 Berlin, Germany
${}^{2}$ Department of Physics and Mathematics,
University of Hull, Kingston-upon-Hull, United Kingdom
(November 23, 2020)
Abstract
The waiting time distribution (WTD) is a common tool for analysing discrete stochastic processes in classical and quantum systems. However, there are many physical examples
where the dynamics is continuous and only approximately discrete, or where it is favourable to discuss the dynamics on a discretized and a continuous level in parallel. An example is the hindered motion of particles through potential landscapes with barriers. In the present paper we propose a consistent generalisation of the WTD from the discrete case
to situations where the particles perform continuous barrier-crossing characterised by a finite duration. To this end, we introduce a recipe to calculate the WTD
from the Fokker-Planck/Smoluchowski equation. As an application, we consider the paradigmatic example of an overdamped particle diffusing through a washboard potential. For sufficiently large energy barriers, the WTD defined via the Smoluchowski equation becomes consistent not only with that obtained from direct simulation of the underlying Langevin equation, but also with the WTD resulting from the analytical solution of a (two-state) master equation model for the short-time dynamics developed previous by us [PRE 86, 061135 (2012)]. Thus, our approach “interpolates” between these two types of stochastic motion. We illustrate our approach for both symmetric systems and systems under constant force.
pacs:
05.10.Gg, 05.60.-k, 05.40.-a
November 23, 2020
I Introduction
Waiting time distributions (WTD) measure the distribution of delay times between subsequent hopping events (“jumps”) in a discrete stochastic process.
The concept of a WTD has been introduced already half a century ago for the description of diffusion processes on lattices
by continuous time random walks montroll65 ; scher73 ; haus87 .
Since then, the analysis of transport processes via WTDs has become common for a wide range of classical and quantum systems
whose dynamics can be viewed, at least approximately, as a discrete stochastic process. Recent applications include, e.g. in the biophysical area,
the opening kinetics of ion channels higgins09 , the (un-)binding of chemical bonds qian13 , and the “turnover” events of enzyme molecules saha12 ; english06 .
In econophysics, WTDs are used to analyse stock market fluctuations mainardi00 . A further important field are diffusion processes of atoms, molecules, or colloids on
corrugated surfaces skaug13 ; prager06 ; ganapathy10 ; hanes12 .
Although the processes considered there are actually continuous (particles on real surfaces are not bound to lattice sites),
the effective barriers characterising rough surfaces are often much larger than the thermal energy; thus
the motion can be approximately viewed as a sequence of jumps.
A similar situation occurs in supercooled liquids and glasses,
where the stochastic processes correspond to hopping of particles between cages (formed by other particles)
jung05 ; chaud08 , to glass rearrangements ahn13 or, on a more abstract level,
to jumps between “metabasins” in configuration space doliwa03 ; heuer05 ; rubner08 .
Finally, within the area of quantum transport, typical applications include
electron transport in quantum-dot nanostructures brandes08 ; flindt09 ; rajabi13 and tunnelling events in dynamical single-electron emitters albert11 .
Despite these wide and crossdisciplinary applications of WTDs there is, for many systems, no unique recipe to actually calculate this quantity.
Indeed, a straightforward definition is only possible for truly discrete stochastic systems where the dynamics is described by a (Markovian)
master equation (ME). Here, the jumps are a priori discrete and have (once they occur) zero duration. Given a sufficiently simple structure of the ME, it is often
even possible to calculate the WTD analytically (see recent studies in quantum transport flindt09 ; brandes08 ; albert11 and ion channel kinetics higgins09 ).
For continuous systems, however, the calculation and
even the very definition of a WTD are not straightforward: and ultimately this is tied to the unavoidable ambiguity in defining a discrete
“jump” for a continuous evolving system.
A frequently-used strategy in such situations is to obtain the WTD numerically from an analysis of (particle) trajectories generated either experimentally or by many-particle computer simulations. Indeed, state-of-the-art experiments allow real time tracking of the motion of colloids on random surfaces hanes12 ; evers13 , on solid-liquid interfaces skaug13 ,
or during surface growth ganapathy10 . Similar trajectory analysis has been performed
in computer simulations (such as Molecular or Brownian Dynamics) of glassy or otherwise frustrated systems (see, e.g., jung05 ; heuer05 ).
For a continuous system with “quasi-hopping” motion, typical trajectories reveal the existence of “traps” (in real or configuration space), where the trajectory spends most time,
and fast motion (barrier crossing) in between. However, to extract from that a WTD, one necessarily needs to discretise the space (i.e., to define regions which trajectories can enter or leave),
yielding some ambiguity in the definition of jumps. In particular, depending on these details of the discretisation,
the resulting jumps can have a finite duration. Moreover, in many practical applications the measured or simulation-generated trajectories render quite noisy WTDs;
extracting smooth results (and relevant time constants) then requires extensive simulation or experimental time. In view of these difficulties it seems desirable to construct a WTD in a yet alternative way, namely
on the basis of a noise-averaged description of the stochastic process. The natural framework is, of course, the Fokker-Planck equation for the one-body probability density.
In this spirit, we propose in the present paper a simple recipe to calculate the WTD from a Fokker-Planck equation.
Our approach is inspired from the calculation risken84 of the first passage time distribution; indeed the two quantities are closely related when we assume the dynamics to be
Markovian (i.e., time-homogeneous). Considering a particular model system we validate our WTD definition by comparing
with the WTD following from direct simulation of the underlying Langevin equation of motion.
Specifically, we consider the paradigm example of an overdamped (“Brownian”) particle in a periodic “washboard” (e.g., sinusoidal) potential,
a generic model with relevance in many areas such as particle transport on topologically or energetically
structured surfaces Tierno10 ; Dalle11 , the motion of motor proteins xing05 , particles in optical lattices Siler10 and optical line traps Lopez08 ,
biased Josephson junctions zapata96 ; shah07 ; augello10 , and more generally in biophysical processes bonnet11 ; xing05 ; kay07 ; Ros05 .
In an earlier study emary12 we have analysed in detail the short-time dynamics of that system, particularly the plateaus appearing in the mean-squared displacement and the corresponding non-Gaussian parameters for large values of the ratio between barrier height and thermal energy.
We have demonstrated that, in this limit of deep wells, the continuous dynamics described by the Smoluchowski equation (SE)
can be well described by a ME approach involving two states per well emary12 . One attractive feature of this ME model is that it is analytically solvable; i.e., properties such as cumulants
of the density distribution can be calculated by minimal effort.
Here we use this ME approach as a third, analytical route to calculate the WTD. We show that the WTD obtained from the SE and BD approach
becomes consistent with that of the discrete case in the appropriate limit (deep wells).
As wells become shallower, the ME approach becomes less good, an indicator being that the duration of the transitions between different wells increases. In summary, we suggest a consistent generalisation of WTD from the discrete, zero-duration jump case
to the continuous, extended-jump one.
The rest of this paper is organised as follows. In Sec. II we specify the equations of motion of our model and discuss representative trajectories. The three approaches to the WTD are introduced in Sec. III, supplemented by two appendices where we present detailed information on the numerical calculation. Numerical results are presented and discussed in Sec. IV, and in Sec. V we summarise our findings.
II Model
We begin by discussing a concrete model as an illustration of the nature of trajectories and jumps in the class of systems we investigate here. We consider a particle in the tilted washboard potential
$$u(x)=u_{0}\sin^{2}\frac{\pi x}{a}-Fx,$$
(1)
where $a$ is the period of the potential, and the parameters $u_{0}$ and $F$
describe the depth and tilt of the potential, respectively. Under overdamped conditions (strong friction),
the equation of motion of the particle in one dimension is given by
$$\gamma\dot{x}(t)=-u^{\prime}(x)+\sqrt{2\gamma k_{\mathrm{B}}T}\xi(t).$$
(2)
The thermal energy is denoted by $k_{\mathrm{B}}T$, $\gamma=k_{\mathrm{B}}T/D_{0}$ is the friction constant (involving the diffusion constant
$D_{0}$ of the free system), $-u^{\prime}(x)=-du/dx$ is the force, and $\xi(t)$ represents Gaussian white noise
with properties $\langle\xi(t)\rangle=0$ and $\langle\xi(t)\xi(t^{\prime})\rangle=\delta(t-t^{\prime})$.
In the present study we focus on cases where the form of the potential is such that it can be meaningfully divided in a number of wells (defined between subsequent maxima).
This occurs when the difference between
the potential minima and maxima, $\Delta u(u_{0},F)$, is large compared to the
thermal energy, i.e., $u_{0}\gg k_{\mathrm{B}}T$, and when the driving force $F$ is smaller than the so-called critical force $F_{\mathrm{c}}=u_{0}\pi/a$, related to the diffusion maximum
Reimann01 (for $F>F_{\mathrm{c}}$ the potential minima vanish).
In order to illustrate the motion in such situations we calculate representative particle trajectories
from Eq. (2) by using a Brownian Dynamics (BD) algorithm ermak75 .
Examples for two different values of $u_{0}/k_{\mathrm{B}}T$ and $F\!=\!0$ are shown in Fig. 1a). The trajectories in Fig. 1a) clearly reflect that the particle is “trapped” for certain times in the regions around the potential minima $x/a\in\mathbb{Z}$. To facilitate the comparison between the trajectories corresponding to different $u_{0}$ they are plotted in Fig. 1b) as functions of the relative time. This latter is defined as the difference between the actual time and the time $t_{\mathrm{jump}}$, which is the arithmetic mean of the times related to the beginning and end of a jump. Here, “beginning” and “end” refer to leaving of a potential minimum (without going back) and arriving at a different minimum (this definition of a jump will later be called minimum-based definition).
From Fig. 1b) we see that even for the largest barrier considered, the particle needs a finite time to cross the barrier. Thus, the motion is still not perfectly discrete.
However, as we may note already here, a different picture on the degree of “discreteness” arises when we rescale the time axis with the corresponding Kramers’ rate $r_{\text{K}}(u_{0})=D_{0}/(2\pi k_{\mathrm{B}}T)\sqrt{-u^{\prime\prime}(x_{%
\mathrm{min}})u^{\prime\prime}(x_{\mathrm{max}})}\,\exp(-\Delta u/(k_{\mathrm{%
B}}T))$ risken84 ; emary12 ; kramers40 , see Fig. 2.
By multiplying the time with this well-known quantity, we take into account that the increase of the potential height alone already leads to an increase of the escape time reimann99 . As a consequence the intervals between jumps in the rescaled time become independent of $u_{0}$ in the long time limit.
Indeed, after rescaling, the data for $u_{0}=15k_{\mathrm{B}}T$ reflects an essentially discrete motion. However, at $u_{0}=5k_{\mathrm{B}}T$ we still observe a finite duration of the jump. This clearly demonstrates that the case $u_{0}=5k_{\mathrm{B}}T$ is at the boundary between discrete and continuous motion.
III Routes to calculate the WTD
III.1 Direct evaluation via BD simulations
To define the WTD we first need to consider the possible types of jumps. To this end, we note that the stochastic process $x(t)$ [see Eq. (2)] is a Markov process. Thus, the jump characteristics are independent of the history. Further, because the motion is one-dimensional, there are only two directions in which the particle can jump. Finally, Eq. (2) obeys the translational symmetry $x\to x\pm a$. Because of these three properties the jumps can be grouped into only two types, namely “to the right” and “to the left”, independent of when or where the jump began.
We label these jump types by the index “J”. The WTD $w^{J}(t)$ is then the probability density for the time between an arbitrary jump and a subsequent jump of type $J$.
In the context of BD simulations [i.e., direct evaluation of Eq. (2)], the WTD $w^{J}(t)$ is extracted from a histogram of waiting times extracted from the trajectories pertaining to a given realisation of noise. We note that the WTDs $w^{J}(t)$ must fulfil the normalisation condition
$$\displaystyle 1=\int_{0}^{\infty}\mathrm{d}t\,\sum_{J}w^{J}(t),$$
(3)
which expresses the fact that the particle leaves its minimum for certain. In terms of the survival probability
$$\displaystyle S(t)=\int_{t}^{\infty}\mathrm{d}t^{\prime}\,\sum_{J}w^{J}(t^{%
\prime})$$
(4)
the normalisation is expressed as $S(0)=1$. The sum runs over all jumps types that leave a potential minimum, in our case “to the right” and “to the left”.
In our BD calculations of $w^{J}(t)$ we consider two definitions of a jump which are illustrated in Fig. 3. Within the first definition, a jump occurs if the particle crosses a potential maximum (maximum-based definition). Note that this automatically defines the jump to be instantaneous.
Within the second definition, a jump involves the entire motion between two neighbouring minima of the potential (minimum-based definition). As a consequence, the jump has a finite duration defined as the time between leaving a minimum and arriving at a neighbouring one.
We call the corresponding probability distribution the jump duration distribution.
Further, we define the waiting time as the time between the end times of two subsequent jumps.
Numerical details of the calculation are given in Appendix A.
III.1.1 Discussion of the jump definitions
To investigate the role of the two jump definitions we consider the resulting WTDs for the symmetric case $F\!=\!0$, $u_{0}\!=\!5k_{\mathrm{B}}T$, plotted in Fig. 4. Because of the symmetry of the potential at $F\!=\!0$, there is only one relevant WTD. Clearly, the choice of the jump definition has a strong impact on the WTD.
In particular, we find that only the minimum-based definition yields a smooth shape of $w^{J}(t)$.
On the contrary, the maximum-based jump definition poses several problems. First, during the analysis of the trajectories we noticed that the particle crosses the maximum many times subsequently which generates a lot of spurious (maximum-based) jumps. We deal with this problem by introducing a “dead time” after recording a jump which must pass until the next jump can be recorded. We set the “dead time” to $1\tau$ (i.e., the time the particle needs to diffuse the distance $\sqrt{\tau/D_{0}}=a$) to ensure that it leaves the maximum during that time. As a consequence of introducing the dead time, the corresponding WTD is defined to be zero for $t<1\tau$.
Further, we see in Fig. 4 that by using the maximum-based jump definition, waiting times in the range $1\tau\lesssim t\lesssim 5\tau$ are more likely than by using the minimum-based jump definition. The reason is that the particle, once it has gone uphill, is counted to have made a jump in the maximum-based jump definition. On the contrary, the same particle is only counted in the minimum-based jump definition if it goes downhill in the same direction (and not backwards). Thus, the maximum-based jump definition frequently records jumps although the particle does not change the potential well. This contradicts our idea of a jump to cross the potential barrier. This problem could be solved by keeping track of the direction in which the particle crosses the maximum. However, the first problem remains (i.e. accessing short waiting times). We therefore choose the minimum-based jump definition for the further investigation.
III.2 Definition from the Smoluchowski equation
The Fokker-Planck equation corresponding to Eq. (2) is given by the Smoluchowski equation (SE)
$$\displaystyle\partial_{t}p(x,t)$$
$$\displaystyle=D_{0}\frac{\partial^{2}}{\partial x^{2}}p(x,t)+\frac{D_{0}}{k_{%
\mathrm{B}}T}\frac{\partial}{\partial x}\Big{(}p(x,t)u^{\prime}(x)\Big{)}\,,$$
(5)
where $p(x,t)\Delta x$ is the probability that the particle is in the interval $[x,x+\Delta x]$ at time $t$.
Again, we are interested in the distribution of waiting times $t$.
In Sec. III.1 we defined $w^{J}(t)$ as the WTD for the jump $J$. Here we associate $J$ with the jumps from an initial position $B$ to a target position $C$. Typically $B$ and $C$ are neighbouring minima of the potential. In order to realise the requirement that the WTD describes the time between subsequent jumps we need to exclude processes that lead the particle towards the other neighbouring minimum, which we call $A$.
Correspondingly, we describe the WTD by the function $w_{A}^{B\to C}(t)$ where $B\to C$ stands for $J$ and the subscript $A$ indicates the excluded position. The definition is as follows:
Definition: The quantity $w_{A}^{B\to C}(t)\,\Delta t$, where $\Delta t$ is a small time interval, is the probability that
•
the particle was for certain at position $$B$$ at $$t\!=\!0$$,
(6a)
•
the particle reaches position $$C$$ in the interval $$[t,t+\Delta t]$$, and
(6b)
•
the particle neither arrived at $$C$$ nor $$A$$ within the time interval $$[0,t]$$.
(6c)
Note that we do not care about what happened at earlier times $t<0$, that is, whether the particle moved to $B$ via a jump process or whether it was placed there “by hand”.
This crucial assumption reflects that we are working in a Markovian situation, where the motion from $B$ to $C$ is independent of the history.
To formulate a recipe to calculate the WTD, we recall that the SE, Eq. (5), can also be written as a continuity equation, that is,
$$\displaystyle\partial_{t}p(x,t)+\partial_{x}j(x,t)$$
$$\displaystyle=0,$$
(7)
where $j(x,t)=-D_{0}\left(\partial_{x}p(x,t)+p(x,t)u^{\prime}(x)/(k_{\mathrm{B}}T)\right)$ is the current density. Equation (7) further shows that
$j(x,t)\,\Delta t$ (or $-j(x,t)\,\Delta t$, respectively) can be interpreted as the probability that the particle crosses position $x$ into positive (negative) $x$-direction during the time interval $[t,t+\Delta t]$.
Consequently $|j(C,t)\Delta t|$ is the probability that the particle crosses position $C$ in forward or backward direction.
This probability is similar in character to that mentioned in condition (6b). However, $|j(C,t)\,\Delta t|$ includes the probability that the particle went across $C$ (or $A$), came back, and crosses $C$ again. Such a process violates condition (6c).
Based on these considerations, we propose the following recipe to calculate the WTD $w_{A}^{B\to C}(t)$:
We initialise the probability density as
$$\displaystyle p(x,t=0)$$
$$\displaystyle=\delta(x-B)$$
(8)
according to condition (6a).
Then, we let $p(x,t)$ evolve in time according to Eq. (5). However, during this time evolution
we exclude processes from the ensemble of realisations (i.e. possible trajectories) where the particle has already reached position $C$ or $A$ [see condition (6c)].
Therefore $p(x,t)$, which is an average over this reduced set of realisations, fulfils the boundary conditions
$$\displaystyle p(x,t)$$
$$\displaystyle=0,\,x\geq\max(A,C)\quad\forall\,t$$
(9a)
$$\displaystyle p(x,t)$$
$$\displaystyle=0,\,x\leq\min(A,C)\quad\forall\,t\,.$$
(9b)
Equation (9) expresses what is often called “absorbing wall boundary conditions”. Clearly, there is some arbitrariness in defining a suitable position $A$ (the boundary “on the other side”) for a continuous system in general. However, given that we consider a continuous potential with well-defined wells (as it is the case here) one can give a clear physical meaning to the positions $A$, $B$, and $C$.
To summarise, we calculate the WTD from the relation
$$\displaystyle w_{A}^{B\to C}(t)=|j^{B}_{A,C}(C,t)|\,,$$
(10)
where the notation $j^{B}_{A,C}$ expresses the dependency of the current density on the initial and boundary conditions given in Eqs. (8) and (9), respectively.
By using the absolute value in Eq. (10) we take into account the fact that the sign of $j$ depends on the direction of motion. Indeed, motion into the positive (negative) x-direction
implies a positive (negative) sign of $j$. The numerical calculation of the WTD via the SE route is described in appendix B.
Finally, we note that the WTDs defined according to Eq. (10) also fulfil the normalisation condition Eq. (3), that is
$$\displaystyle\int_{0}^{\infty}\mathrm{d}t\left(w_{A}^{B\to C}(t)+w^{B\to A}_{C%
}(t)\right)=1.$$
(11)
By definition risken84 , the first passage time distribution (FPTD) is the probability distribution for the time $t$ the particle needs to leave a given domain for the first time. This definition implicitly assumes that the spatial probability distribution $q(x,\tau)$ to find the particle at position $x$ at an earlier time $\tau<0\leq t$ is known. In the present context, where we are considering a Markovian system, the precise value of $\tau$ is irrelevant. We thus consider the FPTD $F_{D}(t;q)$ where the domain $D$, in an one-dimensional system, is given by an interval, e.g. $D=[A,C]$. Further, if the particle was for certain at a position $B$ at $t=0$, we can specialise $q(x)=q^{0}_{B}(x)\colonequals\delta(x-B)$. The relation between the FPTD and the WTDs defined before is then given by
$$\displaystyle F_{[A,C]}(t;q^{0}_{B})=w_{A}^{B\to C}(t)+w_{C}^{B\to A}(t).$$
(12)
Equation (12) reflects the fact that the FPTD does not contain information about the direction in which the particle left the domain, whereas the WTD does.
III.3 Master equation
In Ref. emary12 , we introduced a simple master equation model of this system.
In this model, the location of the particle is described by two discrete indices: an index $n$ that describes which of the potential wells the particle finds itself in, and an internal index $\alpha=L,R$ that corresponds to whether the particle is localised in the left or right side of the well.
Relative to the centre of the well at $x=0$, the loci of these localised states are denoted $x_{L}$ and $x_{R}$.
Transitions between these states are assumed to occur at rates $\gamma^{\pm}$ to describe hopping within a single well, and $\Gamma^{\pm}$ to describe hopping between the wells. In both cases, superscript $+$ indicates movement to the right and $-$, movement to the left.
We choose the inter-well rates, $\Gamma^{\pm}$, to be twice the corresponding Kramers’ rate emary12 ; risken84 ; kramers4thorder ; the intra-well rates, $\gamma^{\pm}$, as well as the two positions, $x_{L,R}$, are set by fitting the short-time behaviour of the the first two position cumulants to the behaviour of the SE. See Ref. emary12 for details.
With the probabilities $p_{\alpha}^{(n)}$ that the particle is in state $\alpha=L,R$ of the $n$th well arranged into the vector
$$\displaystyle|\rho^{(n)}\rangle\!\rangle\equiv\left(\begin{array}[]{c}p^{(n)}_%
{L}\\
p^{(n)}_{R}\end{array}\right),$$
(13)
the system can be described with the master equation
$$\displaystyle\!\!\!\!\!\!\!\!\!\frac{d}{dt}|\rho^{(n)}\rangle\!\rangle=%
\mathcal{W}_{0}|\rho^{(n)}\rangle\!\rangle+\mathcal{W}_{+}|\rho^{(n-1)}\rangle%
\!\rangle+\mathcal{W}_{-}|\rho^{(n+1)}\rangle\!\rangle,$$
(14)
where matrices
$$\displaystyle\mathcal{W}_{+}=\left(\begin{array}[]{cc}0&\Gamma^{+}\\
0&0\end{array}\right)~{}~{}\text{and}~{}~{}\mathcal{W}_{-}=\left(\begin{array}%
[]{cc}0&0\\
\Gamma^{-}&0\end{array}\right)$$
(15)
describe jump processes between the wells and
$$\displaystyle\mathcal{W}_{0}$$
$$\displaystyle=$$
$$\displaystyle\left(\begin{array}[]{cc}-\gamma^{+}-\Gamma^{-}&\gamma^{-}\\
\gamma^{+}&-\gamma^{-}-\Gamma^{+}\end{array}\right)$$
(16)
describes intra-well processes and normalisation.
The waiting time distribution(s) in the ME approach are easy to define but before we do so, we need some more notation. We define the ‘expectation value’
$\langle\!\langle\mathcal{A}\rangle\!\rangle\equiv\langle\!\langle 1|\mathcal{A%
}|\rho_{0}\rangle\!\rangle$
where trace and intra-well stationary vectors are defined by
$$\displaystyle\langle\!\langle 1|\left(\mathcal{W}_{0}+\mathcal{W}_{+}+\mathcal%
{W}_{-}\right)$$
$$\displaystyle=$$
$$\displaystyle 0$$
$$\displaystyle\left(\mathcal{W}_{0}+\mathcal{W}_{+}+\mathcal{W}_{-}\right)|\rho%
_{0}\rangle\!\rangle$$
$$\displaystyle=$$
$$\displaystyle 0.$$
(17)
Following brandes08 , we then obtain the WTDs as
$$\displaystyle w_{ss^{\prime}}(\tau)\equiv\frac{\langle\!\langle\mathcal{W}_{s}%
e^{\mathcal{W}_{0}\tau}\mathcal{W}_{s^{\prime}}\rangle\!\rangle}{\langle\!%
\langle\mathcal{W}_{s^{\prime}}\rangle\!\rangle};\quad s,s^{\prime}=\pm$$
(18)
There are four different waiting times defined here, corresponding to the four different combinations of jumps: $s,s^{\prime}=\pm$.
In comparing with the results from BD/SE, we will only consider the diagonal WTDs $w_{ss}(\tau)$ as these are found to be the closest analogues of the definitions used in the continuous system. The reason for this is that in off-diagonal WTDs, the second jump can occur immediately after the first, and thus $\lim_{\tau\to 0}w_{ss^{\prime}}(\tau)$ for $s^{\prime}\neq s$ is finite. Clearly, this is not the case in the full dynamics and so we consider only the diagonal WTDs.
This difference between on- and off-diagonal WTDs only occurs at very short times and, as we will see in the next section, the ME description is unreliable in this regime, anyway.
IV Numerical results
The main new finding obtained so far is our definition of the WTD via the continuous SE route. In the following we compare results obtained numerically from this definition with those from BD simulations (using the minimum-based jump definition), as well as with results from the ME approach.
To be specific, we choose the positions $A$, $B$, and $C$ to be adjacent minima of the potential $u(x)$ [see Eq. (1)].
In the subsequent section IV.1,
we examine the case of zero drive ($F\!=\!0$) for two amplitudes of the periodic potential. Both symmetric and asymmetric initial (and boundary)
conditions are considered. In Sec. IV.2
we fix $u_{0}$ to $15k_{\mathrm{B}}T$ and study driven systems for two values $F<F_{c}$ where $F_{c}=u_{0}\pi/a$ is the “critical” driving force beyond which the minima in $u(x)$ are eliminated.
IV.1 Zero drive ($F=0$)
We start by considering fully “symmetric” cases, where the potential is untilted ($F=0$), and the particle jumps from the centre of a minimum (say, $x=0$) to the right ($0\to a$) or to the left ($0\to-a$).
Clearly, these jumps are characterised by the same WTD, i.e.,
$w_{-a}^{0\to a}(t)=w_{a}^{0\to-a}(t)$. Moreover, in such fully symmetric situations [with respect to the initial- and boundary conditions, and to $u(x)$], the WTD is proportional to the first passage distribution, i.e. $w_{-a}^{0\to a}(t)=1/2\,F_{[-a,a]}(t;q_{0}^{0})$ [see Eq. (12)]. However, as we will demonstrate, the SE approach is more versatile when we deviate from the symmetric case.
We thus discuss the symmetric case as a reference.
Results for the corresponding WTD at $u_{0}=5k_{\mathrm{B}}T$ and $u_{0}=15k_{\mathrm{B}}T$ are plotted in Figs. 5
and 6, respectively.
These values of $u_{0}$ are representative since, as we recall from our earlier discussion of particle trajectories (see Figs. 1 and 2), the case $u_{0}=5k_{\mathrm{B}}T$ is at the boundary between continuous and discrete motion while at $u_{0}=15k_{\mathrm{B}}T$, the motion is essentially discrete.
Figures 5 and 6 contain data from all three approaches (SE, BD, ME). All curves share the same general structure in that
the WTD is essentially (but not strictly) zero at very short times ($t\ll\tau$) and then grows with time up to a value which remains nearly constant over a range of intermediate times
(note the logarithmic time axis). Finally, at very large times the WTD decays rapidly to zero. The extension of the different regions and the actual values of the WTD depend on the potential height,
as we will analyse in detail below. From a methodological perspective we can state already here that the SE approach represents at all times a very accurate,
smooth “fit” of the (somewhat noisy) BD data. The ME approach displays deviations at short times, to which we will come back at the end of this subsection.
The growth region of the WTD can be further analysed by inspection of the jump duration distribution obtained from BD, which is included in Figs. 5
and 6. At both potential amplitudes this distribution displays a peak located at the (finite) time where the WTD grows most strongly. Moreover, the width
of the peak corresponds approximately to that of the entire growth region. We can interpret these findings as follows.
First, the fact that the peak occurs at a finite time already signals that we are looking at (more or less) continuous stochastic processes where the particles needs a finite time
to cross a barrier (see Fig. 1b)). This is also the reason that we find a growth region in the WTD at all. As we have seen from the trajectories in Fig. 1,
the jump time tends to decrease with increasing $u_{0}$. This explains the shift of the peak of the corresponding distribution towards earlier times in Fig. 6.
Second, regarding the width of the jump duration distribution, we note that only realisations of the random force with a strong bias can push the particle against the potential ascent. The larger $u_{0}$, the smaller the fraction of appropriate noise realisations and hence the sharper the distribution. This also has a direct influence on the width of the growth region of the WTD: For very short times only very short jumps can contribute to the WTD, hence the WTD rises coincidently with the jump duration distribution.
Interestingly, it is also possible to calculate the jump duration distribution via the SE approach.
To this end we just need to adjust the boundary and initial conditions. During a jump the particle leaves a minimum, say $x\!=\!0$, crosses the barrier and arrives at the next minimum, $x\!=\!a$. The difference to the jump $0\to a$ we considered above is that the particle actually leaves the minimum at $t\!=\!0$, i.e., it does not return. The essential step to the jump duration distribution is to consider an ensemble of realisations without processes where the particle comes back. Following Sec. III.2 this condition corresponds to an absorbing boundary at $x\!=\!0$. The second absorbing boundary must be located at the point of arrival, $x\!=\!a$. This suggests the construction of the WTD $w_{0}^{\epsilon\to a}(t)$ where the starting position $\epsilon$ is close to $0$ because the particle just left the minimum.
Through the normalisation Eq. (11) $w_{0}^{\epsilon\to a}(t)$ is connected with $w_{a}^{\epsilon\to 0}(t)$, which states how likely the particle comes back to $x\!=\!0$. We do not have any information about this quantity in the BD simulation, with which we compare. Therefore we need to remove this information. The total probability that the particle returns is given by $\int_{0}^{\infty}\mathrm{d}t\,w_{a}^{\epsilon\to 0}(t)$. This leads us to the following definition of the normalised jump duration distribution.
$$\displaystyle w_{j}(t)$$
$$\displaystyle=\frac{w_{0}^{\epsilon\to a}(t)}{1-\int_{0}^{\infty}\mathrm{d}t^{%
\prime}\,w_{a}^{\epsilon\to 0}(t^{\prime})}=\frac{w_{0}^{\epsilon\to a}(t)}{%
\int_{0}^{\infty}\mathrm{d}t^{\prime}\,w_{0}^{\epsilon\to a}(t^{\prime})}$$
(19)
A comparison between the BD and SE results for the distribution is given in Fig. 7. Clearly, the SE route yields very good results.
At intermediate times the WTD has a broad maximum which, for deep wells, takes the form of a plateau (see Fig. 6).
Numerical values for the WTD maxima at the two potential amplitudes are given in Table 1, where we have included values of Kramers’ escape rate $r_{\text{K}}$ in the most precise form risken84 , i.e., without the Gaussian approximation of the integrals,
$$\displaystyle r_{\text{K},\infty}^{-1}=D_{0}^{-1}\int_{-a/2}^{a/2}\mathrm{d}x%
\,e^{-\beta u(x)}\int_{0}^{a}\mathrm{d}y\,e^{\beta u(y)}\,.$$
(20)
This close relation can be understood on the basis of our SE approach where we have identified the WTD as a current
(i.e., a rate in one dimension). This current has been calculated with absorbing boundary conditions. A very similar calculation, namely by using one absorbing boundary and an infinite soft potential barrier on the other side together with the stationarity approximation risken84 ; haenggi90 , leads to Kramers’ rate. Of course, for short and long times the probability distribution is not stationary; therefore the connection between the values of the WTD and Kramers’ rate only holds at intermediate times.
At times beyond the maximum (or plateau, respectively), the WTD rapidly decays to zero. In Fig. 8 we replot the corresponding behaviour with a linear time axis and a
logarithmic $y$-axis. From that it is clearly seen that the decay can be described by a (single) exponential, i.e., $w_{-a}^{0\to a}(t)\sim e^{-\lambda t}$ in this range of times.
The corresponding decay constants $\lambda$
(as obtained from the SE approach) are listed in Table 1. We find that, for both values of $u_{0}$ considered, $\lambda$ is approximately
twice as large as Kramers’ escape rate (or plateau height, respectively). This is expected since the probability density is petering out simultaneously into both directions. Therefore the decay constant is expected to be the sum of Kramers’ rates for every barrier, in the symmetric case $\lambda\approx 2r_{\text{K}}$.
We now turn to the ME approach. As described in Sec. III.3, this approach assumes a discretised
two-state-per-well model, that is, the particle can (only) take the positions $x_{L}$ and $x_{R}$ within each well. Leaving, e.g., position
$x_{L}$, the particle performs either an intra-well jump (to $x_{R}$) or an inter-well jump to $x_{R}-a$ or to $x_{L}+a$ (via $x_{R}$). Here we are only interested in processes of the latter type, where the particle actually crosses a barrier. Further, because of translational symmetry and $F\!=\!0$ we only need to distinguish between the WTD for a “long” jump
$w_{\mathrm{long}}(t)=w_{x_{R}-a}^{x_{L}\to x_{L}+a}(t)=w_{x_{L}+a}^{x_{R}\to x%
_{R}-a}(t)$, and the WTD for a “short” jump, $w_{\mathrm{short}}(t)=w_{x_{R}-a}^{x_{R}\to x_{L}+a}(t)=x_{x_{L}+a}^{x_{L}\to x%
_{R}-a}(t)$.
The ME results shown in Figs. 5, 6, and Fig. 8 pertain to a long jump. It is seen that the
ME data become indeed consistent with those from the SE and BD approach, if one considers times beyond the growth region of the WTD.
Regarding the short-time behaviour, we find from Figs. 5 and 6
that the growth region of the WTD occurs at smaller times in the ME approach as compared to the SE and BD results. In other words, the WTD
predicts non-zero waiting times already below the mean jump duration predicted by BD. More precisely, the inflexion point of the WTD occurs at (roughly) the inverse
of the transition rate $\gamma\!=\!\gamma^{+}+\gamma^{-}$ for intra-well transitions. In a linear approximation with respect to $x_{R}$ and $x_{L}$ of Eq. (25) in emary12 one has $1/\gamma\!\approx\!\tau/(2u_{0}\pi^{2})$,
which yields a good estimate of the inflexion point. We conclude that it is the intra-well relaxation
(which, in turn, is governed by the potential amplitude) which is the main ingredient determining the short time behaviour of the WTD given by the ME. Indeed, because the ME is based on
a discrete model (and thus neglects travel times) we would not expect that the corresponding WTD is connected to the mean jump duration (as it was the case within the continuous SE approach).
We finally note that asymmetric initial conditions such as those used in ME ($x_{L/R}\neq 0$)
can also be incorporated into the SE approach. As described before, we can distinguish between the WTD for a “long” jump,
$w_{\mathrm{long}}(t)$, and the WTD for a “short” jump $w_{\mathrm{short}}(t)$.
In Fig. 9 we show these two WTDs together with our earlier SE result for $w_{-a}^{0\to a}(t)$.
At short times, $w_{\mathrm{short}}(t)$ has a global maximum
which exceeds the corresponding values of $w_{\mathrm{long}}(t)$ and the maximum of $w_{-a}^{0\to a}(t)$, that is, Kramers’ escape rate. At subsequent time
the two WTDs $w_{\mathrm{long/short}}$ then merge at a value slightly below $w_{-a}^{0\to a}$. This is because the blip of $w_{\mathrm{short}}$ reduces the survival probability which in turn reduces the escape rate. The inset of Fig. 9 shows the probability densities $p(x,t)$ corresponding to $w_{\mathrm{long}}$ and $w_{\mathrm{short}}$ at $t\!=\!0.02\tau$ where the WTDs deviate from each other. At $t\!=\!0.02\tau$ the asymmetrically initialised densities have been broadened by diffusion to reach the boundaries, but yet not strong enough to eliminate the asymmetry.
IV.2 Driven system ($F>0$)
We now consider driven systems. Again, minima of the potential occur every $a$, the first being located at $x=m\colonequals a/(2\pi)\arcsin(F/F_{c})$. Because of the tilt of the potential,
we have to distinguish between jumps to the left, $w^{-}(t)=w_{m+a}^{m\to m-a}(t)$, and jumps to the right, $w^{+}(t)=w_{m-a}^{m\to m+a}(t)$. Clearly, these WTDs provide
directional information, which cannot be extracted from the first passage time distribution.
Results for the WTDs in the “deep well” case are shown in Fig. 10.
It is seen that the general structure of the WTD (consisting of a growth region, a plateau, and then a rapid decay) is the same as that at $F=0$ (see Fig. 6).
One main effect of increasing $F$ from zero is that the global maximum of $w^{+}$ increases, while the time range corresponding to the plateau shortens. In other words,
the average waiting time decreases and it occurs with larger probability. This is plausible, since a non-zero driving force leads to a lower effective barrier
$\Delta u(u_{0},F)=u_{0}(\sqrt{1-(F/F_{c})^{2}}-F/F_{c}\arccos(F/F_{c}))$ in the driving direction. The opposite effects occurs against the driven direction, as reflected by the decreasing maxima
in $w^{-}(t)$. We also note that, for each $F$, the effects in the two WTDs are coupled via the the normalisation condition, i.e., the maximum in one of the WTDs can only grow
at the expense of the other one.
Comparing the different methods, we see from Fig. 10 that the WTDs calculated with the SE agree with the BD results at all times. The WTDs given by the ME are consistent with SE and BD results for intermediate and long times. Again we observe a deviation for short times because, as discussed in the previous section, the short time scale in the two-state-per-well model describes intra-well relaxation and there is no time scale connected to the mean jump duration.
V Conclusion
In present paper we have introduced and compared several routes to calculate the WTD in a system
with continuous, Markovian dynamics. Specifically, we have considered
the example of a Brownian particle in an one-dimensional (tilted) washboard potential, and focused on cases where the potential barriers are large
against the thermal energy.
Traditionally, the WTD in such a situation is calculated by analysing trajectories obtained from BD simulations of the Langevin equation.
One main new result of the present paper is a definition of the WTD on the basis of the corresponding Smoluchowski equation (SE).
Here, the WTD is identified as a current calculated from the SE with adsorbing boundaries.
The resulting WTD is closely related to the first passage time distribution; however, the WTD contains directional information which is absent in the FPTD. This becomes particularly important in spatially asymmetric situations. Moreover,
our definition of the WTD is more versatile in that it allows for unusual (asymmetric) initial and/or boundary conditions.
Analysing a variety of systems with different initial conditions, as well as with and without external drive, we find in all cases full quantitative agreement
between the SE and BD results for the WTD. We also stress that, due to our rather general definition of the WTD in the SE approach, we are able
to calculate additional quantities such as the jump duration distribution. The latter is crucial for understanding the growth behaviour of the WTD at short times.
In addition to the SE (and BD) method, we have also presented a fully analytical master equation approach to the WTD. The ME approach is based on the discrete,
two-state-per well model introduced by us in an earlier study emary12 . Comparing the results with those from SE and BD it turns out that the ME yields a very accurate
WTD at intermediate times, where it exhibits a plateau, as well as at long times, where it has an exponential tail. Only the short-time behaviour differs due to the
fact that the ME model involves an “intra-well” relaxation time which is absent in the continuous approaches.
From a computational point of view the BD route to the WTD seems, at first glance, to be the most straightforward one. However, closer inspection shows that the results
are quite dependent on the definition of “jumps”. Furthermore, the results are often quite noisy. Here, the SE approach where noise is averaged out a priori, is clearly superior.
We stress, however, the solution is computationally expensive and “fragile” when using a standard solver. In Appendix B
we have therefore sketched an alternative route to obtain the WTD from the SE based on a Fourier transform.
Finally, our SE approach to the WTD can be easily generalised to systems characterised by a different potential, to interacting systems,
and to systems with higher dimensionality. For example, for processes involving more than one spatial dimension, one would
simply replace the absorbing boundary value by an absorbing surface and calculate the WTD
as an integral over the probability current over a (part of the) surface. Another generalisation concerns the character of the dynamics, which we here assumed to be Markovian.
We claim that even for a non-Markovian SE (see guill99 ; lichtner12 ; loos14 for examples), which would involve memory kernels and thus has higher dimension in time,
the identification of the WTD with the probability current remains intact. Finally, our generalisation of the WTD for continuous systems opens the route to calculate other quantities typically reserved for discrete (Markovian or non-Markovian) systems, such as the idle-time distribution albert12 . Work in these directions is in progress.
Acknowledgements.
We gratefully acknowledge stimulating discussions with T. Brandes.
This work was supported by the Deutsche Forschungsgemeinschaft
through SFB 910 (projects B2 and A7).
Appendix A WTD from BD simulations
To obtain a reliable WTD the recorded histograms must have a high statistical quality at all times where the WTD is nonzero. Any deviation influences the normalisation and therefore shifts the entire WTD. Typically, the domain of non-zero WTD spans several orders of magnitude. Moreover, for large times the WTD decays exponentially (see Fig. 8) which further enhances the computational effort due to the need of many long simulations. These considerations imply that the histogram bins must be sufficiently small to capture the time variations of the WTD and sufficiently large to yield a good statistical average. Here we use a piece-wise constant bin size distribution, which we adjust manually.
To achieve the necessary accuracy in the generated trajectories we choose a time step of $\Delta t=10^{-5}\tau$. This sets the mean size of a fluctuation to $4\cdot 10^{-3}a$ which is necessary to sample the potential sufficiently accurate.
The simulation starts with the particle at the first minimum $m=a/(2\pi)\arcsin(F/F_{c})$. During the simulation the trajectory is analysed to record leavings and arrivals of the particle at other minima $m\pm ka,\,k\in\mathbb{Z}$. Because one simulation alone cannot produce enough jumps in a reasonable computation time, we run several simulations in parallel. The least amount of jumps occur at $u_{0}\!=\!15k_{\mathrm{B}}T$ and $F\!=\!0$. In this case we run $832$ simulations in parallel or successively, each computing $500$ jumps, to achieve a reasonable resolution for the long time part of the WTD.
Short waiting times occur even less frequently than large ones. To calculate the WTD at these short times a new set of simulations is started. Contrary to the regular trajectory simulation, the new simulations stop if the jump does not happen in a given time interval $u$ (with $\log_{10}(u/\tau)\in\{-1,\dots,2\}$). Each histogram for the Figs. 5, 6 and 10 is made of about $10^{4}$ of those short and $10^{5}$ long waiting times.
Appendix B WTD from the SE
To calculate the WTD from the SE Eq. (5) we mainly use the FTCS (forward-time-centered-space) discretisation scheme fletcher91 . Because the whole process is determined by a very small current $j$ over the barrier and, correspondingly, small values for the probability density (see inset in Fig. 9) we need very small discretisation steps $\Delta x$, a typical example being $\Delta x=0.0016a$. Nevertheless, the FTCS method works for all parameter sets except for $u_{0}\!=\!15k_{\mathrm{B}}T$, $F\!=\!0$ where the solution of the SE just takes too long. To calculate the long-time part of the WTD in this particular case we use a different method involving a Fourier transform. To this end we write the probability density as
$$\displaystyle p(x,t)$$
$$\displaystyle=\sum_{n=1}^{N}p_{n}(t)\,s_{n}(x)\,.$$
(21a)
The sum involves the first $$N$$ modes
$$\displaystyle p_{n}(t)$$
$$\displaystyle=\frac{1}{a}\int_{-a}^{a}\mathrm{d}x\,s_{n}(x)\,p(x,t)\,,$$
(21b)
where we exploit that for the special case $$F\!=\!0$$ with $$A$$, $$B$$, $$C$$ being successive minima of $$u(x)$$ the basis functions $$s_{n}$$ can be chosen as
$$\displaystyle s_{n}(x)$$
$$\displaystyle=\sin\left(\pi n\frac{x+a}{2a}\right)\,.$$
(21c)
Applying the Fourier transform (21) to Eq. (5) yields
$$\displaystyle\dot{p}_{n}(t)=\frac{\pi^{2}}{4a^{2}}\Big{(}-n^{2}p_{n}$$
$$\displaystyle+nU(p_{n-4}\Theta(n\!-\!4)-p_{n+4}-p_{4-n}\Theta(4\!-\!n))\Big{)}\,,$$
(22)
where $\Theta(x)$ is the Heaviside-step function. The numerical solution of Eq. (22) takes even longer than FTCS. Therefore we introduce the matrix based formulation $\dot{\mathbf{p}}(t)=A\cdot\mathbf{p}$
where $\mathbf{p}(t)$ contains all $p_{n}(t)$. The solution reads
$$\displaystyle\mathbf{p}(t)=\exp(tA)\cdot\mathbf{p}(0)\,,$$
(23)
which we calculate by decomposing the matrix $A$ into $S\cdot D\cdot S^{-1}$. In the latter expression the diagonal matrix $D$ consists of the eigenvalues of $A$ and $S$ contains the corresponding eigenvectors column by column. Inserting the diagonalisation in Eq. (23) we arrive at
$$\displaystyle\mathbf{p}(t)$$
$$\displaystyle=e^{t\,S\cdot D\cdot S^{-1}}\mathbf{p}(0)=S\cdot e^{tD}\cdot S^{-%
1}\cdot\mathbf{p}(0)\,,$$
(24)
where $e^{tD}$ is also diagonal. The computation of $\mathbf{p}(t)$ via Eq. (24) is fast and reliable. Specifically, for the case $u_{0}\!=\!15k_{\mathrm{B}}T$ we find that $100$ Fourier modes are sufficient (computations finish in minutes even for $10^{4}$ modes).
A further advantage of the diagonalisation is that it provides a simple access to the long time dynamics. This is because the limit $\lim_{t\to\infty}\partial_{t}\ln w(t)$ gives the largest eigenvalue $d$
, and thus $w(t)\sim e^{dt}$. Note that $d$ is negative and close to zero.
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Matrix Theory on ALE Spaces and Wrapped Membranes
David Berenstein and Richard Corrado
Research supported in part by the Robert A. Welch
Foundation and NSF Grant PHY 9511632.
[email protected]
[email protected]
(March 5, 1998)
††preprint: UTTG–02–98
hep-th/9803048
\oneaddress
Theory Group, Department of Physics
University of Texas at Austin
Austin TX 78712 USA
\Abstract
We study the properties of wrapped membranes in matrix
theory on ALE spaces. We show that the only BPS bound states of wrapped
membranes that can form are roots of the $A$-$D$-$E$ group. We
determine a bound on the energy of a bound state and find the correct
dependence on the blow-up parameters and longitudinal momentum
expected from M-Theory. For the $A_{n-1}$ series, we
construct explicit classical solutions for the wrapped membrane bound
states. These states have a very rich structure and have a natural
interpretation in terms of noncommutative geometry.
In the $A_{1}$ case, we examine the spectrum of excitations
around the wrapped membrane solution and provide an explicit
calculation of their energies. The results agree exactly with supergravity
calculations.
1 Introduction
The M(atrix)-Theory [1] proposal for a
non-perturbative description of M-Theory has been demonstrated to
properly capture M-Theory physics in a variety of settings (see
[2, 3, 4] for
recent reviews). However, several important areas where our
understanding of matrix theory is incomplete still remain. For
example, the fact that matrix theory captures the physics of
linearized supergravity is fairly well
understood [5]. The reasons for its apparent
failure to capture supergravity results in other
situations [6, 7, 8, 9] seem to be subtleties in our understanding
of the discrete light-cone quantization (DLCQ) of M-Theory, namely
that the low-energy description of DLCQ M-Theory is not quite
supergravity [2, 10, 3].
Another area which requires further study is the
compactification of matrix theory, in particular on curved manifolds.
One promising case is that of “compactification” on ALE spaces,
which has been conjectured to be described by the theory of D0-brane
partons moving on the ALE space [11, 6, 12, 13].
In this paper we continue the study of matrix theory on ALE spaces. In
Section 2, we give a very brief review of ALE matrix
theory. Our emphasis will be to further examine the characteristics of
the description of membranes which are wrapped around homology
2-cycles as recently described
in [14, 15, 16]. In
Section 3, we focus on the existence of wrapped membrane
BPS bound states in the wrapped membrane matrix model presented
in [16]. We demonstrate that such bound states
must be roots under the $A$-$D$-$E$ group. We then derive a bound on
the energy of such a bound state. For the case of the $A_{n-1}$
series, we explicitly construct the bound states for all roots. In
Section 4, we
briefly discuss how the membrane solution fits into the framework of
noncommutative geometry and thereby satisfies the properties of
spherical membranes [17, 18]. In
Section 5, we discuss the spectrum of excitations
of the wrapped membrane and calculate the energies of excitations
around the $A_{1}$ solution, finding agreement with supergravity expectations.
2 A Brief Review of ALE Matrix Theory
A matrix description of M-Theory on an ALE space must possess several
crucial ingredients if it is to be considered both correct and
useful. A candidate formalism for such a description has been provided
by the worldvolume effective theories describing D0-branes moving on
an ALE
space [19, 20, 21].
Let us briefly review the construction of these models and examine the
spacetime features of M-Theory that they capture.
The ALE matrix models are given by the dimensional reduction to
quantum mechanics of the six-dimensional gauged supersymmetric sigma
models appearing in the hyperkähler quotient
construction [22, 23] of the ALE
space. The field content is summarized by a quiver diagram, which is
based on the $A$-$D$-$E$ extended Dynkin diagram, such as the
$A_{n-1}$ diagram shown in Figure 1.
For each vertex of the diagram there is a gauge group $U(N_{i})$, where
$N_{i}=Nk_{i}$, with $N$ the number of D0-branes and $k_{i}$ the Dynkin
label of the vertex, as well as a vector multiplet, $(A_{i},a_{i})$ (in
terms of $d=4$, $\CN=1$ vector and chiral superfields), in the
$(1,\ldots,\text{ad}(U(N_{i})),\ldots,1)$ representation. For each edge
of the diagram there is an associated hypermultiplet,
$(x_{i,i+1},y_{i+1,i})$, in the fundamental–anti-fundamental
representation, $(1,\ldots,1,N_{i},\overline{N}_{i+1},1,\ldots,1)$, of
the neighboring gauge groups.
The Lagrangian is the most general one compatible with
the gauge symmetry and $d=6$, $\CN=1$ SUSY. However, since we are in
$d=1$ dimensions, the question of the mass dimension of the fields and
coupling constants is an issue. We will choose string units, with
$T_{A}=1$. All fields have mass dimension $-1$ and the coupling constant
has dimension $-\frac{1}{2}$. This requires us to scale the
hypermultiplets so that an inverse square of a coupling constant
appears out in front of the terms in the Lagrangian that they appear
in. We will choose this to be the “average” coupling constant, defined by
$$\frac{1}{g^{2}}=\sum_{i}\frac{k_{i}}{g_{i}^{2}}.$$
(2.1)
We also scale the D and F-terms by $2/g^{2}$. In terms of
$d=4$, $\CN=1$ superfields, our Lagrangian is
$$\begin{split}\displaystyle\CL=\sum_{i}&\displaystyle\left[\left(\frac{1}{16N_{%
i}g_{i}^{2}}\int d^{2}\theta\,W_{i}^{2}+\text{c.c.}\right)+\frac{1}{g_{i}^{2}}%
\int d^{4}\theta\,\bar{a}_{i}e^{A_{i}}a_{i}e^{-A_{i}}\right.\\
&\displaystyle+\frac{1}{g^{2}}\int d^{4}\theta\,\left(\bar{x}_{i,i+1}e^{A_{i}}%
x_{i,i+1}e^{-A_{i+1}}+\bar{y}_{i+1,i}e^{A_{i+1}}y_{i+1,i}e^{-A_{i}}\right)\\
&\displaystyle+\left(\frac{1}{g^{2}}\int d^{2}\theta\,\left(y_{i+1,i}a_{i}x_{i%
,i+1}-x_{i-1,i}a_{i}y_{i,i-1}\right)+\text{c.c.}\right)\\
&\displaystyle\left.+\frac{2}{g^{2}}\int d^{4}\theta\,d_{i}A_{i}+\left(\frac{2%
}{g^{2}}\int d^{2}\theta\,f_{i}a_{i}+\text{c.c.}\right)\right].\end{split}$$
(2.2)
The coupling constant satisfies
$$\frac{1}{g^{2}}=T_{D0}.$$
(2.3)
Then it is clear that the $1/g_{i}^{2}$ are the masses of the fractional
D0-branes [20, 11, 15] associated to each vertex.
The coefficients of the D and F-terms are constrained by the
requirement that a supersymmetric ground state exists. This requires
that
$$\sum_{i}k_{i}f_{i}=\sum_{i}k_{i}d_{i}=0.$$
(2.4)
For $N_{i}=Nk_{i}$, there are zero-energy ground states. Under the
$SU(2)_{R}$ R-symmetry, the $\vec{\zeta}_{i}=(f_{i},d_{i})$ form
triplets. In fact, according to the hyperkähler quotient
construction, the $\vec{\zeta}_{i}$ are the blow-up parameters for the
ALE space. With the normalization chosen in (2.2), the
area of the $i^{th}$ $\BP^{1}$ is given by $4\pi|\vec{\zeta}_{i}|^{2}$.
For simplicity, one typically chooses all the gauge
couplings to be the same, but this is not required. As we will see
later, these coupling constants have an interpretation as Wilson lines
for the $A$-$D$-$E$ gauge group in the DLCQ M-Theory description.
Applying the matrix prescription [1], one is lead
to conjecture that, in the limit $N\rightarrow\infty$, $g\rightarrow\infty$, the D0-branes described by (2.2) are the
partons of the infinite-momentum frame description of M-Theory on the
ALE space [11, 6]. On the other hand,
the finite $N$, finite $g$ matrix models would be conjectured to
provide a description of the DLCQ of
M-Theory [24] on the ALE
space [8].
Several pieces of evidence support the M-Theory interpretation of
these ALE matrix models. First, the models contain the geometry of the
ALE space. For $N=1$, we have the standard $U(1)$ gauged
supersymmetric sigma model. After taking the hyperkähler
quotient [22],
we recover the target space $\BR^{5}\times\CM_{\vec{\zeta}}$.
Here we use $\CM_{\vec{\zeta}}$ to denote the ALE space parameterized
by the blowup parameters $\vec{\zeta}$. For larger $N$,
the moduli space of the theory is
$$(\BR^{5}\times\CM_{\vec{\zeta}})^{N}/S_{N},$$
(2.5)
so that the ALE geometry is recovered along the flat directions of the
classical ground states of the system.
The models have the equivalent of $\CN=1$ supersymmetry in six
dimensions, or eight supercharges, which is the same amount of
supersymmetry present in light-cone M-Theory on an ALE space. The
gauge group $\prod_{i}U(N_{i})$ contains an overall $U(1)$ factor which is a
linear combination of the $U(1)$s for each $U(N_{i})$ factor. The vector
associated to this $U(1)$ is decoupled from the rest of the dynamics. In the
matrix model, these decoupled degrees of freedom are associated to the
center-of-mass motion in the transverse space. The amount of
supersymmetry provides for a 256-fold degeneracy in the continuum
spectrum of the theory, which allows these states to be
identified with the gravity multiplet. The SUSY
might also allow for the existence of non-renormalization theorems
that could lead to correct results for interactions.
It is crucial that each finite mass BPS object in M-Theory have an
explicit description as some state in the matrix quantum
mechanics. Furthermore, whenever BPS branes come together, or a
2-cycle which has a membrane wrapped around it shrinks to zero-size,
an enhanced gauge symmetry must appear in the quantum dynamics.
The mechanism by which wrapped membranes and enhanced gauge symmetry
appear in the model was introduced by
Douglas [11] and elaborated upon
in [6, 14, 8, 15, 16].
From the Kronheimer construction [23], the vertices
of the extended Dynkin diagram are associated to the homology
2-cycles (which are $\BP^{1}$s) of the ALE
space. In [14], it was proposed that, in
the blow-down, states which correspond to wrapped membranes exist on
the Coulomb branch of the quiver gauge theory. In particular,
a configuration describing a membrane wrapped on the $i^{th}$ $\BP^{1}$
would be described by a metastable state for which the hypermultiplets
corresponding to that vertex vanished, $x_{i-1,i}=x_{i,i+1}=\cdots=0$. This picture was
further discussed in [15], where a
proposal was made for its extension to the blown-up ALE spaces.
A different approach was taken in [16], where
it was proposed that the matrix description of membranes wrapped on
these $\BP^{1}$s could be obtained by considering the ALE matrix models
described above, but now relaxing the condition that $N_{i}=Nk_{i}$. Instead, one should take
$$N_{i}=Nk_{i}+r_{i},$$
(2.6)
where $N$ is chosen such that the set of $r_{i}>0$ are as small as
possible. The resulting matrix model describes $N$ D0-branes
propagating on a ALE space with $r_{i}$ membranes wrapped around the
$i^{th}$ $\BP^{1}$. It is this latter approach which we will consider in
the rest of the paper.
If all of the $r_{i}=nk_{i}$ for some integer $n$ (we can include the
possibility that $n=0$, in which case $N_{i}=Nk_{i}$), then the
corresponding homology 2-cycle is trivial, and the configuration of
wrapped membranes can decay into a collection of D0-branes. A single
wrapped membrane corresponds to a simple root of the spacetime gauge
group, while the other roots have been conjectured to be given by
bound states of these simple roots [11]. We will
demonstrate this result explicitly in Section 3.
The objective of the following sections is to give a detailed account
of several features of wrapped membranes in the ALE matrix models. We
give a complete description of what BPS bound states of wrapped membranes
form in the quantum mechanical system and then construct explicit bound
state solutions for the $A_{n-1}$ series. We give a short discussion
of the noncommutative geometry properties of the solutions and exhibit
the spherical membrane properties for the explicit $A_{1}$
solution. For the $A_{1}$ ALE space, we examine the spectrum of
excitations of the wrapped membrane, providing an explicit calculation
of their energies. We find that the large $N$ behavior of the excitations
matches supergravity expectations.
3 Bound States of Wrapped Membranes
We would like to determine the conditions under which a
(non-threshold) BPS bound state of membranes can form. These are ground
states of the interacting part of the theory that, classically,
completely break the gauge symmetry group. Since the argument is
classical, any solution of the F-terms for the hypermultiplets can be
deformed to set the gauge fields equal to zero. In this way, the
symmetry breaking pattern can be chosen to preserve the $SO(5)$
symmetry of the directions transverse to the ALE.
In order to determine the masses of the gauge fields, the argument can
be extended to the six-dimensional quiver gauge theory. The Higgs
mechanism requires that the extra degrees of freedom that give mass to
a vector multiplet come from “eating” a hypermultiplet. The number
of vector multiplets that can be Higgsed can be obtained by counting
the total number of vector multiplet degrees of freedom
($4\sum_{i}N_{i}^{2}$) and subtracting the number of hypermultiplet
($4\sum_{i,j\,\text{adj.}}N_{i}N_{j}$) and decoupled $U(1)$ degrees of
freedom (4). The resulting number must be smaller than zero or otherwise
there remains a flat direction corresponding to a vector multiplet, in
which case the membrane configuration can be separated classically
into two separate bound states. We therefore need $N_{i}$ such that
$$0\geq 4\left(\sum_{i}N_{i}^{2}-\sum_{i,j\,\text{adj.}}N_{i}N_{j}-1\right)=2%
\left(\sum_{i,j}\hat{C}_{ij}N_{i}N_{j}-2\right),$$
(3.1)
where $\hat{C}_{ij}$ is the extended Cartan matrix of the
corresponding $A$-$D$-$E$ group. One immediately recognizes the
quantity $\sum_{i,j}\hat{C}_{ij}N_{i}N_{j}$ as the squared norm of the
vector $N_{i}$ in the lattice of the algebra. Since the
inequality (3.1) is satisfied whenever
$\sum_{i,j}\hat{C}_{ij}N_{i}N_{j}\leq 2$, we see
that the state must either be neutral under the Cartan subalgebra (and
is therefore a collection of D0-branes) or it is a root.
It is fairly straightforward, in general, to
determine a bound on the energy of the state. Wrapped membrane
bound states should form massive vector multiplets in seven spacetime
dimensions. Since the fermionic zero-modes corresponding to the
decoupled $U(1)$ center-of-mass motion already give rise to a 16-fold
degeneracy, the ground state of the interacting part of the theory must be
non-degenerate. This is difficult to prove in general, but we can
still obtain a bound on the energy rather easily. We will then show
that these states exist explicitly in the case of the $A_{n-1}$ ALE spaces.
Let us assume that the F-terms are zero, then, after computing the traces,
the Hamiltonian for the D-terms we obtain from (2.2) is
$$H_{D}=\sum_{i}\left(\frac{N_{i}D_{i}^{2}}{2g_{i}^{2}}-\frac{2N_{i}d_{i}D_{i}}{%
g^{2}}\right).$$
(3.2)
Since the decoupled $U(1)$ is the sum of the $U(1)$s at each vertex, the
corresponding D-term, $D_{\text{dec.}}$, is given by the sum
$$D_{\text{dec.}}=\frac{1}{\sum_{i}\frac{N_{i}}{g_{i}^{2}}}\sum_{i}\frac{N_{i}D_%
{i}}{g_{i}^{2}}.$$
(3.3)
Therefore, each D-term has an expansion as
$$D_{i}=D_{\text{dec.}}+D_{i,\text{int.}},$$
(3.4)
where the $D_{i,\text{int.}}$ are orthogonal to each other and
$D_{\text{dec.}}$ and satisfy
$$\sum_{i}\frac{N_{i}D_{i,\text{int.}}}{g_{i}^{2}}=0.$$
(3.5)
Using the expansion (3.4) in (3.2), we
find that the Hamiltonian for the decoupled $U(1)$ is
$$H_{\text{dec.}}=D_{\text{dec.}}\sum_{i}N_{i}\left(\frac{D_{\text{dec.}}}{2g_{i%
}^{2}}-\frac{2d_{i}}{g^{2}}\right).$$
(3.6)
This is minimized by
$$D_{\text{dec.}}=-\frac{2\sum_{i}N_{i}d_{i}}{g^{2}\sum_{i}\frac{N_{i}}{g_{i}^{2%
}}},$$
(3.7)
which leads to a bound on the energy
$$E\geq\frac{2\left(\sum_{i}N_{i}d_{i}\right)^{2}}{g^{4}\sum_{i}\frac{N_{i}}{g_{%
i}^{2}}}.$$
(3.8)
From this expression for the bound, we see that the
D0-brane charge of the state is proportional to
$\sum_{i}\frac{N_{i}}{g_{i}^{2}}$. If we take the $N_{i}$ as in (2.6)
and apply (2.4), we find that
$$E\geq\frac{2\left(\sum_{i}r_{i}d_{i}\right)^{2}}{g^{4}\sum_{i}\frac{N_{i}}{g_{%
i}^{2}}}.$$
(3.9)
A similar calculation works when the F-terms are non-zero. Defining
$F_{\text{dec.}}$ in a manner analogous to (3.3), we find
the solution
$$F_{\text{dec.}}=-\frac{2\sum_{i}N_{i}f_{i}}{g^{2}\sum_{i}\frac{N_{i}}{g_{i}^{2%
}}}.$$
(3.10)
Putting the results together, we find a bound
$$E\geq\frac{2}{g^{4}\sum_{i}\frac{N_{i}}{g_{i}^{2}}}\left(\bigl{(}\sum_{i}r_{i}%
d_{i}\bigr{)}^{2}+\big{|}\sum_{i}r_{i}f_{i}\bigr{|}^{2}\right)=\frac{2\bigl{|}%
\sum_{i}r_{i}\vec{\zeta}_{i}\bigr{|}^{2}}{g^{4}\sum_{i}\frac{N_{i}}{g_{i}^{2}}}.$$
(3.11)
This is the light-cone energy of an object with mass
$$m=\frac{2\bigl{|}\sum_{i}r_{i}\vec{\zeta}_{i}\bigr{|}}{g^{2}}$$
(3.12)
and longitudinal momentum
$$p^{+}=\sum_{i}\frac{N_{i}}{g_{i}^{2}}.$$
(3.13)
In particular, it is
clear that by tuning the $g_{i}$, one assigns different amounts of
longitudinal momentum to each simple root. In the DLCQ, the parameters
responsible for this are the Wilson lines of the spacetime gauge group
around the light-like circle.
In the large $N$ limit, we obtain from (3.11)
$$E\geq\frac{2\bigl{|}\sum_{i}r_{i}\vec{\zeta}_{i}\bigr{|}^{2}}{Ng^{2}}\left(1-O%
(1/N)\right).$$
(3.14)
We will show explicitly below that there are solutions that satisfy
the bound for any $N$. Moreover, from the DLCQ interpretation
of the coupling constants, we know that at large $N$ the effects of
the Wilson lines should disappear. In particular this implies that the
ratios $g_{i}/g_{j}$ become unimportant in the large $N$ limit and
therefore that some of these coupling constants decouple from the
dynamics in the large $N$ limit. This can serve as a very useful
calculational tool, and might serve to produce some
non-renormalization theorems in the large $N$ limit.
As a check on our results, we can calculate the tension of
the membrane from (3.12). For a single membrane state
$$m=\frac{2\bigl{|}\vec{\zeta}\bigr{|}}{g^{2}}=T^{(2)}A,$$
(3.15)
Since the area of the 2-sphere is $A=4\pi|\vec{\zeta}|$,
from (2.3)
we recover (in string units) the well known
result [25]
$$T^{(2)}=\frac{T_{D0}}{2\pi}.$$
(3.16)
Now we will construct the bound states for all roots in the $A_{n-1}$
matrix models. First, as the $k_{i}=1$ for the $A_{n-1}$, the
$r_{i}=0~{}\text{or}~{}1$. For a state to be a root and non-trivial in
homology, all of the $r_{i}=1$ must be adjacent and there must be at
least one $r_{i}=0$.
We will begin with the simplifying assumption that the F-terms are set
to zero. Later, we will describe how to obtain solutions when both
F and D-terms are present.
In order to solve the F-term equations of the hypermultiplets, we set
all of the $a_{i}=0$ in the vector multiplets. To solve the F-term
equations of the vector multiplets, we make the ansatz
$$x_{i,i+1}y_{i+1,i}=y_{i+1,i}x_{i,i+1}=0.$$
(3.17)
Now we must minimize the D-terms. The states we are interested in
should be excited states under the decoupled $U(1)$, but should form
supersymmetric bound states of the internal part of the
theory. Therefore, we will set $D_{i,\text{int.}}=0$ and seek a
solution to the equations
$$\bar{x}_{i-1,i}x_{i-1,i}-y_{i,i-1}\bar{y}_{i,i-1}-x_{i,i+1}\bar{x}_{i,i+1}+%
\bar{y}_{i+1,i}y_{i+1,i}=\left(2d_{i}+\frac{g^{2}}{g_{i}^{2}}D_{\text{dec.}}%
\right)\,\Bid_{N_{i}}.$$
(3.18)
With our ansatz, the operators $\bar{x}_{i-1,i}x_{i-1,i}$ and
$y_{i,i-1}\bar{y}_{i,i-1}$ commute. As they are self-adjoint, they
can be simultaneously diagonalized, and they moreover multiply to
zero. Therefore they are fully determined as the projections to the
positive and negative spectra of the operator
$$O_{i}=\bar{x}_{i-1,i}x_{i-1,i}-y_{i,i-1}\bar{y}_{i,i-1}$$
(3.19)
Now, $O_{i}$ and the operator
$$O^{\prime}_{i}=x_{i-1,i}\bar{x}_{i-1,i}-\bar{y}_{i,i-1}y_{i,i-1}$$
(3.20)
which appears in the D-term at the $(i-1)^{th}$ vertex, are
isospectral, except for an extra zero eigenvalue on one of them
whenever $r_{i}=r_{i-1}\pm 1$, so that $N_{i}=N_{i-1}\pm 1$. Imposing an
ordering on the eigenvalues, $O_{i}=\text{diag}(O_{i}^{1},\ldots,O^{N_{i}}_{i})$,
with $O_{i}^{1}\geq\cdots\geq O^{N_{i}}_{i}$, the D-term
equations (3.18) become
$$O_{i}^{\ell}-O_{i+1}^{\prime\ell}=\left(2d_{i}+\frac{g^{2}}{g_{i}^{2}}D_{\text%
{dec.}}\right)$$
(3.21)
Now, if the state is nontrivial, then, without loss of generality, we
can label the first vertex where there is a membrane wrapped as the
$0^{th}$, so that on the quiver diagram we will have the sequence
$\ldots,r_{n-1}=0,r_{0}=1,r_{1}=1,\ldots,r_{\iota}=1,r_{\iota+1}=0,\ldots$
(i.e., there are $\iota$ 1s and $n-\iota$ 0s on the diagram). Then
$O_{0}^{N+1}=0$, so that (3.21) can be solved to find
$O_{1}^{\prime N+1}=-(2d_{0}+\frac{g^{2}}{g_{0}^{2}}D_{\text{dec.}})$. But then
$O_{1}^{N+1}=-(2d_{0}+\frac{g^{2}}{g_{0}^{2}}D_{\text{dec.}})$ as well. By
repeated use of (3.21), we can follow these
eigenvalues until we get to the transition $r_{\iota}=1\rightarrow r_{\iota+1}=0$, where we jump to the next non-zero eigenvalue, so that
$$O_{\iota+1}^{N}=O_{\iota+1}^{\prime N+1}=-\sum_{j=1}^{\iota+1}\left(2d_{j-1}+%
\frac{g^{2}}{g_{j-1}^{2}}D_{\text{dec.}}\right).$$
(3.22)
By iterating this process around to the $0^{th}$ vertex, we
find
$$\begin{split}\displaystyle O_{0}^{N}&\displaystyle=-\sum_{j=1}^{n}\left(2d_{j-%
1}+\frac{g^{2}}{g_{j-1}^{2}}D_{\text{dec.}}\right)=-{D_{\text{dec.}}}\\
\displaystyle O_{1}^{N}&\displaystyle=-2d_{0}-\left(1+\frac{g^{2}}{g_{0}^{2}}%
\right)D_{\text{dec.}}.\end{split}$$
(3.23)
By continuing this process of circuiting the diagram, we determine all
of the eigenvalues. We find the general expression
$$O^{\ell}_{k}=-\sum_{j=1}^{k}\left(2d_{j-1}+\frac{g^{2}}{g_{j-1}^{2}}D_{\text{%
dec.}}\right)-\begin{cases}(N+1-\ell)D_{\text{dec.}}&\text{for}~{}0\leq k\leq%
\iota\\
(N-\ell)D_{\text{dec.}}&\text{for}~{}\iota<k<n.\end{cases}$$
(3.24)
The explicit form for $D_{\text{dec.}}$ is given by the
solution (3.7). For $n=1$, this formula is in agreement
with the $A_{1}$ solution presented in [16].
Now let us consider the situation when both F and D-terms are
present. Here, in addition to solving the D-terms (3.18), we
must also solve the F-term equations, which are of the form
$$y_{i,i-1}x_{i-1,i}-x_{i,i+1}y_{i+1,i}=\left(2f_{i}+\frac{g^{2}}{g_{i}^{2}}F_{%
\text{dec.}}\right)\,\Bid_{N_{i}}.$$
(3.25)
The simplest ansatz for a solution would clearly be one for which the
above method of tracing eigenvalues around the quiver diagram would
work. This will require that the individual terms in (3.18)
and (3.25) commute amongst themselves, so that they are
simultaneously diagonalizable. We must therefore demand that
$$\begin{split}\displaystyle\bar{y}_{i,i-1}y_{i,i-1}x_{i-1,i}&\displaystyle=x_{i%
-1,i}y_{i,i-1}\bar{y}_{i,i-1}\\
\displaystyle y_{i,i-1}x_{i-1,i}\bar{x}_{i-1,i}&\displaystyle=\bar{x}_{i-1,i}x%
_{i-1,i}y_{i,i-1}.\end{split}$$
(3.26)
If we then define
$$\begin{split}\displaystyle P_{i}&\displaystyle=y_{i,i-1}x_{i-1,i}\\
\displaystyle P_{i}^{\prime}&\displaystyle=x_{i-1,i}y_{i,i-1},\end{split}$$
(3.27)
then it is a straightforward exercise to show that
$$[O_{i},P_{i}]=[P_{i},\bar{P}_{i}]=0.$$
(3.28)
From (3.26), we see that $x_{i-1,i}$ generates a sort of
“supersymmetry” for the matrices $\bar{y}_{i,i-1}y_{i,i-1}$ and
$y_{i,i-1}\bar{y}_{i,i-1}$. That is, it acts as an intertwiner between
their eigenspaces, so that given an eigenstate, $|\alpha\rangle$, of
$y_{i,i-1}\bar{y}_{i,i-1}$, the state $x_{i}|\alpha\rangle$ is an
eigenstate of $\bar{y}_{i,i-1}y_{i,i-1}$ with the same eigenvalue.
Similarly,
$y_{i,i-1}$ is a supersymmetry for $x_{i-1,i}\bar{x}_{i-1,i}$ and
$\bar{x}_{i-1,i}x_{i-1,i}$, while both $x_{i-1,i}$ and $y_{i,i-1}$
are supersymmetries for $P_{i}$ and $P_{i}^{\prime}$. We can therefore
conclude that, as before, $O_{i}$ and
$O_{i}^{\prime}$ have the same spectrum, except, perhaps, for a
zero-mode. Similarly, the spectra of $P_{i}$ and $P_{i}^{\prime}$ can differ
only by a zero-mode.
Solutions may be explicitly obtained by solving the
eigenvalue equations (3.21) and
$$P_{i}^{\ell}-P_{i+1}^{\prime\ell}=\left(2f_{i}+\frac{g^{2}}{g_{i}^{2}}F_{\text%
{dec.}}\right),$$
(3.29)
via following the eigenvalues around the quiver diagram, as outlined
above. Then we find that the $O_{i}$ are given by (3.24),
as before, while the
$$P^{\ell}_{k}=-\sum_{j=1}^{k}\left(2f_{j-1}+\frac{g^{2}}{g_{j-1}^{2}}F_{\text{%
dec.}}\right)-\begin{cases}(N+1-\ell)F_{\text{dec.}}&\text{for}~{}0\leq k\leq%
\iota\\
(N-\ell)F_{\text{dec.}}&\text{for}~{}\iota<k<n,\end{cases}$$
(3.30)
where $F_{\text{dec.}}$ is given by (3.10).
We note that if the F-terms are in fact absent, we recover our old
ansatz (3.17).
Within the ansatz (3.26) the solutions obtained
from (3.24) and (3.30) are
unique. Explicit expressions for the $x_{i,i+1}$ and $y_{i+1,i}$ can
be obtained by choosing, say, the $x_{i,i+1}$ to have positive real
entries. This can
always be achieved by a gauge transformation. The solution obtained in
this manner is a classical solution to the equations for a
supersymmetric vacuum (of the internal theory), and therefore there is
a corresponding quantum mechanical state whose wavefunction is
localized near the classical solution. It is clear that all of the
hypermultiplets have a mass gap and it is reasonable to believe that
the solution breaks the gauge group completely.
4 Noncommutative Geometry and Spherical Membranes
It is worthwhile to illustrate the
geometric nature of the membrane solutions we have constructed. We
will show in this section that the natural set of gauge-invariant
coordinates derived from the quiver theory are elevated by these
solutions to noncommuting coordinates111For a recent discussion
of noncommutative geometry and matrix theory,
see [26] and references therein.. These
noncommuting coordinates
will satisfy the constraints imposed by the ALE space geometry to
leading order in the $1/N$ expansion.
In the case of the $A_{n-1}$, we know that $\BZ_{n}$-invariant
coordinates are given by invariant products of the coordinates of the
$\BC^{n}$ being quotiented. In terms of the quiver theory describing a
single membrane wrapped on the $0^{th}$ $\BP^{1}$, we can
consider the $N_{0}\times N_{0}$ matrices
$$\begin{split}\displaystyle U&\displaystyle=x_{01}\cdots x_{n-1,0}\\
\displaystyle V&\displaystyle=y_{0,n-1}\dots y_{10}\\
\displaystyle W&\displaystyle=x_{01}y_{10}.\end{split}$$
(4.1)
Coordinates $(u,v,w)$ on an ALE space of the same shape will satisfy
$$uv=P(w),$$
(4.2)
where $P(w)$ is an $n^{th}$-order polynomial whose coefficients are
determined by the $f_{i}$.
Now, when the $f_{i}=0$ (e.g. on the blow-down), $P(w)=w^{n}$. Also,
from (3.30) we see that
$W=0$. Since (3.24) tells us that $O_{0}$ is
positive-definite (for $\sum_{i}r_{i}d_{i}>0$, $D_{\text{dec.}}$ is
negative-definite), we know that
$\bar{y}_{01}y_{01}=0$, whence $y_{01}=V=0$. On the other hand,
$O_{1}$ is negative-definite, so that $x_{01}\bar{x}_{01}=0$ and
$x_{01}=U=0$ as well. Therefore the membrane is indeed localized at
the singularity, as $u=v=w=0$ is the singular locus of (4.2).
By means of the F-term equations (3.25), one can show that
for large $N$
$$UV=P(W)+O(1/N).$$
(4.3)
Moreover, one can also show that
$$\begin{split}\displaystyle[W,U]&\displaystyle=F_{\text{dec.}}U=-\frac{2\sum_{i%
}d_{i}f_{i}}{N}U+O(1/N^{2})\\
\displaystyle[W,V]&\displaystyle=-F_{\text{dec.}}V=\frac{2\sum_{i}d_{i}f_{i}}{%
N}V+O(1/N^{2}).\end{split}$$
(4.4)
This is reminiscent of the angular momentum
commutation relations, and shows that the intrinsic geometry of the
wrapped membranes that we have constructed is noncommutative. The
membranes are spherical, but if we were to probe the membrane locally,
where it approaches the flat membrane
solution, we would find an effective “Planck constant” of order
$1/N$ times the area of the wrapped membrane. This comes as no
surprise when we consider the way membranes were first constructed in
[17, 1], as well as the discussion
of spherical membranes in matrix theory by Kabat and
Taylor [18].
As an illustration of the noncommutative properties of the solutions,
it is illuminating to check that single membranes have properties
which are analogous to those discussed
in [18]. In the $A_{1}$ case, we can set the F-terms
to zero and use (3.24) to solve for the hypermultiplets.
Then $X_{01}=Y_{01}=0$, while the other hypermultiplet components are
off-diagonal (up to a $U(N)\times U(N+1)$ gauge transformation)
$$\begin{split}\displaystyle X_{10}&\displaystyle=\sqrt{\frac{4d}{2N+1}}\begin{%
pmatrix}\sqrt{N}&0&\cdots&\cdots&0\\
0&\sqrt{N-1}&0&\cdots&\vdots\\
\vdots&0&\ddots&0&\vdots\\
0&\cdots&0&1&0\end{pmatrix}\\
\displaystyle Y_{10}&\displaystyle=\sqrt{\frac{4d}{2N+1}}\begin{pmatrix}0&1&0&%
\cdots&0\\
\vdots&0&\ddots&0&\vdots\\
\vdots&\cdots&0&\sqrt{N-1}&0\\
0&\cdots&\cdots&0&\sqrt{N}\end{pmatrix},\end{split}$$
(4.5)
where we have set $g_{0}=g_{1}$ for convenience. As coordinates on the ALE
space are gauge-invariant combinations of the quiver theory
hypermultiplet components, we may consider the quadratic combinations
$$\begin{split}\displaystyle J_{0}&\displaystyle=\frac{1}{8\sqrt{d}}\bigl{(}X_{1%
0}\bar{X}_{10}-Y_{10}\bar{Y}_{10}\bigr{)}\\
&\displaystyle=\frac{\sqrt{d}}{2N+1}\begin{pmatrix}N-1&0&\cdots&\cdots&0\\
0&N-3&0&\cdots&\vdots\\
\vdots&0&\ddots&0&\vdots\\
\vdots&\cdots&0&-(N-3)&0\\
0&\cdots&\cdots&0&-(N-1)\end{pmatrix}\end{split}$$
(4.6)
$$J_{+}=\frac{1}{4\sqrt{d}}Y_{10}\bar{X}_{10}=\frac{2\sqrt{d}}{2N+1}\begin{%
pmatrix}0&\sqrt{N-1}&0&\cdots&0\\
\vdots&0&\sqrt{2(N-2)}&0&\vdots\\
\vdots&\cdots&0&\ddots&0\\
\vdots&\cdots&\cdots&0&\sqrt{N-1}\\
0&\cdots&\cdots&\cdots&0\\
\end{pmatrix}$$
(4.7)
$$J_{-}=\frac{1}{4\sqrt{d}}X_{10}\bar{Y}_{10}=\frac{2\sqrt{d}}{2N+1}\begin{%
pmatrix}0&\cdots&\cdots&\cdots&0\\
\sqrt{N-1}&0&\cdots&\cdots&\vdots\\
0&\sqrt{2(N-2)}&0&\cdots&\vdots\\
\vdots&0&\ddots&0&\vdots\\
0&\cdots&0&\sqrt{N-1}&0\\
\end{pmatrix}.$$
(4.8)
These matrices are proportional to the generators of
the N of $SU(2)$, so they commute according to the
$SU(2)$ algebra
$$\begin{split}&\displaystyle[J_{+},J_{-}]=2\left(\frac{2\sqrt{d}}{2N+1}\right)J%
_{0}\\
&\displaystyle[J_{\pm},J_{0}]=\mp\left(\frac{2\sqrt{d}}{2N+1}\right)J_{\pm}%
\end{split}$$
(4.9)
and the sum over their squares is
$$\sum_{i}J_{i}^{2}=J_{0}^{2}+\frac{1}{2}\{J_{+},J_{-}\}=\frac{4(N^{2}-1)d}{(2N+%
1)^{2}}\Bid_{N}=r^{2}\Bid_{N}.$$
(4.10)
Evidently, (4.6)–(4.8) are the redundant set
of coordinates
which parameterize the surface of the membrane. We note that the set
of operators $\tilde{J}_{+}=\bar{X}_{10}Y_{10}/4\sqrt{d},\ldots$
also form a set of membrane coordinates, this time in
the $\mathbf{N+1}$ representation. That there is a pair of good
membrane coordinates can be useful in calculations.
As in [18], we can ask if there is a regime in which
these membranes approach the flat membrane solution. At large $N$,
$$[J_{+},J_{-}]=\frac{d}{N}\begin{pmatrix}\Bid&0\\
0&-\Bid\end{pmatrix}+O(1/N^{2}),$$
(4.11)
which, when restricted to the upper-left quadrant, resembles the
commutation relations of the coordinates of the flat membrane.
In the semi-classical correspondence between Poisson brackets and
commutators of [17, 1], we find
the “Planck constant”
$$\hbar=\frac{d}{N},$$
(4.12)
which, as promised above, is proportional to the area of the membrane.
In the large $N$ limit, from (4.10), we also find that
$r$ is the radius of the membrane,
$$r=\sqrt{d}\left(1-\frac{1}{2N}\right).$$
(4.13)
Our choice of normalization in (4.6)–(4.8)
was made specifically
to obtain this leading-order behavior. These results agree nicely with
those of [18], once the different $N$ dependence of
the membrane longitudinal momentum here is taken into account. We note
that the corrections to the radius (4.13) are stronger than
those found for the membranes in [18], being of
$O(1/N)$ as compared to $O(1/N^{2})$.
5 The Spectrum of Excitations of the Wrapped Membrane
We will now concentrate our efforts on finding the spectrum of
excitations around the membrane solution for the simplest possible
case, namely the $A_{1}$ singularity. We will first examine a toy model,
in order to establish a framework for understanding the problem. Then
we utilize the rotational invariance of the $A_{1}$ solution to
determine the allowed representations for perturbations around the
membrane. We then calculate the mass spectrum for the P-wave
excitations and, finally, for all modes.
We note that the perturbative approach we take to the calculation of
the spectrum of fluctuations is well-suited only for membranes which
are large as compared to the 11-dimensional Planck length. It is only
in this limit that we may reliably expect that the fluctuations have
energies small compared to the mass of the membrane and are thereby
long-lived.
We want to describe the linearized spectrum of excitations of a large
membrane wrapped around a sphere. Consider a toy model consisting of a
massless free scalar field on the space $S^{2}\times\BR$, where $\BR$
is time and the sphere is of radius $r$. It is a straightforward
exercise to show that the normal modes are the spherical harmonics,
with masses given by
$$M_{\ell}=\frac{\sqrt{\ell(\ell+1)}}{r}$$
(5.1)
for $\ell=0,1,$ etc. This toy model reproduces the linearized degrees
of freedom for a BPS membrane wrapped on a 2-sphere, if we give the
action maximal supersymmetry. Instead of a single scalar, we can
consider $U(1)$ SYM with maximal supersymmetry as our starting point.
This is the natural effective worldvolume theory of a D2-brane (see,
e.g., [27, 28]) and,
in particular, is valid for the large radius limit we are considering.
As we are considering a free $U(1)$ theory, there is no need to
quantize the magnetic flux through the sphere. This quantum number is
associated to the compactness of the 11th dimension. In order to get
an explicitly $O(6)$-invariant Lagrangian, as one would expect from
M-Theory, one can dualize the vector into a scalar.
Since half of the supersymmetries are broken in the ALE space, we must also
explicitly break half of the supersymmetry of our toy model. This can
be achieved by twisting one complex scalar. In particular, we identify
one of the scalars as a section of the normal bundle of the
2-sphere sitting in the ALE space. The resulting theory is
supersymmetric if the fermions are sections of the bundle
$\CO(-1)\oplus\CO(1)$. There are six bosonic zero modes, which
correspond to the motion of the membrane in the transverse $\BR^{6}$.
The corresponding eight fermionic zero modes generate the required
16-fold degeneracy of the ground state. For the massive modes, there
are eight bosonic and eight fermionic modes for each spherical
harmonic. This gives us the
information we require in order to interpret the calculation we will
now make in the matrix model.
For the $A_{1}$ singularity there is an $SU(2)$
rotational invariance which aids in the solution. Namely, since there
is a pair of hypermultiplets connecting
the same two vertices, the action (and so too the D and F-terms) is
invariant under the $SU(2)_{r}$ transformation
$$\begin{pmatrix}x_{01}\\
y_{01}\end{pmatrix}\rightarrow\begin{pmatrix}x_{01}^{\prime}\\
y_{01}^{\prime}\end{pmatrix}=R\begin{pmatrix}x_{01}\\
y_{01}\end{pmatrix},~{}~{}~{}\begin{pmatrix}x_{10}\\
y_{10}\end{pmatrix}\rightarrow\begin{pmatrix}x_{10}^{\prime}\\
y_{10}^{\prime}\end{pmatrix}=R\begin{pmatrix}x_{10}\\
y_{10}\end{pmatrix},$$
(5.2)
for $R\in SU(2)_{r}$, which mixes the hypermultiplets.
Clearly there is a $U(N)\times U(N+1)$ gauge
transformation,
$$\begin{pmatrix}U&0\\
0&U\end{pmatrix}R\begin{pmatrix}X_{10}\\
Y_{10}\end{pmatrix}V=\begin{pmatrix}X_{10}\\
Y_{10}\end{pmatrix},$$
(5.3)
where $U\in U(N)$ and $V\in U(N+1)$, that undoes the action
of (5.2) on the membrane ground
state (4.5). In other words,
the solution (4.5) is invariant under the product
$U\otimes R\otimes V$, which is a consequence of the fact that the
ground state is spherically symmetric. Obviously $U$ and $V$ must
themselves correspond to (inverse) elements of $SU(2)_{r}$.
Now the action of $U\otimes R\otimes V$ on any perturbations
around (4.5) should be faithful. Since $R$ acts as
the 2 of $SU(2)_{r}$ on
$\Bigl{(}\begin{smallmatrix}\delta x_{10}\\
\delta y_{10}\end{smallmatrix}\Bigr{)}$
and $U$ and $V$ act as the N and N+1,
respectively, $U\otimes R\otimes V$ must act as the representation
$$\textbf{2}\otimes\textbf{N}\otimes(\textbf{N+1})=\textbf{1}\oplus\textbf{3}%
\oplus\textbf{3}\oplus\textbf{5}\oplus\textbf{5}\oplus\cdots\oplus\textbf{2N--%
1}\oplus\textbf{2N--1}\oplus\textbf{2N+1}.$$
(5.4)
On the other hand, under this gauge transformation,
the vector multiplets transform in the
$$\begin{split}\displaystyle\textbf{N}\otimes\textbf{N}&\displaystyle=\textbf{1}%
\oplus\textbf{3}\oplus\cdots\oplus\textbf{2N--1}\\
\displaystyle\textbf{(N+1)}\otimes(\textbf{N+1})&\displaystyle=\textbf{1}%
\oplus\textbf{3}\oplus\cdots\oplus\textbf{2N+1}\end{split}$$
(5.5)
representations.
One immediately notices that the dimension of the representation of
the vectors (5.5) is greater by a singlet than that
of the hypermultiplets in (5.4). This singlet is
simply the decoupled $U(1)$. The absence of a singlet for the
decoupled $U(1)$ in the hypermultiplet representation protects against
the possibility of Higgsing away this element of the gauge
symmetry. Another important observation
about (5.4) and (5.5) is that
only even spherical harmonics will appear in the multipole
expansion of the perturbations. This is a considerable simplification,
and we will be able to compute the full spectrum of excitations of the
wrapped membrane.
As a start, let us now calculate the mass spectrum for the P-wave
excitations, i.e., those in the 3 of $SU(2)_{r}$. The
calculation will be illustrative of the techniques we will use to
compute the spectrum for generic $j$.
Since the masses are generated via the Higgs mechanism, it will be
sufficient to calculate the masses of the vector multiplets, since
these are related to those
of the hypers in an obvious way. Furthermore, we can exploit the
$SO(5)$ invariance of the scalars in the vector multiplet, so that we
have $a_{0i}=a^{\alpha}_{0}\lambda^{(N+1)}_{\alpha}$,
$a_{1i}=a^{\alpha}_{1}\lambda^{(N)}_{\alpha}$, for each $i$, where
the $\lambda^{(N)}_{\alpha}$ generate the N
of $SU(2)_{r}$. Finally, we note from the $SU(2)_{r}$ invariance that we
can take $a^{\alpha}_{0}=a_{0}$, $a^{\alpha}_{1}=a_{1}$, for each $\alpha$. We
then can compute the mass of $a_{0}$ and $a_{1}$ for the simplest case,
namely the Cartan generator, $\lambda_{0}$.
The quadratic part of the Lagrangian for $a_{0}$ and $a_{1}$ is
$$\begin{split}\displaystyle\CL=&\displaystyle\frac{1}{2g_{0}^{2}}\,\text{Tr}%
\bigl{(}\lambda^{(N+1)}_{0}\bigr{)}^{2}\dot{a}_{0}^{2}+\frac{1}{2g_{1}^{2}}\,%
\text{Tr}\bigl{(}\lambda^{(N)}_{0}\bigr{)}^{2}\dot{a}_{1}^{2}\\
&\displaystyle-\frac{1}{g^{2}}\left(\text{Tr}\left[\bigl{(}\lambda^{(N+1)}_{0}%
\bigr{)}^{2}(\bar{X}_{10}X_{10}+\bar{Y}_{10}Y_{10})\right]a_{0}^{2}+\text{Tr}%
\left[\bigl{(}\lambda^{(N)}_{0}\bigr{)}^{2}(X_{10}\bar{X}_{10}+Y_{10}\bar{Y}_{%
10})\right]a_{1}^{2}\right.\\
&\displaystyle\phantom{+\frac{1}{g^{2}}}~{}\left.+2\,\text{Tr}\left[\lambda^{(%
N)}_{0}X_{10}\lambda^{(N+1)}_{0}\bar{X}_{10}+\lambda^{(N)}_{0}Y_{10}\lambda^{(%
N+1)}_{0}\bar{Y}_{10}\right]a_{0}a_{1}\right).\end{split}$$
(5.6)
From (4.5), if we generalize to arbitrary coupling
constants at each vertex, we have
$$\begin{split}\displaystyle X_{10}\bar{X}_{10}+Y_{10}\bar{Y}_{10}&\displaystyle%
=(N+1)|D_{\text{dec.}}|\,\Bid_{N}\\
\displaystyle\bar{X}_{10}X_{10}+\bar{Y}_{10}Y_{10}&\displaystyle=N|D_{\text{%
dec.}}|\,\Bid_{N+1}\\
\displaystyle X_{10}\bar{X}_{10}-Y_{10}\bar{Y}_{10}&\displaystyle=|D_{\text{%
dec.}}|\,\lambda^{(N)}_{0}\\
\displaystyle\bar{X}_{10}X_{10}-\bar{Y}_{10}Y_{10}&\displaystyle=|D_{\text{dec%
.}}|\,\lambda^{(N+1)}_{0}.\end{split}$$
(5.7)
To simplify the cross-terms
in (5.6), we use the “commutation” relations
$$\begin{split}\displaystyle\lambda^{(N)}_{0}X_{10}-X_{10}\lambda^{(N+1)}_{0}+X_%
{10}&\displaystyle=0\\
\displaystyle\lambda^{(N)}_{0}Y_{10}-Y_{10}\lambda^{(N+1)}_{0}-Y_{10}&%
\displaystyle=0.\end{split}$$
(5.8)
These relations encode, in part, the invariance of the membrane
solution (4.5) under the rotation group of the
sphere (5.2).
One may also readily calculate
$$\text{Tr}\bigl{(}\lambda^{(N)}_{0}\bigr{)}^{2}=\frac{(N-1)N(N+1)}{3}.$$
(5.9)
Using these relations in (5.6) yields
$$\CL=\frac{1}{2}\bigl{(}\dot{\tilde{a}_{0}}^{2}+\dot{\tilde{a}_{1}}^{2}\bigr{)}%
-\frac{|D_{\text{dec.}}|}{g^{2}}\left[Ng_{0}^{2}\tilde{a}_{0}^{2}+2\sqrt{(N-1)%
(N+2)}\,g_{0}g_{1}\tilde{a}_{0}\tilde{a}_{1}+(N+1)g_{1}^{2}\tilde{a}_{1}^{2}%
\right],$$
(5.10)
where we have defined the normalized variables
$$\begin{split}\displaystyle\tilde{a}_{0}&\displaystyle=\sqrt{\text{Tr}\bigl{(}%
\lambda^{(N+1)}_{0}\bigr{)}^{2}}\,\frac{a_{0}}{g_{0}}\\
\displaystyle\tilde{a}_{1}&\displaystyle=\sqrt{\text{Tr}\bigl{(}\lambda^{(N)}_%
{0}\bigr{)}^{2}}\,\frac{a_{1}}{g_{1}}.\end{split}$$
(5.11)
The eigenvalues of the mass matrix are the energies for the oscillator modes
$$\begin{split}\displaystyle\omega_{\pm}^{2}&\displaystyle=\left[\left(Ng_{0}^{2%
}+(N+1)g_{1}^{2}\right)\pm\sqrt{\left(Ng_{0}^{2}+(N+1)g_{1}^{2}\right)^{2}-8g_%
{0}^{2}g_{1}^{2}}\right]\frac{|D_{\text{dec.}}|}{g^{2}}\\
&\displaystyle\sim\frac{2g_{0}^{2}g_{1}^{2}d}{g^{4}}\left[1\pm\left(1-\frac{4g%
^{4}}{g_{0}^{2}g_{1}^{2}}\,\frac{1}{N^{2}}+\cdots\right)\right].\end{split}$$
(5.12)
Therefore, in the large $N$ limit, we find two modes, one whose energy
is independent of $N$ at leading order (and therefore decouples) and
another whose energy is
$$\omega_{-}=\frac{2\sqrt{2d}}{N}$$
(5.13)
and survives as the membrane excitation as $N\rightarrow\infty$.
Notice that the energy of this mode only depends on $d$,
providing some evidence for the hypothesis made in
Section 3 that the ratio $g_{0}/g_{1}$
decouples in the large $N$ limit . The mass of this mode, $\delta m$,
can be calculated from
$$\omega_{-}=\frac{(m+\delta m)^{2}-m^{2}}{2p^{+}}\sim\frac{m\,\delta m}{p^{+}},$$
(5.14)
where $m$ and $p^{+}$ are given by (3.12)
and (3.13), respectively. In this way, we find
$$\delta m=\sqrt{\frac{2}{d}}=\frac{\sqrt{2}}{r},$$
(5.15)
where $r$ is the radius of the membrane. This result agrees precisely
with our toy model result (5.1) for the P-wave, $\ell=1$.
Now, for generic excitations appearing in the $\mathbf{2j+1}$
representation under the decomposition (5.4), we may
again exploit symmetry arguments. In general, these reduce the problem
to computing the masses of the scalars in the vector multiplets, which
may be taken to be of the form
$$\begin{split}\displaystyle a_{0i}&\displaystyle=a_{0}\bigl{(}\lambda^{(N+1)}_{%
+}\bigr{)}^{j}+\bar{a}_{0}\bigl{(}\lambda^{(N+1)}_{-}\bigr{)}^{j}\\
\displaystyle a_{1i}&\displaystyle=a_{1}\bigl{(}\lambda^{(N)}_{+}\bigr{)}^{j}+%
\bar{a}_{1}\bigl{(}\lambda^{(N)}_{-}\bigr{)}^{j},\end{split}$$
(5.16)
for each $i=1,\ldots,5$. As a motivation for why we have chosen this
particular form, we note that these are the highest weight states for
their respective $SU(2)_{r}$ representations. This stems from the fact
that, as the representations are irreducible, all of the operators
acting in a given representation are constructed from polynomials in
the generators.
After dropping the terms which vanish after
taking traces, the quadratic Lagrangian for these modes is
$$\begin{split}\displaystyle\CL=&\displaystyle 2\,\text{Tr}\left[\bigl{(}\lambda%
^{(N+1)}_{+}\bigr{)}^{j}\bigl{(}\lambda^{(N+1)}_{-}\bigr{)}^{j}\right]\,\left(%
\frac{1}{g_{0}^{2}}|\dot{a}_{0}|^{2}-\frac{N|D_{\text{dec.}}|}{g^{2}}|a_{0}|^{%
2}\right)\\
&\displaystyle+2\,\text{Tr}\left[\bigl{(}\lambda^{(N)}_{+}\bigr{)}^{j}\bigl{(}%
\lambda^{(N)}_{-}\bigr{)}^{j}\right]\,\left(\frac{1}{g_{1}^{2}}|\dot{a}_{1}|^{%
2}-\frac{(N+1)|D_{\text{dec.}}|}{g^{2}}|a_{1}|^{2}\right)\\
&\displaystyle-\frac{2}{g^{2}}a_{0}\bar{a}_{1}\left[\bigl{(}\lambda^{(N)}_{+}%
\bigr{)}^{j}X_{10}\bigl{(}\lambda^{(N+1)}_{-}\bigr{)}^{j}\bar{X}_{10}+\bigl{(}%
\lambda^{(N)}_{+}\bigr{)}^{j}Y_{10}\bigl{(}\lambda^{(N+1)}_{-}\bigr{)}^{j}\bar%
{Y}_{10}\right]\\
&\displaystyle-\frac{2}{g^{2}}\bar{a}_{0}a_{1}\left[\bigl{(}\lambda^{(N)}_{-}%
\bigr{)}^{j}X_{10}\bigl{(}\lambda^{(N+1)}_{+}\bigr{)}^{j}\bar{X}_{10}+\bigl{(}%
\lambda^{(N)}_{-}\bigr{)}^{j}Y_{10}\bigl{(}\lambda^{(N+1)}_{+}\bigr{)}^{j}\bar%
{Y}_{10}\right].\end{split}$$
(5.17)
The cross-terms here can be simplified through the use of the
“commutation” relations
$$\begin{split}\displaystyle X_{10}\lambda^{(N+1)}_{+}-\lambda^{(N)}_{+}X_{10}-Y%
_{10}&\displaystyle=0\\
\displaystyle X_{10}\lambda^{(N+1)}_{-}-\lambda^{(N)}_{-}X_{10}&\displaystyle=%
0\\
\displaystyle Y_{10}\lambda^{(N+1)}_{+}-\lambda^{(N)}_{+}Y_{10}&\displaystyle=%
0\\
\displaystyle Y_{10}\lambda^{(N+1)}_{-}-\lambda^{(N)}_{-}Y_{10}-X_{10}&%
\displaystyle=0.\end{split}$$
(5.18)
As in the case of the relations (5.8), these encode the
rotational invariance of the membrane solution. By repetitive
application of (5.18), one may establish the useful relationship
$$(N-j)\,\text{Tr}\left[\bigl{(}\lambda^{(N+1)}_{+}\bigr{)}^{j}\bigl{(}\lambda^{%
(N+1)}_{-}\bigr{)}^{j}\right]=(N+j+1)\,\text{Tr}\left[\bigl{(}\lambda^{(N)}_{+%
}\bigr{)}^{j}\bigl{(}\lambda^{(N)}_{-}\bigr{)}^{j}\right].$$
(5.19)
In terms of the normalized variables
$$\begin{split}\displaystyle\tilde{a}_{0}&\displaystyle=\sqrt{2\,\text{Tr}\left[%
\bigl{(}\lambda^{(N+1)}_{+}\bigr{)}^{j}\bigl{(}\lambda^{(N+1)}_{-}\bigr{)}^{j}%
\right]}\,\frac{a_{0}}{g_{0}}\\
\displaystyle\tilde{a}_{1}&\displaystyle=\sqrt{2\,\text{Tr}\left[\bigl{(}%
\lambda^{(N)}_{+}\bigr{)}^{j}\bigl{(}\lambda^{(N)}_{-}\bigr{)}^{j}\right]}\,%
\frac{a_{1}}{g_{1}},\end{split}$$
(5.20)
the Lagrangian (5.17) becomes
$$\begin{split}\displaystyle\CL=&\displaystyle|\dot{\tilde{a}_{0}}|^{2}+|\dot{%
\tilde{a}_{1}}|^{2}-\frac{|D_{\text{dec.}}|}{g^{2}}\left(Ng_{0}^{2}|\tilde{a}_%
{0}|^{2}+(N+1)g_{1}^{2}|\tilde{a}_{1}|^{2}\right)\\
&\displaystyle-\frac{|D_{\text{dec.}}|}{g^{2}}\sqrt{(N-j)(N+j+1)}\,g_{0}g_{1}%
\left(\tilde{a}_{0}\bar{\tilde{a}_{1}}+\bar{\tilde{a}_{0}}\tilde{a}_{1}\right)%
,\end{split}$$
(5.21)
From this, we find the energies of the modes as the eigenvalues of
the mass matrix
$$\begin{split}\displaystyle\omega_{\pm}^{2}&\displaystyle=\left[\left(Ng_{0}^{2%
}+(N+1)g_{1}^{2}\right)\pm\sqrt{\left(Ng_{0}^{2}+(N+1)g_{1}^{2}\right)^{2}-4j(%
j+1)g_{0}^{2}g_{1}^{2}}\right]\frac{|D_{\text{dec.}}|}{g^{2}}\\
&\displaystyle\sim\frac{2g_{0}^{2}g_{1}^{2}d}{g^{4}}\left[1\pm\left(1-\frac{2j%
(j+1)g^{4}}{g_{0}^{2}g_{1}^{2}}\,\frac{1}{N^{2}}+\cdots\right)\right].\end{split}$$
(5.22)
We note that, in the case $j=1$, this reduces properly
to (5.12).
As in the case of the P-waves above, we can consider the behavior
at large $N$. There are again two modes, one which decouples and
another with energy
$$\omega_{-}=\frac{\sqrt{2j(j+1)d}}{N}$$
(5.23)
and mass
$$\delta m=\sqrt{\frac{j(j+1)}{d}}=\frac{\sqrt{j(j+1)}}{r},$$
(5.24)
which agrees exactly with the toy model result (5.1). In the
limit of large radius, the effective field theory on the membrane and
the perturbation expansion in the matrix model agree.
6 Conclusions
We have given quite a bit of evidence that the matrix model proposed
in [19, 16] captures many of the
essential features of M-Theory membranes which are wrapped around the
homology 2-cycles of an ALE space. In [16], we
gave an explicit derivation of the solution describing the wrapped
membrane in the $A_{1}$ case. We found that its energy, as well as the
leading order membrane-antimembrane potential, had the properties
necessary for its interpretation as a wrapped membrane.
In Section 3, we gave a counting argument that set the
correspondence between BPS bound states of membranes and roots of the
$A$-$D$-$E$ group. We derived a general bound on the light-cone energy
of a membrane bound state and found that it had the necessary
dependence on the blow-up parameters and longitudinal momentum. We
also gave an explicit solution for all membrane bound states for the
$A_{n-1}$ series. The 16-fold degeneracy of these solutions is
consistent with the requirement that they form BPS vector multiplets
in seven-dimensional spacetime physics.
We were able to obtain all of the solutions that we expected from the
M-Theory interpretation of the model, but we have not been able to
show that we have given a unique solution. The existence of other
solutions would probably lead to a greater than 16-fold degeneracy
that would be hard to reconcile with the expected seven-dimensional
physics.
The solutions we found exhibit a very rich
structure, and fit naturally into a noncommutative geometry framework.
In section 4, we showed that there exist gauge-invariant
coordinates for the membrane in the $A_{1}$ model that satisfy the
relations appropriate to spherical membranes [18].
We then characterized the representations of excitations around the $A_{1}$
membrane in Section 5, finding that, as expected,
only even spherical harmonics contribute in the multipole
expansion. We then calculated the energy spectrum of the excitations
and found the exact agreement with the expected result from the
consideration of a large membrane, where we could trust both a
calculation in a toy model, as well as in matrix model perturbation theory.
The end result is very promising. The perturbation analysis gives the
correct results in the large radius limit, and some coupling constants
seem to decouple in the large $N$ limit. If this decoupling is a
generic property of the large $N$ dynamics, this opens up the possibility of
proving non-renormalization theorems.
Despite our success in solving for the membrane bound state
solutions for the $A_{n-1}$ series, we have not been able to
generalize them to the $D$ and $E$ series. This is basically due to
the fact that our method of following the eigenvalues around the
quiver diagram, while perfect for closed quivers, does not work for
the $D$ and $E$ open quivers. By inspection of a few cases where $N$
is taken very small, one may also see that not all of
the matrices in these cases can be diagonalized simultaneously. This
is bound to complicate finding membrane solutions.
However, one should obtain similar results for the membrane properties
in these cases. For example, the association of the BPS bound states with
roots of the $A$-$D$-$E$ algebra and the bound on the energy of a
bound state discussed in Section 3 are both completely
general, so they apply equally well to the $D$ and $E$ series. Also,
at large $N$, one expects that the couplings $g_{i}/g_{j}$
should still decouple from the low energy degrees of freedom (whose
energies scale as $1/N$), as the states that feel these couplings
should have energies of order 1. Of course, even given a solution, in
these cases the mass matrix is more complicated and should prove
difficult to diagonalize.
There are still more tests that the matrix models for the ALE spaces should
be put to. For example, it is particularly interesting to reproduce
the Coulomb
and velocity-dependent potentials between membranes and
gravitons [29, 30, 5, 18].
These and related matters are currently under investigation.
Acknowledgements
We would like to thank Philip Candelas, Dan Kabat, and especially
Willy Fischler for enlightening discussions about these and related
issues. We thank Jacques Distler for his collaboration at an early
stage of this work and many important conversations.
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LC-PHSM-2004-006
March 2004
Study of Higgs Boson Pair Production
at Linear Collider
K. Desch${}^{a}$, T. Klimkovich ${}^{a,b}$, T. Kuhl${}^{b}$,
A. Raspereza${}^{b}$
${}^{a}$ Universität Hamburg, Institut für Experimentalphysik
Notkestrasse 85, D-22607 Hamburg, Germany.
${}^{b}$ DESY, Notkestrasse 85, D-22607 Hamburg, Germany.
We study the potential of the TESLA linear collider operated at
a center-of-mass energy of 500 to 1000${\mathrm{\ Ge\kern-1.0ptV}}$
for the measurement of the neutral Higgs boson
properties within the framework of the MSSM. The process of associated
Higgs boson production with subsequent decays of Higgs
bosons into b-quark and $\tau$-lepton pairs is considered.
An integrated luminosity of 500 fb${}^{-1}$ is assumed at each energy.
The Higgs boson masses and production cross sections
are measured by reconstructing the $\mathrm{b\bar{b}b\bar{b}}$ and $\mathrm{b\bar{b}\tau^{+}\tau^{-}}$ final states.
The precision of these measurements
is evaluated in dependence of the Higgs boson masses.
Under the assumed experimental conditions
a statistical accuracy ranging from 0.1 to 1.0 GeV is achievable
on the Higgs boson mass.
The topological cross section $\mathrm{\sigma(\mathrm{e^{+}e^{-}}\rightarrow HA\rightarrow b\bar{b}b\bar{b})}$ can be determined with
the relative precision of 1.5 – 6.6 % and cross sections $\mathrm{\sigma(\mathrm{e^{+}e^{-}}\rightarrow HA\rightarrow b\bar{b}\tau^{+}%
\tau^{-})}$ and $\mathrm{\sigma(\mathrm{e^{+}e^{-}}\rightarrow HA\rightarrow\tau^{+}\tau^{-}b%
\bar{b})}$ with precision of 4 – 30 %.
Constraints on the Higgs boson widths can be set exploiting
$\mathrm{b\bar{b}\tau^{+}\tau^{-}}$ channel. The 5$\sigma$ discovery limit corresponds to the Higgs mass of around 385 GeV for the degenerate Higgs boson masses in
the $\mathrm{HA\to b\bar{b}b\bar{b}}$ channel at $\sqrt{s}$ = 800 GeV with integrated luminosity
of 500 fb${}^{-1}$. The potential of the Higgs mass
determination for the benchmark point SPS 1a for the process $\mathrm{HA\to b\bar{b}b\bar{b}}$
at $\sqrt{s}$ = 1 TeV and luminosity 1000 fb${}^{-1}$ is investigated.
1 Introduction
Elucidating the mechanism of electroweak symmetry breaking will be one of the central
tasks at a future linear $\mathrm{e^{+}e^{-}}$ collider. In the minimal
Standard Model (SM) breaking of electroweak symmetry
is provided by one Higgs doublet, which introduces an
additional spin-0 particle, the Higgs boson [1].
Among models with an extended Higgs sector, two Higgs doublet
models (2HDM) [2] are of particular
interest, since this structure of the Higgs sector is required
in the Minimal Supersymmetric Standard Model (MSSM) [3], the
most seriously considered extension of the SM.
The Higgs sector of the CP-conserving MSSM comprises the
following physical states:
two CP-even Higgs bosons, the lighter of which is denoted h and
the heavier H, one CP-odd Higgs boson A and two charged
bosons H${}^{\pm}$. Two crucial parameters,
which influence the Higgs boson phenomenology, are $\tan\beta$, the ratio of vacuum expectation values
of the two Higgs doublets, and the angle $\alpha$, which
controls mixing in the CP-even sector.
The cross sections of the Higgs-strahlung
and fusion processes, involving CP-even Higgs bosons,
$$\mathrm{e^{+}e^{-}\to hZ(HZ),\hskip 19.916929pte^{+}e^{-}\to h(H)\nu_{e}\bar{%
\nu}_{e},\hskip 19.916929pte^{+}e^{-}\to h(H)e^{+}e^{-},}$$
(1)
are defined in terms of the strength of the Higgs couplings to
weak bosons at tree level. These depend on the parameters $\alpha$
and $\beta$ in the following way:
$$\mathrm{g_{hZZ,hWW}\sim\sin(\beta-\alpha),\hskip 56.905512ptg_{HZZ,HWW}\sim%
\cos(\beta-\alpha).}$$
(2)
In the MSSM, the set of the tree level couplings between
neutral Higgs particles and weak bosons is extended by
two additional couplings,
$$\mathrm{g_{hAZ}\sim\cos(\beta-\alpha),\hskip 56.905512ptg_{HAZ}\sim\sin(\beta-%
\alpha).}$$
(3)
As a consequence, in $\mathrm{e^{+}e^{-}}$ collisions, the Higgs-strahlung
and fusion processes will be complemented by the associated
Higgs boson pair production mechanisms:
$$\mathrm{e^{+}e^{-}\to hA(HA).}$$
(4)
To fully explore the MSSM Higgs sector, the processes (4)
must be studied, since
they directly probe $\mathrm{g_{hAZ}}$ and $\mathrm{g_{HAZ}}$ trilinear couplings,
provide access to the A boson properties
and supplement the study of the CP-even Higgs boson
properties via Higgs-strahlung and fusion mechanisms.
The study of the Higgs pair production process is motivated
by a so-called decoupling limit of the MSSM in which
the h boson approaches the properties of the SM Higgs
boson. The closer MSSM scenario moves towards the decoupling
limit the more difficult it becomes to distinguish the Higgs sector
from the SM. In such a scenario the detection of heavier neutral
Higgs bosons would be crucial for establishing an extended Higgs sector.
The decoupling limit is approached relatively fast for
large values of the H and A boson masses, $m_{\mathrm{A}},m_{\mathrm{H}}$
$>$ 200 GeV, in a large portion of the MSSM parameter space.
The distinct feature of this scenario is vanishing coupling of heavy CP-even
Higgs boson to weak bosons, $\cos(\beta-\alpha)\rightarrow 0$.
As a consequence, the H boson production via
the fusion and Higgs-strahlung processes is significantly
suppressed, whereas the cross section of the $\mathrm{e^{+}e^{-}\to HA}$ process
reaches its maximal value making associated heavy Higgs pair
production a promising channel for the detection of the H and A
bosons at a future linear $\mathrm{e^{+}e^{-}}$ collider.
It should be also emphasized that in the decoupling limit the H and A
bosons are almost degenerate in mass and have similar decay
properties111
These features will be exploited later on in our analysis..
In this note the study of Higgs boson pair production
at a future linear $\mathrm{e^{+}e^{-}}$ collider is presented. This analysis
extends the previous studies [4, 5]. The analyzed topologies
include the $\mathrm{b\bar{b}b\bar{b}}$ and $\mathrm{b\bar{b}\tau^{+}\tau^{-}}$ final states.
In the following we will assume the $\mathrm{e^{+}e^{-}\to HA}$ production process.
However, this assumption does not restrict the generality
of our study and the analyses developed in this
note are applicable to the process $\mathrm{e^{+}e^{-}\to hA}$ as well as to the processes
of associated Higgs pair production in CP-violating MSSM scenarios
where the neutral Higgs bosons are no longer CP eigenstates [6].
This note is organized as follows.
In Section 2 the assumed experimental conditions and
main features of the detector are discussed.
Section 3 describes the simulation of the signal and the
relevant background processes.
In Section 4 the analysis tools, employed in our study, are briefly
reviewed. Section 5 describes the analysis
of the $\mathrm{b\bar{b}b\bar{b}}$ and $\mathrm{b\bar{b}\tau^{+}\tau^{-}}$ final states.
In Section 6, we discuss the prospects of
measuring the Higgs boson properties using
the $\mathrm{b\bar{b}b\bar{b}}$ and $\mathrm{b\bar{b}\tau^{+}\tau^{-}}$ topologies. In Appendix the interpretation of this analysis is done for the SPS 1a benchmark point.
2 Experimental Conditions and Detector
Simulations
The study is performed for a linear collider operated
at a center-of-mass energies of 500${\mathrm{\ Ge\kern-1.0ptV}}$ to 1000${\mathrm{\ Ge\kern-1.0ptV}}$ and
an event sample corresponding to an integrated luminosity
of 500 fb${}^{-1}$.
The detector used in the simulation follows the proposal
for the TESLA collider presented in
the TDR [4]. Studies were also performed in Asia and North America [7, 8].
The interaction region is surrounded by a tracker consisting
of a multi-layered pixel micro-vertex detector (VTX) as the innermost part
and a large time projection chamber (TPC) supplemented with additional silicon strip detector in the forward region (FTD) and two bands between VTX and TPC. In radial direction follow
an electromagnetic calorimeter, a hadron calorimeter,
the coils of a superconducting magnet
and an instrumented iron flux return yoke.
The solenoidal magnetic field is 4 Tesla.
The envisaged impact parameter resolution
provided by the vertex detector is
$$\delta(IP_{r\phi,z})\leq 5\mu m\oplus\frac{10\mu m\hskip 2.845276ptGeV/c}{p%
\sin^{3/2}\theta},$$
(5)
where $p$ is the track momentum and $\theta$ is the polar angle.
The overall central tracker momentum resolution is
$$\sigma\left(\frac{1}{p_{t}}\right)\leq 5\cdot 10^{-5}~{}[{\mathrm{\ Ge\kern-1.%
0ptV}}/c]^{-1},$$
(6)
where $p_{t}$ is the transverse momentum. To enable efficient separation
of neutral and charged particles, highly granular
electromagnetic and hadronic calorimeters are foreseen for the
TESLA detector. Their energy resolutions are:
$$\frac{\sigma_{E_{el}}}{E_{el}}=\frac{10\%}{\sqrt{E}_{el}}\oplus 1\%,\hskip 56.%
905512pt\frac{\sigma_{E_{h}}}{E_{h}}=\frac{50\%}{\sqrt{E}_{h}}\oplus 4\%,$$
(7)
where $E_{el}$ and $E_{h}$ are the energies
measured for electrons and hadrons in the corresponding
calorimeters. The polar angle coverage of the central tracker
maintaining the resolution is
$|\cos\theta|<0.85$, above this range the tracking
resolution deteriorates. The electromagnetic and hadron
calorimeters cover $|\cos\theta|<0.996$
maintaining the resolution over the whole angular range.
The detector response is simulated with parametric fast
simulation program SIMDET [9].
Beamstrahlung is accounted for using CIRCE [10].
The event reconstruction is done in terms of energy flow objects.
First, tracks are measured with tracking system and associated
to calorimetric clusters to define charged energy flow objects
(electrons, muons and charged hadrons). Since momentum measurement
with tracking system is much more accurate than direction and energy
measurements with calorimeters, the tracking information
is used for estimation of the four-momentum of the charged objects.
Calorimetric clusters with no associated tracks are regarded as
neutral energy flow objects (photons and neutral hadrons).
Measurements of the four-momentum of neutral objects are solely
based on the calorimetric information.
3 Physics Processes and Monte Carlo
Samples
Samples of $\mathrm{e^{+}e^{-}\to HA}$ events are generated
for several Higgs boson mass hypotheses
with PYTHIA 6.2 [11], including initial state radiation.
The tree level cross section of the HA production
is related to the Higgs-strahlung cross section in the SM,
$\mathrm{\sigma_{HZ}^{SM}}$, in the following way [2]:
$$\mathrm{\mathrm{\sigma_{HA}}=g^{2}_{HAZ}\bar{\lambda}_{HA}\mathrm{\sigma_{HZ}^%
{SM}},}$$
(8)
where
$$\mathrm{\bar{\lambda}_{HA}=\frac{\lambda^{3/2}_{HA}}{\lambda^{1/2}_{HZ}[12m^{2%
}_{Z}/s+\lambda_{HZ}]}}$$
(9)
accounts for the correct suppression of the P-wave cross section near the kinematic threshold. The quantity $\lambda_{ij}=[1-(m_{i}+m_{j})^{2}/s][1-(m_{i}-m_{j})^{2}/s]$ is the usual momentum factor of the two particle phase space. The Higgs-strahlung cross section in the SM is computed using
the program HPROD [12].
For the case of maximal
allowed $\mathrm{g_{HAZ}}$ coupling, $\sin(\beta-\alpha)$ = 1, which is
assumed for these studies, signal cross sections are given in Table 1.
The branching fractions of the Higgs bosons into b-quarks and $\tau$-lepton
pairs are set to their typical values in the MSSM :
$\mathrm{Br(H\to b\bar{b})}$ = $\mathrm{Br(A\to b\bar{b})}$ = 90%, $\mathrm{Br(H\to\tau\tau)}$ = $\mathrm{Br(A\to\tau\tau)}$ = 10%.
For the background estimation, the following processes are generated
using PYTHIA 6.2: $\mathrm{\mathrm{e^{+}e^{-}}\to WW}$, $\mathrm{\mathrm{e^{+}e^{-}}\to ZZ}$,
$\mathrm{\mathrm{e^{+}e^{-}}\to\mathrm{q\bar{q}}}$222The $\mathrm{\mathrm{e^{+}e^{-}}\to t\bar{t}}$ events are also taken into account,
$\mathrm{\mathrm{e^{+}e^{-}}\to Z\mathrm{e^{+}e^{-}}}$ and $\mathrm{\mathrm{e^{+}e^{-}}\to We\nu}$. The cross sections
for the most important background processes
are given in Table 2.
4 Analysis tools
4.1 B-tagging
Identification of b-quarks plays a crucial role in this analysis. Efficient tagging of jets containing
heavy flavour hadrons will be achieved with a highly granular micro-vertex
detector, allowing for precise reconstruction of track parameters in the
vicinity of the primary interaction point.
The procedure of tagging b-jets exploits information from the single track as well as secondary vertex information.
Secondary vertices are searched within jets using the
package ZVTOP [13]
developed for the SLD experiment, where tracks are described
as probability tubes and seed vertices are defined as regions
where these tubes overlap. Afterwards an attempt is made to assign
additional tracks by an iterative procedure. For each found vertex
the invariant total mass and momentum are calculated from the four-momenta of energy flow
objects assigned to the vertex. Three dimensional decay length and
decay length significance are also computed.
For a jet flavour separation a neural network is developed.
The detailed description of the neural network implementation
for the full simulation of the TESLA detector with the
BRAHMS program [14]
can be found in [15]. Three different neural networks
are introduced for jets with no, one and more than one secondary
vertices found. For jets without secondary vertex,
the impact parameter joint probability [16] and the two
highest impact parameter
significances are used as inputs for the neural network. If jet contains
at least one secondary vertex, an additional information,
including vertex masses, momenta, decay lengths and decay
length significances, is fed into the neural network.
The neural networks are trained on event samples simulated with SIMDET
using the same variables and jet classification as in BRAHMS. For the
analysis, a jet-wise tag, referred hereafter as jet b-tag variable,
is used. For a jet with neural network output $x$ it is defined as
$$B(x)=\frac{f_{b}(x)}{f_{b}(x)+f_{udsc}(x)},$$
(10)
where
$f_{b}(x)$ and $f_{udsc}(x)$ are probability density functions
of neural network output in a samples of b– and udsc–jets, respectively.
Tagging of c-jets proceeds in a similar way.
Fig. 1 shows b-tag and c-tag
purity versus efficiency curves for $\mathrm{Z\rightarrow q\bar{q}}$
events simulated at
a center-of-mass energy of $\sqrt{s}=91.2\,\mathrm{GeV}$.
Results obtained with SIMDET and BRAHMS are compared in this figure.
The c-tag performance agrees within 5% over the entire range of efficiency.
Some discrepancy in modelling of the b-tag
is observed in the region of high efficiency and low purity.
This discrepancy occurs due to not adequate modelling of the
resolution tails in the impact parameter joint probability
distribution by SIMDET.
However, both $\mathrm{b\bar{b}b\bar{b}}$ and $\mathrm{b\bar{b}\tau^{+}\tau^{-}}$ analyses impose b-tag requirements
strong enough not to be sensitive to this discrepancy.
Some flavour tag related systematic studies have been performed.
It has been found that the b-tag performance is nearly independent of
the center-of-mass energy in the range from $\sqrt{s}=91.2\,\mathrm{GeV}$
to $\sqrt{s}=500\,\mathrm{GeV}$. Furthermore, possible changes in
micro-vertex detector configuration are found to have
marginal impact on the b-tag performance. For example,
removing innermost silicon layer changes the selection efficiency by not
more than 5%. These studies confirm the stability of b-tagging.
However, c-tag and e.g. b-quark charge tagging are depending on the
innermost layer.
4.2 Kinematic Fit
The mass resolution of the reconstructed Higgs bosons is improved
by means of a kinematic fit. In the $\mathrm{b\bar{b}b\bar{b}}$ analysis,
conservation of four-momentum is required, leading
in total to four constraints. In the case of the $\mathrm{b\bar{b}\tau^{+}\tau^{-}}$ topology,
the energies of $\tau$-leptons are not measurable due to undetectable
neutrinos. However, the flight direction of highly boosted $\tau$-leptons
can be approximated with good accuracy by the direction of the
total momentum of its visible decay products. Exploiting
this approximation together with the the four-momentum conservation
requirement, we arrive at a two constraint fit.
The jet angular and momentum resolution functions and the
angular resolution of $\tau$-leptons
are derived from Monte Carlo studies.
The kinematic fit is performed
using the code developed by DELPHI [17].
For the samples of the $\mathrm{HA\to b\bar{b}b\bar{b}}$ and $\mathrm{HA\to b\bar{b}\tau^{+}\tau^{-}}$ events
with ($m_{\mathrm{H}},m_{\mathrm{A}}$) = (150,100) GeV, the performance of the kinematic
fit is illustrated in Fig. 2 and 3,
respectively.
5 Analysis Procedures
5.1 The $\mathrm{HA\to b\bar{b}b\bar{b}}$ Channel
Events of the $\mathrm{b\bar{b}b\bar{b}}$ topology are characterized by four high
multiplicity hadronic jets, containing decay products
of b-hadrons. A cut-based technique is employed to separate signal
from background. Selection criteria are optimized separately for
500 GeV and 800 GeV center-of-mass energies. Each event is required
to pass the following cuts:
1.
Total energy deposited in the detector, the visible
energy $\mathrm{E_{vis}}$, must be greater than 340 GeV (600 GeV)
for 500 GeV (800 GeV) center-of-mass energies.
2.
Each event is forced into four jets using the DURHAM algorithm [18] and the number of tracks per jet is required to be greater than three.
3.
To separate centrally produced H and A bosons from the
WW and ZZ events, peaking in forward/backward direction,
we apply a cut on the polar angle of the thrust vector [11], $|\cos\theta_{T}|<0.8$.
4.
Further suppression of the WW and ZZ backgrounds is achieved by
requiring the event thrust value to be less than 0.88.
5.
Two fermion background is suppressed by applying a cut on
the
DURHAM jet resolution parameter, for which the event changes from four
to three jets, $\log_{10}{y_{34}}\geq-2.9$.
6.
High multiplicity six-jet events originating from the $\mathrm{\mathrm{e^{+}e^{-}}\to t\bar{t}}$
production are reduced by requiring the number of enflow (energy flow) objects in the event to be less than 130. This cut is applied only at $\sqrt{s}$ = 500 GeV.
7.
The background from $\mathrm{\mathrm{e^{+}e^{-}}\to t\bar{t}}$ events is further reduced by applying a
cut on the jet resolution parameter, for which the event changes from
six to five jets, $\log_{10}{y_{56}}$ $\leq$ -3.1 (-2.8) at
$\sqrt{s}$ = 500 GeV (800 GeV).
8.
Finally, we make use of the b-tag information to enhance the purity
of the selected event sample. First, the b-tag variable for each jet
is calculated as described in Section 4. The four b-tag variables are sorted in descending order,
$B_{1}>B_{2}>B_{3}>B_{4}$. Two quantities $B_{12}$, $B_{34}$ are
then defined as
$$B_{12}=\frac{B_{1}B_{2}}{B_{1}B_{2}+(1-B_{1})(1-B_{2})},$$
$$B_{34}=\frac{B_{3}B_{4}}{B_{3}B_{4}+(1-B_{3})(1-B_{4})}.$$
The value of $B_{12}$ has to be greater than 0.75 (0.6)
at $\sqrt{s}$ = 500 GeV (800 GeV). The value of $B_{34}$ is required to be greater
than 0.05 independent of the center-of-mass energy.
The numbers of expected signal and background events, retained after selection, and signal efficiency for the example points with ($m_{\mathrm{H}}$,$m_{\mathrm{A}}$) = (150,100) GeV at $\sqrt{s}$ = 500 GeVand ($m_{\mathrm{H}}$,$m_{\mathrm{A}}$) = (300,250) at $\sqrt{s}$ = 800 GeVare presented in Tables 3, 4. Fig. 4 and 5 show the distributions of
the selection variables for the different sources of
background and for the signal at $\sqrt{s}$ = 500 GeV. The cutflow table for this case shows that such cuts like those on visible energy, number of tracks per jet, polar angle on the thrust vector and b-tag cuts are very effective to separate signal from the background. Especially should be stressed that after all kinematic selection i.e. before b-tag cuts the signal efficiency is 54% while the background is drastically reduced (1.6% from the initial value). After b-tag cuts only 1.3% of the background events, left after kinematic selection, remains, while the signal efficiency is 43%. This confirms, that the new tool for b-tagging is very powerful. The signal efficiencies, number of signal events and total background for different Higgs boson masses after selection cuts are given in Table 5.
Events accepted in the final sample are subjected to a 4C kinematic fit.
For each of the three possible di-jet pairings, the di-jet mass sum and
the di-jet mass difference are reconstructed. For the example point the mass sum and the mass difference presented in Fig. 2 show that due to the kinematic fit the mass reconstruction resolution can be improved significantly.
5.2 The $\mathrm{HA\to b\bar{b}\tau^{+}\tau^{-},\tau^{+}\tau^{-}b\bar{b}}$ Channels
The signature of the $\mathrm{b\bar{b}\tau^{+}\tau^{-}}$ final state consists of two high-multiplicity hadronic jets enriched with the decay
products of b-hadrons and two low-multiplicity jets initiated
by $\tau$-leptons.
Initially, a preselection of high-multiplicity events compatible
with the four-jet topology is applied. Events are required to fulfill
the following criteria.
1.
The leptonic and two photon events are rejected by applying
lower cut on the number of energy flow objects, $\mathrm{N_{enflow}}$ $>$ 30.
2.
Genuine six-jet events, resulting from the $\mathrm{\mathrm{e^{+}e^{-}}\to t\bar{t}}$
production and characterized by very large particle multiplicity,
are partially suppressed by an upper cut on the number of energy flow
objects, $\mathrm{N_{enflow}}$ $<$ 120.
3.
The visible energy $\mathrm{E_{vis}}$ must be greater than
250 GeV (400 GeV) at $\sqrt{s}$ = 500 GeV (800 GeV). This cut suppresses background processes
characterized by large missing energy: $\mathrm{ZZ\to q\bar{q}\nu\bar{\nu}}$ events, radiative
returns to the Z resonance with a photon escaping into the beam pipe and $\mathrm{\mathrm{e^{+}e^{-}}\to We\nu}$, $\mathrm{\mathrm{e^{+}e^{-}}\to Z\mathrm{e^{+}e^{-}}}$ processes with electrons escaping undetected
in the forward direction. An upper cut $\mathrm{E_{vis}}$ $<$ 500 GeV (760 GeV) at
$\sqrt{s}$ = 500 GeV (800 GeV) allows partial
rejection of the fully hadronic $\mathrm{WW\to q\bar{q}q\bar{q}}$ and $\mathrm{ZZ\to q\bar{q}q\bar{q}}$ events without
significant loss in the signal efficiency.
4.
The event thrust value has to be less than 0.95. This cut disentangles
more spheric HA events from the WW, ZZ and $\mathrm{q\bar{q}}$ backgrounds in
which the high boost of the final state bosons/quarks results in large values
of the event thrust.
5.
To separate centrally produced H and A bosons from the WW
and ZZ processes in which weak bosons tend to peak in the forward/backward
directions, a cut on the polar angle of the thrust vector,
$|\cos\theta_{T}|$ $<$ 0.9, is applied.
6.
The event is discarded if it contains an $\mathrm{e^{+}e^{-}}$ or $\mathrm{\mu^{-}\mu^{+}}$ pair
with invariant mass compatible with the Z mass within 5 GeV.
This cut suppresses the $\mathrm{ZZ\to q\bar{q}\ell^{+}\ell^{-}}$ background.
Events satisfying these criteria are resolved into four-jet topology
and $\tau$-lepton identification is performed. Two identification
categories for $\tau$-leptons are introduced. A low multiplicity jet is assigned
to the first category if it fulfills one of the following tight conditions:
•
A jet contains only one charged track,
identified as an electron or muon. No neutral clusters are assigned to the jet.
•
A jet contains one charged track, identified
as hadron, and not more than four neutral clusters
(one-prong hadronic decay of $\tau$-lepton).
•
A jet contains three charged tracks with unit total charge and
has not more than two neutral calorimetric clusters (three-prong
hadronic decay of $\tau$-lepton).
To enhance the acceptance of signal events, the second identification
category, imposing looser criteria, is introduced. According to these
criteria, a jet is identified as $\tau$-lepton if it has not more than two
charged tracks and less than five neutral clusters.
An event is accepted if at least one of the low multiplicity jets satisfies the tight identification criteria.
Signal efficiencies and numbers of signal and background
events remaining after preselection are given in
Table 6.
In the next step, events are selected into the final sample
by means of a binned likelihood technique. The signal likelihood, $\mathrm{L_{HA}}$
is built from the following variables:
•
event thrust, T;
•
the polar angle of the thrust vector, $\cos\theta_{T}$;
•
minimal opening angle between any two of the four jets;
•
b-tag variables of the two hadronic jets, $\mathrm{B_{1}}$ and $\mathrm{B_{2}}$
(hadronic jets are ordered by their energy);
•
missing energy;
•
number of energy flow objects, $\mathrm{N_{enflow}}$.
Fig. 6 and 7 show
the distributions of these variables. The distribution of
$\mathrm{L_{HA}}$ is shown in Fig. 8 for
($m_{\mathrm{H}}$,$m_{\mathrm{A}}$) = (150,100) GeV at $\sqrt{s}$ = 500 GeV. An event is accepted in the
final sample if the value of $\mathrm{L_{HA}}$ is greater than certain
threshold optimized separately for each Higgs boson
mass hypothesis considered to yield maximal significance of the
signal. The number of signal and background events and the
signal efficiencies after final selection are given in
Table 7. Signal efficiency ranges between 30 and
45%, and the number of background events contributing
to the final sample between 2000 and 6000, depending on the Higgs mass.
For events accepted in the final sample,
the di-jet and di-tau invariant masses are reconstructed
exploiting the 2C kinematic fit, described in Section 4.
6 Results
In the final step of the analysis, the spectra
of the di-jet mass sum and difference, obtained in the $\mathrm{HA\to b\bar{b}b\bar{b}}$ channel,
and distributions of the di-jet and di-tau masses,
reconstructed in the $\mathrm{HA\to b\bar{b}\tau^{+}\tau^{-},\tau^{+}\tau^{-}b\bar{b}}$ channels, are used to determine
Higgs boson properties.
6.1 Cross section and Mass
First, the analysis is performed assuming that the natural widths
of the Higgs bosons are small compared to the detector
resolution.
As an example Fig. 9 shows the distributions of the di-jet mass sum and di-jet mass difference obtained after selection cuts and kinematic fit in the $\mathrm{HA\to b\bar{b}b\bar{b}}$
channel for the Higgs boson mass hypothesis of
($m_{\mathrm{H}}$,$m_{\mathrm{A}}$) = (300,250) GeV at $\sqrt{s}$ = 800 GeV. Three entries per event contribute to these distributions, corresponding to three possibilities to
pair jets in the four-jet events. Two entries form a so-called combinatorial background. Fig. 10 demonstrates the final di-jet mass sum and di-jet mass difference after the cut on the di-jet mass difference sum, respectively, as indicated by arrows in Fig. 9. The signal efficiencies, number of signal events and total background for different Higgs boson masses after cuts on di-jet mass sum and difference are given in Table 5. The mass distributions are fitted separately
with the signal normalization, the sum and the difference
of the Higgs boson masses as free parameters. The shapes of the
signal distributions are parametrised using high statistics signal samples.
Background is approximated with a polynomial function. From the fit,
the sum and the difference of the Higgs boson masses
and errors on these quantities are determined.
The di-jet mass sum and difference are found to be weakly correlated
quantities. Hence, the statistical errors on the masses of the
Higgs bosons can be calculated as follows:
$$\delta m_{\mathrm{H}}=\delta m_{\mathrm{A}}=\frac{1}{2}\sqrt{(\delta\Sigma^{2}%
+\delta\Delta^{2})},$$
where $\delta\Sigma$ and $\delta\Delta$ are statistical
uncertainties in determination of the Higgs boson mass sum and difference.
The error on the topological cross section is calculated as
$$\delta\sigma/\sigma=\sqrt{N_{S}+N_{B}}/N_{S}.$$
The notations $N_{S}$ and $N_{B}$ stand for
the number of background and signal entries
within the windows in the di-jet mass sum and di-jet mass
difference distributions, where the signal is accumulated. The boundaries
of these windows are optimized to yield a minimal relative error on
the topological cross section.
In the $\mathrm{HA\to b\bar{b}\tau^{+}\tau^{-},\tau^{+}\tau^{-}b\bar{b}}$ channels, in the case of large mass splitting between
Higgs bosons, the signal exhibits itself as two peaks in the
reconstructed di-jet and di-tau mass spectra as illustrated in
Fig. 11.
The spectra are fitted simultaneously with the superposition of background
and signal distributions. The fit is performed with
four free parameters : two normalization factors for the
$\mathrm{HA\to b\bar{b}\tau^{+}\tau^{-}}$ and $\mathrm{HA\to\tau^{+}\tau^{-}b\bar{b}}$ samples and two Higgs boson masses.
Background distribution is assumed to be well measured and therefore fixed.
The shapes of signal peaks are parametrised using high statistics
MC samples. When H and A are degenerate in mass,
signal distributions overlap and
the contributions from the $\mathrm{HA\to b\bar{b}\tau^{+}\tau^{-}}$ and $\mathrm{HA\to\tau^{+}\tau^{-}b\bar{b}}$ samples
cannot be disentangled. To stabilize the fitting procedure we assume that
the numbers of the $\mathrm{HA\to b\bar{b}\tau^{+}\tau^{-}}$ and $\mathrm{HA\to\tau^{+}\tau^{-}b\bar{b}}$ events, selected
into the final sample, are equal.
This assumption is validated by the properties of the
decoupling limit. As was mentioned in the introductory section,
in the decoupling limit the H and A bosons have almost identical
properties: $m_{\mathrm{H}}$ $\approx$ $m_{\mathrm{A}}$, $\mathrm{Br(H\to b\bar{b})}$ $\approx$ $\mathrm{Br(A\to b\bar{b})}$,
$\mathrm{Br(H\to\tau\tau)}$ $\approx$ $\mathrm{Br(A\to\tau\tau)}$, $\mathrm{\Gamma_{H}}$ $\approx$ $\mathrm{\Gamma_{A}}$.
Consequently, the $\mathrm{HA\to b\bar{b}\tau^{+}\tau^{-}}$ and $\mathrm{HA\to\tau^{+}\tau^{-}b\bar{b}}$ events are expected
to contribute equally to the final selected sample. This
reduces the number of free parameters to three : a common normalization factor for the two signal samples and two Higgs
boson masses. As an example, Fig. 12
shows the result of the fit for
($m_{\mathrm{H}}$,$m_{\mathrm{A}}$) = (300,300) GeV.
Table 8 summarizes the
statistical accuracy of the mass measurements for the $\mathrm{HA\to b\bar{b}b\bar{b}}$ and $\mathrm{HA\to b\bar{b}\tau^{+}\tau^{-},\tau^{+}\tau^{-}b\bar{b}}$ channels and their combination. The statistical accuracy of the combined
mass measurement is $\sim$ 100 MeV for the Higgs pair production far
above kinematic threshold and degrades to $\sim$ 1 GeV with aproaching
kinematic limit. At the same time the statistical errors for the $\sqrt{s}$ = 800 GeV are in general twice as bigger than those for the 500 GeV for the both channels. The combination of the two channels helps to improve the mass determination accuracy.
The statistical uncertainty on the topological cross
sections is reported in Table 9. In the $\mathrm{HA\to b\bar{b}b\bar{b}}$ channel, the topological
cross sections can be measured with relative precision between 1.5 and 6.6% while in the
$\mathrm{HA\to b\bar{b}\tau^{+}\tau^{-},\tau^{+}\tau^{-}b\bar{b}}$ channels between 5 to 30%.
A big part of SUSY parameter space leads to degenerate H and A Higgs boson masses. For this case the discovery significance as a function of $m_{H}$ (=$m_{A}$) (Fig. 13) is calculated for the $\mathrm{HA\to b\bar{b}b\bar{b}}$ channel at $\sqrt{s}$ = 800 GeV. Approaching the kinematic limit, the significance drops below 5$\sigma$ between 380 GeV to 390. The whole range of the significances for the Higgs masses is from 28.2 to 3.4. The 5$\sigma$ discovery reach and $\eta^{2}$ as a function of $m_{H}$ (=$m_{A}$) where $\eta^{2}$ is assumed $\mathrm{e^{+}e^{-}\to HA}$ cross section relative to that for $\sin^{2}(\beta-\alpha)=1$ (as shown in Fig. 13).
6.2 Width
We utilize the $\mathrm{HA\to b\bar{b}\tau^{+}\tau^{-},\tau^{+}\tau^{-}b\bar{b}}$ channels to measure the Higgs boson widths.
Fig. 14 illustrates the impact of the natural widths of
the Higgs bosons on the reconstructed di-jet and di-tau mass spectra.
We consider separately two cases :
1.
large mass splitting between H and A,
2.
mass degeneracy.
In the first case, the strategy of the measurement is the following.
The masses of Higgs bosons are fixed to the values measured in the
$\mathrm{HA\to b\bar{b}b\bar{b}}$ channel.
The reconstructed di-jet and di-tau mass line shapes are parametrised
as a function
of $\mathrm{\Gamma_{H}}$ and $\mathrm{\Gamma_{A}}$ using MC samples of the $\mathrm{HA\to b\bar{b}\tau^{+}\tau^{-}}$ and
$\mathrm{HA\to\tau^{+}\tau^{-}b\bar{b}}$ events generated with different Higgs boson widths. A simultaneous
log-likelihood fit of the parametrised mass line shapes
to the actual reconstructed di-jet and di-tau mass distributions is performed
with four free parameters : $\mathrm{\Gamma_{H}}$, $\mathrm{\Gamma_{A}}$
and two normalization factors for the $\mathrm{HA\to b\bar{b}\tau^{+}\tau^{-}}$ and $\mathrm{HA\to\tau^{+}\tau^{-}b\bar{b}}$ samples.
Fig. 15 shows the dependence of the log-likelihood
on the probed Higgs boson widths for the case of $\mathrm{\Gamma_{H}}$ = $\mathrm{\Gamma_{A}}$ = 5 GeV and
($m_{\mathrm{H}}$,$m_{\mathrm{A}}$) = (200,150) GeV at $\sqrt{s}$ = 500 GeV. The dependence is shown for the case when only di-jet mass or di-tau mass spectrum are fitted and for the simultaneous fit to the two distributions. The feasible accuracies of the
width measurements for three representative Higgs boson mass hypotheses
are given in Table 10. The precision on the width determination ranges from 600 MeV to 4.0 GeV.
In the case of very close Higgs boson masses, the broadening
of the signal peak in the mass distributions can be caused by two factors :
the finite natural widths of the Higgs bosons and non-zero mass
difference, $\delta M=m_{\mathrm{H}}-m_{\mathrm{A}}$. The width measurement in this case proceeds
as follows. The sum of the Higgs boson masses is fixed to the value
measured in the $\mathrm{HA\to b\bar{b}b\bar{b}}$ channel, $m_{\mathrm{H}}+m_{\mathrm{A}}=2M$.
From the $\mathrm{HA\to b\bar{b}\tau^{+}\tau^{-}}$ and $\mathrm{HA\to\tau^{+}\tau^{-}b\bar{b}}$ MC
samples with $\mathrm{\Gamma_{H}}$ = $\mathrm{\Gamma_{A}}$ = 0 GeV, the experimental mass resolution functions
$R_{bb,\tau\tau}(x)$ are derived, which approximate the reconstructed
di-jet and di-tau mass distributions in the limit of infinitely small
Higgs widths. To ensure stability of the fitting procedure, the H and A
boson are assumed to have identical decay properties,
$\mathrm{Br(H\to b\bar{b})}$ = $\mathrm{Br(A\to b\bar{b})}$, $\mathrm{Br(H\to\tau\tau)}$ = $\mathrm{Br(A\to\tau\tau)}$, $\mathrm{\Gamma_{H}}$ = $\mathrm{\Gamma_{A}}$ = $\Gamma$,
as expected in the decoupling limit.
The distributions of the invariant di-jet and di-tau
masses are fitted simultaneously with the functions
$$\begin{array}[]{ccccccc}F_{bb,\tau\tau}(x)&=&B_{bb,\tau\tau}(x)&+&N\cdot BW(x,%
M\pm\delta M/2,\Gamma)\otimes R_{bb,\tau\tau}(x)&+\\
&&&&N\cdot BW(x,M\mp\delta M/2,\Gamma)\otimes R_{bb,\tau\tau}(x)&\\
\end{array}$$
The fitting functions consist of three terms. The first one,
$B_{bb,\tau\tau}(x)$, describes the background spectrum. The other
two describe two signal distributions and represent
the convolution of the Breit-Wigner function
$$BW(x,M,\Gamma)=\frac{xM\Gamma}{(x^{2}-M^{2})^{2}+M^{2}\Gamma^{2}}$$
with the experimental resolution functions $R_{bb,\tau\tau}(x)$.
The fit is performed with the common normalization factor, $N$,
the Higgs boson width, $\Gamma$, and the Higgs boson mass difference,
$\delta M$, as free parameters. The Higgs width errors
obtained from the fitting procedure are presented in
Table 11 for three representative Higgs mass hypotheses. The precision on the width determination ranges from 1 GeV to 4.5 GeV.
7 Conclusion
We examined the potential of a future linear collider detector for the
determination of the MSSM Higgs boson properties exploiting
associated Higgs boson pair production followed by the Higgs decays
to b-quarks and $\tau$-leptons. It is shown that combining
the $\mathrm{HA\to b\bar{b}b\bar{b}}$ and $\mathrm{HA\to b\bar{b}\tau^{+}\tau^{-},\tau^{+}\tau^{-}b\bar{b}}$ channels the Higgs boson
masses can be measured with an accuracy of up to several hundred MeV for
Higgs pair production far above the kinematic threshold.
The precision deteriorates to about 1 GeV with approaching the kinematic
threshold. The topological
cross section $\mathrm{\sigma(\mathrm{e^{+}e^{-}}\rightarrow HA\rightarrow b\bar{b}b\bar{b})}$ can be measured with a relative precision
varying between 1.5 and 6.6%. The error on the topological cross sections
$\mathrm{\sigma(\mathrm{e^{+}e^{-}}\rightarrow HA\rightarrow b\bar{b}\tau^{+}%
\tau^{-})}$ and
$\mathrm{\sigma(\mathrm{e^{+}e^{-}}\rightarrow HA\rightarrow\tau^{+}\tau^{-}b%
\bar{b})}$ is estimated to range between 4 and 30%.
Moderate constraints on the Higgs boson widths can be set from
the analysis of the reconstructed di-jet and di-tau mass spectra in the
$\mathrm{HA\to b\bar{b}\tau^{+}\tau^{-}}$ and $\mathrm{HA\to\tau^{+}\tau^{-}b\bar{b}}$ channels.
The 5$\sigma$ discovery limit corresponds to the Higgs mass of around
385 GeV for the degenerate Higgs boson masses in
the $\mathrm{HA\to b\bar{b}b\bar{b}}$ channel at $\sqrt{s}$ = 800 GeV with integrated luminosity
of 500 fb${}^{-1}$.
Appendix A SPS 1a
The present analysis is applied to one of the so-called benchmark points SPS 1a for SUSY searches [19]. SPS 1 is a typical mSUGRA scenario which consists of a point with an intermediate value of $\tan\beta$ and a model line attached to it (SPS 1a) and of a ”typical” mSUGRA point with relatively high $\tan\beta$ (SPS 1b). The parameters for the SPS 1a point are $m_{0}$ = 100 GeV, $m_{1/2}$ = 250 GeV, $A_{0}$ = -100 GeV, $\tan\beta$ = 10, $\mu>0$. For this point the Higgs masses are $m_{h^{0}}$ = 113.7 GeV, $m_{A^{0}}$ = 394.65 GeV, $m_{H^{0}}$ = 394.9, $m_{H^{\pm}}$ = 403.6 GeV according to the Hdecay and Feynhiggsfast programs [20, 21].
The analysis is done for the center-of-mass energy of $\sqrt{s}=1\,\mathrm{TeV}$, at which the cross section for the process $\mathrm{e^{+}e^{-}\to HA}$ is 2.5 fb. The luminosity assumed is 1000 fb${}^{-1}$. The branching ratio for the H (A) Higgs boson to $b\bar{b}$ is 0.64 (0.40), $\Gamma_{tot}$ = 0.785 GeV (1.251 GeV).
The results presented in Fig. 16 are the mass sum and the mass difference after the selection cuts, kinematic fit and the final cuts for the mass difference and the mass sum respectively. The masses can be measured with precision of 1.3 GeV. The signal efficiency is 29% after selection cuts and 24% after cuts on di-jet mass sum and difference. The cross section can be measured with the relative uncertainty of 9%.
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Subsets and Splits